bluuebunny/crossref_metadata_2025 huggingface dataset:
Maximum Likelihood Item Response Theory Estimation
Authors: Rupp André A.
Published: September 2005

An overview and short didactic for three common estimation techniques for parameters in item response theory models are described: (a) joint maximum likelihood, (b) conditional maximum likelihood, and (c) marginal maximum likelihood. Specifically, likelihood equations fundamental to all three estimation approaches are presented first and joint as well as conditional maximum likelihood estimation are briefly described next. The main part of the entry focuses on an extended description of marginal maximum likelihood estimation and illustrates, in a step‐by‐step fashion, how it makes use of the EM algorithm and a fully Bayesian estimation framework to estimate item parameters. A brief description of how person parameters are subsequently estimated follows. Finally, a few words on alternative estimation approaches are offered.
K
endall's Coefficient of Concordance
Authors: Field Andy P.
Published: September 2005

Kendall's Coefficient of Concordance,
W
, is a measure of the agreement between several judges who have rank ordered several entities. It represents the ratio of the variability of the total ranks for the ranked entities to the maximum possible variability of the total ranks; a small ratio implies disagreement between judges.
Kernel Smoothing
Authors: Hazelton Martin L.
Published: September 2005

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.
K
olmogorov–
S
mirnov Tests
Authors: Berger Vance W. , Zhou YanYan
Published: September 2005

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.
Kruskal–Wallis Test
Authors: Sawilowsky Shlomo , Fahoome Gail
Published: September 2005

The nonparametric Kruskal‐Wallis test is an extension of the Wilcoxon‐Mann‐Whitney test. The null hypothesis is that thekpopulations sampled have the same average (median). The alternative hypothesis is that at least one sample is from a distribution with a different average (median). This test is an alternative to the parametric one‐way analysis of variance F test.
K
urtosis
Authors: Wuensch Karl L.
Published: September 2005

Kurtosis is defined with respect to the fourth moment about the mean. It is related to the shape of a distribution, but not in a simple fashion. Platykurtic distributions have less kurtosis than does a normal distribution, are heavy in their shoulders, tend to be light in their tails, and may be flatter on top than a normal distribution, or may be U‐shaped. Leptokurtic distributions have more kurtosis than does a normal distribution, are light in their shoulders, tend to be heavy in their tails, and may have a more peaked top than does a normal distribution. High kurtosis can signal the presence of outliers in one or both tails.
Latent Variable
Authors: Vermunt Jeroen K. , Magidson Jay
Published: September 2005

Latent variable techniques are used as scaling tools when multiple responses related to the same construct are available. The differences between the four main types of latent variable models – factor analysis, latent trait analysis, latent profile analysis, and latent class analysis are described.
L
atin Squares Designs
Authors: Cotton John W.
Published: September 2005

Latin squares designs have an equal number of values for three variables. The designs provide information about the variables with fewer observations than three‐variable factorial designs, but confound interactions with main effects. Models for Latin squares are displayed and computational examples described, together with information about replicated Latin squares.
Laws of Large Numbers
Authors: Kay Jim W.
Published: September 2005

The laws of large numbers concern when the average of
n
random variables converges to a fixed value as
n
tends to infinity. The
n
random variables could arise in various ways. They could, for example, be mutually independent and identically distributed and represent a simple random sample of observations taken from some fixed population, or they could represent a sequence of
n
realizations of a stationary stochastic process. The laws are defined under the assumptions that the random variables are mutually independent and follow the same probability distribution; generalizations obtained under more general conditions are mentioned. Illustrations are provided, which demonstrate the usefulness of the laws in providing theoretical justification for: the frequentist concept of probability; the strong consistency of the sample mean and variance estimators; the consistent estimation of the
P
value associated with a Monte Carlo test.
Leverage Plot
Authors: Lovie Pat
Published: September 2005

A leverage plot (or residual versus leverage plot) is a scatterplot of studentized residuals versus hat values. It is one of the many tools available for checking the adequacy of data when fitting regression models and is especially valuable for gauging the combined effect of high leverage and regression outliers.
Liability Threshold Models
Authors: Neale Ben
Published: September 2005

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.
Index Plots
Authors: Lovie Pat
Published: September 2005

