bluuebunny/crossref_metadata_2025 huggingface dataset:
Single‐Case Designs
Authors: Onghena Patrick
Published: September 2014

A single‐case design is an experimental design for a study in which one entity is observed repeatedly during a certain period under different levels of at least one independent variable. Single‐case designs have a long tradition and growing impact in behavioral science. They are the designs of first choice if demonstration of an effect in a single case is aimed at or if a test of a theory in a single case is sufficient (or the only possibility), and they are most compatible with clinical practice. The systematic and purposive planning of interventions in single‐case research represents the main difference with both observational time series research and qualitative case study research. In addition, single‐case research can gain internal and statistical‐conclusion validity by including randomization in the design. Two types of randomized single‐case designs are presented: randomized alternation designs and randomized phase designs. Finally, external validity of single‐case results can be demonstrated or tested by simultaneous or sequential replication and the corresponding simultaneous randomization tests and meta‐analytic procedures.
Nonequivalent Group Design
Authors: Reichardt Charles S.
Published: September 2014

A nonequivalent group design is a quasi‐experiment used to assess the relative effects of treatments that have been assigned to groups of participants nonrandomly. Because the participants have been assigned to treatments nonrandomly, differences in the composition of the treatment groups can bias the estimates of the treatment effects. A variety of statistical methods are available for taking account of this selection bias. Each method imposes different assumptions about the nature of the selection effects, but it can be difficult to determine which set of assumptions is most appropriate in a given research setting. Researchers are advised to minimize selection differences by using groups of participants that are initially as similar as possible.
Alternating Treatment Designs
Authors: Herrera Gina Coffee , Kratochwill Thomas R.
Published: September 2014

The alternating treatments design (ATD) is a type of single‐participant design that is commonly used to determine the effectiveness of two or more treatments. Conditions in the ATD are alternated rapidly and randomly/semirandomly, and effects can by analyzed statistically (e.g., randomization tests) and/or by visual inspection. Using the ATD can be advantageous because treatment effects can be found rather quickly and a withdrawal phase and baseline data are not required.
Percentiles
Authors: Clark‐Carter David
Published: September 2014

Percentiles tell you what percentage of a set of numbers, which have been put in ascending order (an ordered set), are above and below a particular number in the set. They can be useful for showing a particular score's position relative to the remainder of the set, which can either be a sample or a population.
Catalogue of Probability Density Functions
Authors: Everitt Brian S.
Published: September 2014

Probability density functions are the foundations of almost all statistical methodology. Here we give a brief account of the density functions most often encountered in practice along with a summary of their most important properties.
Effect Size Measures
Authors: Kirk Roger E.
Published: September 2014

A variety of measures of effect magnitude are described. Many of the measures can be classified as either measures of effect size (typically, standardized mean differences) or measures of strength of association. There also is a large group of statistics that do not fit into either category. Measures of effect magnitude are used for a variety of purposes: (a) to estimate the sample size required to achieve an acceptable power, (b) to integrate the results of empirical research studies in meta‐analyses, (c) to supplement the information provided by null hypothesis significance tests, and (d) to determine whether research results are practically significant.
Trimmed Means
Authors: Wilcox Rand R.
Published: September 2014

A trimmed mean discards a fixed proportion of the largest and smallest observations and averages the rest. With a proper choice for the amount of trimming, good efficiency can be obtained under normality as well as under nonnormal distributions where the mean performs poorly.
Binomial Effect Size Display
Authors: Rosenthal Robert
Published: September 2014

The binomial effect size display (BESD) was designed to indicate the practical importance of any particular effect size estimate. The BESD displays the difference between two proportions such as the difference between the success rate of a new intervention and the success rate of the standard intervention. For example, an
r
of 0.20 is said to account for ‘only 4% of the variance’, but the BESD shows that this proportion of variance accounted for is equivalent to increasing the success rate (e.g., survival rate, improvement rate, etc.) from 40 to 60%.
Three Dimensional (3
D
) Scatterplots
Authors: Wright Daniel B. , Williams Siân E.
Published: September 2014

A three‐dimensional scatterplot displays the relationship between three continuous variables at once. The data are represented as a cloud of points within the three‐dimensional space formed by the three axes of the plot. Such plots can be difficult to interpret and other methods of representing the third variable are generally more effective.
Graphical Presentation of Longitudinal Data
Authors: Wainer Howard , Spence Ian
Published: September 2014

There are some ideas that are so ubiquitous that we cannot imagine the world without them: language and numbers come to mind as immediate examples. Data‐based graphics provide a third example. The graphical representation and communication of data is such a simple and clear process that it is no easier to think of a world before graphs than a world before arithmetic. Yet, graphs were invented and, moreover, invented relatively recently. The very first plots were graphs of longitudinal data. In this essay, we will trace some of the development of graphs for the display of longitudinal data.
Trellis Graphics
Authors: Everitt Brian S.
Published: September 2014

