bluuebunny/crossref_metadata_2025 huggingface dataset:
Number Needed to Treat
Authors: Lesaffre Emmanuel
Published: September 2005

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.
Number of Clusters
Authors: Wishart David
Published: September 2005

Various tests for the best number of clusters in a data set are reviewed, following an exhaustive validation exercise using artificial data sets with hierarchical agglomerative clustering methods. The chapter concludes with a discussion about whether it is appropriate to apply statistical tests or to evaluate a classification at several levels of usability.
Odds and Odds Ratios
Authors: Rudas Tamás
Published: September 2005

After defining the concepts of odds, logits, conditional odds, and odds ratios, the asymptotic variances of the estimates of some of these quantities are given. The properties of odds ratio as a measure of association, and among these the importance of variation independence from the marginal distributions, are discussed. The reasons and correct interpretation of the possibly different directions of marginal and conditional associations (Simpson's paradox) are described. Conditional odds ratios are used to give an interpretation of log‐linear models.
One Way Designs: Nonparametric and Resampling Approaches
Authors: Hayes Andrew F.
Published: September 2005

Resampling‐based variants of the one‐way analysis of variance (ANOVA) are discussed, including their assumptions, computation, and interpretation. Although resampling approaches to comparing groups offer some appealing advantages to ANOVA, they are not a panacea to assumption violations. Caution must be exercised in the interpretation of a resampling version of ANOVA because the null hypothesis tested depends on the test statistic and the method of resampling used.
Directed Alternatives in Testing
Authors: Cohen Arthur
Published: September 2005

Consider a hypothesis‐testing problem concerned with a one‐dimensional parameter θ. In testing a null hypothesis
H
 : θ = θ

o

, a two‐sided alternative is
K
 : θ ≠ θ

o

. A one‐sided alternative is
K
 : θ > θ

o

. This latter alternative is appropriate when it is reasonably certain that θ cannot be less then θ

o

. The advantage of posing a one‐sided alternative is that the appropriate one‐sided test is much more powerful than the corresponding two‐sided test. The idea of one‐sided and two‐sided alternatives extends to the case of testing a
k
‐dimensional parameter
θ
. If the null hypothesis is
H
 : 
θ
 = 
θ
o
, then the analogue to the two‐sided alternative is
K
 : 
θ
 ≠ 
θ
o
. This latter alternative is said to be unrestricted or not directed. An example of the analogue to the one‐sided alternative is
K
 : θ

i

 ≥ θ

io

, for
i
 = 1, …,
k
, with strict inequality for at least one index
i
. Here
θ
 = (θ
1
, …θ

k

). Such an alternative is called a
directed alternative
.

We introduce the notion of directed alternatives for multidimensional parameters. Among the special cases treated are the simple order alternative in which parameters are nondecreasing, the tree order alternative in which case one mean parameter is assumed to be less than or equal to each of the other mean parameters, and the stochastic order alternative in which one multinomial distribution is more concentrated at higher categories than another distribution. We indicate appropriate tests for these directed alternatives.
Optimal Design for Categorical Variables
Authors: Berger Martijn P. F.
Published: September 2005

Optimal design research is important for behavioral research because it enables efficient estimation of parameters and maximum power with minimum sample size and costs of performing a study. In this paper, an elementary introduction to the concepts of optimal design research is given and illustrated by means of four different cases, namely, linear and logistic models, with and without dummy coding. Although different models have different optimal designs, a rule‐of‐thumb may be that if the independent variables are treated as being quantitative, efficiency will increase if the variance of the independent variable increases, and if the design is as orthogonal as possible. For qualitative independent variables, a design that has equal sample sizes for all categories is often optimal.
Optimal Scaling
Authors: Takane Yoshio
Published: September 2005

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.
Optimization Methods
Authors: Lewis Robert Michael , Trosset Michael W.
Published: September 2005

Optimization is concerned with identifying the best element in a set of competing possibilities. Except in special circumstances, one must use numerical methods to search for solutions iteratively. If one's problem is well‐behaved, for example, the objective to be minimized is smooth and the feasible possibilities are smoothly defined by finite numbers of equality and inequality constraints, then one can use a number of sophisticated methods to identify local solutions, that is, possibilities that are as good or better than any other nearby possibilities. However, finding global solutions is much harder.
Outliers
Authors: Wilcox Rand R.
Published: September 2005