An index plot is a scatterplot of values on some measure plotted serially against the associated observation (case) number within the sample. It can help in highlighting extreme values in a sample and in indicating trends when the data are, for instance, in spatial or time order.
Linear Multilevel Models
Authors: de Leeuw Jan
Published: September 2005

Multilevel models are regression models designed for observation units from different nested levels. Predictors at the various levels are combined into a single linear mixed model, in which some of the regression coefficients are random variables and others are fixed constants. The class of models is described, and examples from school effectiveness research and growth curve analysis are given.
Linear Model
Authors: Cortina Jose
Published: September 2005

The term ‘Linear Model’ is often used and often misunderstood. The purpose of this entry is to explain the concept and its relationship to the broader term, ‘Intrinsic Linearity’.
Linkage Analysis
Authors: Sham P.
Published: September 2005

Genetic linkage refers to the nonindependent segregation of the alleles at genetic loci close to one another on the same chromosome. This entry discusses alternative analyses to measure genetic linkage.
Logistic Regression
Authors: Christensen Ronald
Published: September 2005

Logistic regression is a method for predicting the outcomes of ‘either‐or’ trials. Either‐or trials occur frequently in research. A person responds appropriately to a drug or does not; the dose of the drug may affect the outcome. A person may support a political party or not; the response may be related to their income. A person may have a heart attack in a 10‐year period; the response may be related to age, weight, blood pressure, or cholesterol. We discuss basic ideas of modeling in the section titled ‘Basic Ideas’ and basic ideas of data analysis in the section titled ‘Fundamental Data Analysis’. Subsequent sections examine model testing, variable selection, outliers and influential observations, methods for testing lack of fit, exact conditional inference, random effects, and Bayesian analysis.
Log‐Linear Models
Authors: Vermunt Jeroen K.
Published: September 2005

Log‐linear analysis is a widely used method for the analysis of multivariate frequency tables obtained by cross‐classifying sets of nominal, ordinal, or discrete interval level variables. Attention is paid to the standard hierarchical log‐linear modelling framework and to more advanced types of log‐linear models that make it possible to impose interesting restrictions on the model parameters, as well as to ‘regression analytic’, ‘path‐analytic’, and ‘factor‐analytic’ variants of log‐linear analysis.
Log‐linear
R
asch Models for Stability and Change
Authors: Meiser Thorsten
Published: September 2005

This entry presents Rasch models for the analysis of longitudinal data and log‐linear representations of Rasch models to test specific hypotheses about change. The use of log‐linear Rasch models is illustrated by an empirical example concerning the development of arithmetic competence in elementary school. The final section discusses model extensions for the analysis of polytomous items, multidimensional latent constructs, and several latent subpopulations in longitudinal research.
Longitudinal Data Analysis
Authors: Everitt Brian S.
Published: September 2005

Longitudinal data are collected in many investigations in the behavioral sciences and require special methods for their analysis, primarily because the repeated measurements of the response are not independent. During the last decade, several new and powerful methods of analysis have been developed, which can deal with both continuous and categorical responses.
L
ord's Paradox
Authors: Holland Paul W.
Published: September 2005

Frederic Lord presents a statistical paradox regarding the analysis of pre‐post data often used in program evaluation. His paradox cast doubt on the value of certain uses of the analysis of covariance. The resolution of the paradox requires careful attention to the ambiguity of the implicit causal questions that underlie it.
M
Estimators of Location
Authors: Wilcox Rand R.
Published: September 2005

An approach to developing measures of location is to search for some value, say
c
, that is close to the values in a sample, say
X
1
, …,
X

n

. If we measure closeness with the sum of squared differences between each of the
n
observations and
c
, this leads to taking
c
to be the sample mean. But the sample mean can perform poorly in terms of achieving a small standard error when sampling from nonnormal distributions. M‐estimators attempt to correct this problem by replacing squared error with some other measure of closeness. Certain M‐estimators have standard errors that compete well with the sample mean under normality, but they continue to perform well in situations in which the sample mean performs poorly.
Mahalanobis Distance
Authors: Huberty Carl J.
Published: September 2005

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.
Mail Surveys
Authors: Mangione Thomas W.
Published: September 2005