In trellis graphics two or more variables are plotted for (conditioned on) different values or categories of a given variable (or variables). Component graphs for each level of the selected variable are arranged in a single display in order to make comparisons. Many different types of plot lend themselves to this treatment, including
scatterplots
and
three‐dimensional scatterplots
. Trellis graphics are helpful help in understanding both the structure of the data and how well proposed models for the data actually fit.
Cohen, Jacob
Authors: Cohen Patricia
Published: September 2014

Jacob Cohen's contributions to data analysis in the behavioral sciences include identifying the problem of low statistical power of much research, proposing power analysis methods and rough effect size norms to be used in planning research, development of kappa to assess chance‐corrected agreement, and promotion of multiple regression as a flexible general analytic system.
Urban,
F M
Authors: Ross Helen
Published: September 2014

Urban was born in Austro‐Hungary, worked in the USA, suffered under the First and Second World Wars, and lived in various countries. He contributed probability weightings to the calculation of the psychometric function.
History of Behavioral Statistics
Authors: Lovie Sandy
Published: September 2014

This article covers the growth of psychological statistics and the background to it from the nineteenth century to the twentieth. The general theme is that the nineteenth century was a time when statistics took up the challenge of a fundamentally uncertain world by embracing probability theory and providing an array of structuring principles to reduce the uncertainty.
Mid‐

P

Values
Authors: Berger Vance W.
Published: September 2014

Generally, hypothesis testing is conducted by way of a comparison of the value of a test statistic that summarizes the extent to which the data deviate from the null hypothesis with a reference distribution. The
P
value is the tail area of reference distribution relative to the value of the test statistic. Often, the reference distribution will be discrete, and when it is, the test will be conservative, as only certain
P
values can be attained. This conservatism will result in a loss of power. One approach to dealing with this is the mid‐
P
value, which essentially credits back half of the lost power.
Deductive Reasoning and Statistical Inference
Authors: Cussens James
Published: September 2014

Connections and contrasts between deductive reasoning and statistical inference are given. Bayesian statistical inference is analyzed, including an account of its relation to deductive logic and the status of prior distributions. Classical statistical inference is investigated with a focus on hypothesis testing.
MonteCarlo Goodness of Fit Tests
Authors: Besag Julian
Published: September 2014

Monte Carlo and Markov chain Monte Carlo goodness of fit tests are described and an application of the latter in testing the Rasch model is given.
Robust Testing Procedures
Authors: Wilcox Rand R.
Published: September 2014

Robust testing procedures are hypothesis testing techniques that are relatively insensitive to violations of assumptions such as normality and homoscedasticity. Major advances on how to study and develop robust testing techniques have appeared during the last forty years. The result has been a deeper understanding of why conventional ANOVA and regression methods can be unsatisfactory, plus a collection of new methods that have practical value.
A Priori v Post Hoc Testing
Authors: Berger Vance W.
Published: September 2014

In many situations, statistical analyses are given more credibility when they are planned prospectively, that is, in advance of the collection of the data. Such prespecified analyses cannot be said to have been suggested by the data, as post hoc analyses would be. A simple measure to prevent the difficult interpretation that can arise from post hoc analyses is to simply avoid them. This approach may be overly conservative, however, as the need may arise to test a hypothesis that the data suggested, and use the same data to test it. Several approaches exist for doing so, although not all of them have been cast as tests of post hoc analyses. In the analysis of variance framework, Fisher's least significant difference (LSD) and Scheffe's method can both be viewed as a means of performing post hoc linear comparisons, although they may not always be explicitly referred to as post hoc analyses. The Smirnov test may be viewed as a method to allow for the data to select the cut point of an ordinal outcome measure. Certain adaptive tests for ordinal data may be viewed as taking the Smirnov test to the next level, and allowing for the optimal combination of cut points, based on the data. The information‐preserving composite end point is a method to allow for data‐dependent outcome measure selection.
Event History Analysis ‐ Models
Authors: Vermunt Jeroen K. , Moors Guy
Published: September 2014

The purpose of event history analysis is to explain why certain individuals are at a higher risk of experiencing the event(s) of interest than others. This can be accomplished by using special types of methods which are usually referred to as hazard models. Two special features of hazard models are that they can deal with censored observations and with covariates that change their values during the observation period.
Classical Test Models
Authors: Muñiz José
Published: September 2014