Outliers are values or points that are relatively far from the bulk of the observations. They are important because modern outlier detection methods suggest that they are relatively common, they can distort commonly used summaries of data, and they can result in relatively low efficiency and low power when using methods based on means or least squares regression.
Overlapping Clusters
Authors: Leese Morven
Published: September 2005

The ability to identify overlapping clusters in a cluster analysis is important in subject areas such as market research and social network research. A number of methods are available, most of which require special software. Some allow only limited overlap whereas others are more flexible, but the latter entail balancing the complexity of the model against goodness of fit.
Page's Ordered Alternatives Test
Authors: Sawilowsky Shlomo , Fahoome Gail
Published: September 2005

Page's test provides for a distribution‐free randomized blocks test where the researcher can order the
k
treatments in terms of the expected magnitudes of response to those treatments.
Paired Observations, Distribution Free Methods
Authors: Berger Vance W. , Li Zhen
Published: September 2005

Data are said to be paired if theith observation on the first sample is naturally paired with theith observation on the second sample. The objective in paired observations is to eliminate, to the extent possible, sources of extraneous variation. In behavioral studies, for example, paired observations may be employed to assess the effectiveness of a treatment or experimental procedure. Techniques often used to analyze paired observations include the McNemar Chi‐square test, the pairedtTest, the sign test, and the Wilcoxon Signed‐Rank test.
Panel Study
Authors: Billiet Jaak
Published: September 2005

Panel surveys are most suitable for the study of attitude change over time and for arriving to causal conclusion about phenomena in the social world. The opportunities and main weaknesses of panel studies are introduced. A selected bibliography may help the reader to gain more in‐depth knowledge about this issue.
Paradoxes
Authors: Berger Vance W. , Durkalski Valerie
Published: September 2005

A paradox is a statement that apparently derives self‐contradictory conclusions. While the consideration of paradoxes in statistics can be instructional, they are also important, because they often have implications that should be kept in mind when designing studies and analyzing data. This entry reviews some of the more common paradoxes encountered in statistics, and their implications.
Partial Correlation Coefficients
Authors: Brown Bruce L. , Hendrix Suzanne B.
Published: September 2005

Partial correlation is a method for identifying the correlation between two variables, with the effects of a third variable held constant. It is mathematically equivalent to the correlation between the residual scores of the two variables regressed upon the control variable. Partial correlation is conceptually related to, and an integral part of, a number of other useful statistical methods, including path analysis, structural equation modeling, factor analysis, and stepwise multiple regression.
Partial Least Squares
Authors: Sampson Paul D. , Bookstein Fred L.
Published: September 2005

Partial Least Squares (PLS), as described in this article, is a method of statistical analysis for characterizing and making inferences about the relationship(s) among two or more causally related blocks of variables in terms of pairs of latent variables. The methods of computation and inference for 2‐block PLS analysis are described and illustrated for an application in behavioral teratology relating a block of 9 subscale scores on the WAIS‐R IQ test measured on 419 subjects to a block of 13 measures of prenatal maternal alcohol consumption.
Path Analysis and Path Diagrams
Authors: Boker Steven M. , McArdle John J.
Published: September 2005

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.
Pattern Recognition
Authors: Kuncheva Ludmila I. , Whitaker Christopher J.
Published: September 2005

Pattern recognition concerns assigning objects to classes. The objects are described by features (variables or measurements) organized as
p
‐dimensional points in some feature space. A classifier is a formula, an algorithm or a technique that can assign a class label to any given point in the feature space. Pattern recognition comprises supervised learning (predefined class labels) and unsupervised learning (unknown class labels). Supervised learning includes choosing a classifier model, training and testing the classifier using a data set and selecting the relevant features. Unsupervised learning is usually approached by cluster analysis. Multiple classifier systems or classifier ensembles are a recent branch of pattern recognition whereby the outputs of several classifiers (regarded as experts) are combined for improved accuracy.
P
earson Product Moment Correlation
Authors: Kornbrot Diana
Published: September 2005

Pearson's product moment correlation,
r
, describes the strength of the linear relation between two metric variables, such as extraversion and intelligence. Values range from −1 to +1. A scatterplot is necessary to interpret
r
. The proportion of variability accounted for by the linear relation is given by
r
2
.
Percentiles
Authors: Clark‐Carter David
Published: September 2005