Mail surveys are much less costly to conduct than telephone surveys or in‐person surveys and therefore are an attractive method of data collection for researchers. However, in general, mailed surveys have suffered from low response rates which limits their applicability. Extensive research conducted over the past few decades has documented a variety of procedural elements, which if followed, can overcome low response rates. In addition to postage‐paid return envelopes, strong respondent letters, and clear promises of respondent confidentiality or anonymity, the two most important procedures to incorporate in the study design are using as many as three reminders and including financial incentives for participation. Additional elements such as personalization, prenotification, modest length, and utilization of soft deadlines also have been shown to increase response rates slightly. By following these procedures, data collected through mail surveys can be as valid as data collected through telephone or in‐person methods.
Mallows'CpStatistic
Authors: Lance Charles E.
Published: September 2005

Mallows Cp statistic is used in multiple regression analysis to select models that contain smaller numbers of predictors from a larger number that is available for inclusion. Low values of Cp indicate regression models that are parsimonious yet still have good predictive ability.
Mantel–Haenszel Methods
Authors: Fidalgo Ángel M.
Published: September 2005

Mantel–Haenszel (MH) methods constitute a highly flexible methodology for assessing the degree of association between two categorical variables, whether they are nominal or ordinal, controlling the effect of other variables. These procedures, based on minimal assumptions, provide both statistical tests of the null hypothesis of no‐association and estimators of the degree of association. The present entry considers the MH statistics most commonly used for the particular case ofQs contingency tables with dimensions 2 × 2 and the general case ofQtables with dimensionsR × C.
Marginal Independence
Authors: Berger Vance W. , Zhang Jialu
Published: September 2005

Two variables, say
X
and
Y
, are marginally independent if their joint distribution is the product of the two marginal distributions. In the discrete case, inspecting only the marginal table of
X
‐by‐
Y
may lead to false conclusions if the association between
X
and
Y
is not the same across different levels of another covariate, say
Z
. To prevent such misleading conclusions, partial tables of
X
and
Y
at each level of the confounding covariate
Z
should be displayed.
M
arkov Chains
Authors: Laming Donald
Published: September 2005

The notion of a Markov chain is described and illustrated with examples, both everyday examples and some from the psychological literature. Those examples are used to explain the meanings of ‘absorbing’, ‘transient’, ‘identifiable’, ‘periodic’, and ‘ergodic’ states, and of an ‘irreducible’ chain.
High‐Dimensional Regression
Authors: de Leeuw Jan
Published: September 2005

Very large data sets are becoming more and more common in statistics, even in social and behavioral statistics. Such data require new types of statistical techniques. We discuss some of the many adaptions of linear regression analysis to these highly multivariate situations.
Matching
Authors: Berger Vance W. , Liu Li , Thach Chau
Published: September 2005

Evaluation is generally inherently comparative. For example, a treatment is generally neither ‘good’ nor ‘bad’ in absolute terms, but rather better than or worse than another treatment. The studies that allow for such comparative evaluations must then also be comparative. To isolate the effects of the treatments under study, avoid confounding treatment effects with unit effects, and possibly increase the power of the study, the units under study must be carefully matched across treatment groups. There are many matching techniques in practice, including crossover designs and case‐control studies. While randomization is often considered to be a competitor to matching, it actually may be considered to be a type of matching, except that here the matching is group‐to‐group.
Mathematical Psychology
Authors: Chechile Richard A.
Published: September 2005

Mathematical psychology is a branch of theoretical psychology that uses mathematics as a tool for psychological measurement of latent processes as a means to characterize the regular behavioral patterns that emerge from experimental studies. An example of the measurement of latent processes is research that decomposes memory loss into separate components due to storage and retrieval changes. Other models in mathematical psychology provide a detailed description of the results from many memory experiments. Mathematical psychologists have addressed a wide range of psychological problems and have utilized a diverse set of mathematical tools.
Scatterplot Matrices
Authors: Wright Daniel B. , Williams Siân E.
Published: September 2005

A scatterplot matrix is a display for viewing the bivariate relationships among several pairs of variables. Variations, like trellis plots, show bivariate relationships at a different level of a third variable.
Arithmetic Mean
Authors: Clark‐carter David
Published: September 2005