The term ‘Classical Test Theory’ covers an entire set of psychometric models and procedures whose main objective is to solve the psychometric problems arising in the measurement of variables in the social sciences. Classical test theory permits us to estimate the psychometric properties of the tests, questionnaires, and scales used in social science measurement, guaranteeing that measurements are made with the appropriate scientific and technical rigor. The first work in this area appeared at the beginning of the twentieth century, and by the 1960s, the core of the classical approach was well established. The classical model assumes that a person's empirical score in a test is made up of two additive components, the true score corresponding to the assessed person in that test, and a certain measurement error. From this simple model, and with the addition of some assumptions and the definition of parallel tests, we obtain the principal classical formulas that allow the estimation of the psychometric properties of tests, such as reliability and validity.
Item Analysis
Authors: Luecht Richard M.
Published: September 2014

This entry presents a general database framework for scoring and analysis many types of selected‐response and constructed‐response item types. Specific item analysis methods and reporting techniques, including conditional statistics and discrimination indices, are presented within the context of this framework. A brief discussion of item response theory (IRT) item analysis methods is included.
Test Bias Detection
Authors: Beller Michal
Published: September 2014

Much of the criticism of psychological tests is derived from the observation that certain groups differ extensively in test performance. However, much of the polemic concerning the issues of test bias rests on two main areas of confusion: (a) differences in test performance among groups are often regarded, in and of themselves, as an indication of test bias, ignoring performance on the external criterion that the test is designed to predict. Often, groups that perform poorly on tests also tend to perform poorly on measures of the criterion. Furthermore, analysis of the relationship between tests and criteria often reveals similar regression lines for the various groups; and (b) the issue of test bias is often confused with the possibility of bias in the content of some individual items included in a test. Group differences in test performance are attributed to the content of specific items, and rather than eliminating problematic items in a systematic way (i.e., checking all items for differential performance), this confusion has, in some cases, led to suggestions of a wholesale rejection of reliable and valid test batteries.
Item Response Theory (IRT) Models for Dichotomous Data
Authors: Hambleton Ronald K. , Zhao Yue
Published: September 2014

This entry provides an introduction to the topic of item response theory. Shortcomings of classical test models are considered first. Second, current item response models for the analysis of dichotomously scored item response data are introduced. Estimation of model parameters, assessment of model fit, and available software, are described next. Finally, applications of item response theory IRT models to test development, item bias, equating, and computer‐adaptive testing are briefly described.
Item Response Theory: Cognitive Models
Authors: Hendrickson Amy B. , Mislevy Robert J.
Published: September 2014

Advances in cognitively based assessments have increased the need for inferences that can address complex task performance and instructionally useful information. These assessments integrate empirically based models of student learning and cognition with methods for designing tasks and observing student performance, with procedures for interpreting the meaning of those observations. Several psychometric models have been introduced to handle claims cast in such information‐processing terms, explicitly modeling performance in terms of theory‐based predictions of performance. Cognitively based item response theory (IRT) models incorporate features of items that influence persons' responses, and relate these features to response probabilities. Examples of IRT models that incorporate cognitively motivated structures are presented, including the Linear Logistic Test Model, multidimensional cognitive IRT models, and latent class models.
Computer‐Adaptive Testing
Authors: Luecht Richard M.
Published: September 2014

This entry presents an overview of computer‐adaptive testing (CAT). The basic CAT algorithm is introduced, followed by a discussion of item response theory (IRT) test information functions and their role in adaptive testing. Issues such as test efficiency and security risks under CAT are also discussed. The entry concludes with a description of four promising CAT variants, including CAT shadow tests, a‐stratified CAT, testlet‐based CAT, and computer‐adaptive multistage testing.
Latent Transition Models
Authors: Pol Frank , Langeheine Rolf
Published: September 2014

Flows between states for discrete periods of time can be described with a Markov chain, a process without memory. More realistic for social science data is a heterogeneous mixture of Markov chains, of movers and stayers. Alternatively, or in conjunction with heterogeneity in the tendency to change, one can take measurement unreliability into account, using latent or indirectly measured Markov models. The basic model variants are explained with data examples. References to overview papers are given.
Liability Threshold Models
Authors: Neale Ben
Published: September 2014

The liability threshold model holds that for binary traits influenced by multiple factors of small effect, an underlying liability distribution exists, with a threshold that divides the population into the two categories for the trait. This principle extends to include ordered polychotomous (also known as
ordinal
) variables, where the number of thresholds is one less than the number of levels.
Additive Tree
Authors: Corter James E.
Published: September 2014

Additive trees are useful models for the analysis of proximity data. In such analyses, the data dissimilarities are represented by path‐length distances in the additive tree graph. The choice of a root for an additive tree can affect its interpretation in terms of clusters. Effective algorithms exist to fit additive trees to behavioral data.
Optimal Scaling
Authors: Takane Yoshio
Published: September 2014