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.
Permutation Based Inference
Authors: Lunneborg Clifford E.
Published: September 2005

Permutation tests were developed to free analysts of unwarranted assumptions about the mathematical form of response distributions and are nearly synonymous with nonparametric or distribution‐free tests.
Linear Models: Permutation Methods
Authors: Cade Brian S.
Published: September 2005

Permutation tests of hypotheses about parameters in linear models derive their validity from exchangeability of the errors in the null model. These errors are unobservable but differ from the observable residuals by a constant for any hypothesis that posits a null, reduced parameter model with just an intercept term. For these hypotheses, exact permutation inferences are possible for iid error distributions of any form. Other hypotheses on subsets of parameters in multiple regression models are not exact because the residuals are not exchangeable with equal probability. Several options to improve exchangeability for these hypotheses have been developed and are presented for permutation tests on quantile regression estimates of the linear model.
Person Misfit
Authors: Meijer Rob R.
Published: September 2005

Methods are discussed that can be used to investigate the fit of an item score pattern to a test model. Person‐fit statistics that are formulated in the context of item response theory(IRT) take a prominent place in the literature. Also, methods formulated outside the IRT context, and methods investigating particular types of response behavior are discussed. Finally, the usefulness of person‐fit statistics to solve practical measurement problems is considered.
Pie Chart
Authors: Wright Daniel B. , Williams Siân E.
Published: September 2005

Pie charts are used for displaying frequency or categorical data. The slices of a pie chart represent the proportion of values in a sample falling within given categories.
P
itman Test
Authors: Lunneborg Clifford E.
Published: September 2005

The two‐group Pitman Test, proposed by E. J. G. Pitman in 1937, is one of the earliest nonparametric or distribution‐free tests. The test has applications both to random samples drawn independently from two (large) populations, and to the randomization of available cases between two treatments. Although the mechanics of the two applications are identical, the hypotheses tested and the interpretation of those tests differ markedly between the two applications.
Point Biserial Correlation
Authors: Kornbrot Diana
Published: September 2005

The point biserial correlation is the value of Pearson's product moment correlation when one of the variables is dichotomous and the other variable is metric. Values range from +1, a perfect positive relation; through zero, no association at all; to −1, a perfect negative correlation. The square of this correlation,

, is a measure of effect size in terms of the proportion of variability accounted for by the relation between the two variables.
Polychoric Correlation
Authors: Hershberger Scott L.
Published: September 2005

The polychoric correlation is used to estimate the Pearson product‐moment correlation between two continuous, bivariate‐normally distributed variables from categorized versions of those variables. The polychoric correlation may also be thought of as the correlation between the normal scores
Y

i

*,
Y

j

* as estimated from the category scores
Y

i

,
Y

j

. These normal scores are based on estimated threshold values that represent transitions from one category to the other.
Polynomial Model
Authors: Goldsmith L. Jane , Berger Vance W.
Published: September 2005

A special case of the linear model, a polynomial model allows representation and estimation of curvilinear relationships while using the theory and computation efficiency of the linear model.
Power
Authors: Howell David C.
Published: September 2005

The power of a test refers to the probability with which that test will reject the null hypothesis in those situations in which it is false. This article focuses on the variables that influence power and discusses the different conceptions of power.A prioripower is computed before the experiment is conducted and is critical to selecting an appropriate sample size. Retrospective power is calculated on the basis of the completed study. This article outlines a number of concerns with retrospective power. Compromise power is based on the relative seriousness of Type I and Type II errors, and should be considered seriously for exploratory research. Finally, the article discusses the importance of considering confidence intervals on effect sizes, in addition to power.
Power Analysis for Categorical Methods
Authors: Erdfelder Edgar , Faul Franz , Buchner Axel
Published: September 2005

This article shows how to assess and control the power of chi‐square tests for categorical data. It discusses both the case of simple null hypotheses (i.e., the category or cell probabilities are completely specified
a priori
) and the case of composite null hypotheses for parameterized multinomial models requiring parameter estimation (e.g., log‐linear models). Power calculations are illustrated for chi‐square tests of association and chi‐square goodness‐of‐fit tests using standard examples from behavioral research.
Power and Sample Size in Multilevel Linear Models
Authors: Snijders Tom A. B.
Published: September 2005