The arithmetic mean is a measure of central tendency or location. It is the sum of a set of numbers divided by the number of numbers in the set. It is an important descriptive measure and is also the basis of many inferential statistics for comparing the averages of two or more samples of data.
Geometric Mean
Authors: Clark‐Carter David
Published: September 2005

The geometric mean is a measure of central tendency or location. It is defined as the
n
th root of the product of
n
numbers. It has advantages over the arithmetic mean by being less affected by extreme values in skewed data and is an appropriate measure for data that involve ratios, such as percentages.
Harmonic Mean
Authors: Clark‐Carter David
Published: September 2005

The harmonic mean is a measure of central tendency or location. It is defined as the reciprocal of the mean of the reciprocals of a set of numbers. It plays a role in studies employing between‐subjects designs where the various groups have different sample sizes; in such cases, it produces an appropriate sample size for use in calculations of statistical power and in analysis of variance.
Measurement: Overview
Authors: Michell Joel
Published: September 2005

Measurement in behavioral science includes practices over and above those used in physical and biological science. In part because of this, the understanding of measurement within behavioral science is different from thetraditional concept of measurement. This latter concept is built upon that of a quantitative attribute (i.e., one possessing internal additive structure and sustaining ratios) and it defines measurement as estimation of the ratio between a magnitude of a quantitative attribute and a unit. However, it is not clear whether the attributes specific to behavioral science are quantitative. Consequently, some behavioral scientists favor arepresentational concept of measurement, according to which measurement depends upon isomorphisms or homomorphisms between empirical and numerical systems. This understanding extends the concept of measurement to ordinal and nominal scaling. However, within the framework of the representational concept, the status of one major tradition of quantification in behavioral science, namely psychometrics, is unclear. This has led many behavioral scientists to endorse theoperational concept of measurement, according to which measurement is any precisely specified operation that yields a number. Most behavioral scientists agree that measurement is the assignment of numerals to objects and events according to rules, a definition open to representational and operational interpretations.
History of Intelligence Measurement
Authors: Weidman Nadine
Published: September 2005

The history of intelligence measurement can be roughly divided into three chronological periods: the first (1870–1920s), a period of initial interest in defining intelligence and the establishment and use of intelligence tests; the second (1930s–1960s), a period of repudiation of the hereditarian assumptions underlying the tests; and the third (1969 to present), a period of resurgence of belief in the connection between intelligence and heredity.
Median
Authors: Clark‐Carter David
Published: September 2005

The median is a measure of central tendency (or location). It is the score in an ordered set of data that has 50% of values below it and 50% above it. It is unaffected by extreme scores and is one of the basic elements of a box plot.
Median Absolute Deviation
Authors: Howell David C.
Published: September 2005

The median deviation is a measure of scale based on the median of the absolute deviations from the median of the distribution. For a normal distribution, the median deviation is somewhat less efficient than the standard deviation or the average deviation as a measure of scale, but this advantage quickly reverses for distributions with heavier tails.
Median Test
Authors: Sawilowsky Shlomo , Lunneborg Clifford E.
Published: September 2005

The Median Test is a procedure for testing the null hypothesis that
k
 ≥ 2 independent samples have been drawn from populations with a common median.
Mediation
Authors: Kenny David A.
Published: September 2005

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.
M
endelian Inheritance and Segregation Analysis
Authors: Sham P.
Published: September 2005

This entry examines the basic Mendelian model of inheritance. The model is based upon the familiar inheritance ratios of 1:1 and 3:1, which are known as ‘Mendelian segregation ratios’. The entry then discusses classical segregation analysis and then complex segregation analysis.
Meta‐Analysis
Authors: van den Noortgate Wim , Onghena Patrick
Published: September 2005

A meta‐analysis combines or compares in a systematic and quantitative way the results from several studies in which a similar research question is investigated. If differences in research outcomes are entirely due to sampling variation, combining the outcomes results in more precise answers to the research questions. In case of systematic inconsistencies in study outcomes, a meta‐analysis offers the possibility to look for an explanation for the heterogeneity, by investigating whether study outcomes are related to one or more study characteristics. In this article, the historical and theoretical context of meta‐analysis and the successive steps in a meta‐analysis are described shortly, and several models, techniques and caveats are discussed.
Microarrays
Authors: Schalkwyk Leonard C.
Published: September 2005