A data matrix typically represents some kind of relationship between row and column entities. The relationship represented by the data may be described by a model presumed to have generated the data. Observed data, on the other hand, may be measured on one of a variety of scale levels: nominal, ordinal, interval, or ratio. In such cases, we may attempt to do two things simultaneously: (a) we transform the data by a transformation appropriate for the scale level, and (b) we fit a model to the transformed data to account for the data. This process of simultaneous data transformations and data representations is called optimal scaling. In this article, we briefly discuss some of the key features of optimal scaling.
Ultrametric Inequality
Authors: Murtagh Fionn
Published: September 2014

The triangular inequality is a defining property of a metric space, while the stronger ultrametric inequality is a defining property of an ultrametric space. Ultrametric distance is defined from p‐adic valuation.
Ultrametric Trees
Authors: Murtagh Fionn
Published: September 2014

The triangular inequality is a defining property of a metric space, while the stronger ultrametric inequality is a defining property of an ultrametric space. Any hierarchical agglomerative clustering can be considered as the mapping of points in a metric or indeed nonmetric space into an ultrametric space.
Cross‐Lagged Panel Design
Authors: Kenny David A.
Published: September 2014

The minimal cross‐lagged panel design is two variables measured at two times. Multiple regression analyses, in which the time two measures are regressed on the time one measures, have become the standard method of analyzing the design. Currently, methods to rule out third‐variable causation are little used. Several analysts have argued that two waves are insufficient to obtain much information about how processes unfold over time.
Multivariate Normality Test in Practice
Authors: Keselman H. J.
Published: September 2014

Most classical multivariate procedures (e.g., multivariate analysis of variance, multivariate measures of effect size, classification procedures, maximum likelihood factor analysis) require that the data follow a multivariate normal density function. Behavioral science researchers risk committing many more Type I errors, quantifying inaccurately the magnitude of effect sizes, missing treatment effects, establishing inaccurate confidence intervals, and so on by failing to consider whether their data conform to multivariate normality. This paper discusses a number of options for assessing and dealing with nonnormal multivariate data including: (a) testing for univariate normality among thepmeasures, (b) transforming the data to achieve normality, (c) univariate normal probability plots, (d) multivariate measures of skewness and kurtosis, (e) computing squared distance statistics to locate outlying values, and (f) adopting robust estimators with robust test statistics to circumvent the biasing effects of nonnormality.
Independent Pathway Model
Authors: Rijsdijk Frühling
Published: September 2014

While traditional multivariate statistical methods, can describe correlational patterns of a set of variables (e.g., psychiatric symptoms or problem behaviors in children), they cannot provide insight into why certain symptoms or behaviors tend to co‐occur in a population. This can be achieved using methods of
multivariate genetic analysis
. In multivariate genetic models 2 main classes can be distinguished, which differ in the way the common factors are assumed to influence the different observed variables. The way unique factors influence the observed variables is the same in both classes. The first class of models is referred to as
common pathway models
or psychometric models. Common genetic and environmental factors influence all observed variables via a single psychometric factor, or underlying latent liability. The second class of models is referred to as independent pathway or biometric models. The common genetic and environmental factors influence the observed variables directly, without an intermediate higher order factor. As a result, the common genetic and environmental do not necessarily cause similar groupings. This means that different clusters of variables can be identified for shared genetic and environmental effects. For this reason biometric models can yield different results compared to higher order phenotypic factor models, which implicitly assume identical groupings for genetic and environmental effects.
Structural Equation Modeling: Nonstandard Cases
Authors: Marcoulides George A.
Published: September 2014

Model assumptions are often violated in typical data gathering conditions. This article discusses the consequences of violating model assumptions and presents strategies that can be used to cope with the violations.
Covariance Matrices: Testing Equality of
Authors: Krzanowski Wojtek J.
Published: September 2014

Parametric multivariate inferential techniques make distributional assumptions about the data, and a common assumption in multisample problems is that dispersion matrices are equal in all the populations from which the samples have come. There is therefore need for a hypothesis test that can check on the adequacy of this assumption. In this article, we provide an appropriate test statistic, give its approximate (large sample) null distribution, illustrate its use on a small data set, and provide some cautionary comments about interpretation.
Correlation and Covariance Matrices
Authors: Macdonald Ranald R.
Published: September 2014

A covariance matrix measures the shared variability within a set of variables assuming that all the relationships between them are linear. The correlation matrix is the covariance matrix of a set of variables after they have been standardized. Standardization means that the measures of association are scale independent.
Mahalanobis Distance
Authors: Huberty Carl J.
Published: September 2014

The Mahalanobis distance is a generalization of the distance concept to
p
‐dimensional correlated data. The measure uses an appropriate covariance matrix to take account of differences in variable variances and correlations between variables. It is frequently employed in multivariate statistical methods, for example, in multivariate analysis of variance, discriminant analysis, and cluster analysis.
Minimum Spanning Tree
Authors: Michailidis George
Published: September 2014

