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Mehmet Tevfik DORAK, MD, PhD

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Absolute risk: Probability of an event over a period of time; expressed as a cumulative incidence like 10-year risk of 10% (meaning 10% of individuals in the group of interest will develop the condition in the next 10 year period). It shows the actual likelihood of contracting the disease and provides more realistic and comprehensible risk than relative risk/odds ratio.

Absolute risk reduction (ARR): The difference in the absolute risk values in the experimental/treatment and control groups in an interventional study. ARR is the favored effect size in a randomized clinical trial (RCT). The reciprocal of the ARR gives number needed to treat (NNT).

Accuracy: The degree to which a parameter (like the mean) is immune systematic error or bias. Accuracy is increased by f=good experimental design. Accuracy is different from precision (which has to do with random variation).

Addition rule: The probability of any of one of several mutually exclusive events occurring is equal to the sum of their individual probabilities. A typical example is the probability of a baby to be homozygous or heterozygous for a Mendelian recessive disorder when both parents are carriers. This equals to 1/4 + 1/2 = 3/4. A baby can be either homozygous or heterozygous but not both at the same time; thus, these are mutually exclusive events (see also multiplication rule).

Adjusted odds ratio: In a multiple logistic regression model where the response variable is the presence or absence of a disease, an odds ratio for a binomial exposure variable is an adjusted odds ratio for the levels of all other risk factors included in a multivariable model. It is also possible to calculate the adjusted odds ratio for a continuous exposure variable. An adjusted odds ratio results from the comparison of two strata similar at all variables except exposure (or the marker of interest). It can be calculated when stratified data are available as contingency tables by Mantel-Haenszel test.

Affected Family-Based Controls (AFBAC) Method: One of several family-based association study designs (Thomson, 1995). AFBAC uses affected siblings as controls and examines the sharing between two affected family members. The parental marker alleles not transmitted to an affected child or never transmitted to an affected sib pair form the so-called affected family-based controls (AFBAC) population. See also HRR and TDT and Genetic Epidemiology.

Age-standardized rate: An age-standardized rate is a weighted average of the age-specific rates, where the weights are the proportions of a standard population in the corresponding age groups. The potential confounding effect of age is removed when comparing age-standardized rates computed using the same standard population.

Alternative explanations (of observed relationships): A correlation does not mean causation (even if it is statistically significant). An observed relationship (association/correlation) can only be considered causal when alternative explanations can be ruled out. These are chance/random variation; bias/confounding and reverse causation. See Bias & Confounding.

Alternative hypothesis: In practice, this is the hypothesis that is being tested in an experiment. It is the conclusion that is reached when a null hypothesis is rejected. It is the opposite of null hypothesis, which states that there is a difference between the groups or something to that effect.

Analysis of molecular variance (AMOVA): A statistical (analysis of variance) method for analysis of molecular genetic data. It is used for partitioning diversity within and among populations using nucleotide sequence or other molecular data. AMOVA produces estimates of variance components and F-statistic analogs (designated as phi-statistics). The significance of the variance components and phi-statistics is tested using a permutational approach, eliminating the normality assumption that is inappropriate for molecular data (Excoffier, 1992). AMOVA can be performed on Arlequin. For examples, see Roewer, 1996; Stead, 2003; Watkins, 2003).

ANCOVA: See covariance models.

ANOVA (analysis of variance): A test for significant differences between multiple means by comparing variances. It concerns a normally distributed response (outcome) variable and a single categorical explanatory (predictor) variable, which represents treatments or groups. ANOVA is a special case of multiple regression where indicator variables (or orthogonal polynomials) are used to describe the discrete levels of factor variables. The term analysis of variance refers not to the model but to the method of determining which effects are statistically significant. Major assumptions of ANOVA are the homogeneity of variances (it is assumed that the variances in the different groups of the design are similar) and normal distribution of the data within each treatment group. Under the null hypothesis (that there are no mean differences between groups or treatments in the population), the variance estimated from the within-group (treatment) random variability (residual sum of squares = RSS) should be about the same as the variance estimated from between-groups (treatments) variability (explained sum of squares = ESS). If the null hypothesis is true, mean ESS / mean RSS (variance ratio) should be equal to 1. This is known as the F test or variance ratio test (see also one-way and two-way ANOVA). The ANOVA approach is based on the partitioning of sums of squares and degrees of freedom associated with the response variable. ANOVA interpretations of main effects and interactions are not so obvious in other regression models. An accumulated ANOVA table reports the results from fitting a succession of regression models to data from a factorial experiment. Each main effect is added on to the constant term followed by the interaction(s). At each level an F test result is also reported showing the extra effect of adding each variable so it can be worked out which model would fit best. In a two-way ANOVA with equal replications, the order of terms added to the model does not matter, whereas this is not the case if there are unequal replications. When the assumptions of ANOVA are not met, its non-parametric equivalent Kruskal-Wallis test may be used (a review by ; a tutorial on ANOVA posttest; online calculators for ANOVA (1), (2), (3) and (4; for analysis of summary data). See also MANOVA.

Arithmetic mean: M = (x1 + x2 + .... xn) / n (n = sample size).

Association: A statistically significant correlation or relationship between an environmental exposure or a biochemical/genetic marker and a disease or condition. An association may be an artifact (due to random sampling error-chance, bias, confounding) or a real one. In population genetics, an association may be due to population stratification, linkage disequilibrium, or direct causation. A significant association should be presented together with a measure of the strength of association called effect size (odds ratio, relative risk or hazard ratio and its 95% confidence interval) and when appropriate a measure of potential impact (attributable risk, prevented fraction, attributable fraction/etiologic fraction).

Assumptions: Certain conditions of the data that are required to be met for validity of a statistical test. ANOVA generally assumes normal distribution of the data within each treatment group, homogeneity of the variances in the treatment groups, and independence of the observations. In regression analysis, main assumptions include the normal distribution of the response variable, constant variance across fitted values, independence of error terms, and the consistency of underlying hazard rate over time (proportionality assumption) in Cox Proportional Hazard Models.

Asymptotic: Refers to a curve that continually approaches either the x or y axis but does not actually reach it until x or y equals infinity. The axis so approached is the asymptote. An example is the normal distribution curve.

Asymptotically unbiased: In point estimation, the property that the bias approaches zero as the sample size (N) increases. Therefore, estimators with this property improve as N increases. See also bias.

Attributable risk (AR): Also called excess risk or risk difference. A measure of potential impact of an association. It quantifies the additional risk of disease following exposure over and above that experienced by individuals who are not exposed. It shows how much of the disease is eliminated if no one had the risk factor (unrealistic). The information contained in AR combines the relative risk and the risk factor prevalence. The larger the AR, the greater the effect of the risk factor on the exposed group. See also prevented fraction, Walter, 1978 and Attributable Risk Applications in Epidemiology. For online calculation, see EpiMax Table Calculator.

Attributable fraction (etiologic fraction): It shows what proportion of disease in the exposed individuals is due to the exposure.

Balanced design: An experimental design in which the same number of observations is taken for each combination of the experimental factors.

Bartlett’s test: A test for homogeneity of variance.

Bayesian inference: An inference method radically different from the classical frequentist approach which takes into account the prior probability for an event. Established as a new method by Reverend Thomas Bayes. See +Plus: Bayesian Statistics Explained; MathWorld: Bayesian Analysis; Bayesian Calculator (1) & (2).

Bayes' method in genetic counseling: This method uses available additional information to modify risks calculated purely by Mendelian probabilities. It combines prior and conditional probabilities to give joint and posterior probabilities of unknown events. See Bayesian Analysis and Risk Assessment in Genetic Counseling and Testing.

Bernoulli distribution models the behavior of data taking just two distinct values (0 and 1).

Bias: In statistics, an estimator for a parameter is unbiased if its expected value is the true value of the parameter. Otherwise, the estimator is biased. It is the quantity E = (q-hat) - q. If the estimate of q is the same as actual but unknown q, the estimate is unbiased. If bias tends to decrease as n gets larger, this is called asymptotic unbiasedness. In epidemiology, bias is any deviation from truth (intentional bias is scientific fraud). See reviews on epidemiologic meaning of bias: Bias & Confounding in Molecular Epidemiology; Bias and Confounding Lecture Note; a Guide to Bias; ERIC Notebook; Online Catalog of (Epidemiologic) Bias; S4B: Bias.  

Binary (dichotomous) variable: A discrete random variable that can only take two possible values (success or failure).

Binomial distribution: The binomial distribution gives the probability of obtaining exactly r successes in n independent trials, where there are two possible outcomes one of which is conventionally called success (Online Binomial Test for observed vs expected value; Binomial Probability Calculator).

Blocks: Homogeneous grouping of experimental units (subjects) in experimental design. Groups of experimental units that are relatively homogeneous in that the responses on two units within the same block are likely to be more similar than the responses on two units in different blocks. Dividing the experimental units into blocks is a way of improving the accuracy between treatments. Blocking will minimize variation between subjects that is not attributable to the factors under study. Blocking is similar to matching in two-sample tests or stratification to control for confounding.

Blocking: When the available experimental units are not homogeneous, grouping them into blocks of homogeneous units (stratification) will reduce the experimental error variance. This is called blocking where differences between experimental units other than those caused by treatment factors are taken into account. This is like comparing age-matched groups (blocks) of a control group with the corresponding blocks in the patients group in an investigation of the side effects of a drug as age itself may cause differences in the experiment. Block effects soak up the extra, uninteresting and already known variability in a model. Blocking is preferable to randomization when the factors that might affect the outcome are known. 

Bonferroni Correction: This is a multiple comparison technique used to adjust the (type I) a error level. See also HLA and Disease Association Studies, Online Bonferroni Correction & a commentary by Perneger, 1998).

Bootstrap: An application of resampling statistics. It is a data-based simulation method used to estimate variance and bias of an estimator and provide confidence intervals for parameters where it would be difficult to do so in the usual way (Online Resampling Book).

Canonical: Something that has been reduced to its simplest form.

Carryover effect: Any effect of a drug that lasts beyond the period of treatment. This is a worry in drug trials with crossover design and the reason for the washout period between treatments.

Case-control study: A design preferred over cohort studies for relatively rare diseases in which cases with a disease or exposure are compared with controls randomly selected from the same study base. This design yields odds ratio as opposed to relative risk from cohort studies. See Case-control Studies Chapter in Epidemiology for the Uninitiated.

Causal relationship: It does not matter how small it is, a P value does not signify causality. To establish a causal relationship, the following non-statistical evidence is required: consistency (reproducibility), biological plausibility, dose-response, temporality (when applicable) and strength of the relationship (as measured by an effect size such as odds ratio/relative risk/hazard ratio). See Hills's criteria of causality; Seven Common Errors in Statistics; and Causality by DR Cox, JR Stat Soc A 1992;155:291 (JSTOR-UK link). The original reference for Hill's criteria is Hill AB: The environment and disease: association or causation. Proc R Soc Medicine 1965; 58:295-300.

Categorical (nominal) variable: A variable that can be assigned to categories. A non-numerical (qualitative) variable measured on a (discrete) nominal scale such as gender, drug treatments, disease subtypes; or on an ordinal scale such as low, median or high dosage. A variable may alternatively be quantitative (continuous or discrete). See GraphPad QuickCalc: Categorical Data Analysis.

Censored observation: Observations that survived to a certain point in time before dropping out from the study for a reason other than having the outcome of interest (lost to follow up or not enough time to have the event). Thus, censoring is simply an incomplete observation that has ended before time-to-event. These observations are still useful in survival analysis.

Central limit theorem: The means of a relatively large (>30) number of random samples from any population (not necessarily a normal distribution) will be approximately normally distributed with the population mean being their mean and variance being the (population variance / n). This approximation will improve as the sample size (the number of samples) increases. See Mathematical Basis; QuickTime Demonstration; JAVA Demonstration; Simulation.

Chi-squared distribution: A distribution derived from the normal distribution. Chi-squared (C2) is distributed with v degrees of freedom with mean = v and variance = 2v (Chi-Square to P Calculator).

Chi-squared test: The most commonly used test for frequency (categorical) data analysis and as a goodness-of-fit test. In theory, it is nonparametric but because it has no parametric equivalent, it is not classified as such. It is not an exact test and with the current level of computing facilities, there is not much excuse not to use Fisher’s exact test for 2x2 contingency table analysis instead of Chi-squared test. Also, for larger contingency tables, the G-test (log-likelihood ratio test) may be a better choice. The Chi-square value is obtained by summing up the values (residual2/expected) for each cell in a contingency table. In this formula, residual is the difference between the observed value and its expected counterpart. See Statistical Analysis in HLA and Disease Association Studies for assumptions and restrictions of the Chi-squared test (Chi-squared tests of association; Tables of critical values of t, F and Chi-square; Chi-square to P calculator; Chi-squared test with GraphPad QuickCalc; 2x2 Table Analysis at SISA).

Cochran's Q Test: A nonparametric test examining change in a dichotomous variable across more than two observations. If there are two observations, McNemar's test should be used.

Coefficient of determination (R2): See Multiple regression correlation coefficient.

Coefficient of variation (CV): It is a measure of spread for a set of data (variation in relation to the mean). Calculated as standard deviation divided by the mean (x100), which expresses the standard deviation as a percentage of the mean. It allows the comparison of the variation in two different datasets. For example, comparing two datasets for the same temperature data measured in Fahrenheit and Celsius would show greater variation for Fahrenheit data if standard deviation is used, but shows equal variation with CV. Similarly, if one wonders whether height shows more variation in males or females, standard deviation would be misleading, but CV will provide the correct comparison. (Online Calculator for Coefficient of Variation and Other Descriptive Statistics).

Cohort effect: The tendency for persons born in certain years to carry a relatively higher or lower risk of a given disease. This may have to be taken into account in case-control studies.

Collider bias: A type of bias that arises from restricting an epidemiologic research study to cases with a condition (the collider variable). It also occurs when statistical adjustment is applied using the collider variable. A collider is different form a confounder and there is and statistical method to adjust for a collider. A collider variable correlates with at least two other variables (the correlating variables “collide” in the collider). Examples in the literature include paradoxical findings such as the association between postmenopausal hormone treatment and coronary heart disease, the birth weight paradox, and the obesity paradox (see Hernan & Monge, 2023 for details). If a conditional analysis yielded an unexpected result, it is worth considering whether the variable used for conditioning may be a collider. Collider bias can also occur in a randomized clinical trial through differential loss to follow-up in comparison groups. See Griffith, 2020 for a discussion of how collider bias may result in paradoxical results in Covid research due to using hospitalized cases. Collider bias results in one of the best-known statistical fallacies known as Berkson’s paradox/bias (see Statistical Literacy). For reviews, see Holmberg & Henderson, 2022; Tonnies, 2022 and Hernan & Monge, 2023.

Concomitant variable: See covariance models.

Conditional (fixed-effects) logistic regression: The conditional logistic regression (CLR) model is used in studies where cases and controls can be matched (as pairs) with regard to variables believed to be associated with the outcome of interest. The model creates a likelihood that conditions on the matching variable. It is the preferred method for the analysis of nested case-control studies when matching is done at the individual level (there may be more than one control per case). In economic analysis, it is called fixed-effects logit for panel data. See Preisser & Koch, 1997.