Microarray studies have often been viewed as a survey method for finding large gene expression differences between paired samples. Experimental designs amenable to more detailed statistical analysis are increasingly common, particularly with the advent of highly standardized, industrially produced arrays and gradually diminishing costs.
Mid‐

P

Values
Authors: Berger Vance W.
Published: September 2005

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.
Minimum Spanning Tree
Authors: Michailidis George
Published: September 2005

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.
Missing Data
Authors: Little Roderick J.
Published: September 2005

Missing data are a common problem in the social and behavioral sciences. Here we present an overview of the problem and possible solutions. We begin by distinguishing between the pattern of missing data and the mechanism that creates the missing data. We then consider common, but limited, approaches: complete‐cases, available cases, weighting analyses, and single imputation. More principled methods, namely, multiple imputation, maximum likelihood, and inference from Bayesian posterior distributions, are then presented.
Model Evaluation
Authors: Navarro Daniel J. , Myung Jay I.
Published: September 2005

Assessing the viability of a quantitative model given observed data is generally done in three steps: model fitting, testing, and selection. In model fitting, we find the parameter values that best fit observed data. This is followed by model testing in which null hypothesis significance tests allow us to evaluate the descriptive adequacy of a model's fit to data. Model selection concerns the issue of choosing the model, among a set of competing models, that generalizes best to as yet unseen data samples.
Model Identifiability
Authors: Huang Guan‐Hua
Published: September 2005

A model is identifiable if there is a one‐to‐one correspondence between the probability distribution of the data and the values of model parameters. When applying a nonidentifiable model, different people may draw different conclusions from the same model of the observed data. Before one can meaningfully discuss the estimation of a model, model identifiability must be verified.
Models for Matched Pairs
Authors: Berger Vance W. , Zhang Jialu
Published: September 2005

When randomization is impractical or impossible, matched studies are often used to enhance the comparability of comparison groups, so as to facilitate the assessment of the association between an exposure and an event while minimizing confounding. There are several approaches to the analysis of matched data, including the matched pairs
t
Test, stratified analyses, the McNemar test, and conditional logistic regression. Each approach may be best in a given situation.
Moderation
Authors: Beaubien J. Matthew
Published: September 2005

Moderator variables, also known as
interaction effects
, alter the strength or direction of a predictor–criterion correlation. The identification of important moderator variables helps social scientists to interpret conflicting research results and, by extension, build more accurate theories of complex human behavior.
Monotonic Regression
Authors: de Leeuw Jan
Published: September 2005

Monotone regression is a technique for finding the best fitting monotone function in a scatterplot. We discuss Kruskal's up‐and‐down blocks algorithm.
MonteCarlo Goodness of Fit Tests
Authors: Besag Julian
Published: September 2005

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.
Markov ChainMonteCarlo Item Response Theory Estimation
Authors: Keller Lisa A.
Published: September 2005

Item response theory (IRT) is the cornerstone of modern measurement. The benefits of using IRT have been well documented. In order to use IRT models, it is necessary to estimate item parameters. As assessments become more complex, more sophisticated models need to be developed. The difficulty in making models more complex is the issue of parameter estimation, and, without the ability to estimate parameters, the models are not useful. With the growth of computer technology, new estimation techniques are being developed for IRT. One such technique is Markov Chain Monte Carlo (MCMC). MCMC provides parameter estimates by simulating draws from the posterior distribution of the parameters, and uses the constructed samples to provide point estimates of the parameters (e.g., means, modes). Both Gibbs sampling and Metropolis–Hastings within Gibbs are described, and examples of estimating parameters of dichotomous IRT models are provided.
M
onte
C
arlo Simulation
Authors: Gentle James E.
Published: September 2005

Monte Carlo simulation is one of the most important tools in all fields of science. Monte Carlo methods use samples of numbers that appear to be random, although the numbers are generated by deterministic computer algorithms. These “pseudorandom” samples can be used in solving a wide range of problems, including both deterministic problems such as evaluating a definite integral and problems in which there is a random component such as studying and comparing the performance of statistical procedures under various assumptions about the underlying distribution.
Multidimensional Item Response Theory Models
Authors: Ackerman Terry A.
Published: September 2005