Sample size determination in multilevel designs requires attention to the fact that statistical power depends on the total sample sizes for each level. It is usually desirable to have as many units as possible at the top level of the multilevel hierarchy. Some formulas are given to obtain insight in the design aspects that are most influential for standard errors and power.
Probability Plots
Authors: Howell David C.
Published: September 2005

Probability plots are most often used to assess the shape of a distribution of data. Probability–probability plots display the cumulative percentage of the distribution against the cumulative percentages of a reference distribution (often the normal distribution). Quantile–quantile plots display the obtained and expected quantiles of the distribution.
Random Effects in Multivariate Linear Models: Prediction
Authors: Longford Nicholas T.
Published: September 2005

When randomness is interpreted as variation across hypothetical replications, predicting random effects may appear like aiming for a moving target. This contradiction is resolved by considering the conditional distributions of predictors given that the target is held constant over the replications. This entry focuses on predictors based on linear random‐effects models.
Prediction Analysis of Cross‐Classifications
Authors: Szabat Kathryn A.
Published: September 2005

Prediction analysis is an approach to the analysis of cross‐classified, cross‐sectional or longitudinal data. It is a method for predicting events based on specification of predicted relations among qualitative variables and includes techniques for both stating and evaluating those event predictions. This article presents the basic methodology employed in prediction analysis and highlights approaches developed by Hildebrand, Laing, and Rosenthal (6), and by von Eye, Brandtstadter & Rovine, (12, 13). The Hildebrand et al. approach involves measuring prediction success of a set of predictions via a proportionate reduction in error (PRE) measure, known as
del
. The von Eye et al. approach models the prediction hypothesis via nonstandard log‐linear modeling and evaluates the extent to which the model adequately describes the data. Both approaches to prediction analysis provide a viable, valuable data analysis tool that researchers in the social and behavioral sciences can use in a wide range of research settings.
Prevalence
Authors: Grietens Hans
Published: September 2005

Prevalence is determined as the proportion of individuals in a population who have a specified disease. Depending on the period of time that is taken into account, point, period, or lifetime prevalences can be calculated. Prevalence studies are part of epidemiological research. They are conducted in medicine, psychiatry, and behavioral science. They are useful to determine the needs for services in a certain area and allow the study of variations of outcomes over time and place. There is a dynamic relationship between prevalence and incidence. Prevalence rates are a function of both the incidence and the duration of a disease.
Principal Component Analysis
Authors: Jolliffe Ian
Published: September 2005

When large multivariate datasets are analyzed, it is often desirable to reduce their dimensionality. Principal component analysis is one technique for doing this. It replaces the
p
original variables by a smaller number,
q
, of derived variables, the principal components, which are linear combinations of the original variables. Often, it is possible to retain most of the variability in the original variables with
q
very much smaller than
p
. Despite its apparent simplicity, principal component analysis has a number of subtleties, and it has many uses and extensions. A number of choices associated with the technique are briefly discussed, namely, covariance or correlation, how many components, and different normalization constraints, as well as confusion with factor analysis. Various uses and extensions are outlined.
Probits
Authors: Stefanescu Catalina , Berger Vance W. , Hershberger Scott L.
Published: September 2005

The Probit model is frequently used as a transformation of probabilities from an observed categorical variable's distribution into scale values. These scale values (‘probits’) are derived from the normally distributed, continuous latent variable that is assumed to underlie an observed categorical variable. Probit models are especially useful for modeling censored or truncated variables. Binary, ordinal, and multinominal categorical variables can be modeled by probit transformations.
Procrustes Analysis
Authors: Borg Ingwer
Published: September 2005

Procrustes analysis is concerned with fitting one or more configurations of points to some target configuration. In the simplest case, one configuration is rigidly rotated and/or reflected in space so that it matches a fixed target as closely as possible. A common application is to assess how closely the solution of a factor analysis for one sample corresponds to the solution for another sample or to a hypothesized pattern, respectively. The fitting problem can be generalized in several ways by admitting wider classes of transformations that may indicate empirically meaningful patterns of difference among the configurations.
Projection Pursuit
Authors: Stuetzle Werner
Published: September 2005