Confounding variable: A variable that is associated with both the outcome and the exposure variable (also called a lurking variable). A classic example is the relationship between heavy drinking and lung cancer. Here, the data should be controlled for smoking as it is related to both drinking and lung cancer. A positive confounder is related to exposure and response variables in the same direction (as in smoking); a negative confounder shows an opposite relationship to these two variables (age in a study of association between oral contraceptive use and myocardial infarction is a negative confounder). The data should be stratified before analyzing it if there is a confounding effect. Mantel-Haenszel test is designed to analyze stratified data to control for a confounding variable. Alternatively, a multivariable regression model can be used to adjust for the effects of confounders. Despite all efforts to control for unknown confounders, there may always be some residual confounding left due to unknown confounders. To minimize the amount of confounding due to known (like age, sex, ethnicity) or unknown (genetic, behavioral, cultural) confounders, randomization is an option (and the reason for randomized controlled trials to be the gold standard in establishing causality). See Bias & Confounding Lecture Note; a review by Greenland, 2001; a Guide to Confounding; ERIC Notebook; S4BE: Confounding.

Conservative test: A test where the chance of type I error a (false-positivity) is reduced and type II error b (false-negativity) risk is increased. Thus, these tests tend to give larger (less significant) P values compared to non-conservative (liberal) tests for the same comparison.

Contrast: A contrast is combinations of treatment means, which is also called the main effect in ANOVA. It measures the change in the mean response when there is a change between the levels of one factor. For example, in an analysis of three different concentrations of a growth factor on cell growth in cell culture with means m1, m2, m3, against a control value (mo) without any growth factor, a contrast would be:

q = mo - 1/3 (m1+ m2+ m3)

The important point is that the coefficients sum to zero (1/1 - 1/3 - 1/3 - 1/3). If the value of the contrast (q) is zero or not significantly different from zero, there is no main effect, i.e., the combined growth factor mean is not different (positive or negative) from the no growth factor mean.

Controlled experiment: In a controlled experiment, all external variables are fixed and a single exposure (independent) variable is varied among the participants to measure its effect on the outcome (dependent) variable. This way, the researcher can be more confident that the observed relationship is attributable to the exposure (and not confounded). Since complete control of external variables is not always possible in human studies, random allocation (randomization) can be used to attempt equal distribution of uncontrollable and unmeasured variables between the comparative groups. A typical example of a human controlled experiment is a randomized controlled/clinical trial (RCT). The examination of the effect of sun exposure on plant growth by placing the same plants grown in the same soil in a cupboard and another on the window sill, and keeping other exposures the same (like watering) is a prototype controlled experiment.

Cook statistics: A diagnostic influence statistics in regression analysis designed to show the influential observations. Cook's distance considers the influence of the i th value on all n fitted values and not on the fitted value of the i th observation. It yields the shift in the estimated parameter from fitting a regression model when a particular observation is omitted. All distances should be roughly equal; if not, then there is reason to believe that the respective case(s) biased the estimation of the regression coefficients. Relatively large Cook statistics (or Cook's distance) indicates influential observations. This may be due to a high leverage, a large residual or their combination. An index plot of residuals may reveal the reason for it. The leverages depend only on the values of the explanatory variables (and the model parameters). Cook statistics depends on the residuals as well. Cook statistics may not be very satisfactory in binary regression models. Its formula uses the standardized residuals, but the modified Cook statistics uses the deletion residuals.

Confidence intervals: The range of plausible values of a statistic (like an effect size) estimated from a sample that would include the population parameter. If the statistical significance threshold (alpha value) is 0.05, 95% confidence interval is used. A 95% confidence interval indicates that 95% of samples of the size used in the study from the same population would contain the actual (but unknown) population parameter. Margin of error that accompanies the survey results is the same concept.

Correlation coefficient: See Pearson's correlation coefficient (r), Spearman’s rank correlation (rho) and Multiple regression correlation coefficient (R2). (Online Correlation & Regression Calculators at Vassar College.)

Correspondence analysis: In population genetics, a complementary analysis to genetic distances and dendrograms. It displays a global view of the relationships among populations (Greenacre MJ, 1984; Greenacre & Blasius, 1994; Blasius & Greenacre, 1998). With its visual output, it supplements more formal inferential analyzes. This type of analysis tends to give results similar to those of dendrograms as expected from theory (Cavalli-Sforza & Piazza, 1975), but is more informative and accurate than dendrograms especially when there is considerable genetic exchange between close geographic neighbors (Cavalli-Sforza et al. 1994). Cavalli-Sforza et al concluded in their enormous effort to work out the genetic relationships among human populations that two-dimensional scatter plots obtained by correspondence analysis frequently resemble geographic maps of the populations with some distortions (Cavalli-Sforza et al. 1994). Using the same allele frequencies that are used in phylogenetic tree construction, correspondence analysis can be performed on ViSta, VST, Statistica, SAS but most conveniently on MultiVariate Statistical Package (MVSP). Link to a Tutorial; StatSoft Textbook: Correspondence Analysis Chapter.

Cox proportional hazards model: A regression method described by D.R. Cox (J Royal Stat Soc, Series B 1972;34:187-220; JSTOR-UK) for modeling survival times (for significance of the difference between survival times, log-rank test is used). It is also called proportional hazards model because it estimates the ratio of the risks (hazard ratio or relative hazard). As in any regression model, there are multiple predictor variables (such as prognostic markers whose individual contribution to the outcome is being assessed in the presence of the others) and the outcome variable (e.g., whether the patients survived five years, or died during follow-up, etc). The model assumes that the underlying hazard rate (rather than survival time) is a function of the independent variables and consistent over time (proportionality assumption, i.e. the survival functions of the groups are approximately parallel). There is no assumption for the shape and nature of the underlying survival function. Cox's regression model has been the most widely used method in survival data analysis regardless of whether the survival time is discrete or continuous and whether there is censoring (Lee & Go, 1997). Cox regression uses the maximum likelihood method rather than the least squares method (Superlectures on survival analysis).

Covariate (covariable): Generally used to mean any explanatory variable, less generally, an additional explanatory variable that is not of main interest but included in the model to adjust the statistical association of the main explanatory variable. The intention is to produce more precise and adjusted estimates of the association of the explanatory variable of main interest. In the analysis, a model is first fitted using the covariate. Then the main explanatory variable is added and its additional effect is assessed statistically. Whether the use of a covariate is wise (i.e., whether it has a statistically significant influence) can be judged by checking its effect on the residual (error) mean square (variance). If the addition of covariate reduces it remarkably, it will improve the analysis. See also covariance models.

Covariance (covariation): It is a measure of the association between a pair of variables: the expected value of the product of the deviation of two random variables from their respective means. It is also called a measure of ‘linear dependence’ between the two random variables. If the two variables are independent (no linear correlation), then their covariance is zero but for non-zero values covariance is unstandardized (unlike correlation coefficient); there is no limit to possible values. Because of this, it is difficult to compare covariances. A negative value means that for small values of X, there are large values of Y (inverse association). It is calculated as the mean sum-of-products using each xi and yi values, their means mx and my, and (n-1). (The covariance standardized to lie between -1 and +1 is Pearson’s correlation coefficient.) See Wikipedia: Covariance; NetMBA Statistics: Covariance.

Covariance models: Models containing some quantitative and some qualitative explanatory variables, where the chief explanatory variables of interest are qualitative and the quantitative variables are introduced primarily to reduce the variance of the error terms. [Models in which all explanatory variables are qualitative are called analysis of variance -ANOVA- models.] Analysis of covariance -ANCOVA- combines features of ANOVA and regression. It augments the ANOVA model containing the factor effects with one or more additional quantitative variables that are related to the response variable. The intention is to make the analysis more precise by reducing the variance of the error terms. Each continuous quantitative variable added to the ANOVA model is called a concomitant variable (and sometimes also covariates). If qualitative variables are added to an ANOVA model to reduce error variance, the model remains to be ANOVA. By adding extra variables, the results are said to be controlled or adjusted for the additional variables (like age or sex).

Apart from the above use of the term, analysis of covariance is more generally used for almost any analysis assessing the relationship between a response variable and a number of explanatory variables. In a multiple regression model, additional variables, which are known not to have any effect on the response variable, such as age and sex, are sometimes added to the model to adjust the response for these variables (age and sex in this case). Such variables are called confounders (or covariates). When the response is normally distributed, this is the preferred method over a simple t-test when the two groups compared differ, say, in their age and sex distribution (or any other confounding variable). The result is then controlled (or adjusted) for age and sex. When such adjustments are made, the regression coefficient for the significant effect variable will be (most probably) different from the one obtained from a univariable model involving only that variable (say the effect of a disease on pulse rate as compared to healthy controls).

Cramer’s coefficient of association (C): Also known as contingency coefficient. While Chi-squared is used to determine significance of an association (and varies by sample size for the same association), Cramer’s C is a measure of association varying from 0 (no association) to 1 (perfect association) that is independent of the sample size. However, it seldom reaches its upper limit. It allows direct comparison of the degree of association between different contingency tables. It is calculated directly from the Chi-squared value and the total sample size as (C2/C2+N) ½.

Cramer’s V: A measure of the strength association for any size of contingency tables. It can be seen as a correction of the Chi-squared value for sample size. The transformation of the chi-squared value provides a value between 0 and 1 for relative comparison of the strength of the association. For a 2x2 table, Cramer's V is equal to the Phi coefficient. Cramer’s V is most useful for large contingency tables. It can also be used as a global linkage disequilibrium value for multiallelic loci (See GOLD-Disequilibrium Statistics; Online Cramer’s V calculation.)

Crossover design: A clinical trial design during which each subject crosses over from receiving one treatment to another one.

Cross-sectional data: Data collected at one point in time (as opposed to longitudinal/cohort data for example). See Cross-Sectional Studies Chapter in Epidemiology for the Uninitiated.

Degrees of freedom (df): The number of independent units of information in a sample used in the estimation of a parameter or calculation of a statistic. In the simplest example of a 2x2 table, if the marginal totals are fixed, only one of the four cell frequencies is free to vary and the others will be dependent on this value not to alter the marginal totals. Thus, the df is only 1. Similarly, it can easily be worked out that in a contingency table with r rows and c columns, the df = (r-1)(c-1). In parametric tests, the idea is slightly different that the n bits of data have n degrees of freedom before we do any statistical calculations. As soon as we estimate a parameter such as the mean, we use up one of the df, which was initially present. This is why in most formulas, the df is (n-1). In the case of a two-sample t-test with n1 and n2 observations, to do the test we calculate both means. Thus, the df = (n1 + n2 - 2). In linear regression, when the linear equation y = a + bx is calculated, two parameters are estimated (the intercept/constant and the slope). The df used up is then 2: df = (n1 + n2 - 2). Non-parametric tests do not estimate parameters from the sample data, therefore, df is not an issue with them them.

In simple linear regression, the df is partitioned similar to the total sum of squares (TSS). The df for TSS is N-k. Although there are n deviations, one df is lost due to the constraint they are subject to: they must sum to zero. TSS equals to RSS + ESS. In one-way ANOVA, the df for RSS is N-2 because two parameters are estimated in obtaining the fitted line. ESS has only one df associated with it. This is because the n deviations between the fitted values and the overall mean are calculated using the same estimated regression line, which is associated with two df (see above). One of them is lost because the of the constraint that the deviations must sum to zero. Thus, there is only one df associated with ESS. Just like TSS = RSS + ESS, their df have the same relationship: N-1 = (N-2) + 1.

Deletion (or deleted) residual: A modified version of the standardized residual, which uses an estimate of s2 from a regression in which point i has been deleted. It is used in calculation of the modified Cook's distance instead of standardized residuals. Deletion residuals are also known as likelihood residuals.

Descriptive statistics: Summary of available data. Examples are male-to-female ratio in a group; numbers of patients in each subgroup; the mean weight of male and female students in a class, etc. Only when the distribution is symmetric, mean and standard deviation can be used. Otherwise (as in survival data), mean and percentiles/range should be used to describe the data. See GraphPad Guide to Interpreting Descriptive Statistics.

Deviance: A measure for judging the degree of matching of the model to the data when the parameter estimation is carried out by maximizing the likelihood (as in GLMs). The deviance has asymptotically a Chi-squared distribution with df equal to the difference in the number of parameters in the two models being compared. The total deviance compares the fit of the saturated model to the null model, thus, expresses the total variability around a fitted line which can be decomposed to explained and unexplained (error) variability. The residual deviance in a GLM analysis of deviance table corresponds to RSS in an ANOVA table, and regression deviance corresponds to ESS. The residual deviance measures how much fit to the data is lost (in likelihood terms) by modeling compared to the saturated model. This will be a small value if the model is good (and it will be zero for the saturated model containing all main effects and all interactions). The regression deviance measures how much better is the model taking into account the explanatory variables compared to the simplest model ignoring all of them and only containing a constant (the mean of the responses). This measurement is again made in terms of log-likelihood and the bigger the regression deviance the better fits the model (i.e., the regression effect is strong). In likelihood terms, the residual deviance can be expressed as follows:

D = -2 [ln Lc - ln Ls] or D = -2 ln [Lc / Ls]

where Lc is the likelihood of the current model, and Ls is the likelihood of the saturated model.

Similarly, the regression deviance can be expressed as:

D = -2 [ln Lc - ln Ln] or D = -2 ln [Lc / Ln]

where Ln is the likelihood of the null model. The bigger the regression deviance the better the model including this particular variable.

For purposes of assessing the significance of an independent variable, the value of D with and without the independent variable is compared (note the nested character of the two sets). This is called the deviance difference. If a variable is dropped, the residual deviance difference, if a variable is added, the regression deviance difference is compared with the C2 distribution.

Deviance difference: In generalized linear modeling, models that have many explanatory variables may be simplified, provided information is not lost in this process. This is tested by the difference in deviance between any two nested models being compared:

G = D (for the model without the variable) - D (for the model with the variable)

The smaller the difference in (residual) deviance, the smaller the impact of the variables removed. This can be tested by the C2 test.

In the simplest example (in simple linear regression), if the log-likelihood of a model containing only a constant term is L0 and the model containing a single independent variable along with the constant term is L1, multiplying the difference in these log-likelihoods by -2 gives the deviance difference, i.e., G = -2 (L1 - L0). G statistics (likelihood ratio test) can be compared with the C2 distribution on df = 1, and it can be decided whether the model with the independent variable has a regression effect (if P < 0.05, it has). The same method can be used to detect the effect of the interaction by adding any interaction to the model and obtaining the regression deviance. If this deviance is significantly higher than the one without the interaction in the model, there is interaction [the coefficient for the interaction, however, does not give the odds ratio in logistic regression].

Discrete variable: A variable of countable number of integer outcomes. Examples include -ordinal multinomial- several prognostic outcomes (such as poor, median and good) as a function of treatment modalities, stage of the disease, age etc., or -multinomial- people’s choices of hospitals (hospital A, B or C) as a function of their income level, age, education etc. A discrete variable may be binomial: diseased or non-diseased in a cohort or case-control study. The nature of the outcome variable as discrete or continuous is crucial in the choice of a regression model (see the generalized linear model).

Dummy variables: A binary variable that is used to represent a given level of a categorical variable. In genetic data analysis, for example, it is created for each allele at a multiallelic locus. The most common choices as dummy variables are 0 and 1. For each level of the variable, 1 denotes having the feature and 0 denotes all other levels. Also called indicator variables. If an indicator variable is used, the regression coefficient gives the change per unit compared to the reference value. Creating dummy variables on Stata: Stata @ UCLA Statistics: STATA Dummy Variables.

Dunn's Test: This test is used when a difference between the groups is found in a non-parametric ANOVA test. Dunn's test is a post hoc test that makes pairwise (multiple) comparisons to identify the different group. See GraphPad Prism Guide: K-W and Dunn's Test; StatTools: Dunn's Test).

Dunnett's test: When ANOVA shows a significant difference between groups, if one of the groups is a control (reference) group, Dunnett's Test is used as a post hoc test. This multiple comparison test can be used to determine the significant differences between a single control group mean and the remaining treatment group means in an analysis of variance setting. It is one of the least conservative post hoc tests.