A minimum spanning tree (MST) connects all the vertices of a weighted graph by existing edges whose total weight is minimum. It has been the object of intensive study due to its importance in combinatorial optimization. In this paper, we discuss the formulation of the MST problem, present some of the most commonly used algorithms for its construction, and discuss several of its applications in statistics, probability, and data analysis.
Multidimensional Unfolding
Authors: Leeuw Jan
Published: September 2014

Multidimensional unfolding applies distance models and scaling techniques to rectangular matrices of preference and attitude data. We show with examples that unfolding can be applied to numerical data, to rank orders, and to binary data matrices.
Psychophysical Scaling
Authors: Irtel Hans
Published: September 2014

Psychophysical scaling investigates methods for mapping physical stimulus properties into sensation scales. The two major types of subject tasks used are either based on discriminability of rather similar stimulus intensities, or on direct judgments about perceived differences of clearly distinct stimuli. Classical discrimination scaling tries to find a stimulus transformation such that equally discriminable stimuli are mapped onto constant differences on the sensation scale. Direct scaling methods include operations on the set of stimuli that represent certain numerical ratios of perceived stimulus intensity, or require the subject to actually assign these numerical ratios to given pairs of stimuli. In many cases, the power law is an acceptable description of the empirical data.
Scaling of Preferential Choice
Authors: Böckenholt Ulf
Published: September 2014

Scaling models for preferential choice data serve the dual purpose to summarize choice data and to facilitate the forecasting of choices by decision makers facing possibly new or different variants of the choice options. The summary representation obtained by scaling methods can provide useful information for identifying both option characteristics that influence choice and systematic sources of individual differences in the evaluation of these option characteristics. Many scaling models are based on Thurstone's random‐utility approach and allow the analysis of different types of both revealed and stated choice data.
Path Analysis and Path Diagrams
Authors: Boker Steven M. , McArdle John J.
Published: September 2014

Path analysis refers to the calculation of moment structures within and between variables, for example, covariances or correlations, implied by a set of simultaneous linear regression equations – one type of structural equation model. Path diagrams of structural equation models offer a graphical representation with exact one‐to‐one mapping to the simultaneous linear regression equations. Path analysis allows one to decompose the covariance between two variables in a structural equation model into additive components, thus helping to understand how the interrelationships between many variables in a model predict the covariance between two selected variables. By performing path analysis calculations on all variables and pairs of variables in a model, one may calculate the expected moment structure implied by that model. Path analysis preceded and led to the development of modern structural equation modeling methods of parameter estimation.
Random Forests
Authors: Cutler Adele
Published: September 2014

Random forests are collections of trees that give accurate predictions for regression, classification, and clustering problems. They handle missing values, can be used for continuous and/or categorical input variables, and allow class weighting to adjust for unequal sampling schemes. Random forests do not overfit the data so they can be used for problems in which the number of input variables is much larger than the number of observations. Random forests provide measures of variable importance and proximities that can be used to interpret the data.
R
&
Q
Analysis
Authors: Jackson J. Edward
Published: September 2014

This entry discusses five alternative criteria for performing principal components analysis on a data matrix. These criteria differ in whether the data are corrected for the variable means, the observation means, both variable and observation means, or neither of them.
Robust Statistics for Multivariate Methods
Authors: Victoria‐Feser Maria‐Pia
Published: September 2014

Robust statistics deals with slightly misspecified models. When the data have not been generated exactly from the assumed models, robust procedures are by definition not (or very little) influenced by these model deviations. This is, in general, not the case with classical procedures such as least squares, maximum likelihood, and so on. Numerous multivariate models are based on the multivariate normal distribution, for which a relatively large number of robust procedures have been proposed. Most of the robust estimators are based on weighted means and covariances, and the weights are decreasing functions of the Mahalanobis distances. For nonnormal multivariate models, weights are more difficult to define, and more research is needed to develop robust procedures in these cases.
Factor Score Estimation
Authors: Hershberger Scott L.
Published: September 2014

Factors scores are measures of principal components or common factors. Under the principal components model, the factor scores are uniquely determined; under the common factor model, they are not. In the latter situation, the factor scores are indeterminate, potentially having an infinite number of solution sets, and thus their true values can only be estimated. Three methods of factor score estimation are discussed: (a) regression, (b) Bartlett's method, and (c) Anderson and Rubin's method. Although the properties and values of the factor score estimates produced by the three methods differ, the estimates are in general highly correlated.
Two‐Way Factorial: Distribution‐Free Methods
Authors: Toothaker Larry E.
Published: September 2014

