A priori tests in repeated measures designs: Effects of nonsphericity

The validity conditions for univariate repeated measures designs are described. Attention is focused on the sphericity requirement. For av degree of freedom family of comparisons among the repeated measures, sphericity exists when all contrasts contained in thev dimensional space have equal variances. Under nonsphericity, upper and lower bounds on test size and power of a priori, repeated measures,F tests are derived. The effects of nonsphericity are illustrated by means of a set of charts. The charts reveal that small departures from sphericity (.97 <1.00) can seriously affect test size and power. It is recommended that separate rather than pooled error term procedures be routinely used to test a priori hypotheses.Appreciation is extended to Milton Parnes for his insightful assistance.     

Tests of sphericity of normal distributions and the analysis of repeated measures designs

The locally best invariant test statistic for testing sphericity of normal distributions is shown to be a simple function of the Box/Geisser-Greenhouse degrees of freedom correction factor in a repeated measures design. Because of this relationship it provides a more intuitively appealing test of the necessary and sufficient conditions for valid F-tests in repeated measures analysis of variance than the likelihood ratio test. The properties of the two tests are compared and tables of the critical values of the Box/Geisser-Greenhouse correction factor are given.     

The mixed model for multivariate repeated measures: validity conditions and an approximate test

Repeated measures on multivariate responses can be analyzed according to either of two models: a doubly multivariate model (DMM) or a multivariate mixed model (MMM). This paper reviews both models and gives three new results concerning the MMM. The first result is, primarily, of theoretical interest; the second and third have implications for practice. First, it is shown that, given multivariate normality, a condition called multivariate sphericity of the covariance matrix is both necessary and sufficient for the validity of the MMM analysis. To test for departure from multivariate sphericity, the likelihood ratio test can be employed. The second result is an approximation to the null distribution of the likelihood ratio test statistic, useful for moderate sample sizes. Third, for situations satisfying multivariate normality, but not multivariate sphericity, a multivariate correction factor is derived. The correction factor generalizes Box's and can be used to construct an adjusted MMM test.I am grateful to an anonymous referee for carefully attending to the mathematical details of this paper.     

Analysis of multivariate mixed longitudinal data: A flexible latent process approach

Multivariate ordinal and quantitative longitudinal data measuring the same latent construct are frequently collected in psychology. We propose an approach to describe change over time of the latent process underlying multiple longitudinal outcomes of different types (binary, ordinal, quantitative). By relying on random‐effect models, this approach handles individually varying and outcome‐specific measurement times. A linear mixed model describes the latent process trajectory while equations of observation combine outcome‐specific threshold models for binary or ordinal outcomes and models based on flexible parameterized non‐linear families of transformations for Gaussian and non‐Gaussian quantitative outcomes. As models assuming continuous distributions may be also used with discrete outcomes, we propose likelihood and information criteria for discrete data to compare the goodness of fit of models assuming either a continuous or a discrete distribution for discrete data. Two analyses of the repeated measures of the Mini‐Mental State Examination, a 20‐item psychometric test, illustrate the method. First, we highlight the usefulness of parameterized non‐linear transformations by comparing different flexible families of transformation for modelling the test as a sum score. Then, change over time of the latent construct underlying directly the 20 items is described using two‐parameter longitudinal item response models that are specific cases of the approach.     

Structured latent curve models for the study of change in multivariate repeated measures

This article considers a structured latent curve model for multiple repeated measures. In a structured latent curve model, a smooth nonlinear function characterizes the mean response. A first-order Taylor polynomial taken with regard to the mean function defines elements of a restricted factor matrix that may include parameters that enter nonlinearly. Similar to factor scores, random coefficients are combined with the factor matrix to produce individual latent curves that need not follow the same form as the mean curve. Here the associations between change characteristics in multiple repeated measures are studied. A factor analysis model for covariates is included as a means of relating latent covariates to the factors characterizing change in different repeated measures. An example is provided.     