Ecological study: Analyses based on data grouped to the municipal, provincial or national level. See Ecological Studies Chapter in Epidemiology for the Uninitiated.

Ecological fallacy: The aggregation bias, which is the unfortunate consequence of making inferences for individuals from aggregate data. It results from thinking that relationships observed for groups necessarily hold for individuals. The problem is that it is not valid to apply group statistics to an individual member of the same group. See an essay on Ecological Fallacy.

Edwards' test: A statistical test for seasonality that looks for a one-cycle sinusoidal deviation from the null distribution (see Westerbreek et al, 1998).

Effect modification: The situation in which a measure of effect changes over values of another variable (the association estimates are different in different subpopulations of the sample). The relative risk or odds ratio associated with exposure will be different depending on the value of the effect modifier. For example if in a disease association study, the odds ratios are different in different age groups or in different sexes, age or sex are effect modifiers. Effect modification is highly related to statistical interaction in regression models. If where an exposure decreases risk for one value of the effect modifier and increases risk for another value of effect modifier, this is called crossover. See Thompson, 1991 and Effect Modification in Encyclopedia of Biostatistics.

Effect size: In statistics, effect size is the measure of the strength of an association or effect. It is the measure of clinical/biological significance. In epidemiologic research, it is usually quantified by calculation of relative risk, odds ratio or hazard ratio. Effect size (together with its 95% confidence interval) complements the P value and should always accompany it when an association is reported. The practice of exclusive reporting of the P values for associations without an effect size is unacceptable. Other effect sizes include absolute risk reduction (ARR), number needed to treat (NNT) and difference in means. Effect size is one of the determinants of statistical power (in addition to sample size, variability and the statistical significance threshold or alpha value).

Eigenvalues (latent values): In multivariate statistics, eigenvalues give the variance of a linear function of the variables. Eigenvalues measure the amount of the variation explained by each principal component (PC) and will be largest for the first PC and smaller for the subsequent PCs. An eigenvalue greater than 1 indicates that PCs account for more variance than accounted by one of the original variables in standardized data. This is commonly used as a cut-off point for which PCs are retained.

EM Algorithm: A method for calculating maximum likelihood estimates with incomplete data. E (expectation)-step computes the expected values for missing data and M (maximization)-step computes the maximum likelihood estimates assuming complete data. It was first used in genetics (Ceppellini R et al, 1955) to estimate allele frequency for phenotype data when genotypes are not fully observable (this requires the assumption of HWE and calculation of expected genotypes from phenotype frequencies). Arlequin implements EM algorithm in haplotype construction and frequency analysis.

Empirical P value: A P value obtained by a simulation program such as Monte Carlo statistics. This is the preferred method to rule out chance findings in the context of multiple comparisons.

Empirical rule: In variables normally distributed, 68% of the data values are within 1SD of the mean; 95% are within 2SD of the mean; and 99.7% (nearly all) are within 3SD of the mean.

Epidemiologic flaws and fallacies: Beware of confounders, selection bias, response bias, variable observer, Hawthorne effect (changes caused by the observer in the observed values), diagnostic accuracy bias, regression to the mean (see also Barnett, 2005), significance Turkey, nerd of non-significance, cohort effect, ecological fallacy, Berkson bias (selection bias in hospital-based studies) and others (discussed in M Michael III et al. Biomedical Bestiary. Little, Brown and Company, 1984; and in Bias & Confounding in Molecular Epidemiology). See also Online Catalog of (Epidemiologic) Bias.

Epi-Info: An epidemiologic data management and analysis package freely available from the CDC website. Originally a DOS program, later versions are designed for Microsoft Windows (User Guide; Tutorial).

Error terms: The difference between an observed/predicted value and the true value is the error or epsilon (e) in statistics. Error terms are shown as Wi or ei. Their expected value is zero (which is the assumption of many statistical tests especially in regression analysis), thus, they vary around zero with a variance equal to s2: N(0, s2). As a major assumption of the regression analysis, they are assumed to be normally distributed, have equal variance for all fitted values, and independent. Normality is a reasonable assumption in many cases. The assumption of equal variance implies that every observation on the dependent variable contains the same amount of information. The impact of heterogeneous variances is a loss of precision of estimates compared to the precision that would have been realized if the heterogeneous variances had been taken into account. Transformation of the dependent variable may help to homogenize the unequal variances. Correlated errors are most frequent in time sequence data and they also cause the loss in precision in the estimates. See also residuals.

Explained (regression) sum of squares (ESS): The measure of between treatments sum of squares (variability) in ANOVA. If the means of treatment groups are different, the ESS would be greater than RSS to yield a high variance ratio. The bigger the ESS, the better explained the data by the model.

Exploratory data analysis: An initial look at the data with minimal use of formal mathematics or statistical methods, but more with an informal graphical approach. Scatter plots, correlation matrices and contingency tables (for binary data) can be used to get an initial idea for relationships between explanatory variables (for collinearity) or between an explanatory variable(s) and a response variable(s) (correlation). In ANOVA, normality can be checked by box-plots. It gives some indication of which variables should be in the model and which one of them should be put into the model first, and whether linear relationship is adequate.

Exponential distribution: The (continuous) distribution of time intervals between independent consecutive random events like those in a Poisson process.

Exponential family: A family of probability distributions in the form

f(x) = exp {a(q )b(x) + c(q ) + d(x)}

where q is a parameter and a, b, c, d are known functions. This family includes the normal distribution, binomial distribution, Poisson distribution and gamma distribution as special cases.

Exposure: In an epidemiologic study, exposure may represent an environmental exposure, an intervention or the presence of a marker (biomarker/genetic marker).

Factor: A categorical explanatory variable with small number of levels such that each item classified belongs to exactly one level for that category. If the factor is 'sex', the levels are 'male' and 'female'; if the factor is 'drug received', the levels are 'drug A', 'drug B', 'drug C', etc. A set of factor levels, uniquely defining a single treatment, is called a cell. A cell may have just one observation (no replication) or multiple observations (replications).

Factorial experiments: In some data, the explanatory (predictor) variables are all categorical (i.e., factors) and the response variable is quantitative. When there are two or more categorical predictor variables, the data are called factorial. The different possible values of the factors are often assigned numerical values known as levels.

Factorial analysis of variance: An analysis in which the treatments differ in terms of two or more factors (with several levels) as opposed to treatments being different levels of a single factor as in one-way ANOVA.

False discovery rate (FDR): False discovery rate estimates the proportion of false positive results among those tests that are being reported as statistically significant. Link to a calculator (for 2x2 tables).

Family-wise error rate (FWER; designated as z): The type I error probability (a) for a single, individual endpoint estimates the occurrence of a false-positive result for a single analysis. The family-wise error level is about the occurrence of at least one type I error in the entire set of analyses. Thus, FWER is the probability that at least one type I error has occurred across all of the analyses. Bonferroni correction for the number of comparisons is an example of adjustment for the FWER (if corrected P values smaller than 0.05 are expected as statistically significant, FWER is set at 0.05).

F distribution: A continuous probability distribution of the ratio of two independent random variables, each having a Chi-squared distribution, divided by their respective degrees of freedom. The commonest use is to assign P values to mean square ratios (variance ratios) in ANOVA. In regression analysis, the F-test can be used to test the joint significance of all variables of a model. (Tables of critical values of t, F and Chi-square).

Fisher's exact test: An exact significance test to analyze 2x2 tables for any sample size. It is a misconception that it is suitable only for small sample sizes. This arises from the demanding computational procedure for large samples, which is no longer an issue. It is the only test for a 2x2 table when an expected number in any cell is smaller than 5 (Online Fisher's Test (1); (2); Calculator 3 in Clinical Research Calculators at Vassar). For an exact test for larger contingency tables, see Vassar Online or download RxC by Mark Miller.

F test: The F test for linear regression tests whether the slope is significantly different from 0, which is equivalent to testing whether the fit using non-zero slope is significantly better than the null model with 0 slope. See also mean squares.

Gambler's ruin: A classical topic in probability theory. It is a game of chance related to a series of Bernoulli trials. There are variations of the game theory associated with problems of the random walk and sequential sampling.

Game theory: The theory of contests between two or more players under specified sets of rules. The statistical aspect is that the game proceeds under a chance scheme such as throwing a die.

Gaussian distribution: Another name for the normal distribution (GraphPad Gaussian Distribution Calculator).

G Statistics: An application of the log-likelihood ratio statistics for the hypothesis of independence in an r x c contingency table. It can also be used to test goodness-of-fit. The G-test should be preferred over Chi-squared test when for any cell in the table, ½ O-E½ > E. The Chi-squared distribution is usually poor for the test statistics G2 when N/rc is smaller than five (preferable to the Chi-squared test in Hardy-Weinberg Equilibrium (HWE) test as long as this condition is met). HyperStat and StatXact perform G statistics (Online G Statistics).

General linear model: A group of linear regression models in which the response variable is continuous and normally distributed, the response variable values are predicted from a linear combination of predictor variables, and the linear combination of values for the predictor variables is not transformed (i.e., there is no link function as in generalized linear models). Linear multiple regression is a typical example of general linear models whereas simple linear regression is a special case of generalized linear models with the identity link function.

Generalized linear model (GLM): A model for linear and non-linear effects of continuous and categorical predictor variables on a discrete or continuous but not necessarily normally distributed dependent (outcome) variable. (Note that in the general linear model, the dependent (outcome) variable should be normally distributed). Normal, binary (or linear logistic; when the outcome variable is a proportion), binomial or Poisson (when the outcome variable is a count), exponential and gamma (when the outcome variable is continuous and non-negative) models are different versions of generalized linear models. Particular types of models arise by specifying an appropriate link function, variance and distribution. For example, normal linear regression corresponds to an identity link function, constant variance and a normal distribution. Logistic regression arises from a logit link function and a binomial distribution (the variance of the response (npq) is related to its mean (np): variance = mean (1 - (mean/n)). Loglinear models are used for binomial or Poisson counts. Standard techniques for analyzing censored survival data, such as the Cox regression, can also be handled within the GLM framework (Online GLM, GLM Website, Lecture Notes of a GLM Course).

Genetic distance: A measurement of genetic relatedness of populations. The estimate is based on the number of allelic substitutions per locus that have occurred during the separate evolution of two populations. Link to a lecture on Estimating Genetic Distance and GeneDist: Online Calculator of Genetic Distance. The software Arlequin, PHYLIP, GDA, PopGene and SGS are suitable to calculate population-to-population genetic distance from allele frequencies. See Basic Population Genetics.

Genetic Distance Estimation by PHYLIP: The most popular (and free) phylogenetics program PHYLIP can be used to estimate genetic distance between populations. Most components of PHYLIP can be run online. One component of the package GENDIST estimates genetic distance from allele frequencies using one of the three methods: Nei's, Cavalli-Sforza's or Reynold's (see papers by Nei et al, 1983, Nei M, 1996 and a lecture note for more information on these methods). GENDIST can be run online using the default options (Nei's genetic distance) to obtain genetic distance matrix data. The PHYLIP program CONTML estimates phylogenies from gene frequency data by maximum likelihood under a model in which all divergence is due to genetic drift in the absence of new mutations (Cavalli-Sforza's method) and draws a tree. The program comes as a freeware as part of PHYLIP or this program can be run online with default options. If new mutations are contributing to allele frequency changes, Nei's method should be selected on GENDIST to estimate genetic distances first. Then a tree can be obtained using one of the following components of PHYLIP: NEIGHBOR also draws a phylogenetic tree using the genetic distance matrix data (from GENDIST). It uses either Nei's "Neighbor Joining Method," or the UPGMA (unweighted pair group method with arithmetic mean; average linkage clustering) method. Neighbor Joining is a distance matrix method producing an unrooted tree without the assumption of a clock (UPGMA does assume a clock). NEIGHBOR can be run online. Other components of PHYLIP that draw phylogenetic trees from genetic distance matrix data are FITCH / online (does not assume evolutionary clock) and KITSCH / online (assumes evolutionary clock).

Geometric mean: G = (x1.x2...xn)1/n where n is the sample size. This can also be expressed as antilog ((1/n) S log x), which means the antilog of the mean of the logs of each value. See Applications of the Geometric Mean; Spizman, 2008: Geometric Mean in Forensic Economy.

Greek letters: It is customary to use Greek letters for (usually unknown) population parameters and Roman letters/numerals for estimated sample statistic(s). Note that most Greek letters used in statistics/mathematics correspond to the initial of English word for the same parameter (like mu (µ) and mean; sigma (lowercase) (s) and standard deviation; Sigma (uppercase) (S) and Sum; Pi (P) and Product; epsilon (€) and error etc). The first two Greek letters alpha (a) and beta (b) are reserved for type I (a) and type II (b) errors in statistics. See complete lists of statistical symbols at link1; link2; link3; link4; link5.

Half-normal plot: A diagnostic test for model inadequacy or revealing the presence of outliers. It compares the ordered residuals from the data to the expected values of ordered observations from a normal distribution. While the full-normal plots use the signed residuals, half-normal plots use the absolute values of the residuals. Outliers appear at the top right of the plot as distinct points, and departures from a straight line mean that the model is not satisfactory. It is appropriate to use a half-normal plot only when the distribution is symmetrical about zero because any information on symmetry will be lost.

Haplotype Relative Risk method: This method uses non-inherited parental haplotypes of affected persons as the control group and thus eliminates the risks and bias associated with using unrelated individuals as controls in case-control association studies, as well as the higher cost (see Falk & Rubinstein, 1987; Knapp, 1993; Terwilliger & Ott, 1992).

Hardy-Weinberg equilibrium (HWE): In an infinitely large population, gene and genotype frequencies remain stable as long as there is no selection, mutation, or migration. For a bi-allelic locus where the gene frequencies are p and q: p2+2pq+q2 = 1 (see Hardy-Weinberg parabola). HWE should be assessed in controls in a case-control study and any deviation from HWE should alert for genotyping errors (Lewis, 2002) unless there are biological reasons for any deviation (see  Ineichen & Batschelet, 1975 for the effect of natural selection on Hardy-Weinberg equilibrium). (Online HWE Analysis; HWE and Association Testing for SNPs in Case-Control Studies; OEGE: HWE Calculator; Population Genetics Notes).

Harmonic mean: Of a set of numbers (y1 to yn), the harmonic mean is the reciprocal of the arithmetic mean of the reciprocal of the numbers [H = N / (1/(y1 + y2 + .... yn))]. The harmonic mean is either smaller than or equal to the arithmetic mean. It is a measure of position.

Hazard function (instantaneous failure rate, conditional failure, intensity, or force of mortality function): The function that describes the probability of failure during a very small time increment (assuming that no failures have occurred prior to that time). Hazard is the slope of the survival curve – a measure of how rapidly subjects are having the event (dying, developing an outcome etc).

Hazard rate: It is a time-to-failure function used in survival analysis. It is defined as the probability per time unit that a case that has survived to the beginning of the respective interval will fail in that interval. Specifically, it is computed as the number of failures per time units in the respective interval, divided by the average number of surviving cases at the mid-point of the interval.

Hazard ratio (relative hazard): Hazard ratio is an effect size which compares two groups differing in exposures (including treatments), prognostic variables or other (binary) characteristics in survival analysis. HR is the relative risk/hazard, associated with an exposure, of suffering an event (like relapse or death) in a fixed period of time. The response variable is a survival time (time-to-event), and the coefficients of the (regression) model correspond to logs of hazard ratios. If the hazard ratio is 2.0, then the rate of failure (adverse outcome) in one group is twice the rate in the other group. The computation of the hazard ratio assumes that the ratio (i.e., proportional hazard) is consistent over time (so that at least the survival curves do not cross), and that any differences are due to random sampling. When performing any tests of hypotheses to compare survival curves, the proportionality of hazards assumption should be checked (and should hold for the validity of Cox's proportional hazard models). (See also Log-rank test).