Projection pursuit was conceived as a method for finding ‘interesting’ one‐ or two‐dimensional projections of multivariate data revealing unanticipated structure such as clusters, skewness, or presence of outliers. The initial idea of exploring high‐dimensional data using low‐dimensional marginals was later extended into a general paradigm for modeling multivariate data that has been applied to density estimation, regression, and classification.
Propensity Score
Authors: D'Agostino Ralph B.
Published: September 2005

In observational studies, treatments cannot be randomly assigned. The treated and nontreated (i.e., control) groups may have large differences on their observed covariates, and these differences can lead to biased estimates of treatment effects. The propensity score, defined as the conditional probability of being treated given the observed covariates, can be used to balance the covariates in the two groups, and therefore reduce this bias. In order to estimate the propensity score, one must model the distribution of the treatment indicator, given the observed covariates. Once estimated, the propensity score can be used to reduce bias through matching, subclassification, regression adjustment, or some combination of all three. In this entry, we briefly discuss the uses of propensity score methods for bias reduction.
Proximity Measures
Authors: Bock Hans‐Hermann
Published: September 2005

Proximities describe the similarity or dissimilarity between items or objects in a numerical way. General properties and special definitions are presented. Proximities are typically used for analyzing and visualizing hidden similarity structures by graphical displays or when looking for homogeneous clusters in a set of data points. Various types of dissimilarity measures are uniquely related to special types of clusterings or visualizations. Finally, we characterize the proximity between probability distributions by various divergence measures.
Psychophysical Scaling
Authors: Irtel Hans
Published: September 2005

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.
Quantitative Methods in Personality Research
Authors: Fraley R. Chris , Marks Michael J.
Published: September 2005

The science of personality documents and explains ways that people differ with respect to their thoughts, feelings, social relationships, and experiences. Quantitative methods such as factor analysis played an important role in facilitating the discovery of the fundamental dimensions of personality. Recent developments in the application of multiple regression, structural equation modeling, hierarchical linear modeling, item response theory, and taxometrics allow personality researchers to test increasingly sophisticated models of personality processes.
Evaluation Research
Authors: Lipsey Mark W. , Tidd Simon T.
Published: September 2005

Quantitative methods are widely used in the evaluation of social programs. The most methodologically challenging type of evaluation attempts to determine if a program actually has the intended effects on the social conditions it addresses. Variations on experimental and quasi‐experimental research designs are used for this purpose, but their application to social programs involves many distinctive issues, problems, and pitfalls.
Quasi‐experimental Designs
Authors: Shadish William R. , Luellen Jason K.
Published: September 2005

Quasi‐experiments are used to test hypotheses about the effects of manipulable treatments but lack the process of random assignment that occurs with true experiments. Consequently, researchers must be particularly concerned with reducing threats to the validity of causal inferences. Quasi‐experiments can be thought of as a collection of design elements aimed at reducing threats to validity. A number of threats to validity and basic quasi‐experimental designs are discussed.
Quasi‐Independence
Authors: Hershberger Scott L.
Published: September 2005

Independence models test the null hypothesis that no association is present between two variables, whereas quasi‐independence models test the same null hypothesis while correcting for the presence of structural zeros in the data. It is conceptually and analytically important to treat structural and sampling zeros differently. Not doing so can result in the false rejection or acceptance of the null hypothesis of independence.
Quasi‐Symmetry in Contingency Tables
Authors: Hershberger Scott L.
Published: September 2005

Symmetry models require that corresponding cells above and below the main diagonal of a square contingency table be equal, a requirement that is rarely met because symmetry models also require marginal homogeneity. A less restrictive model, quasi‐symmetry, permits marginal heterogeneity while requiring equal odds ratios above and below the main diagonal. Ordinal quasi‐symmetry models specify a parameter that represents a constant difference between adjacent levels of the response categories.
R
‐Squared, Adjusted
R
‐Squared
Authors: Miles Jeremy
Published: September 2005

R
‐squared and adjusted
R
‐squared are statistics derived from analyses based on the general linear model (e.g., regression, ANOVA). It represents the proportion of variance in the outcome variable which is explained by the predictor variables in the sample (
R
‐squared) and an estimate in the population (adjusted
R
‐squared).
Random Effects and Fixed Effects Fallacy
Authors: Smith Philip T.
Published: September 2005