It is common for educational and psychological tests to be inherently multidimensional, or measuring two or more traits, skills, or dimensions. If assessment data are determined to be multidimensional testing practitioners, researchers can apply multidimensional item response theory (MIRT) to help them better understand what their tests are measuring, how accurately the different composites of ability are being assessed, and use this information to inform the test development process. In this entry, several different aspects of MIRT analyses are discussed: obtaining evidence that the test is multidimensional, estimating the item and ability parameters for a multidimensional model, and illustrating the estimated parameters graphically to gain a better understanding about the composites of traits or skills that are being assessed.
Multidimensional Scaling
Authors: Groenen Patrick J. F. , van de Velden Michel
Published: September 2005

Multidimensional scaling is a statistical technique to visualize dissimilarity data. In multidimensional scaling, objects are represented as points in a usually two dimensional space, such that the distances between the points match the observed dissimilarities as closely as possible. Here, we discuss what kind of data can be used for multidimensional scaling, what the essence of the technique is, how to choose the dimensionality, transformations of the dissimilarities, and some pitfalls to watch out for when using multidimensional scaling.
Multidimensional Unfolding
Authors: de Leeuw Jan
Published: September 2005

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.
Multilevel and
SEM
Approaches to Growth Curve Modeling
Authors: Hox Joop , Stoel Reinoud D.
Published: September 2005

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.
Multiple Baseline Designs
Authors: Ferron John , Scott Heather
Published: September 2005

Multiple‐baseline designs are a type of single‐case design used to study treatment effects across multiple participants, multiple behaviors, or multiple settings. Baselines are established by repeated observation. Interventions are then implemented at different times for the different participants, behaviors, or settings. Effects are demonstrated when changes are observed that coincide with the interventions.
Multiple Comparison Procedures
Authors: Keselman H. J. , Holland Burt , Cribbie Robert A.
Published: September 2005

Multiple comparison procedures (MCPs) are frequently adopted by applied researchers to locate specific differences between treatment groups. That is, omnibus test statistics, such as the analysis of varianceFtest, can only signify that effects are present, not which specific groups differ from one another (when there are more than two groups). In our paper, we discuss MCPs that can be used to investigate simple pairwise differences between treatment group means, as well as MCPs that can be used to examine complex comparisons (i.e., nonpairwise comparisons) between treatment group means. In particular, we discuss simultaneous as well as stepwise MCPs, emphasizing procedures that can be utilized when the derivational assumptions of the classical procedures of normality and variance homogeneity do not hold.
Multiple Comparison Tests: Nonparametric and Resampling Approaches
Authors: Keselman H. J. , Wilcox Rand R.
Published: September 2005

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.
Multiple Informants
Authors: Saudino Kimberly J.
Published: September 2005

The use of multiple informants permits a better understanding of the behavior being examined. Quantitative genetic models allow the researcher to examine genetic and environmental overlap between different informants' ratings, and test assumptions regarding whether different informants assess the same phenotype and the presence of rater biases.
Multiple Testing
Authors: Darlington Richard B.
Published: September 2005

We examine methods designed to correct for the fact that when many significance tests are performed, some are likely to be significant just by chance.
Multitrait–Multimethod Analyses
Authors: Pardy Susan A. , Fabrigar Leandre R. , Visser Penny S.
Published: September 2005

The multitrait‐multimethod (MTMM) framework originally proposed by Campbell and Fiske has long guided efforts to assess the discriminant and convergent validity of measures. Numerous statistical approaches for analyzing MTMM matrices have been proposed. Surprisingly, little consensus has developed on the most appropriate method for analyzing these matrices or what criteria should be used as a basis for choosing the most appropriate method. We provide a nontechnical review of statistical approaches for analyzing MTMM matrices. We conclude with a set of general recommendations regarding the analysis of MTMM data.
Multivariate Analysis:
B
ayesian
Authors: Press Jim
Published: September 2005