Hetereoscedastic data: Data that have non-constant (heterogeneous) variance across the predicted values of y. In this case, residual graph will show varying variability across the fitted values. This is a regression diagnostic problem and should be fixed by transforming the data. See also Homoscedasticity.

Heuristics: A term in computer science that refers to guesses made by a program to obtain approximately accurate results. Frequently used in phylogenetics and computational biology.

Hierarchical model: In linear modeling, models which always include all the lower-order interactions and main effects corresponding to any interaction they include.

Hierarchy of strength of evidence: The ranking of epidemiologic research designs for the strength of the evidence they provide for causality. The highest ranking design is the controlled interventional studies (prototype: randomized controlled/clinical trials/RCTs). This design minimizes bias/confounding (due to random allocation of subjects to comparative groups), reduces chance effects due to large sample sizes and sufficient statistical power to begin with, and eliminates reverse causation by virtue of being longitudinal studies (and simply by mainly being drug trials). Thus, by minimizing alternative explanations (chance/random variations; bias/confounding; reverse causality), randomized controlled trials provide the strongest evidence for causality of the observed relationship.

Historical fallacy: The mistake of assuming that an association observed in cross-sectional data will be similar to that observed in longitudinal data or vice versa.

Homoscedasticity (homogeneity of variance): Normal-theory-based tests for the equality of population means such as the t-test and analysis of variance, assume that the data come from populations that have the same variance, even if the test rejects the null hypothesis of equality of population means. If this assumption of homogeneity of variance is not met, the statistical test results may not be valid. Heteroscedasticity refers to lack of homogeneity of variances.

Hotelling's T2 test: This is a generalization of Student's t-test for multivariate data. Designed to provide a global significance test for the difference between two groups with simultaneously measured multiple dependent/outcome variables and multiple explanatory/independent variables. It can also be used for one group with simultaneously measured multiple dependent outcome variables (another test similar to Hotelling's T2 test is Mahalanobis's D2 test). See multivariate analysis.

Hypergeometric distribution: A probability distribution of a discrete variable generally associated with sampling from a finite population without replacement. An example may be that given a lot with 25 good units and five faulty. The probability that a sample of five will yield not more than one faulty item follows a hypergeometric distribution.

Index plot: An index plot plots each residual, leverage, or Cook's distance against the corresponding observation (row) number (i or index) in the dataset. In many cases, the row number corresponds to the order in which the data were collected. If this is the case, this would be similar to plotting the residuals (or another diagnostic quantity) against time. The index plot is a helpful diagnostic test for normal linear and particularly generalized linear models. Both outliers and influential points can be detected by the index plot. It is particularly useful when the data is in time order so that pattern in the residuals, etc. over time can be detected. If a residual index plot is showing a trend in time, then they are not independent (violation of a major assumption of linear regression).

Inferential statistics: Making inferences about the population from which a sample has been drawn and analyzed.

Influential points: Observations that actually dominate a regression analysis (due to high leverage, high residuals or their combination). The method of ordinary least squares gives equal weight to every observation. However, every observation does not have equal impact on the least squares results. The slope, for example, is influenced most by the observations having values of the independent variable farthest from the mean. An observation is influential if deleting it from the dataset would lead to a substantial change in the fit of the generalized linear model. High-leverage points have the potential to dominate a regression analysis but not necessarily exert an influence (i.e., a point may have high leverage but low influence as measured by Cook statistics). Cook statistics is used to determine the influence of a data point on the model.

Interaction: If the effect of one factor depends on the level of another factor, the two factors involved are said to interact, and a contrast involving all these levels is called their interaction. Factors A and B interact if the effect of factor A is not independent of the level of factor B. For example, when there are two main effects on a response variable, if their combined effect is higher than the sum of their main effects due to a bonus (say, the effects of a kind of exercise and a kind of diet on blood lipid levels), they have an interaction (meaning a simple additive model is not sufficient to account for the observed data and a multiplicative term must be added). Briefly, interaction is a deviation from additivity. Also, there would be an interaction between the factors sex and treatment if the effect of treatment was not the same for males and females in a drug trial. Interaction is closely linked with effect modification in epidemiology (see Genetic Epidemiology Glossary; Wikipedia: Statistical Interaction).

Intercept: In linear regression, the intercept (also called the constant) is the mean value of the response variable when the explanatory variable takes the value of zero (the value of y when x=0). See Statistics by Jim: Intercept.

Interpolation: Making deductions from a model for values that lie between data points. Making deductions for values beyond the data points is called extrapolation and the results are not valid.

Interquartile range (dQ or IQR): dQ / IQR is a measure of spread and is the counterpart of the standard deviation for skewed distributions. It is the distance between the upper (75th percentile) and lower (25th percentile) quartiles (Q­U-QL).

Interval variable (equivalent to continuous variable): A quantitative variable measured on a scale with constant intervals (like days, milliliters, kilograms, miles so that equal-sized differences on different parts of the scale are equivalent) where the zero point and unit of measurement are arbitrary. When temperature is measured on two scales, Fahrenheit and Centigrade, the zero points in these two scales do not correspond, and a 10% increase in Fahrenheit (from 500 to 550) is not a 10% increase in the corresponding Centigrade scale (10o to 12.8o = 2.8%); these two measurements cannot be mixed or compared. For estimation of correlation coefficients, data should be interval type (See also ratio variable and variable).

Kendall's Coefficient of Concordance: An estimate of the degree of correlation (a correlation effect size). This estimate is useful when the dependent variable and all the independent variables are ordinal.

Kolmogorov-Smirnov two-sample test: A non-parametric test applicable to continuous frequency distributions. It is considered to be the equivalent of the C2-test for quantitative data and has greater power than the G-statistics or C2-test for goodness of fit especially when the sample size is small. It can be used to compare two independent groups. The test is based on differences between two cumulative relative frequency distributions (it compares the distributions not the parameters). Thus, the Kolmogorov-Smirnov test is also sensitive to differences in the general shapes of the distributions in the two samples such as differences in dispersion, skewness. Its interpretation is similar to that of the Wald-Wolfowitz runs test. Online Kolmogorov-Smirnov “One-Sample” Test at Vassar.

Kruskal-Wallis test (One-way ANOVA by ranks): It is one of the non-parametric tests equivalent to one-way ANOVA that are used to compare multiple (k > 2) independent samples. This test assesses the hypothesis that the different samples in the comparison were drawn from the same distribution or from distributions with the same median. It can be used to analyze ordinal variables. It is an extension of the Mann-Whitney (U) test (for two independent samples). The interpretation of the Kruskal-Wallis test is identical to that of one-way ANOVA, but is based on ranks rather than means. 

Kurtosis: Kurtosis is a measure of whether the data are peaked or flat in its distribution relative to a normal distribution (whose kurtosis is zero). Positive kurtosis indicates a ‘peaked’ distribution and negative kurtosis indicates a ‘flat’ distribution (data sets with high kurtosis have a distinct peak near the mean and decline rapidly; data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak) (Definition of Kurtosis and Skewness; Online Skewness-Kurtosis Calculator). See also skewness.

Large sample effect: In large samples, even small or trivial differences can become statistically significant. This should be distinguished from biological/clinical importance.

Least squares method: A method of fitting a straight line or curve based one minimization of the sum of squared differences (residuals) between the predicted and the observed points. Given the data points (xi, yi), it is possible to fit a straight line using a formula, which gives the y=a+bx. The gradient of the straight line b is given by [S(xi - mx)(yi-my)] / [(S(x-mx))2], where mx and my are the means for xi and yi. The intercept/constant “a” is obtained by my - bmx. See Wikipedia: Least Squares.

Lead-time bias: Overestimation of survival time due to earlier diagnosis of disease. Actual time of death does not change when lead-time bias is present despite the earlier time of diagnosis, but time-to-death is longer.

Leverage points: In regression analysis, these are the observations that have an extreme value on one or more explanatory variable. The leverage values indicate whether or not X values for a given observation are outlying (far from the main body of the data). A high leverage value indicates that the particular observation is distant from the center of the X observations. High-leverage points have the potential to dominate a regression analysis but not necessarily influential. If the residual of the same data point and Cook's distance are also high, then it is an influential point. See also influential points.

Likelihood: The probability of a set of observations given the value of some parameter or set of parameters. For example, the likelihood of a random sample of n observations (x1 to xn) with probability distribution f(x; q ) is given by: L = P f(xi; q 0). This function, which applies equally to continuous density and discrete mass functions, is the basis of maximum likelihood estimation.

Likelihood distance test (likelihood residual, deletion residual): This test is based on the change in deviance when one observation is excluded from the dataset. It uses the difference between the log-likelihood of the complete dataset and the log-likelihood when a particular observation is removed. A relatively large difference indicates that the observation involved is an outlier (poorly fitted by the model).

Likelihood ratio test: A general purpose test of hypothesis Ho against an alternative H1 based on the ratio of two likelihood functions one derived from each of Ho and H1. The statistics l is given by l = -2 ln (LH0 / LH1) has approximately a C2 distribution with df equal to the difference in the number of parameters in the two hypotheses. One application of this test is the G-test, which is used in categorical data analysis as a goodness-of-fit or independence test (the tests statistics has a Chi-squared distribution).

Linear expression: A polynomial expression with the degree of polynomial being 1. It will be something like, f(x)=2x1+3, but not x2+2x+4.

Linear logistic model: A linear logistic model assumes that for each possible set of values for the independent (X) variables, there is a probability p that an event (success) occurs. Then the model is that Y is a linear combination of the values of the X variables: Y = bo + b1*X1 + b2*X2 + b3*X3 + … bk*Xk, where Y is the logit transformation of the probability p. Logistic in statistical usage stems from logit and has nothing to do with the military use of the word which means the provision of material.

Linear regression models: In the context of linear statistical modeling, 'linear' means linear in the parameters (coefficients), not the explanatory variables. The explanatory variables can be transformed (say, x2), but the model will still be linear if the coefficients remain linear. When the overall function (Y) remains a sum of terms that are each an X variable multiplied by a coefficient, the function Y is said to be linear in the coefficients. A non-linear model is different in that it has a non-constant slope (a tutorial on Simple Linear Regression; see also Vassar College; Excel macro for Linear Correlation & Regression).

Linkage disequilibrium: Also called gametic association, which is more appropriate. It means increased probability for two or more alleles to be on the same chromosome at the population level. In a population at equilibrium, haplotype frequency is obtained by multiplying the allele frequencies x2. When there is linkage disequilibrium, the observed frequency (say by family analysis or sperm typing) is different from the expected frequency. The difference gives the D value (D for difference), which can be tested for significant difference from 0 by a 2x2 table analysis (for two alleles). The D value can be negative or positive. Linkage disequilibrium can derive from population admixture, tight linkage or elapse of insufficient time for the population to reach equilibrium. A classic example in immunogenetics is the HLA-A1-B8-DR3 haplotype which shows significant linkage disequilibrium extending over 6.5 Mb. Software for LD estimation: Genetic Data Analysis, EH, 2LD, MLD, LDA, PopGene, Online Linkage Disequilibrium Analysis, Genotype2LDBlock (online). For more, see Basic Population Genetics.

Link function: A particular function of the expected value of the response variable used in modeling as a linear combination of the explanatory variables. For example, logistic regression uses a logit link function rather than the raw expected values of the response variable; and Poisson regression uses the log link function. The response variable values are predicted from a linear combination of explanatory variables, which are connected to the response variable via one of these link functions. In the case of the general linear model for a single response variable (a special case of the generalized linear model), the response variable has a normal distribution and the link function is a simple identity function (i.e., the linear combination of values for the predictor variable(s) is not transformed).

LOD Score: Stands for the logarithm of odds. It is a statistical measure of the likelihood that two genetic markers occur together on the same chromosome and are inherited as a single unit of DNA (co-segregation). The LOD score serves as a test of the null hypothesis of free recombination versus the alternative hypothesis of linkage. Determination of LOD scores requires pedigree analysis and a score of >3 is traditionally taken as evidence for linkage. Linkage is between two genetic loci but not alleles. An example is the linkage between the hemochromatosis gene (HFE) and HLA-A. This means that within the same family all affected subjects will have the same HLA-A allele but not necessarily a particular one, i.e., there will be no recombination between HFE and HLA-A. LOD score has nothing to do with linkage disequilibrium.

Log 0 is undefined. If we need to use a log transformation but some data values are 0, the usual way to get round this problem is to add a small positive quantity (such as 1/2) to all the values before taking logs.

Log-rank test (of equality across strata): A non-parametric test of significance for the difference between two survival curves (for categorical data). It is a special application of the Mantel-Haenszel test. It can be adjusted for confounders (not preferable to Cox proportional hazard regression which is a semi-parametric model), or performed for trend (for details and parametric alternatives, see Lee & Go, 1997). It was developed by Mantel and Haenszel as an adaptation of the Mantel-Haenszel C2 test. The other commonly used nonparametric tests for comparison of two survival distributions are the Gehan's Generalized Wilcoxon test (Gehan 1965a & 1965b). The log-rank test is appropriate for survival distributions whose hazard functions are proportional over time, i.e. the two survival curves do not cross (proportionality assumption). Otherwise, the Gehan's Wilcoxon test is recommended. The Wilcoxon statistic puts more weight on early deaths compared to the log-rank (Lee & Go, 1997). See UCLA Stata: Survival Analysis & Log-rank Test; BMJ Statistics Notes: Logrank Test.

Log transformation: This transformation pulls smaller data values apart and brings the larger data values together and closer to the smaller data values (shrinkage effect). Thus, it is mostly used to shrink highly positively skewed data.

Logistic (binary) regression: A statistical analysis most frequently models the relationship between a dichotomous (binary) outcome variable (such as diseased or healthy; dead or alive; relapsed or not relapsed), and a set of explanatory variables of any kind (such as age, HLA type, blood pressure, kind of treatment, disease stage etc). It can also be used when the outcome variable is polytomous (several categories of the prognosis; including ordinal response 'ordinal logistic regression' or 'proportional odds ratio model'), and when there are several outcome variables (multinomial logistic regression - a special class of loglinear models). Analysis of data from case-control studies via logistic regression can proceed in the same way as cohort studies. See Logistic Regression Lecture Note, Online Logistic Regression, Logistic Regression with SPSS, STATA and SAS; Power Calculation for Logistic Regression (including Interaction).

Logit transformation: The logit (or logistic) transformation Y of a probability p of an event is the logarithm of the ratio between the probability that the event occurs (p) and that the event does not occur (1-p): Y = ln (p/(1-p)). Thus it is a transformation of a binary (dichotomous) response variable. The logit transformation of p is also known as the log odds of p, since it is the logarithm of the odds. There are other link functions for binary response variables.

Loglinear model: Multinomial count data (from contingency tables of more three or more nominal variables) can be fitted by using a generalized linear model with a Poisson response distribution and a log link function. The resulting models for counts in the cells of a contingency table are known as loglinear models in which the logarithm of the expected value of a count variable is modeled as a linear function of parameters. Loglinear models try to model all important associations among all variables. In this respect, they are related to ANOVA models for quantitative data. Loglinear modeling allows more than two discrete variables to be analyzed and interactions between any combination of them to be identified. Associations between variables in log-linear models are analogous to interactions in ANOVA models. The aim is to find a small model, which achieves a reasonable fit (small residual). Data sets with a binary response (outcome) variable and a set of explanatory variables that are all categorical can be modeled either by logistic regression or by loglinear modeling. More on Loglinear Models and Online Loglinear Test at Vassar College. The loglinear model can be used to find a conditional probability involving the factors in the contingency table for one of the factors chosen as a response variable. When the factor chosen as a response variable is binomial, logistic regression can be used to analyze the same data (a binomial response variable and categorical explanatory variables). Logistic regression substitutes the loglinear model with equal results as long as the fitted loglinear model includes the main effects of the response variable and a saturated model for all the explanatory variables.