The distinction is made between fixed effects and random effects and applied to the analysis of experiments involving linguistic materials, where its use has become a contentious issue. Specifically, some researchers have argued that treating linguistic variables such as words as fixed effects in analysis of variance is fallacious. Methods for analyzing data containing two random‐effect factors are reviewed, and the conclusion reached that the use of quasi‐
F
ratios or their approximation by min
F
′ is often best practice. Alternative legitimate analyses are also discussed.
Random Forests
Authors: Cutler Adele
Published: September 2005

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.
Random Walks
Authors: Laming Donald
Published: September 2005

The notion of a random walk is introduced and illustrated with everyday examples. Random walks with a single absorbing barrier have been used to model simple reaction times, and a simple method of calculation is described for this relatively simple case. Random walks with two absorbing barriers have also been applied to two‐choice reaction times, but calculation is more difficult in this case.
Randomization
Authors: Boik Robert J.
Published: September 2005

Behavioral scientists were among the earliest to use randomization in research. Justifications for using randomization are plentiful and include the following: it ensures that estimators are unbiased and consistent; it ensures that permutation tests and associated confidence intervals are valid; it justifies normal‐theory tests, even in the absence of normality; it protects against subjective biases of the investigator; it helps to elucidate causality; and it enhances the robustness of inferences. Nonetheless, randomization is not universally accepted. Randomization has been criticized because it violates the conditionality and likelihood principles and because it can be less efficient than methods based on optimal sampling and optimal design.
Randomized Block Design: Nonparametric Analyses
Authors: Liu Li , Berger Vance W.
Published: September 2005

In a randomized block design, there are, in addition to the experimental factor or factors of interest, one or more nuisance factors. The role of blocking is to reduce or eliminate that part of the experimental error attributable to these nuisance factors. Standard parametric ANOVA can be used to analyze such designs, but it requires the normality assumption, and may be misleading if the errors are not normally distributed or if there are outliers. In this article, we consider some distribution‐free tests, such as the sign test, the Wilcoxon signed rank test, Friedman's test, aligned rank tests, Durbin's test, and the row mean score test. None of these methods require the normality assumption, and they can all be used to analyze such designs.
Range
Authors: Clark‐Carter David
Published: September 2005

The range is a measure of spread (or dispersion). It is the difference between the largest (the maximum) and the smallest (the minimum) values in a set of data.
Rasch Models for Ordered Response Categories
Authors: Andrich David
Published: September 2005

This article explains the latent response structure and process compatible with the Rasch model (RM) for ordered response categories in standard formats. It considers the rationale for the RM but is not concerned with details of parameter estimation and tests of fit. There are a number of software packages that implement the Rasch model at an advanced level.
Rater Agreement
Authors: Schuster Christof , Smith David
Published: September 2005

Human judgment is prone to error that researchers routinely seek to quantify, understand, and minimize. To quantify the quality of nominal judgments, agreement among multiple raters of the same target is examined using either global summary indices, such as Cohen's kappa, or by modeling important properties of the judgment process using latent class analysis. In this entry, we give an overview of the latent class analysis approach to nominal scale rater agreement data.
Rater Bias Models
Authors: Saudino Kimberly J.
Published: September 2005

Raters' response tendencies, standards, or expectations can result in the tendency to consistently overestimate or underestimate the behavior of targets. Such rater biases can influence the outcomes of behavioral genetic studies. Quantitative genetic models allow researchers to determine the significance of rater biases and estimate genetic and environmental influences on behaviors controlling for bias.
Mendelian Genetics Rediscovered
Authors: Dickins David
Published: September 2005

Three European botanists coincidentally claimed their independent ‘rediscovery’ in 1900 of Mendel's neglected 1865 paper, in which he derived what are still the classic principles of genetics, from breeding experiments with peas in his monastery garden. It is necessary to get a clearer picture of why Mendel was doing what he did, and how the ‘rediscoverers’, and others enthused by them, misinterpreted its significance in some important respects, leading to an eclipse of Darwinism, not to recover for a further two or three decades of dispute and reinterpretation, after which the complementarity of Mendel's work with Darwin's became established.
Regression Artifacts
Authors: Kenny David A.
Published: September 2005

Regression toward the mean is a universal statistical phenomenon that is responsible for many artifacts in the interpretation of data, especially studies of change. Regression toward the mean is also important in both prediction and understanding of biases in human judgment.
Regression Discontinuity Design
Authors: Shadish William R. , Luellen Jason K.
Published: September 2005

