Abstract In this entry, I describe the covariance, variance, and correlation, each in a common framework that allows their interrelationships to be apparent. Specifically, the covariance is the uniting concept, within which the variance and correlation are special cases. Following separate treatments, I present a formula that explicitly shows the relationships between the three indices. Finally, I briefly describe the role of each in Structural Equations Modeling (SEM).

This article has no abstract. Keywords: Facet theory; multidimensional scaling; radex; factor analysis

Abstract Exploratory Data Analysis (EDA) is one of the most influential recent developments in statistics. Led by John Tukey, EDA offered a new inductive orientation to the discovery of knowledge. In detail, this meant the invention of new graphical methods, the development of robust, outlier resistant measurements, and the modernization of regression analysis and related linear model approaches.

Abstract 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.

Abstract An observational study is an empiric investigation that attempts to estimate the effects caused by a treatment when it is not possible to perform an experiment. Random assignment of subjects to treatment or control in an experiment ensures that comparable groups of subjects are compared under alternative treatments. Without random assignment, in an observational study, the treatment and control groups may not be comparable prior to treatment, so that differing outcomes after treatment may not be effects caused by the treatment. Pretreatment differences that have been accurately measured are calledovert biases; they may often be controlled using analytical adjustments, such as matching, stratification, or covariance adjustment. Pretreatment differences that were not recorded are called hidden biases, and they pose substantial problems that must be addressed by other means. A well designed observational study permits detection of one or two of the most plausible hidden biases. The analysis of an observational study can measure the degree of sensitivity of conclusions to hidden biases of plausible magnitudes. Design strategies involving pattern specificity, such as Campbell's ‘multiple operationalism’ or Hill's dose‐response, may reduce sensitivity to hidden bias, and this possibility may be appraised using the design sensitivity of competing designs.

Abstract A nonparametric hypothesis test for the slope in simple linear regression proposed by Theil (1950) also provides the basis for a distribution‐free estimate of that parameter.

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

Abstract A skewed distribution is one that has an asymmetric tail on the left (negative) or the right (positive) side. Skewness is most commonly defined in terms of the third moment about the mean. Measures of skewness may alert the behavioral researcher to the need to investigate outliers, consider using the median rather than the mean, transform the data, and/or choose analytic techniques that do not assume normality.

Abstract 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.

Abstract Yates' correction for continuity adjusts the formula for Pearson's chi‐square test by subtracting 0.5 from each observed value in a 2 × 2 contingency table. This formula is used when at least one cell of the table has an expected frequency less than 5, although some sources say that this correction should be used when the expected frequency is less than 10.