Longitudinal data: Data collected over a period of time as in cohort studies. These data are usually analyzed by using survival analysis techniques. See Longitudinal Studies Chapter in Epidemiology for the Uninitiated.

Lurking variable: See confounding variable.

Mann-Whitney (U) test: A non-parametric test for comparing the distribution of a continuous variable between two independent groups. It is analogous to the independent two-sample t-test, so that it can be used when the data are not normally distributed. The Wilcoxon signed ranks T-test for independent samples is another non-parametric alternative to the t-test in this context (for paired samples, Wilcoxon matched pairs signed rank test should be used) (online Mann-Whitney Test).

Mantel-Haenszel C2 test (also called Cochran-Mantel-Haenszel (CMH) test): Test for a null hypothesis of no overall relationship in a series of 2x2 tables for stratified data derived either from a cohort or a case-control study. It allows analysis of confounding and gives an adjusted odds ratio or relative risk. It can be used on categorical or categorized continuous data. The test is only valid when the variance of observed data is ³ 5. It is inappropriate when the association changes dramatically across strata (heterogeneity is usually tested by Breslow-Day test). It is, however, applicable for sparse data sets for which asymptotic theory does not hold for G2. The test statistics, M2, has approximately a Chi-squared distribution with df = 1 (see a review by Stefano & Ezio, 2007; online Mantel-Haenszel Test for a single table). Mantel & Haenszel’s 1959 JNCI paper is now a citation classic.

Markov Chain Monte Carlo (MCMC, random walk Monte Carlo) methods: See Wikipedia: MCMC; MCMC Applet; Markov Chain Simulation Applets; Buffon's Needle Applet; Monte Carlo Methods Links; MCMC Tests of Genetic Equilibrium (Lazzeroni & Lange, 1997); Markov Chain Monte Carlo in Practice (Book) and Virtual Labs: Markov Chain (requires MathPlayer). See also Metropolis-Hastings algorithms.

Maximum likelihood: This method is a general method of finding estimated values of parameters. It yields values for the unknown parameters, which maximize the probability of obtaining the observed values. The estimation process involves considering the observed data values as constants and the parameter to be estimated as a variable, and then using differentiation to find the value of the parameter that maximizes the likelihood function. First a likelihood function is set up which expresses the probability of the observed data as a function of the unknown parameters. The maximum likelihood estimators of these parameters are chosen to be those values, which maximize this function. The resulting estimators are those, which agree most closely with the observed data. This method works best for large samples, where it tends to produce estimators with the smallest possible variance. The maximum likelihood estimators are often biased in small samples (see maximum likelihood). Another method for point estimation is the method of moments.

McNemar's test: A special form of the Chi-squared test used in the analysis of paired (not independent) proportions. This non-parametric test compares two correlated dichotomous responses and finds its most frequent use in situations where the same sample is used to find out the agreement (concordance) of two diagnostic tests or difference (discordance) between two treatments. If the pairs of data points are the measurements on two matched people (such as affected and unaffected siblings) in a case-control study, or two measurements on the same person, the appropriate test for equality of proportions is the McNemar's test. It can be used to assess the outcome of two treatments applied to the same individuals or the significance of the agreement between two detection methods of a physical sign. If there are more than two periods of data collection (such as pretest, posttest and follow-up), Cochran's Q test should be used (Online McNemar's Test (1) (2) (3).

Mean (or average): A measure of location for a batch of data values; the sum of all data values divided by the number of elements in the distribution. Its accompanying measure of spread is usually the standard deviation. Unlike the median and the mode, it is not appropriate to use the mean to characterize a skewed distribution (see also standard deviation) (Online Calculator for Mean).

Mean squares: A sum of squares divided by its associated df is a mean square. In ANOVA, the regression (explained) mean square is ESS/k-1, and the residual (error) mean square is RSS/N-k. Note that the mean squares are not additive, i.e., they do not add up to TSS/N-1. Importantly, the residual (error) mean square is an unbiased estimator of the variance (s2) in ANOVA. The regression (explained) mean square equals to the variance only when the slope (b) is zero. Their ratio (mean square ratio = regression MS / residual MS) is therefore provides a test for the null hypothesis that b = 0. Large F values support the alternative hypothesis that the slope is not zero. This is the basis of the F test in ANOVA.

Measurement type: The data may be measured in the following scales: nominal, ordinal, interval or ratio scales (known as Stevens' typology). The scale of the measurements may be (other than nominal scale measurements) either continuous or discrete, and either bounded or unbounded.

Measures of association: These measures include the Phi coefficient of association, Cramer's contingency coefficient (C) and V, Kendall's tau-b and (Stuart's) tau-c, Somers' D (a modification of Kendall's tau-b), Yule's Q, gamma, Spearman's rank correlation coefficient (rho), Pearson's correlation coefficient, lambda (symmetric and asymmetric), uncertainty coefficients (symmetric and asymmetric), Guttman's coefficient of predictability (lambda)/Goodman-Kruskal's lambda, Goodman-Kruskal's gamma and Goodman-Kruskal's tau (concentration coefficient). See a review on Measures of Association by Jaeschke, 1995; Measures of Effect-Size & Association; Measures of Association for Cross-tabulations; Measures on SAS, STATA, SYSTAT; Ennis, 2001; Morton, 2001.

Measures of central tendency: These are parameters that characterize an entire distribution. These include mode, median and mean. See S4BE: Measures of Central Tendency.

Measures of variability: These are parameters that characterize an scatter of a distribution. These include range (including interquartile range), standard deviation and variance.

Median: Another measure of location just like the mean. The value that divides the frequency distribution in half when all data values are listed in order. It is insensitive to small numbers of extreme scores in a distribution. Therefore, it is the preferred measure of central tendency for a skewed distribution (in which the mean would be biased) and is usually paired with the interquartile range (dQ) as the accompanying measure of spread. See Martin Bland's page for calculation of confidence intervals for a median.

Median absolute deviation (MAD) or absolute deviation around the mean: This is a robust measure of central tendency insensitive to outliers. The MAD is the median of the absolute values of the residuals (deviations) from the data’s median. MAD can easily be calculated using R. See the article on MAD in R-Bloggers.

Median test: This is a crude version of the Kruskal-Wallis ANOVA in that it assesses the difference in samples in terms of a contingency table. The number of cases in each sample that fall above or below the common median is counted and the Chi-square value for the resulting 2xk samples contingency table is calculated. Under the null hypothesis (all samples come from populations with identical medians), approximately 50% of all cases in each sample are expected to fall above (or below) the common median. The median test is particularly useful when the scale contains artificial limits, and many cases fall at either extreme of the scale. In this case, the median test is the most appropriate method for comparing samples (Online Median Test).

Meta-Analysis: A systematic approach yielding an overall answer by analyzing a set of studies that address a related question. This approach is best suited to questions, which remain unanswered after a series of studies. Meta-Analysis provides a weighted average of the measure of effect (such as odds ratio). The rationale is to increase the power by analyzing the sets of data. The selection of studies to include in a meta-analysis study is the main problem with this approach. Funnel Plot is an informal method to assess the effect of publication bias in this context. See also Introduction to Meta-Analysis by the Cochrane Collaboration; Meta-Analysis in Epidemiology by Stroup et al (2000); Methods for Meta-Analysis in Medical Research by AJ Sutton; Introduction to Meta-Analysis by Borenstein et al (2009), Online Meta-Analysis Tests, Meta-Analysis presentation by MT Dorak.  

Metropolis-Hastings algorithms: These algorithms are a class of Markov chains which are commonly used to perform large scale calculations and simulations in Physics and Statistics. See Metropolis-Hastings Applet. See Markov Chain Monte Carlo methods.

Mode: The observed value that occurs with the greatest frequency. The mode is not influenced by small numbers of extreme values.

Model building: The traditional approach to statistical model building is to find the most parsimonious model that still explains the data. The more variables included in a model (overfitting), the more likely it becomes mathematically unstable, the greater the estimated standard errors become, and the more dependent the model becomes on the observed data. Choosing the most adequate and minimal number of explanatory variables helps to find out the main sources of influence on the response variable, and increases the predictive ability of the model. Ideally, there should be more than 10 observations for each variable in the model. The usual procedures used in variable selection in regression analysis are: univariate analysis of each variable (using C2 test), stepwise method (backward or forward elimination of variables; using the deviance difference), and best subsets selection. Once the essential main effects are chosen, interactions should be considered next. As in all model building situations in biostatistics, biological considerations should play a role in variable selection.

Monte Carlo trial: Studying a complex relationship difficult to solve by mathematical analysis by means of computer simulations. An online book on Resampling Statistics, and software (CLUMP, downloadable from clump22.zip) to do Monte Carlo statistics for case-control association studies.

Multicolinearity: In multiple regression, two or more X variables are colinear if they show strong linear relationships. This makes estimation of regression coefficients impossible. It can also produce unexpectedly large estimated standard errors for the coefficients of the X variables involved. This is why an exploratory analysis of the data should be first done to see if any collinearity among explanatory variables exists. Multicolinearity is suggested by non-significant results in individual tests on the regression coefficients for important explanatory (predictor) variables. Multicolinearity may make the determination of the main predictor variable having an effect on the outcome difficult.

Multiple regression: To quantify the relationship between several independent (explanatory) variables and a dependent (outcome) variable. The coefficients (a, b1 to bi) are estimated by the least squares method, which is equivalent to maximum likelihood estimation. A multiple regression model is built upon three major assumptions:

1. The response variable is normally distributed,

2. The residual variance does not vary for small and large fitted values (constant variance),

3. The observations (explanatory variables) are independent.

Multiple regression is the prototype for general linear models because the response variable should be normally distributed and there is no link function, whereas, simple linear regression is a special case for generalized linear models. The extension of multiple regression to multivariate data analysis is called canonical correlation (Online Multiple Regression; Reference Guide on Multiple Regression).

Multiple regression correlation coefficient (R2 - R-squared): R2 is a measure of the proportion of variability explained by, or due to the regression (linear relationship) in a sample of data (equal to the percentage of variance accounted for). It is a measure of the effect of X in reducing the uncertainty in predicting Y (the distance of data points to the best fir line). The R2 statistic is given by R2 = s.s. (regression) / s.s. (total) [i.e., ESS/TSS] where s.s. is the sum of squares. It is also called the coefficient of determination. When all observations fall on the fitted regression line, then s.s. (residual) = 0; and ESS=TSS; thus, R2 = 1. When the fitted regression line is horizontal (b = 0), then s.s. (regression) = 0 and R2 = 0. The adjusted R2a is different in that it takes into account the sample size and the number of explanatory variables. The fit of a model should never be judged from the R2 value even if the statistical significance is achieved (for example, a high value may arise when the relationship between the two variables is non-linear; or a small R2 may be significant due to large sample effect). The square root of the R2 is the correlation coefficient (r) and has the same sign (minus or plus) as the slope. The r value does not have a clear-cut interpretation as the R2 in linear regression. The R2 and r are simply descriptive measures of the degree of linear association between X and Y in the sample observations (see a Cautionary Note on R2; Logistic Regression and R2; Interpretation of r and R2; Statistical Significance of Correlation Coefficient; GraphPad Guide to Correlation Parameters; Statistical Comparison of Two Correlation Coefficients).

Multiplication rule: The probability of two or more statistically independent events all occurring is equal to the product of their individual probabilities. If the probability of having trisomy 21 is a, and trisomy 6 is b, assuming independence of these two events, for a baby the probability of having both trisomies is (a x b). One of the most critical errors of judgment in the use of independence assumption relates to a court case in the UK (Watkins, 2000). (See also addition rule.)

Multivariable analysis: As opposed to univariable analysis, statistical analysis performed in the presence of more than one explanatory variable to determine the relative contributions of each to a single event is (or should be) called multivariable analysis (in practice, however, it is called univariate and multivariate analysis more frequently). It is a method to simultaneously assess contributions of multiple variables or adjust for the effects of confounders. Multiple linear regression, multiple logistic regression, proportional hazards analysis are examples of multivariable analysis, which has no similarity whatsoever to multivariate analysis (see also Peter TJ, 2009). See a review on Multivariable Methods by MH Katz (book on Multivariable Analysis by MH Katz).

Multivariate analysis: Methods to deal with more than one related 'outcome/dependent variable' (like two outcome measures from the same individual) simultaneously with adjustment for multiple confounding variables (covariates). When there is more than one dependent variable, it is inappropriate to do a series of univariate tests. Hotelling's T2 test is used when there are two groups (like cases and controls) with multiple dependent measures (may be more than two), and multivariate analysis of variance (MANOVA) is used for more than two groups. Wilks' lambda is the most common parameter used to quantitate the significance of the result of a multivariate procedure (others are Roy's largest criterion, Hotelling-Lawley trace, and Pillai-Bartlett trace). Unfortunately, the word 'multivariate' is most frequently used instead of 'multivariable' analysis (which means multiple independent/explanatory variables but one outcome/dependent variable; see also Peter TJ, 2009)). Canonical correlation, cluster analysis, factor analysis (FA) and principal component analysis (PCA) are types of multivariate analysis. PCA aims at reducing a large set of variables to a small set that still contains most of the information in the large set. While PCA summarizes or approximates the data using fewer dimensions (to visualize it, for example), FA provides an explanatory model for the correlations among the data. See an Online Multivariate Statistics Book, Multivariate Statistical Methods: A Primer by BF Manly and MultiVariate Statistical Package-MVSP.

Multivariate analysis of variance (MANOVA): An extension of Hotelling's T2 test to more than two groups with related 'multiple' outcome measures. Groups are compared on all variables simultaneously as a global test (rather than one-by-one as ANOVA does). See also multivariate analysis; MANOVA on Stata.

Natural (raw) residuals: The difference between the observed (Yi) and fitted values (Yi-hat) of the response variable in regression. To obtain the standardized residual, this value is divided by its estimated standard deviation. Other residual types are Pearson residual, deviance residual and deletion residual. In all cases, a relatively large residual indicates an observation poorly fitted by the model. The one that is most closely related to maximum likelihood estimation (i.e., generalized linear models) is the standardized deviance residual (which is equal to the standardized Pearson residual in normal linear models).

Negative predictive value: Probability of a true negative as in a person identified healthy by a test is really free from the disease (see also positive predictive value).

Nested model: Models that are related where one model is an extension of the other.

Nominal variable: A qualitative variable defined by mutually exclusive unordered categories such as blood groups, races, sex etc. (see also ordinal variable).

Nonlinear Regression: Regression analysis in which the fitted (predicted) value of the response variable is a nonlinear function of one or more X variables. GraphPad Guide to Nonlinear Regression, Introduction to Nonlinear Regression, A GraphPad Practical Guide to Curve Fitting.

Nonparametric methods (distribution free methods): Statistical methods to analyze data from populations, which do not assume a particular population distribution. Mann-Whitney U test, Kruskal-Wallis test and Wilcoxon's (T) test are examples. Such tests do not assume a distribution of the data specified by certain parameters (such as mean or variance). For example, one of the assumptions of the Student's t-test and ANOVA is normal distribution of the data. If this is not valid, a non-parametric equivalent must be used. If a wrong choice of test has been made, it does not matter very much if the sample size is large (a non-parametric test can be used where a parametric test might have been used but a parametric test can only be used when the assumptions are met). For a small sample size, non-parametric tests tend to give a larger P value. In general, parametric tests are more robust, more complicated to compute and have greater power efficiency. Parametric tests compare parameters such as the mean in t-test and variance in F-test as opposed to non-parametric tests that compare distributions. Nonparametric methods are most appropriate when the sample sizes are small. In large (e.g., n > 100) data sets, it makes little sense to use nonparametric statistics (reviews of non-parametric tests by Whitley & Ball, 2002 and Bewick, 2004; a tutorial on Parametric vs Nonparametric Methods; Review of Nonparametric Tests in Intuitive Biostatistics; Nonparametric Tests for Ordinal Data at Vassar).