Nonparametric and resampling approaches for the two‐way factorial design include permutation tests, bootstrap tests, the rank transform, aligned ranks tests, tests on trimmed means, and a rank‐based ANOVA method that allows heteroscedastic variances, one known as the BDM method, or combinations of these such as bootstrap tests on trimmed means. Research shows the BDM method, parametric tests on trimmed means, and bootstrap tests on trimmed means to be the most promising methods in terms of α‐control and power.
Kernel Smoothing
Authors: Hazelton Martin L.
Published: September 2014

Kernel smoothing is a nonparametric technique for exposing structure or trend in data. It can be applied in a wide variety of settings, including the problems of density estimation and nonparametric regression on which we focus attention. In each case, the performance of kernel methods is highly dependent on the choice of bandwidth, a parameter that controls the amount of smoothing applied to the data.
Theil Slope Estimate
Authors: Lunneborg Clifford E.
Published: September 2014

A nonparametric hypothesis test for the slope in simple linear regression proposed by Theil (1950) also provides the basis for a distribution‐free estimate of that parameter.
Distribution Free Inference, an Overview
Authors: Lunneborg Clifford E.
Published: September 2014

Distribution‐free or nonparametric techniques provide a basis for statistical inference where parametric assumptions cannot be justified. In particular, such techniques are essential where the scale of measurement is not interval or, if interval, observations are not likely to follow a Normal law. Standard and more recent distribution‐free techniques are surveyed here.
Runs Test: Examples
Authors: Lunneborg Clifford E.
Published: September 2014

The Wald–Wolfowitz runs test dates from 1940, making it one of the earliest nonparametric tests. It provides a test of a common distribution for two independent random samples. However, the test has low power relative to such alternatives as the Kolmogorov–Smirnov or Cramér–von Mises two‐sample tests and has declined in popularity.
Kolmogorov–Smirnov Test: Overview
Authors: Berger Vance W. , Zhou YanYan
Published: September 2014

While often confused, the Kolmogorov–Smirnov test and the Smirnov test are actually distinct. Specifically, the Kolmogorov–Smirnov test is used to test the goodness of fit of a given set of data to a theoretical distribution, making this a one‐sample test. In contrast, the Smirnov test is a two‐sample test, used to determine if two samples appear to follow the same distribution. The intuition behind the two tests is the same, however, in that both compare cumulative distribution functions, either two empirical cumulative distribution functions for the two‐sample Smirnov test, or one empirical cumulative distribution function and one known cumulative distribution function for the one‐sample Kolmogorov–Smirnov test.
Symmetry: Distribution Free Tests for
Authors: Lunneborg Clifford E.
Published: September 2014

Nonparametric tests for the median of a distribution and the related estimation of confidence intervals for that parameter assume that the distribution sampled is symmetric about the median. To avoid obtaining misleading results, a preliminary test for distribution symmetry is advisable.
Scatterplot Smoothers
Authors: Everitt Brian S.
Published: September 2014

Fitting a simple linear regression to describe the relationship between two variables is not always adequate. Polynomial fits can be used, but a more satisfactory solution is often achieved by using a locally weighted regression approach or spline functions.
A
nsari–
B
radley Test
Authors: Lunneborg Clifford E.
Published: September 2014

The Ansari–Bradley procedure provides a distribution‐free test of the equivalence of the variances in two distributions having a common median.
Multiple Comparison Tests: Nonparametric and Resampling Approaches
Authors: Keselman H. J. , Wilcox Rand R.
Published: September 2014

Resampling‐based multiple comparison procedures attempt to circumvent the deleterious effects of nonnormality and/or variance heterogeneity by utilizing empirical rather than theoretical sampling distributions of test statistics. In particular, the original observed data are randomly resampled in order to build an empirical sampling distribution for the test statistic, and the observed test statistic can then be compared to quantiles in this empirical distribution to determine statistical significance or to compute confidence intervals. We present resampling test statistics that can be used to test pairwise and complex contrasts among treatment group trimmed means. Trimmed means are used to circumvent problems due to nonnormality. Moreover, the test statistic is designed to deal with variance heterogeneity. Combining robust estimators (trimmed means) with a heteroscedastic test statistic (e.g., Welch's (1938) two‐sample test) and applying a bootstrap method results in test statistics that should be robust to the combined effects of nonnormality and variance heterogeneity.
Cross‐Classified and Multiple Membership Models
Authors: Rasbash Jon
Published: September 2014

This paper proposes a notation and some diagrammatic tools for describing cross‐classified and multiple‐membership multilevel models. Example structures and analyses are given from the areas of education, psychology, medicine, and veterinary epidemiology.
Fixed and Random Effects
Authors: B. Snijders Tom A.
Published: September 2014