The regression discontinuity design is a quasi‐experimental design in which the experimenter assigns participants to treatment conditions on the basis of a cutoff score on an assignment variable – usually a measure of need or merit. This design yields unbiased estimates of treatment effects when certain assumptions are met. Just as in a randomized experiment, the selection process is both completely known and perfectly measured in the regression discontinuity design. Numerous variations of the design are possible, including combinations with randomized experiments.
Regression Model Coding for the Analysis of Variance
Authors: Cohen Patricia
Published: September 2005

Analyses that are typically carried out by analysis of variance (ANOVA) can often be carried out with multiple regression (MR) methods as well. The MR approach can include direct tests of the study's specific hypotheses and cope efficiently with unbalanced designs. The methods of coding groups for MR analysis should depend on the study design and hypotheses. Alternative methods are illustrated and discussed.
Regression Models
Authors: Lewis‐Beck Michael S.
Published: September 2005

Regression analysis is the most widely employed statistical technique in the study of observational data in the behavioral sciences. A dependent variable is posited as a function of an independent variable(s). The most common procedure for estimating a regression equation is ordinary least squares, although there are others.
Regression to the Mean
Authors: Smith Randolph A.
Published: September 2005

Regression to the mean describes the tendency for extreme pretest scores to become less extreme on a posttest. Regression to the mean is a threat to internal validity in repeated measures experimental design.
Relative Risk
Authors: Everitt Brian S.
Published: September 2005

This entry explains the difference between relative risk and absolute risk.
Family Study and Relative Risk
Authors: McGuffin Peter , Burke Ailbhe
Published: September 2005

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.
Repeated Measures Analysis of Variance
Authors: Everitt Brian S.
Published: September 2005

Repeated measures data arise when each participant in a study has measurements of a response variable of interest made under several different conditions, on different occasions, and so on. A univariate analysis of variance approach can be used if the covariance structure of the repeated measurements satisfies a condition known as sphericity, otherwise a multivariate analysis of variance approach may be needed.
Replicability of Results
Authors: Thompson Bruce
Published: September 2005

Because result replicability is important to researchers, and statistical significance does not evaluate replicability, both external and internal replicability analyses are important. Some researchers find true external replication studies impractical, and therefore turn to ‘internal’ methods. Three internal replicability analyses have been proposed: cross‐validation, the jackknife, and the bootstrap.
Reproduced Matrix
Authors: Lance Charles E.
Published: September 2005

A reproduced matrix is a matrix of correlations or covariances that is calculated from parameter estimats obtained for a path analytic, factor analytic, or latent variable structural equation model. If one or more ‘zero restrictions’ are specified, model goodness‐of‐fit can be evaluated by the extent to whihc the reproduced matrix equals (or approximates) the observed matrix of correlations or covariances calculated directly from the data, for exdample, using the chi‐square statistic.
Resentful Demoralization
Authors: Onghena Patrick
Published: September 2005

Resentful demoralization is a threat to the construct validity of the treatment in intervention studies. This threat may occur in intervention studies in which comparison groups not obtaining a desirable treatment become discouraged or retaliatory and, as a result, perform worse on the outcome measures. Some strategies to avoid, minimize, or deal with resentful demoralization are suggested and illustrated.
Residual Plot
Authors: Miles Jeremy
Published: September 2005

In a regression analysis, a number of assumptions about the data must be made and tested. One of the most powerful ways to do this is to produce a range of graphs of the residuals from the regression. Valuable information may be gained from examining a histogram of the residuals, a plot of the residuals against time, and a plot of the residuals against the predicted values from the regression equation.
Residuals
Authors: Miles Jeremy
Published: September 2005

Residuals are the difference between the predicted values of individual cases in a regression model and the actual observed values. They provide information that helps test the model assumptions, particularly in the detection of outliers. Residuals are difficult to interpret because they have the scale of the original variable; two approaches to aid in the interpretation of residuals are ‘standardizing’ and ‘studentizing’.
Retrospective Studies
Authors: McKenzie Dean P.
Published: September 2005