Normal distribution (Gaussian distribution) is a model for values on a continuous scale. A normal distribution can be completely described by two parameters: mean (m) and variance (s2). It is shown as C ~ N(m, s2). The distribution is symmetrical with mean, mode, and median all equal at m. In the special case of m = 1 and s2 = 1, it is called the standard normal distribution. See Normal Distribution (1), (2), (3) & (4).

Normal probability plot of the residuals: A diagnostic test for the assumption of normal distribution of residuals in linear regression models. Each residual is plotted against its expected value under normality. A plot that is nearly linear suggests normal distribution of the residuals. A plot that obviously departs from linearity suggests that the error distribution is not normal.

Null Hypothesis Statistical Testing (NHST): The most common method of inferential statistical by which an observation is tested against a (null) hypothesis of no effect or no relationship. The opposing hypothesis is the alternative hypothesis.

Null model: A model in which all parameters except the intercept are 0. It is also called the intercept-only model. The null model in linear regression is that the slope is 0, so that the predicted value of Y is the mean of Y for all values of X. The F test for the linear regression tests whether the slope is significantly different from 0, which is equivalent to testing whether the fit using non-zero slope is significantly better than the null model with 0 slope.

Number needed to treat (NNT): The reciprocal of the absolute risk reduction (ARR) between experimental/treated and control groups in a randomized clinical trial (RCT). It is interpreted as the number of patients who need to be treated to prevent one additional adverse event (over and above that obtained in the control group). See also Number Needed to Treat (NNT) Guide by Bandolier (1) (2); Interpreting Diagnostic Tests; NNT Calculator; GraphPad NNT Calculator; EpiMax Table Calculator; Evidence-based Medicine Toolbox and articles by Cook & Sackett, 1995; Wu, 2002; Barratt, 2004.

Observational study: An epidemiologic study design in which subjects select themselves into groups (such as cases and controls) and the investigator is a passive observer (no intervention). An observational study may be descriptive or analytic. In an observational study, causality of the observed associations cannot be established, and residual confounding cannot be entirely ruled out. Bias may also play a role in generating a spurious correlation or deviating the results. Cohort studies are the highest-ranking observational study type in the hierarchy of causality pyramid (but still below interventional studies).

Odds: The odds of a success is defined as the ratio of the probability of a success to the probability of a failure (p/(1-p)). If a team has a probability of 0.6 of winning the championship, the odds for winning is 0.6/(1-0.6) = 3:2. Similarly, the odd in a case-control study is the frequency of the presence of the marker divided by the frequency of absence of the marker (in cases or controls separately). The link function logit (or logistic) is the loge of the odds.

Odds multiplier: In logistic regression, b = loge (odds ratio), thus exp b = odds ratio. For a continuous (explanatory) variable, exp b is called the odds multiplier and corresponds to the odds ratio for a unit increase in the explanatory variable. The odds multiplier of the coefficient is the odds ratio for its level relative to the reference level. If x increases from a to b by c, the odds multiplier becomes exp (cb). The resulting value shows the proportional change in the odds associated with x = b relative to x = a. It follows that for binary variables where x can only get values of 0 and 1, exp b = odds ratio.

Odds ratio (OR): Also known as relative odds and approximate relative risk. It is the ratio of the odds of the risk factor in a diseased group and in a non-diseased (control) group (the ratio of the frequency of presence / absence of the marker in cases to the frequency of presence / absence of the marker in controls). The interpretation of the OR is that the risk factor increases the odds of the disease ‘OR’ times. OR is used in retrospective case-control studies (relative risk (RR) is the ratio of proportions in two groups which can be estimated in a prospective -cohort- study). These two and relative hazard (or hazard ratio) are measures of the strength/magnitude of an association. As opposed to the P value, these do not change with the sample size. OR and RR are considered interchangeable when certain assumptions are met, especially for large samples and rare diseases. Odds ratio is calculated as ad/bc where a,b,c,d are the entries in a 2x2 contingency table (hence the alternative definition as the cross-product ratio). In logistic regression, the coefficient b corresponds to the loge of the odds ratio. There are statistical methods to test the homogeneity of odds ratios (Online Odds-Ratio & 95% CI Calculation; Odds Ratio-Relative Risk Calculation (Calculator 3) in Clinical Research Calculators at Vassar).

Offset: A fixed, already known regression coefficient included in a generalized linear model (which does not have to be estimated).

Omnibus test: If the chi-square test has more than one degree of freedom (larger than 2x2 table), it is called an ‘omnibus’ test, which evaluates the significance of an overall hypothesis containing multiple sub-hypotheses (these multiple sub-hypotheses then need to be tested using follow up tests).

One-way ANOVA: A comparison of several groups of observations, all of which are independent and subjected to different levels of a single treatment (such as cells exposed to different dosage of a growth factor). It may be that different groups were exposed to the same treatment (different cell types exposed to a new agent). The main interest focuses on the differences among the means of groups.

Ordinal variable: An ordered (ranked) qualitative/categorical variable. The degree of HLA matching (one, two, three or four antigen matching in two loci) in transplant pairs, or HLA sharing in parents (one-to-four shared antigens) are ordinal variables although the increments do not have to be equal in magnitude (see interval and ratio variables). An ordinal variable may be a categorized quantitative variable. When two groups are compared for an ordinal variable, it is inappropriate to use ordinary Chi-squared test but the trend test or its equivalents must be used.

Outcome (response, dependent) variable: The observed variable, which is shown on y axis. A statistical model shows this as a function of predictor variable(s).

Outlier: An extreme observations that is well separated from the remainder of the data. In regression analysis, not all outlying values will have an influence on the fitted function. Those outlying with regard to their X values (high leverage), and those with Y values that are not consistent with the regression relation for the other values (high residual) are expected to be influential. The test the influence of such values, the Cook statistics is used.

Overdispersion: Dispersion is a measure of the extent to which data are spread about an average. Overdispersion is the situation that occurs most frequently in Poisson and binomial regression when variance is much higher than the mean (normally, it should be similar). It is evident with a high (>2) residual mean deviance (which should normally be around one) and the presence of too many outliers. The reasons for overdispersion may be outliers, misspecification of the model, variation between the response probabilities and correlation between the binary responses. It distorts standard error and confidence interval estimations. In the analysis, overdispersion may be taken into account by estimating a dispersion parameter.

Overfitting: In a multivariable model, having more variables than can be justified from sample size. The statistical rule of thumb is to have at least ten subjects for each variable investigated. When fitting models, adding more and more parameters increases the likelihood of fitting, but this is called overfitting. The resulting model will be fitting very well (with minimal residuals) but it will be very hard to replicate in following experiments.

Overmatching: When cases and controls are matched by an unnecessary non-confounding variable, this is called overmatching and causes underestimation of an association. For example, matching for a variable strongly correlated with the exposure but not with the outcome will cause loss of efficiency. Another kind of overmatching is matching for a variable which is in the causal chain or closely linked to a factor in the causal chain. This will also obscure evidence for a true association. Finally, if a too strict matching scheme is used, it will be very hard to find controls for the study. See BMJ Statistics Notes: Matching.

Parameter: A numerical characteristic of a "population" specifying a distribution model (such as normal or Poisson distribution). This may be the mean, variance, degrees of freedom, the probability of a success in a binomial distribution, etc. A population parameter is usually unknown and estimated as a statistic in a random sample from the population. Customarily, it is denoted by a Greek letter (as opposed to Roman letters/numerals for statistic(s) in a sample).

Parsimonious: The simplest plausible model with the fewest possible number of variables.

Pearson's correlation coefficient (r): A measure of the strength of the 'linear' relationship between two quantitative variables. A major assumption is the normal distribution of variables. If this assumption is invalid (for example, due to outliers), the non-parametric equivalent Spearman's rank correlation should be used. The r represents C2  obtained from the 2x2 table, corrected for the total sample size. It can then be calculated as ±(C2/N)1/2. This formula is equivalent to covariance divided by the product of the standard deviations of the two variables. The correlation coefficient, r, can take any value between -1 and +1; 0 meaning no "linear" relationship (there may still be a strong non-linear relationship). It is the absolute value of r showing the strength of relationship. An associated P value can be computed for the statistical significance (a small P value does not necessarily mean a strong relationship). The square of the r is r2 (r-squared or coefficient of determination) which corresponds to the variance explained by the correlated variable (see GraphPad Guide to Correlation Parameters and Interpretation of r). R2 is also used in regression analysis (see multiple regression correlation coefficient; Online Correlation & Regression Calculators at Vassar College; Excel Macro for Linear Correlation & Regression).

Pharmacoepidemiology: Application of epidemiological reasoning, methods and knowledge to the study of the uses and effects (beneficial and adverse) of drugs in human populations. A relatively new field in epidemiology becoming more closely related to pharmacogenetics. See a review on statistical analysis of pharmacoepidemiological case-control studies (Ashby, 1998).

Phi coefficient: A measure of association of two variables calculated from a contingency table as (X2 / N) 1/2. Its value varies between 0 (no association) and 1 (strongest association) for 2x2 tables where it is an accurate statistics (for larger tables, Cramer’s V is more accurate). In a way, it is a corrected Chi-squared value for the number of observations. See Calculator 3 in Clinical Research Calculators at Vassar.  

Poisson distribution: The probability distribution of the number of (rare) occurrences of some random event in an interval of time or space. Poisson distribution is used to represent distribution of counts like number of defects in a piece of material, customer arrivals, insurance claims, incoming telephone calls, or alpha particles emitted. A transformation that often changes Poisson data approximately normal is the square root. See Poisson Distribution (QuickTime); GraphPad Poisson Probability Calculator.

Poisson regression: Analysis of the relationship between an observed count with a Poisson distribution (i.e., outcome variable) and a set of explanatory variables. In general it is appropriate to fit a Poisson model to the data if the sample size is > 100 and the mean for the occurrence of the event is <0.10xN.

Polynomial: A sum of multiples of integer powers of a variable. The highest power in the expression (n) is the degree of the polynomial. When n=1, for example, f(x)=2x1+3, this is a linear expression. If n=2, it is quadratic (for example, x2 + 2x + 4); if n=3, it is cubic, if n=4, it is quartic and if n=5, it is quintic.

Polytomous variable: A variable with more than two levels. If there are two levels it is called dichotomous (as in the most common form of logistic regression).

Population: The population is the universe of all the objects from which a sample could be drawn for an experiment.

Population attributable risk: The proportion of a disease in a specified population attributable to a specific factor (such as a genetic risk factor).

Population stratification (substructure): An example of 'confounding by ethnicity' in which the co-existence of different disease rates and allele frequencies within population sub-sections lead to an association between the two at a whole population level. Differing allele frequencies in ethnically different strata in a single population may lead to a spurious association or mask an association by artificially modifying allele frequencies in cases and controls when there is no real association (for this to happen, the subpopulations should differ not only in allele frequencies but also in baseline risk to the disease being studied). Case-control association studies can still be conducted by using genomic controls (Devlin, 1999; Pritchard, 1999) even when population stratification is present. The software STRUCTURE & STRAT or ADMIXMAP can be used to analyze case-control data with genomic control. See Cardon & Palmer, 2003 for an example of spurious association due to population stratification. See also Genetic Epidemiology.

Positive predictive value: Probability of a true positive as in a person identified as diseased by a test is really diseased (see also negative predictive value).

Post hoc test: A test following another one. The most common example is performing multiple comparisons between groups when the overall comparison between groups shows a significant difference. For example, when an ANOVA analysis yields a small P value, post hoc tests (such as Newman-Keuls, Duncan's or Dunnett's tests) are done to narrow down exactly which pairs differ most significantly (similarly, Dunn's test is done in a non-parametric ANOVA setting) (GraphPad Post ANOVA Test Calculator). In genetic association studies, multiple comparisons are justified only when performed as a post hoc test following a significant deviation in overall gene/marker frequencies (see HLA and Disease Association Studies).

Power of a statistical test: See Statistical Power.

Precision: The degree to which a parameter (like the mean) is immune from random variation. Precision is quantified by the confidence interval; precision increases by increasing the sample size. Precision is different from accuracy (which has to do systematic error or bias).

Predictor (explanatory, independent) variable: The variable already in hand in the beginning of an experiment or observation and whose effect on an outcome variable is being modeled.

Predictive value: The probability that a person with a positive test if affected by the disease (positive predictive value) or the probability that a person with a negative test does not have the disease (negative predictive value). Estimation requires sensitivity, specificity and disease prevalence.

Prevented fraction: The amount of a health problem that actually has been prevented by a prevention strategy in real world.

Probability: The ratio of the number of likely outcomes to the number of possible outcomes.

Probability density function: When a curve is used to model the variation in a population and the total area between the curve and the x-axis is 1, then the function that defines the curve is a probability density function. 

Probability distribution function: A function which gives for each number x, the probability that the value of a continuous random variable X is less than or equal to x. For discrete random variables, the probability distribution function is given as the probability associated with each possible discrete value of the variable.

Probability vector: Any vector with non-negative entries whose sum is equal to 1.0. See Wikipedia.

Proportional odds ratio: When the response variable in an ordered/ordinal logistic regression model has more than two ordered response categories, odds ratio obtained for each category is called a proportional odds ratio. See UCLA Stata: Ordinal Logistic Regression; Lecture note on Logistic Regression.

P value (SP = significance probability): The P value gives an idea on the strength of evidence against the null hypothesis, i.e., departure from the null hypothesis when the null hypothesis is true (the larger the P value, the larger the probability of type I error for false positivity). When the P value is larger than 0.05, the result is considered more likely to be due to random error (false-positivity; a) rather than a true effect/difference. The P value does not measure the magnitude of effect (that is measured by the effect size). The P value represents the probability of making an error in accepting the alternative hypothesis (or rejecting the null hypothesis) for the presence of an association/effect. For example, the P level of 0.01 (i.e., 1/100) indicates that assuming there was no relation between those variables whatsoever (the null hypothesis is correct), and we were repeating experiments like ours one after another, we could expect that approximately in every 100 replications of the experiment, there would be one in which the relation between the variables in question would be equal to or more extreme than what has been found (in other words, random sampling from identical populations (i.e., null hypothesis is true) would lead to a difference larger than you observed only in 1% of experiments). Thus, it is the probability of observing a statistic (in the sample) that extreme if the null hypothesis is true (not the probability that the null hypothesis is true). In the interpretation of a P value, it is important to know the accompanying measure of association and the biological/clinical significance of the significant difference. However small, a P value does not indicate the size of an effect (odds ratio/relative risk/hazard ratio/correlation coefficient and any other effect size do that). A P value >0.05 does not necessarily mean lack of association or no effect. It does so only if there is enough statistical power (i.e., at least 80%) to detect an effect/association. Most statistical non-significance is due to lack of power to detect an association (poor experimental design). Statistical significance threshold (the pre-determined P value for statistical significance or a) is one of the determinants of statistical power. Both 'p' and 'P' are used to indicate significance probability, but the international standard is P (capital italic). See Interpreting Statistical P Values; review of hypothesis testing (Guyatt, 1995), and P Value and  The Most Common Misinterpretations of a P Value in GraphPad Statistics Guide.