In performing a multilevel analysis, what is a level? And when should a variable have a random slope? This depends on whether the units in the design should be regarded as being representative of a population and on whether the researcher wishes to draw conclusions primarily about the observed units or primarily about the population.
Multilevel and
SEM
Approaches to Growth Curve Modeling
Authors: Hox Joop , Stoel Reinoud D.
Published: September 2014

Growth curve models are used to model development explicitly as a function of time. Two techniques are commonly used to construct and analyze growth curve models: structural equation modeling, with distinct measurement occasions represented by separate variables, and multilevel regression modeling, with different measurement occasions represented as nested within individuals. Both approaches are essentially the same, but they differ in the way they can be extended into models that are more complex. An example is given, and estimation methods and software are briefly discussed.
Mediation
Authors: Kenny David A.
Published: September 2014

A mediator explains, at least in part, a causal relationship between two variables. The total relationship between the two variables is due to a indirect effect, that part explained by the mediator, and due to an direct effect, that part not explained be the mediator. Strong assumptions must be made to estimate and test mediation. The inclusion of multiple variables complicates mediational analysis. Mediational analysis while seemingly simple is actually quite complex.
Marginal Models for Clustered Data
Authors: Fitzmaurice Garrett M.
Published: September 2014

Clustered data commonly arise in studies in the behavioral sciences. For example, clustered data arise when observations are obtained on children nested within schools. Longitudinal and repeated measures studies also give rise to clustered data. Marginal models provide a broad class of regression models that are suitable for analyzing diverse types of clustered data (e.g., continuous, binary, and count data). The distinctive feature of marginal models is that they focus on each response within the cluster separately, allowing dependence of the mean response on covariates but not on any random effects or other responses within the same cluster. By focusing on the population mean response, averaged over all clusters, the regression coefficients relating the mean response to covariates are said to have
population‐averaged
interpretations.
Transformation: Overview
Authors: Schmitt Neal
Published: September 2014

Transformations are systematic alterations to a set of scores that allow for application of various statistical procedures or better communication to readers. Transformations that often change the mean and standard deviation of a set of scores, but preserve the rank order and relative size of the distance between scores are linear; nonlinear transformations do not preserve these characteristics of the original data. Taking the square or square root or normalizing a set of scores are nonlinear transformations that make multivariate or parametric data analyses procedures more acceptable. Computing percentiles or T scores is often done to provide data in terms that are more easily understood by the statistically unsophisticated audiences.
Hill's Criteria of Causation
Authors: Goodman Karen J. , Phillips Carl V.
Published: September 2014

The term
criteria of causation
(or
causal criteria
) is often applied to Sir Austin Bradford Hill's (8) list of factors to consider before inferring causation from an observed association. This list is widely taught and cited, despite widespread cautions about the limitations of the list, by Hill and others, and despite lack of evidence that invoking such criteria is useful. This entry describes Hill's views on assessing causation and what others have since said on the subject of causal criteria.
Survey Questionnaire Design
Authors: Tourangeau Roger
Published: September 2014

Survey questionnaire design encompasses a variety of activities ranging from library research to formal experiments that estimate the reliability or validity of different versions of a questionnaire. Although handbooks with advice for survey questionnaire designers first appeared more than 50 years ago, it has only been during the last 20 years that researchers have attempted to provide a firm theoretical and empirical foundation for such guidelines, based on a cognitive analysis of the survey response process. The last 20 years or so have also seen an expansion in the methodological and statistical tools used in developing survey questionnaires; these tools include focus groups, cognitive interviews, field tests, behavior coding of respondent‐interviewer exchanges, reinterviews, and experiments.
Epistasis
Authors: Evans David M.
Published: September 2014

Biologically speaking, epistasis refers to the interaction between genes at different loci to produce effects which are different from that expected from the individual loci themselves. In quantitative genetic terms, epistasis refers to a deviation from additivity in the effect of alleles and genotypes at different loci with respect to their contribution to a quantitative phenotype. The proportion of variance due to epistasis may be estimated using careful breeding experiments in experimental organisms, but cannot be estimated using the classical twin design. It is possible to model epistasis explicitly in linkage and association analyses.
Genotype
Authors: Elston Robert C.
Published: September 2014

The genotype of an organism is its genetic make‐up, as opposed to its phenotype, or outward appearance. If the two alleles at a locus are the same, the genotype is homozygous; otherwise it is heterozygous. If a heterozygote has the same phenotypic distribution as one of the corresponding homozygotes, the allele in that heterozygote is dominant and the other allele is recessive with respect to that phenotype. A multilocus genotype is made up of two haplotypes. In the case of a quantitative trait, dominance corresponds to intralocus interaction, epistasis to interlocus interaction.
Number Needed to Treat: Definition and Controversies
Authors: Lesaffre Emmanuel
Published: September 2014