Variable criteria sequential stopping rule: Validity and power with repeated measures ANOVA,multiple correlation,MANOVA and relation to Chi-square distribution

The variable criteria sequential stopping rule (vcSSR) is an efficient way to add sample size to planned ANOVA tests while holding the observed rate of Type I errors, α_o, constant. The only difference from regular null hypothesis testing is that criteria for stopping the experiment are obtained from a table based on the desired power, rate of Type I errors, and beginning sample size. The vcSSR was developed using between-subjects ANOVAs, but it should work with p values from any type of F test. In the present study, the α_o remained constant at the nominal level when using the previously published table of criteria with repeated measures designs with various numbers of treatments per subject, Type I error rates, values of ρ, and four different sample size models. New power curves allow researchers to select the optimal sample size model for a repeated measures experiment. The criteria held α_o constant either when used with a multiple correlation that varied the sample size model and the number of predictor variables, or when used with MANOVA with multiple groups and two levels of a within-subject variable at various levels of ρ. Although not recommended for use with χ² tests such as the Friedman rank ANOVA test, the vcSSR produces predictable results based on the relation between F and χ². Together, the data confirm the view that the vcSSR can be used to control Type I errors during sequential sampling with any t- or F-statistic rather than being restricted to certain ANOVA designs.     

A method for determining the aversive threshold in the rat using repeated measures: Tests with morphine sulfate

The procedure described here provides an objective automated technique for the psychophysical assessment of the aversive threshold and allows the animal Ss to serve as their own controls. Sensitivity to drug effects, as well as comparisons between drugs, using the same animals are possible also. Using a rectangular tilt cage, the aversive threshold to grid shock was defined as that intensity avoided 75% of the time. Each shock intensity (30, 60, 90, 120, and 150 microA) was presented for 5 min on one side of the cage and then switched to the other side for 5 min, thereby forcing the animal to sample each shock intensity. Aversive thresholds were determined daily for each rat. The analgesic effects of morphine sulfate were compared to saline. Animals were tested for 3 days at each morphine dose level (2, 4, 8, and 16 mg/kg). Each 3-day morphine series alternated with 3 days of testing under saline. Significant differences were detected between saline and morphine at 4, 8, and 16 mg/kg.     

Multi-response permutation procedure as an alternative to the analysis of variance: An SPSS implementation

A permutation test typically requires fewer assumptions than does a comparable parametric counterpart. The multi-response permutation procedure (MRPP) is a class of multivariate permutation tests of group difference useful for the analysis of experimental data. However, psychologists seldom make use of the MRPP in data analysis, in part because the MRPP is not implemented in popular statistical packages that psychologists use. A set of SPSS macros implementing the MRPP test is provided in this article. The use of the macros is illustrated by analyzing example data sets.     

Fixed‐links models for investigating experimental effects combined with processing strategies in repeated measures designs: A cognitive task as example

Fixed‐links models enable the investigation of experimental effects in combination with processing strategies in the framework of repeated measures designs. The consideration of different processing strategies is reasonable whenever the task requirements can be met in different ways. Effects due to specific processing strategies become obvious in comparing the models representing these processing strategies and also by investigating the variances of the corresponding latent variables. The usefulness of such models is demonstrated by the reaction time data of a cognitive task including three treatment levels. Two different processing strategies were considered: analytic and holistic. The results indicated that the participants applied both processing strategies. The analytic strategy was primarily applied for performing with respect to the second and third treatment levels and the holistic strategy with respect to the first treatment level.     

G*Power 3: A flexible statistical power analysis program for the social,behavioral, and biomedical sciences

G*Power (Erdfelder, Faul, & Buchner, 1996) was designed as a general stand-alone power analysis program for statistical tests commonly used in social and behavioral research. G*Power 3 is a major extension of, and improvement over, the previous versions. It runs on widely used computer platforms (i.e., Windows XP, Windows Vista, and Mac OS X 10.4) and covers many different statistical tests of thet, F, and χ₂ test families. In addition, it includes power analyses forz tests and some exact tests. G*Power 3 provides improved effect size calculators and graphic options, supports both distribution-based and design-based input modes, and offers all types of power analyses in which users might be interested. Like its predecessors, G*Power 3 is free.     

CTTITEM: SAS macro and SPSS syntax for classical item analysis

This article describes the functions of a SAS macro and an SPSS syntax that produce common statistics for conventional item analysis including Cronbach’s alpha, item difficulty index (p-value or item mean), and item discrimination indices (D-index, point biserial and biserial correlations for dichotomous items and item-total correlation for polytomous items). These programs represent an improvement over the existing SAS and SPSS item analysis routines in terms of completeness and user-friendliness. To promote routine evaluations of item qualities in instrument development of any scale, the programs are available at no charge for interested users. The program codes along with a brief user’s manual that contains instructions and examples are downloadable from suen.ed.psu.edu/~pwlei/plei.htm.