Qualitative: Qualitative (categorical) variables define different categories or classes of an attribute. Examples are gender, blood groups or disease states. A qualitative (categorical) variable may be nominal or ordinal. When there are only two categories, it is termed binary (like sex, dead or alive).

Quantitative: Quantitative variables are variables for which a numeric value representing an amount is measured. They may be discrete (for example, taking values of integers) or continuous (such as weight, height, blood pressure). If a quantitative variable is categorized, it becomes an ordinal variable.

R: A language and environment for statistical computing and graphics. R is an open platform and offers thousands of programs (called libraries) to achieve a wide variety of statistical, graphical, biological tasks. It has a large number of (statistical) genetics and statistics libraries. To learn how to use R (not how to program R!), see this self-paced R course and the links within.

R2 (R-squared): See Multiple regression correlation coefficient and Pearson's correlation coefficient (for r2).

Random sampling: A method of selecting a sample from a target population or study base using simple or systematic random methods. In random sampling, each subject in the target population has equal chance of being selected to the sample. Sampling is a crucially important point in selection of controls for a case-control study. The idea with sampling is to estimate the unknown population parameters from the sample. The sample should, therefore, be a randomly selected one to represent the whole population. Even randomly selected samples will be different from each other than the population itself. This represents the sampling error which can cause false-positive results (type I error rate or a, which is measured by the P value). By randomization, a balancing out effect is expected: known and unknown factors that might influence the outcome are assigned equally to the comparison groups (and minimizes bias and confounding). One disadvantage of randomization is generation of a potentially large error term. This can be avoided by using a block design. See Wikipedia: Random Sampling.

Randomized (complete) block design: An experimental design in which the treatments in each block are assigned to the experimental units in random order. Blocks are all of the same size and each treatment appears in the same number of times within each block (usually once). A different level of the factor is assigned to each member of the block randomly. The data can be analyzed using the paired t-test (when there are two units per block) or by randomized block ANOVA (in blocks of any size). The results are substantially more precise than a completely randomized design of comparable size. In studies with a block design, more assumptions are required for the model: no interactions between treatments and blocks, and constant variance from block to block.

RCT: See randomized controlled trial.

Randomized controlled (or clinical) trial (RCT): The favored clinical research design in which the comparative groups are formed by random allocation of subjects in an attempt to minimize (selection) bias and to distribute measurable and unmeasurable (or, known and unknown) confounding factors equally, and double-blinding is used to conceal allocation of the intervention to reduce a number of biases, leaving the exposure (usually a new drug) whose effect will be measured as the causal factor on the outcome. Intention-to-treat (ITT) analysis is applied to avoid further bias. RCTs are the highest-ranking clinical research design in the epidemiologic hierarchy of strength of evidence for causality. This is because it suffers from minimum bias/confounding, and isolates the (therapeutic) effect of interventions due to being considered a controlled design. It is also always statistically powered for even minimum effect size (Cohen’s d value = 0.2) with a large sample size.    

Ratio variable: A quantitative variable that has a zero point as its origin (like 0 cm = 0 inch) so that ratios between values are meaningfully defined. Unlike the interval variables, which do not have a true zero point, the ratio of any two values in the scales is independent of the unit of measurement. For example, 2/12 cm has the same ratio as the corresponding values in inch (but the same cannot be said for 2/12 Celsius and 2/12 Fahrenheit which are interval variables).

Receiver operating characteristics (ROC) curve analysis: Also called discrimination statistics used in diagnostic test accuracy assessment and the utility of predictive tests (but see Pencina, 2008; Pencina, 2012). See ROC in Clinical Research Calculators and Difference Between the Areas Under Two ROC Curves at Vassar, ROC101 by Tom Fawcett. See reviews on ROC analysis: Hanley & McNeil. 1982; Bewick, 2004; Obuchowski, 2005; Cook NR, 2007; Steyerberg, 2012. See also the Supplementary Data File for Mamtani, 2006 for the use of Stata in ROC analysis.

Regression diagnostics: Tests to identify the main problem areas in regression analysis: normality, common variance and independence of the error terms (epsilon or e); outliers, influential data points, collinearity, independent variables being subject to error, and inadequate specification of the functional form of the model. The purpose of the diagnostic techniques is to identify weaknesses in the regression model or the data. Remedial measures, correction of errors in the data, elimination of true outliers, collection of better data, or improvement of the model, will allow greater confidence in the final product. See also error terms, residuals (including likelihood distance test), leverages and Cook statistics.

Regression modeling: Formulating a mathematical model of the relationship between a response (outcome, dependent) variable, Y, and a set of explanatory (predictor, independent, regressor) variables, x. Depending on the characteristics of the variables, the choice of model can be simple linear regression, multiple regression, logistic (binary) regression, Poisson regression, etc. In any regression problem, the key quantity is the mean value of the outcome variable, given the value of the independent variable(s). This quantity is called the conditional mean and expressed as "E (Y½x)" where Y is the response (outcome), x is the explanatory (predictor) variable. The question is whether the variable(s) in question tells us more about the outcome variable than a model that does not include that variable. In other words, whether the coefficient of the variable(s) is zero and the outcome is equal to a constant (which is the mean for Y) or not. The aim of model building is to arrive at a meaningful (say, biologically relevant) and parsimonious model that explains the data. The model may be linear if the parameters are linear, or nonparametric if the parameters are not linear. No matter how strong is the statistical relationship between x and Y, no cause-and-effect pattern is necessarily implied by the regression models. See Regression Applet.

Regression towards the mean (RTM): A statistical principle (and a kind of bias) based on the fact that unusual events are unlikely to recur. By chance alone, measurements subsequent to an unusual measurement tend to be closer to the mean. See the explanation and a simulation at Rice Virtual Lab in Statistics; a Lecture Note; and Barnett, 2005.

Relative risk (RR): Also known as risk ratio. The RR shows how many times more or less the individuals with the risk factor are likely to get the disease relative to those who do not have the risk factor. RR gives the strength of association in prospective cohort studies. It cannot be estimated in retrospective case-control studies, and its use to describe the cross-product ratio (as frequently done in HLA association studies) is inappropriate. See Calculator 3 in Clinical Research Calculators at Vassar. See also odds ratio.

Repeated measures design: In this design, the same experimental unit is subjected to the different treatments under consideration at different points in time. Each unit, therefore, serves as a block. If for example, two different treatments and placebo treatment are applied to the same patient sequentially, this is a repeated measures design. See also cross-over design.

Resampling statistics: Data-based simulation procedures that sample (with replacement) repeatedly from observed data to generate empirical estimates of results that would be expected by chance. Examples include bootstrapping and permutation tests. See also Online Resampling Book.

Residuals: Residuals reflect the overall badness-of-fit of the model. They are the differences between the observed values of the outcome variable and the corresponding fitted values predicted by the regression line (the vertical distance between the observed values and the fitted line). In a regression analysis, a large residual for a data point indicates that the data point concerned is not close to its fitted value. If there are too many large (standardized) residuals either the model fitted is not adequate or there is overdispersion of the data. Ideally, the residuals should have constant variance along the line. This can be checked by a normal probability plot of the residuals. In the plot of residuals against the explanatory variable (or the fitted values), there should not be any pattern if the assumption of constant variation is met, i.e., residuals do not tend to get larger as the variable values get larger or smaller (see also likelihood distance test).

Residual plot: A graph that plots residuals against fitted values. It is used to check equal variance assumption of the error terms in linear regression. Residual analysis for logistic regression is more difficult than for general linear regression models because the responses Yi can take only the values 0 and 1. Consequently, the residuals will not be normally distributed. Plots of residuals against fitted values or explanatory variables will be uninformative. Residual plots are generally unhelpful for generalized linear models (where index plots and half-normal plots are preferred).

Residual (error) sum of squares (RSS): The measure of within treatment groups sum of squares (variability) in ANOVA. It is the deviation around the fitted regression line. The sum of squared differences between each observed Y value (Yi) and the fitted Y value (Yi -hat) equals to the residual (error) sum of squares. See also degrees of freedom and mean squares.

Regression (explained) sum of squares (ESS): The measure of between treatment groups sum of squares (variability) in ANOVA. It is the deviation of fitted regression value around mean. The sum of squared differences between each fitted Y value (Yi -hat) and the overall mean of the Y values equals to the explained (regression) sum of squares. The sum of ESS and RSS gives the total sum of squares (TSS) which equals to the sum of squared differences between each observed Y value (Yi ) and the overall mean (total deviation). See also degrees of freedom and mean squares.

Risk ratio (relative risk): The risk ratio is the percentage difference in classification between two groups obtained as the ratio of two risks or proportions. For example, the proportion of people recovering after stroke with one treatment equals 0.10, while the proportion after a different treatment equals 0.16. The risk ratio equals 0.625 (0.10/0.16); 37.5% ((1-0.625)*100 or (0.16-0.10)/0.16) fewer patients treated by the first method recover. The risk ratio takes on values between zero ('0') and infinity. One ('1') is the neutral value and means that there is no difference between the groups compared. See also relative risk.

Robustness: A statistical test or procedure is robust when violation of assumptions has little effect on the results. Student's t-test, for example, is robust against departures from normality.

R project for statistical computing: R is a language and environment for statistical computing and graphics which can be seen as a different implementation of the S language. R and a comprehensive set of programs written for a variety of statistical analysis are all available freely. See the R Project Website & List of Contributed R Packages. See also OpenIntro Statistics Book (based on R); OpenIntro Advanced High School Statistics Book; OpenIntro Statistics with Randomization and Simulation.

Rule of Two: In his books entitled "Statistical Methods for Research Workers" (1925) and "Design of Experiments" (1935), Fisher imposed a "Rule of Two": if a result departs from an assumed hypothesis by two or more standard deviations of its own sampling variation, then it is called a “significant” scientific finding. However, this significance is purely statistical and has to be specified as such, i.e., "statistical significance". Statistical significance does not say anything about the practical (or clinical/biological) significance of the finding and should be treated as meaningless unless accompanied by an effect size and its confidence interval.

Sample size determination: Mathematical process of deciding how many subjects should be studied (at the planning phase of a study) to detect the hypothesized effect with statistical significance. If the sample size is insufficient, the effect may be observed, but the statistical significance will not be reached (and will be called not detected [with statistical significance]). Among the factors to consider are the incidence or prevalence of the condition, the magnitude of difference expected between comparison groups (i.e., effect size), the statistical power that is desired (usually 80% due to allowing 20% type II error rate (false-negativity or b)) and the allowable magnitude of type I error (pre-determined significance probability; false-positivity alpha value; usually 0.05). Sample Size Calculator; Sample Size Calculation; and How to Perform a Simple Power Analysis in R.

Sampling: Since studying the whole population is rarely possible, a random sample as much representative of it as possible is used to derive a statistic as an “estimate” of the unknown population “parameter”. The process of extracting a representative sample is called sampling. The values obtained from multiple samples from the same population generate a sampling distribution of the estimate, which is the basis of the confidence interval values. No two samples will yield exactly the same results is one of the fundamental concepts in statistics and at the core of statistical significance. According to “Fisher’s Rule of Two”, if two results are more than two standard deviations away from each other (within the sampling distribution), the difference is called “statistically significant”.

Sampling error: Due to each sample from a population being different, there is a difference between a sample statistic and the population parameter it estimates. This is the sampling error, which is an inherent factor when using samples.

SAS (Statistical Analysis System): A comprehensive computer software system for data processing and analysis. It can be used for almost any type of statistical analysis. Produced by SAS Institute. See SAS Learning Resources (UCLA); SAS Tutorials; Getting Started with SAS Enterprise Guide (Free Online Course); SAS e-Learning; SAS Genetic Software and Genetic Data Analysis.

Saturated model: A model that contains as many parameters as there are data points. This model contains all main effects and all possible interactions between factors. For categorical data, this model contains the same number of parameters as cells and results in a perfect fit for a data set. The (residual) deviance is a measure of the extent to which a particular model differs from the saturated model.

Scales of measurement: The type of data is always one of the following four scales of measurement: nominal, ordinal, interval, or ratio. Each of these can be discrete or continuous.

Schoenfeld residual test: One of the diagnostic tests to check the proportionality assumption (covariates are time independent) in proportional hazard modeling. A variation is the use of scaled Schoenfeld residuals (see Tests of Proportionality in SAS, Stata, R and SPLUS).

Sensitivity: Sensitivity is the proportion of true positives that are correctly identified by a diagnostic test. Those that produce few false negatives have higher sensitivity. See Sensitivity and Specificity by Altman & Bland, BMJ 1994; Interpreting Diagnostic Tests and DAG-STAT; calculators: Vassar: Clinical Calculator 1 & Clinical Calculator 2; SISA: Diagnostic Effectiveness Tests. See also specificity.

Sensitivity analysis: To check the validity and robustness of the results obtained, sensitivity analysis tries to see how results change under different choices for a bias parameter (by altering one or more factors from their best guess or baseline estimates) and examine its impact on the outcome. Leaving out outliers to check their dominance on the outcome is a simple sensitivity analysis. See Mowbray, 2022 for a review.

Sign test: A test based on the probabilities of different outcomes for any number of pluses and minuses, i.e., observations below or above a prespecified value. The sign test can be used to investigate the significance of the difference between a population median and a specified value for it, or between the observed sex/transmission ratio and the 50:50 expected value. It can also be used for paired data. This time, the differences between the pairs will be either negative or positive, and the smaller of the two total negatives or positives plus the total number of pairs will form the test statistics. For example, when the total number is 20, if the number for the less frequent sign is 5 or smaller, P < 0.05 (two-tailed). A sign test in disguise is McNemar's test, which is used for paired data for dichotomous response.

Simple linear regression model: The linear regression model for a normally distributed outcome (response) variable and a single predictor (explanatory) variable. The straight line models the mean value of the response variable for each value of the explanatory variable. The major assumption is constant variation of residuals along the fitted line which points out that the model is equally good across all x values. The null hypothesis stating that the explanatory variable has no effect on the response (in other words, the slope of the fitted line is zero) can be tested statistically. The two main aims of regression analysis are to predict the response and to understand the relationships between variables. As in all linear models, the error term (shown as Wi or ei) is additive (as opposed to multiplicative, i.e., yi = a + bxi + ei) and independent, and they are assumed to have a normal distribution. As an exception, the simple linear regression is a special case for generalized linear models.

Skewness: The degree of (lack of) asymmetry about a central value of a distribution. A distribution with many small values and few large values is positively (right) skewed (long tail in the distribution curve or stem plot is to the right); the opposite (left tail) is negatively (left) skewed. The measures of location median, mid-interquartile range (midQ) and mid-range decrease in this order for a left-skewed distribution. (Definition of Kurtosis and Skewness; Online Skewness-Kurtosis Calculator; see also kurtosis).

Sparseness: A contingency table is sparse when many cells have small values. When N is the total sample size, and r and c are the number of rows and columns, N / rc is an index of sparseness. Smaller values refer to more sparse tables. Sparse tables often contain zero values (empty cell).

Spearman's rank correlation: A non-parametric correlation coefficient (rho) that is calculated by computing the Pearson's correlation coefficient (r) for the association between the ranks given to the values of the variables involved. It is used for ordinal data and interval/ratio data. It is not appropriate to take the square of Spearmen’s correlation coefficient rho to obtain coefficient of determination (r2). It is also possible to compare two correlation coefficients. See also Pearson's correlation coefficient.

Specificity: Specificity is the proportion of true negatives that are correctly identified by the test. Those that produce few false positives have higher specificity. Sensitivity and Specificity by Altman & Bland, BMJ 1994; Interpreting Diagnostic Tests and DAG-STAT. Calculators: Vassar: Clinical Calculator 1 & Clinical Calculator 2; SISA: Diagnostic Effectiveness Tests. See also sensitivity.