This article discusses Bayesian multivariate analysis. It focuses on higher‐dimensional statistical inference based upon the posterior distribution, so inference involves multidimensional prior, posterior, and predictive probability distributions. The article describes credibility intervals and regions, and hypothesis testing. There is reference to both vague and natural conjugate prior distributions. Computer‐intensive methods of numerical evaluation such as MCMC and Metropolis–Hastings algorithms are considered. There is an illustrative example involving Bayesian factor analysis.
Multivariate Analysis: Overview
Authors: Krzanowski Wojtek J.
Published: September 2005

Multivariate analysis is appropriate whenever more than one variable is measured on each sample individual, and overall conclusions about the whole system are sought. Many different multivariate techniques now exist for addressing a variety of objectives. This brief review outlines, in broad terms, some of the more common objectives and sketches out some of the ways in which they are tackled. The techniques are grouped under four headings: visualization and description, extrapolation and inference, discrimination and classification, modeling and explanation. Within each heading, the chief objectives are first identified, and the rationales behind some of the specific techniques are sketched out. The review ends with an outline of current trends and recent developments.
Multivariate Genetic Analysis
Authors: Gillespie Nathan A. , Martin Nicholas G.
Published: September 2005

The goal of multivariate genetic analysis is to estimate the causes of genetic and environmental covariation between phenotypes. As with univariate modeling, when data from genetically informative relatives such as twins are available, the genetic and environmental components of covariation can be separated by a technique which is analogous to factor analysis (Martin & Eaves,). This is achieved by making use of the additional information in the cross‐correlations between relatives for different traits (or same traits measured at different times) and can allow researches to estimate the extent to which genetic and environmental influences are shared in common by several traits or are trait specific. The various forms of multivariate analyses covered in this section include the Cholesky decomposition, common and independent genetic pathway models, and genetic simplex modeling. This section will give the reader an appreciation for the flexibility of SEM approaches to address more complicated questions beyond simpler univariate decompositions.
Multivariate Multiple Regression
Authors: Hartung Joachim , Knapp Guido
Published: September 2005

Multivariate regression is used to explain the relationship betweenp > 1 quantitative dependent variables andqquantitative explanatory variables. When a correlation structure among the dependent variables is present, a single multivariate regression is more efficient than regressions analyses for each dependent variable separately.
Multivariate Normality Tests
Authors: Keselman H. J.
Published: September 2005

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.
Multivariate Outliers
Authors: Fieller Nick
Published: September 2005

Outliers in multivariate data provide particular problems primarily because they may not be revealed as an extreme observation in any single component and are, consequently, easy to miss in routine graphical inspection of the data. This article describes some graphical techniques particularly directed towards the objective of detecting outliers before considering more formal methods of testing for discordancy and overcoming the problems of masking and swamping.
Neural Networks
Authors: Murtagh Fionn
Published: September 2005

Neural network algorithms attempt to model the way a human reaches decisions for such tasks as categorization and finding associations between observed phenomena. Notwithstanding the prosaic name, they are in fact algorithms with very clear procedures for processing given inputs and targeting particular outputs. We will discuss perhaps the dominant paradigm in this area, referred to as the multilayer perceptron. This class of algorithm is used for categorization, and more generally, iteratively learning an association, or mapping between given inputs and given outputs. The second algorithm we will describe, called theKohonen self‐organizing feature map, attempts to find a simplified map or representation of a set of phenomena. The map gives priority to the finding of association or clustering relationships.
Neuropsychology
Authors: Suchy Yana
Published: September 2005

Neuropsychology is a scientific and clinical discipline that is concerned with understanding the brain‐behavior relationship and the application of this understanding to assessment of patients with brain dysfunction. This chapter surveys statistical applications of both the scientific and the clinical aspects of neuropsychology, including
t
Tests; univariate, factorial, and repeated measures analysis of variance; analysis of covariance; correlational analyses; meta‐analysis; factor analysis; multiple regression; discriminant function analysis; and the receiver operating characteristic curve. Different statistical methods have the ability to contribute to different circumscribed components of clinical neuropsychology practice.
N
eyman–
P
earson Inference
Authors: Nickerson Raymond S.
Published: September 2005

