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.

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. By randomization systematic effects are turned into error (term) and there is an expected balancing out effect: known and unknown factors that might influence the outcome are assigned equally to the comparison groups. One disadvantage of randomization is generation of a potentially large error term. This can be avoided by using a block design. See Basic Concepts of Sampling and

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

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.

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).

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.

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

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

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

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.

Predictor (explanatory independent) variable: The variable already in hand

Power of a statistical test: See Statistical Power.

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) (

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).

Population stratification (substructure):

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

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

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).

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)=2×1+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.

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.

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

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).

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 is not met (for example due to outliers) the non-parametric equivalent Spearman’s rank correlation should be used. The r represents

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

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.

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.

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.

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.

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.

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).

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.

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.

Omnibus test: If the chi-square test has more than one degree of freedom (larger than 2×2 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).

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

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 abcd are the entries in a 2×2 contingency table (hence the alternative definition as the cross-product ratio). In logistic regression the coefficient

Odds multiplier: In logistic regression

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.

Number needed to treat:

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.

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.

Normal distribution(Gaussian distribution) is a model for values on a continuous scale. A normal distribution can be completely described by two parameters: mean (

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 (a tutorial on Parametric vs Nonparametric Methods; Review of Nonparametric Tests in Intuitive Biostatistics; Nonparametric Tests for Ordinal Data at Vassar).

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.

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

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

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).

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).

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.

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;

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

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.)

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 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 (

Multiple regression: To quantify the relationship between several independent (explanatory) variables and a dependent (outcome) variable. The coefficients (

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.

Monte Carlo trial: Studying a complex relationship difficult to solve by mathematical analysis by means of computer simulations. An online book on

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

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

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.

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

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).

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.

Measures of central tendency: These are parameters that characterize an entire distribution. These include mode median and mean.

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 also Measures of Effect-Size & Association; Measures of Association for Cross-tabulations; Measures on SAS STATA SYSTAT; Ennis 2001; Morton 2001.

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 (

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) (

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 (

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

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.

Mantel-Haenszel(also called Cochran-Mantel-Haenszel (CMH) test): Test for a null hypothesis of no overall relationship in a series of 2×2 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

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 ordinal or 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).

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.

Loglinear model: Multinomial data (from contingency tables) 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.

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.

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

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.

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

Log 0is 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.

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.

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).

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

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

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 = b

Linear expression: A polynomial expression with the degree of polynomial being 1. It will be something like f(x)=2x

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

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: 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;

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 centre 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.

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 (x

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

Kurtosis:

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 (

Kolmogorov-Smirnov two-sample test: A non-parametric test applicable to continuous frequency distributions. It is considered to be the equivalent of the

Interval variable(equivalent to continuous variable)

Interquartile range (dQ): dQ is a measure of spread and is the counterpart of the standard deviation for skewed distributions. dQ is the distance between the upper and lower quartiles (Q­U-QL).

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.

Intercept: In linear regression the intercept is the mean value of the response variable when the explanatory variable takes the value of zero (the value of y when x=0).

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).

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.

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

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).

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.

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.

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.

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

Hazard Ratio (Relative Hazard): Hazard ratio compares two groups differing in treatments or prognostic variables etc. If the hazard ratio is 2.0 then the rate of failure in one group is twice the rate in the other group. The computation of the hazard ratio assumes that the ratio is consistent over time and that any differences are due to random sampling. Before 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).

Hazard Rate: It is a time-to-failure function used in

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).

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.

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).

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.

Geometric mean: G = (x1.x2…x

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 e

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 (

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.

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

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

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.

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.

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.

Friedman test: This non-parametric test is an extension of Wilcoxon matched pairs signed ranks test which is concerned with more than two time periods of data collection or conditions and groups of three or more matched subjects (

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).

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.

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.

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).

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

Exponential family: A family of probability distributions in the form

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

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.

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.

Error terms: Residuals in regression models. Shown as Wi or

Epi-Info: An epidemiologic data management and analysis package freely available from the CDC website. Originally a DOS program the latest version (3.3.2) is designed for Microsoft Windows 95-XP (release date: Febr 9 2005). There are also tutorials available online: CDC Nebraska University Dalhousie University and Henry Ford Health Systems.

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 significance Turkey nerd of nonsignificance 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).

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.

Empirical P value: A P value obtained by a simulation program such as Monte Carlo statistics. This is less likely to be affected by multiple comparisons.

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). For a brief overview see ARC CIGMR: EM Algorithm. Arlequin implements EM algorithm in haplotype construction and frequency analysis.

Eigenvalues(latent values): In multivariate statistics eigenvalues

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.

Edwards’ test:

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.

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

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.

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 (for an example see

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.

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- peoples 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).

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:

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:

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.

Deletion (or deleted) residual: A modified version of the standardized residual which uses an estimate of

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 2×2 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 =

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.

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

Cramers 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 2×2 table Cramer’s V is equal to the Phi coefficient. Cramers 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 Cramers V calculation.)

Cramers 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) Cramers 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 (

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).

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 x

Covariate: Generally used to mean explanatory variable less generally an additional explanatory variable is of no interest but included in the model. More specifically it denotes a ‘continuous’ explanatory variable which is unaffected by treatments and has a linear relationship to the response. The intention is to produce more precise estimates of the effect of the explanatory variable of main interest. In the analysis a model is first fitted using the covariate (and the blocking factor if any). Then the main explanatory variable is added and its additional effect is assessed statistically. Whether the use of a covariate is wise 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.

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 (Online Cox Proportional Hazards Survival Regression; a Superlectures on survival analysis; Cox’s proportional hazards model; an example of Model Building).

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 (v7.0) VST Statistica SAS but most conveniently on MultiVariate Statistical Package (MVSP). Link to a Tutorial; Course Notes; StatSoft Textbook: Correspondence Analysis Chapter.

Correlation Coefficient: See

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.

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

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

Confounding variable: A variable that is associated with both the outcome and the exposure 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. See Bias & Confounding Lecture Note; a review by Greenland 2001.

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.

Concomitant variable: See covariance models.

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.

Coefficient of variation: It is a measure of spread for a set of data. It is a measure of variation in relation to the mean. Calculated as standard deviation divided by the mean (x100). (Online Calculator for Coefficient of Variation and Other Descriptive Statistics).

Cochran’s Q Test: A nonparametric test examining change in a dichotomous variable across more than

Chi-squared test: The most commonly used test for frequency data and goodness-of-fit. 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 Fishers exact test for 2×2 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/fit) for each cell in a contingency. In this formula residual is the difference between the observed value and its expected counterpart and fit is the expected value. See

Chi-squared distribution: A distribution derived from the normal distribution. Chi-squared (

Central limit theorem: The means of a relatively large (>30) number of random samples from any population (not necessarily a normal distribution) will be

Censored observation: O

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.

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 odds ratio/relative risk/hazard ratio). See Hills’s criteria of causality

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.

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.

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

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).