Square root transformation: Usually used for highly positively skewed data, but especially in transforming Poisson counts to normality.

Standard deviation: Like variance, the standard deviation (SD) is a measure of spread (scatter) of a set of data. Unlike variance, which is expressed in squared units of measurement, the SD is expressed in the same units as the measurements of the original data. It is calculated from the deviations between each data value and the sample mean. It is the square root of the variance. For different purposes, n (the total number of values) or n-1 may be used in computing the variance/SD. If you have a SD calculated by dividing by n and want to convert it to a SD corresponding to a denominator of n-1, multiply the result by the square root of n/(n-1). If a distribution's SD is greater than its mean, the mean is inadequate as a representative measure of central tendency. For normally distributed data values, approximately 68% of the distribution falls within ± 1 SD of the mean, 95% of the distribution falls within ± 2 SDs of the mean, and 99.7% of the distribution falls within ± 3 SDs of the mean (empirical rule). SD should not be confused with the standard error of the mean (SEM), which quantifies how precise the mean is (See Normal Distribution; Online Calculator for Standard Deviation; Standard Deviation calculation by GraphPad QuickCalc). See also: Altman & Bland: Statistical Notes - Standard deviations and standard errors. BMJ 2005.331:903.

Standard error: The standard error (SE) or as commonly called the standard error of the mean (SEM) is a measure of the extent to which the sample mean deviates from the true but unknown population mean. It is the standard deviation (SD) of the random sampling distribution of means (i.e., means of multiple samples from the same population). As such, it measures the precision of the statistic as an estimate of a population. The (estimated) SE/SEM is dependent on the sample size. It is inversely related to the square root of the sample size:

(estimated) SE = SD / (N)1/2

The true value of the SE can only be calculated if the SD of the population is known. When the sample SD is used (as almost always), it is an estimate and should be called estimated standard error (ESE). When the sample size is relatively large (N ³ 100), the sample SD provides a reliable estimate of the SE. See the explanation of the difference between SD and SEM (Standard Error calculation by GraphPad QuickCalc). See also: Altman & Bland: Statistical Notes - Standard deviations and standard errors. BMJ 2005.331:903.

Standard residual: The standardized residual value (observed minus predicted divided by the square root of the residual mean square).

Stata: A powerful statistical package particularly useful for epidemiologic and longitudinal data management and analysis. It is mainly a command driven program produced by Stata Corporation. See the list of Stata CapabilitiesStata Starter Kit with Learning Modules by UCLA; What Stat to Use (Stata); Tutorial by University of Essex; Tutorial by Princeton University; Stata Highlights by Notre Dame University; Tutorial by Carolina Population Center; Stata Refresher by Syracuse University; Stata for Researchers by Wisconsin University.

Statistic: A numerical measure that describes a variable (characteristic) of a representative sample (randomly drawn from a population) to estimate the unknown population parameter. A sample statistic is an estimate of a population parameter.

Statistical Power: The probability that the hypothesized and observed effect will be statistically significant at a given significance level is called the power of the test. This is equal to the probability of rejecting the null hypothesis (i.e., getting a P value less than 0.05) when it is untrue (the hypothesized effect is present), i.e., making the correct decision. It is 1 minus the probability of a type II error (b or false-negativity rate). The true differences between the samples compared (effect size), the sample size and the significance threshold chosen are determinants of the power of a statistical test. Ideally, power should be at least 0.80 to detect a reasonable departure from the null hypothesis (allowing 20% false-negativity rate or b). Insufficient statistical power is one of the major statistical errors contributing to a bad research design. See power calculators: G*Power 3 (User Guide; Tutorial); Statistical Power Calculator for Frequencies; Retrospective Power Calculation; Power Calculation for Logistic Regression (including Interaction); Power Calculation for t-test (SISA); A Beginner’s Tutorial to Power Analysis using jamovi and How to Perform a Simple Power Analysis in R.

Stepwise regression model: A method in multiple regression studies aimed to find the best model. This method seeks a model that balances a relatively small number of variables with a good fit to the data by seeking a model with high R2a (the most parsimonious model with the highest percentage accounted for). The stepwise regression can be started from a null or a full model and can go forward or backward, respectively. At any step in the procedure, the statistically most important variable will be the one that produces the greatest change in the log-likelihood relative to a model lacking the variable. This would be the variable, which would result in the largest likelihood ratio statistics, G (a high percentage accounted for gives an indication that the model fits well). See also multiple regression correlation coefficient - R2.

Stochastic model: A probability model that includes chance events in the form of random measurement error or uncertainty. In a deterministic model, however, random error is inconsequential or nonexistent. See Wikipedia: Stochastic Modeling.

Stratum (plural strata): When data are stratified according to its characteristics, each subgroup is a stratum.

Student's t-test: A parametric test for the significance of the difference between means (two-samples t-test) or between a mean and a hypothesized value (one-sample t-test). One assumption is that the observations must be normally distributed, and the ratio of variances in two samples should not be more than three. If the assumptions are not met, there are non-parametric equivalents of the t-test to use (see for example, Wilcoxon's Test). It is inappropriate to use the t-test for multiple comparisons as a post hoc test. The t-test for independent samples tests whether or not two means are significantly different from each other but only if they were the only two samples taken (Online t-test (GraphPad QuickCalc) [for raw data or for mean, SD, N]; Online t-test for Samples with Unequal Variance at Vassar). (Tables of critical values of t, F and Chi-square & t to P Calculator) and How to Do a t-test in R.

Subgroup analysis: Analysis of subgroups of a sample either because of a prior hypothesis (gender or age-specific effect/association) or as a fishing expedition / data dredging. This practice increases type I error (false-positivity) rates (also called P hacking). See a commentary by Sleight, 2000.

Survival Analysis: See Superlectures on ‘Survival Analysis’; A Primer on Survival Analysis (J Nephrol 2004); Tutorials on Survival Analysis in Br J Cancer 2003: Part I - II - III - IV;  ‘Understanding Survival Curves’ at NMDP website; Survival Curves and Comparing Survival Curves in Intuitive Biostatistics; BMJ Statistics Notes: Time to Event (Survival) Data - Survival Probabilities (the Kaplan-Meier method) - Logrank Test; Survival Analysis by STATA and SAS; Power Calculator for Survival Outcomes, PS: Power and Sample Size Calculation. Online survival analysis at VassarStats. For a comprehensive review, see Lee & Go, 1997.

Survival function: A time to failure function that gives the probability that an individual survives past a time point (does not experience an adverse event like death, metastasis, relapse etc). Where the event is death, the value of the survival function at time T is the probability that a subject will die at some time greater than T. The survival function always has a value between 0 and 1 and is nonincreasing.

Synergism: A joint effect of two treatments being greater than the sum of their effects when administered separately (positive synergism) or the opposite (negative synergism).

Theta (q): Used to denote recombination fraction (in statistical genetics).

Transformations (ladder of powers): Transformation deals with non-normality of the data points and non-homogeneous variance. The power transformations form the following ladder: ..., x-2, x-1, x-1/2, log x, x1/2 ; x1, x2, x3, ..... Provided x > 1, powers below 1 (such as x1/2 or log x) reduce the high values relative to the low values as in positively skewed data, whereas, powers above 1 (such as x2) have the opposite effect of stretching out high values relative to low ones, as in negatively skewed data. All power transformations are monotonic when applied to positive data (they are either increasing or decreasing, but not first increasing and then decreasing, or vice versa). The square root transformation often renders Poisson data approximately normal.

Transmission Disequilibrium Test (TDT): A family-based study to compare the proportion of alleles transmitted (or inherited) from a heterozygous parent to a disease-affected child. Any significant deviation from 0.50 in transmission ratio implies an association (Spielman, 1993 & 1994).

Treatment: In experiments, a treatment is what is administered to experimental units (explanatory variables). It does not have to be a medical treatment. Fertilizers in agricultural experiments; different books and multimedia methods in teaching; and chemotherapy of bone marrow transplantation in the treatment of leukemia are examples of treatments in regression analysis.

Trend test for counts and proportions: A special application of the Chi-squared test (with a different formula) for ordinal data tabulated as a 2xk table. It should be used when the intention is not just to compare the differences between the two groups but to see whether there is a consistent trend towards decrease or increase in the difference between the groups. An example is the association of parental HLA sharing (sharing one-to-four antigens in two loci) with fetal loss in a case-control study (those with recurrent miscarriages and normal fertile couples). In genetic studies, the additive model analysis is done by trend test for counts in cases and controls (2 columns) of three genotypes (3 rows): wildtype homozygous (0 variant allele), heterozygote (1 variant allele) and variant homozygote (2 variant alleles). A frequent application is the analysis of dose-response relationships in toxicology and pharmacology. The Chi-squared test for trend has one degree of freedom. The associated P value obtained by the Chi-squared for trend test (1 df) is always smaller than the corresponding P value of an ordinary Chi-squared test (2 df) for departure if indeed there is a trend. The trend test for counts and proportions is called Cochrane-Armitage trend test. Alternative tests for the analysis of trend are Wilcoxon-Mann-Whitney test or the t-test with use of ordered scores, and the Jonckheere-Terpstra test as a non-parametric test for ordered data (see Trend for Binomial Outcome in the manual of Epi Info; Epi Info Freeware for Trend Test (Trend Test in StatCalc or Open-Epi Online); Trend Tests in Stata.

t-statistics: Defined as difference of sample means divided by standard error of difference of sample means (see Student's t-test).

Two-way ANOVA: This method studies the effects of two factors (with several levels) separately (main effect) and, if desired, their effect in combination (interaction).

Type I error: If the null hypothesis is true but we reject it (wrongly) this is an error of first kind or type I error (also called a error, which is basically false-positivity). This results in a false positive finding (null hypothesis of no effect is true but we rejected it). The probability of (type I) error is assessed by the P value (i.e., the probability of false-positivity). Acceptable type I error rate is usually up to 5%.

Type II error: If the null hypothesis is accepted when it is in fact wrong (missing an existing effect), this is an error of the second kind or type II error (also called b error). This results in a false negative result. Type II error rate is (1 - statistical power), therefore, usually 20% (because statistical power is usually set at 80%). By setting statistical power at 80%, we acknowledge that up to 20% of the time, we will miss an existing effect (i.e., it will be observed, but will not reach statistical significance [i.e., will not be detected]).

Uncertainty: One of the pillars of statistics, based on no two samples will yield the identical statistics estimating the population parameter. Any statistics (from a sample) comes with an uncertainty characterized by some measure of it (like confidence intervals). Acknowledging uncertainty should not be viewed as a negative, and presenting every statistics with a measure of uncertainty (like an effect size with its 95% confidence interval) is good practice.

Unreplicated factorial: A single replicate of a 2k design (where each of k factors of interest has only two levels).

Variable: A characteristic that varies among experimental units (subjects) or from time to time. A variable may be quantitative or categorical. A quantitative variable is either discrete (assigning meaningful numerical values to observations: number of children, dosage in mg) or continuous (such as height, weight, temperature, blood pressure; also called interval variable). A categorical variable is either nominal (assigning observations to categories: gender, treatment, disease subtype, groups) or ordinal (ranked variables: low, median, high dosage). Conventionally, a random variable is shown by a capital letter, and the data values it takes by lower case letters.

Variance: The major measure of variability for a data set. To calculate the variance, all data values, their mean, and the number of data values are required. It is expressed in the squared unit of measurement. Its square root is the standard deviation. It is symbolized by s2 (a parameter) for a population and S2 (a statistic) for a sample (Online Calculator for Variance and Other Descriptive Statistics).

Variance ratio: Mean square ratio obtained by dividing the mean square (regression) by mean square (residual). The variance ratio is assessed by the F-test using the two degrees of freedom (k-1, N-k).

Wald test: A test for the statistical significance of a regression coefficient. It is obtained by comparing the maximum likelihood estimate of the slope parameter (expected b1) to an estimate of its standard error. The resulting ratio (W), under the hypothesis that b1 = 0, will follow a standard normal distribution. The two-tailed P value will be found from the Z table corresponding to P ( ç Z ç > W). It is not more reliable than the likelihood ratio test (deviance difference).

Welch-Satterthwaite t-test: The Welch-Satterthwaite t-test is an alternative to the pooled-variance t-test, and is used when the assumption that the two populations have equal variances seems unreasonable. It provides a t statistic that asymptotically approaches a t-distribution, allowing for an approximate t-test to be calculated when the population variances are not equal (Online Welch's Unpaired t-test; t-Test Assuming Unequal Sample Variances at Vassar).

Wilcoxon matched pairs signed rank T-test: A non-parametric significance test analogous to paired t-test. Most suitable for paired ordinal or interval/ratio variables (Online Wilcoxon's Matched-Pairs Signed-Ranks (W) Test at VassarStats; Online Wilcoxon's Matched-Pairs Signed-Ranks Test). It is normally used to test for the significance of the difference between the distributions of two independent samples (repeated measures or matched pairs) instead of t-test when normality is violated or in doubt (Wilcoxon Tests). See also Non-parametric tests on two independent samples in XLStat and IBM-SPSS.

William's correction (for G statistics): This is equivalent to Yates' continuity correction for Chi-squared test but used in likelihood ratio (G) statistics for 2x2 tables (Online G Statistics with William's correction).

Woolf-Haldane analysis: A method first described by Woolf and later modified by Haldane for the analysis of 2x2 table and relative incidence (relative risk) calculation. It is the preferred method for relative risk calculation when one of the cells has a zero using the formula: RR = (2a+1)(2d+1) / (2b+1)(2c+1). Since it is a modification of the cross-product ratio, it should be called odds ratio. For details and references, see Statistical Analysis in HLA and Disease Association Studies.

Yates's correction: The approximation of the Chi-square statistic in small 2x2 tables can be improved by reducing the absolute value of differences between expected and observed frequencies by 0.5 before squaring. This correction, which makes the estimation more conservative, is usually applied when the table contains only small observed frequencies (<20). The effect of this correction is to bring the distribution based on discontinuous frequencies nearer to the continuous Chi-squared distribution. This correction is best suited to the contingency tables with fixed marginal totals. Its use in other types of contingency tables (for independence and homogeneity) results in very conservative significance probabilities. This correction is no longer needed since exact tests are available.

Z score: The Z score or value expresses the number of standard errors by which a sample mean lies above or below the true population mean. Z scores are standardized for a distribution with mean = 0 and standard deviation = 1 (Corresponding P values for Z; Vassar: Z to P Calculator). The Z-statistics is defined as difference of sample proportions divided by standard error of difference of sample proportions (Online Two-Sample Z-Test, Online One-Sample Z-Test).



Armitage P & Colton T. Encyclopedia of Biostatistics. Volumes 1-8. John Wiley & Sons, 2005

Bland M. An Introduction to Medical Statistics. 3rd Edition. Oxford Medical Publications, 2000

Campbell MJ & Machin D. Medical Statistics: A Common Sense Approach. Wiley, 2002

Daly LE & Bourke GJ. Interpretation and Uses of Medical Statistics. 5th Edition. Blackwell Scientific Publications, 2000

Motulsky H. Intuitive Statistics. 4th Edition. OUP, 2017

Norman GR & Streiner DL. PDQ Statistics. 3rd Edition. BC Decker, 2003

Rosner B. Fundamentals of Biostatistics. 8th Edition. Cengage, 2015

Sokal RR, Rohlf FJ. Biometry. 4th Edition. WH Freeman & Company, 2011

Zar JH. Biostatistical Analysis. 5th Edition. Pearson, 2010

Elston, Olson & Palmer. Biostatistical Genetics and Genetic Epidemiology. Wiley, 2002



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Mehmet Tevfik DORAK, MD, PhD

Last updated on 19 August 2023

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