The approach to statistical hypothesis testing developed by Jerzy Neyman and Egon Pearson built upon and extended that of Ronald Fisher, and differed from the latter in several important ways. Ideas that distinguish their approach include choosing between possible hypotheses (in contrast to deciding whether or not to reject a single hypothesis); setting, before experimentation, a criterion for deciding which hypothesis to accept; recognizing the two possible ways to draw an incorrect conclusion from a statistical test and taking account of the fact that they are not likely to be equally desirable; and defining the power of a test as the probability of detecting a real effect if there is one. They saw statistical tests as attempts to quantify commonsense notions about the interpretation of probabilistic data and stressed the importance of the exercise of judgment in their application.
Nonlinear Mixed Effects Models
Authors: Lindstrom Mary J. , Vorperian Houri K.
Published: September 2005

Nonlinear mixed‐effects (NLME) models are a generalization of LME and nonlinear regression models and are used when data have a hierarchical growth‐curve structure where an LME model might be used but a nonlinear model is required to fit the data. Both LME and NLME models require some care in estimating and testing model parameters. NLME models have additional difficulties due to the lack of a closed form for the marginal distribution of the data (see following sections).
Nonlinear Models
Authors: Goldsmith L. Jane , Berger Vance W.
Published: September 2005

Nonlinear models relate a response or dependent variable to one or more predictor variables. The relationship is more complicated than that of the familiar straight‐line linear model, although the least‐squares error criterion is usually used for estimation. The methodology of nonlinear models is especially valuable when special relationships among response and predictor variables are known, resulting from solutions of differential equations or graphical interpretations. Estimates must be found through a complicated iterative process, which can fail to converge. Choice of a model is critical not only for adequate description of the nonlinear relationship but also to obtain possible advantageous estimation properties.
Nonparametric Item Response Theory Models
Authors: Sijtsma Klaas
Published: September 2005

Nonparametric item response theory serves two purposes. The first is to provide a framework for flexible, exploratory analysis of item scores. Several procedures are available for investigating the dimensionality of test data. Estimated item response functions are checked for monotonicity and other order restrictions. Nonparametric item response models imply ordinal scales for persons and items. Ordinal scales are sufficient for many practical applications in psychology, education, sociology, and so on. The second purpose is to provide a mathematical framework for studying the potentials and the limitations of item response theory. Observable data properties implied by nonparametric item response theory may also be valuable for fitting parametric models, and relationships between item response models may be clarified. Present research in nonparametric IRT involves preference data analysis and Bayesian methods for model fitting.
Nonparametric Regression
Authors: Fox John
Published: September 2005

Nonparametric regression analysis traces the dependence of a response variable on one or several predictors without specifying in advance the function that relates the predictors to the response. This article discusses several common methods of nonparametric regression, including kernel estimation, local polynomial regression, and smoothing splines. Additive regression models and semiparametric models are also briefly discussed.
Nonresponse in Sample Surveys
Authors: Levy Paul S. , Lemeshow Stanley
Published: September 2005

This expository article discusses issues related to unit nonresponse in sample surveys including various definitions commonly used to report response rates; effects of nonresponses on the validity and reliability of estimates; strategies used to improve response rates in sample surveys; and statistical methods used to adjust for unit nonresponse in analyzing data from sample surveys. References are given to ‘classic’ as well as recently published books on this topic.
Nonrandom Samples
Authors: Wainer Howard
Published: September 2005

Drawing causal inferences from observational studies in which the data are not generated from a process of random allocation to treatment yields the often‐likely outcome of substantial nonrandom error. In this essay, we illustrate several examples of the kind and magnitude of such errors that have been made in the past and indicate how, if random assignment to treatment is not possible, we can at least estimate the level of our uncertainty.
Nonshared Environment
Authors: Towers Hilary , Neiderhiser Jenae M.
Published: September 2005

Findings from decades of research show that the environmental experiences of individuals reared in the same home can make them different from one another as well as similar. This is true for a wide range of individual characteristics and personal relationships. The phenomenon in which environmental influences cause family members to differ from one another is known in the behavioral genetic literature as nonshared environment.
Structural Equation Modeling: Nonstandard Cases
Authors: Marcoulides George A.
Published: September 2005

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.