The number needed to treat (NNT) is the inverse of the difference between two probabilities. Clinicians prefer the NNT over classical measures like the odds‐ratio to evaluate the impact of a new treatment versus a control treatment. On the other hand, statisticians have pointed out that the NNT has undesirable statistical properties. Despite the dispute between clinicians and statisticians, the NNT is becoming increasingly important in Evidence‐based Medicine. The use of NNT will be illustrated with a medical example. Further, we will review the claims that have been made by the clinicians and the criticism of the statistical community.
Prospective and Retrospective Studies
Authors: Dykacz Janice Marie
Published: September 2014

Prospective and retrospective studies differ in whether the explanatory variable or the outcome variable is used to select the individuals in the study. The relative risk is frequently used to measure association for a prospective study. The odds ratio is frequently used to measure association for a retrospective study.
Historical Controls and Clinical Trials
Authors: Berger Vance W. , Shafran Randi
Published: September 2014

Randomized trials, in which a single group of units is randomly allocated to two or more treatment conditions and then studied in parallel over time, is the gold standard methodology for reducing biases. Yet not every comparison is amenable to a randomized trial. For example, one cannot randomize subjects to their genders or races. Furthermore, even some comparisons that could theoretically be amenable to study with a randomized trial would not be, for ethical reasons. This would include studies of tobacco. Nonrandomized studies may still be parallel over time, in that all treatment groups are treated and studied over the same time interval. In some cases, though, one needs to use historical controls, and perform a before‐and‐after comparison. Strong conditions are required for such comparisons to be reliable.
Single and Double‐Blind Procedures
Authors: Well Arnold D.
Published: September 2014

When an experiment is conducted, there are many factors other than the independent variable that may influence the behavior of both participants and researchers, possibly leading to biased results. Sources of bias that depend on knowledge about the purpose of the experiment or the details of the research design may be removed by keeping participants and researchers ‘blind’ to such information.
Family Study and Relative Risk
Authors: McGuffin Peter , Burke Ailbhe
Published: September 2014

Family studies provide the starting point for investigation of the genetics of common complex disorders. Families are ascertained via systematic series of proband (index cases). Relative risk is a useful statistic in gauging familiality, but estimating the size of familial effect requires age correction to take into account variation in age of onset.
Receiver Operating Characteristics Curves
Authors: Wright Daniel B.
Published: September 2014

The receiver operating characteristic (ROC) curve is the principal graphical device for signal detection theory (SDT). It is generally used when there is a right or a wrong answer and participants provide some measure of confidence in their responses. The ROC is a scatterplot between the hit rate and the false alarm rate. Exploring this space allows the researcher to consider the differential value of hits and correct rejections (or of misses and false alarms). This is where ROCs have a big advantage over the alternatives. The main alternative,
logistic regression
, is based on similar mathematics. Historically, SDT/ROCs assumed normal distributions, but this is very similar to the logistic distributions, and most statistical packages allow either. The choice of approach rests largely on the form of graphical communication the researcher wants to use. While logistic regression focuses on discrimination, in SDT the researcher explicitly differentiates two types of accuracy: sensitivity and specificity. The ROC curve is an excellent tool for comparing these.
Industrial/Organizational Psychology
Authors: Landis Ronald S. , Kaplan Seth A.
Published: September 2014

Industrial and organizational (IO) psychologists utilize a variety of statistical techniques in their research. A nine‐year review of two leading journals (
Journal of Applied Psychology
and
Personnel Psychology
) in the field reveals that correlational techniques appear in the largest percentage of studies. More complex methods such as structural equation modeling (SEM) and hierarchical linear modeling (HLM) appear in proportionally fewer studies. Representative examples of how various statistical techniques are used in IO research are provided, and possible reasons for the observed results are presented. In addition, IO psychologists' contributions to statistics and the role of statistics in IO theory development are discussed.
Cross‐Sectional Design
Authors: Busk Patricia L.
Published: September 2014

Cohorts of individuals are measured at the same time in order to research temporal or age differences. By taking a cross section of a population, researchers avoid the disadvantages of longitudinal research, specifically loss of subjects over time.
Correlation Issues in Genetics Research
Authors: Cherny Stacey S.
Published: September 2014

Correlation is of central importance in behavioral genetic research. A basic goal of behavioral genetic research is to partition variance in a trait into variance due to genetic differences among people (commonly referred to as
heritability
(or
h
2
), variance because of environmental influences shared among family members (shared environmental variance, or
c
2
), and nonshared environmental variance as a result of unique individual experiences (
e
2
) (
see
ACE Model
). This is accomplished by analyzing correlational structure within different familial relationships. Although in practice we perform
maximum‐likelihood
model‐fitting analyses on raw data and partition (unstandardized) covariance structure, examining familial correlations directly is the best way of quickly getting an idea of what is going on in the data.