A retrospective study is an observational study that deals with events that have occurred in the past. Information may be obtained from historical records or by asking individuals to remember previous experiences or exposures. Case‐control studies, in which previous exposures of those individuals with (cases) and without (controls) a particular outcome or disease are examined, are generally retrospective. Historical cohort studies are also retrospective. Cohort members are identified on the basis of past characteristics, and then studied from a previous time‐point up to the present. Retrospective studies are often used when prospective cohort studies (where cohorts are selected on the basis of current characteristics and followed into the future) would be impractical, for example, if the event of interest takes a long time to occur.
Risk Perception
Authors: Everitt Brian S.
Published: September 2005

Quantifying and assessing risk involves the calculation and comparison of probabilities. Research shows that people tend to overestimate the probability of unfamiliar, catastrophic, and well‐publicized events and to underestimate the probability of unspectacular or familiar events that claim one victim at a time. The ability to assess risk in a rational manner would benefit both individuals and society, but there is considerable evidence that persuading people to be rational about risk is not easy.
Robust Testing Procedures
Authors: Wilcox Rand R.
Published: September 2005

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.
Robustness of Standard Tests
Authors: Wilcox Rand R.
Published: September 2005

Conventional hypothesis‐testing methods such as student's
t
, the ANOVA
F
test, and methods based on the ordinary least squares regression estimator, are not robust to violations of assumptions. In fact, there are general conditions under which these methods can provide poor control over the probability of a type I error and inaccurate confidence intervals, no matter how large the sample sizes might be. Relatively poor power is yet another concern.
R
&
Q
Analysis
Authors: Jackson J. Edward
Published: September 2005

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.
Runs Test
Authors: Lunneborg Clifford E.
Published: September 2005

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.
Sampling Distributions
Authors: Landau Sabine
Published: September 2005

A sampling distribution is the probability distribution of a statistic that is based on a random sample from some population. The sampling distribution is a key concept in statistics because it quantifies the precision of estimators in parameter estimation and is the distribution from which
P
values are derived in statistical tests.
Sampling Issues in Categorical Data
Authors: Hershberger Scott L.
Published: September 2005

The sampling method used to collect response on categorical variables determines both the probability of the specific configuration of a contingency table, and, more importantly, determines the interpretation of the data. Sampling methods leading to three probability distributions; the hypergeometric, the binomial, and the negative binomial, are discussed. The three sampling methods indicate the same probability of observing an example contingency table, but lead to different interpretations of the data.
Saturated Model
Authors: Hershberger Scott L.
Published: September 2005

A saturated model is a model that perfectly fits the data because it has as many parameters as there are values to be fitted. Saturated models are also just‐identified models, and can provide useful information in the search for a more parsimonious, unsaturated model in which some of the parameters of the saturated are not estimated. Saturated and unsaturated models must be hierarchically nested in order to evaluate the unsaturated model's fit to the data.
Scales of Measurement
Authors: Wuensch Karl L.
Published: September 2005

Scales of measurement, as conceived by S. S. Stevens, are defined with respect to (1) the extent to which empirical relationships among the measured objects correspond to numerical relationships among the measurements, and (2) the types of transformations that can be applied to the scale without distorting the structure of the scale. Stevens argued that whether or not a particular statistical analysis is ‘permissible’ depends upon the scale of measurement of the data being analyzed. While this assertion has been defended by some statisticians, it has been attacked by others. Many statistical textbooks for social scientists are organized around the assertion that scale of measurement is a critical factor in determining which statistical analysis is appropriate. Here, the authors argue that parametric statistical methods are appropriate only for interval level data and that nonparametric methods should be employed with noninterval data. One's attitude about the relationship between scale of measurement and choice of statistical analysis may follow from one's more basic ideas about measurement and the nature of reality. Ultimately, the distinction between interval and ordinal scales depends on whether one believes the relationship between the measurements and the true amounts of the measured attribute is positive linear (interval) or merely positive monotonic (ordinal), and such beliefs are largely a matter of faith.
Scaling Asymmetric Matrices
Authors: Takane Yoshio
Published: September 2005

Square tables in which some of the elements located symmetrically about the diagonal are not equal to each other are called asymmetric tables. Asymmetric tables arise in a variety of different ways. They can be roughly classified into two groups, one arising from measurements of antisymmetric relationships, and the other, of proximity relationships. A variety of models have been proposed to capture the asymmetry in such tables. In this article, we briefly discuss some of the representative models of asymmetry for each of the two types of data.