Use of a general analysis of variance program in missing data situations

Procedures are described which enable researchers to easily modify a general N-way analysis of variance program so that it can be used in unequal N cases. Advantages in terms of range of application, storage requirements, and accuracy are presented. FORTRAN instructions illustrating the general approach are given.     

The analysis of variance and covariance techniques in relation to the conventional formulas for the standard error of a difference

In this paper it is demonstrated that the analysis of variance techniques yield results equivalent to the calculation oft by means of expressions based on the short or the long formula. It is also shown that the covariance technique gives results equivalent to those obtained by means of the formula fort which should be used with matched groups. The conditions necessary for equivalent results are such that the conventional formulas fort would normally be used rather than the variance or covariance techniques. However, a knowledge of the relationships described in this paper should contribute to one's understanding of the variance and covariance techniques.The relationships described in this paper were brought to the attention of the author by the excellent article of Eugene Shen (8). The proofs given here and the examples are the work of the present author.     

Accounting for common method variance in cross-sectional research designs

Cross-sectional studies of attitude-behavior relationships are vulnerable to the inflation of correlations by common method variance (CMV). Here, a model is presented that allows partial correlation analysis to adjust the observed correlations for CMV contamination and determine if conclusions about the statistical and practical significance of a predictor have been influenced by the presence of CMV. This method also suggests procedures for designing questionnaires to increase the precision of this adjustment.     

A simple test for heterogeneity of variance in complex factorial designs

A simple procedure for testing heterogeneity of variance is developed which generalizes readily to complex, multi-factor experimental designs. Monte Carlo Studies indicate that the Z-variance test statistic presented here yields results equivalent to other familiar tests for heterogeneity of variance in simple one-way designs where comparisons are feasible. The primary advantage of the Z-variance test is in the analysis of factorial effects on sample variances in more complex designs. An example involving a three-way factorial design is presented.     

Robust techniques for testing heterogeneity of variance effects in factorial designs

Several ways of using the traditional analysis of variance to test heterogeneity of spread in factorial designs with equal or unequaln are compared using both theoretical and Monte Carlo results. Two types of spread variables, (1) the jackknife pseudovalues ofs ² and (2) the absolute deviations from the cell median, are shown to be robust and relatively powerful. These variables seem to be generally superior to the Z-variance and Box-Scheffé procedures.This research was sponsored by Public Health Service Training Grant MH-08258 from the National Institute of Mental Health. The author thanks Mark I. Appelbaum, Elliot M. Cramer, and Scott E. Maxwell for their helpful criticisms of this paper. An earlier version of this work was presented at the Annual Meeting of the Psychometric Society, Murray Hill, New Jersey, April, 1976.     

Analysis of variance of one-, two-, and three-treatment designs for a PDP-8

Operating instructions for a series of analysis of variance programs for one-, two-, and three-treatment experimental designs is described. The emphasis is on versatility, speed, accuracy, and sufficiency of output. The on-line aspect of FOCAL allows extensive transformations of raw data. Procedures and terminology conform to Kirk (1968) to provide information for pooling error terms for various models, mean comparisons, and trends analysis. Patches are given for data tape input.     

Using analysis of covariance (ANCOVA) with fallible covariates

Analysis of covariance (ANCOVA) is used widely in psychological research implementing nonexperimental designs. However, when covariates are fallible (i.e., measured with error), which is the norm, researchers must choose from among 3 inadequate courses of action: (a) know that the assumption that covariates are perfectly reliable is violated but use ANCOVA anyway (and, most likely, report misleading results); (b) attempt to employ 1 of several measurement error models with the understanding that no research has examined their relative performance and with the added practical difficulty that several of these models are not available in commonly used statistical software; or (c) not use ANCOVA at all. First, we discuss analytic evidence to explain why using ANCOVA with fallible covariates produces bias and a systematic inflation of Type I error rates that may lead to the incorrect conclusion that treatment effects exist. Second, to provide a solution for this problem, we conduct 2 Monte Carlo studies to compare 4 existing approaches for adjusting treatment effects in the presence of covariate measurement error: errors-in-variables (EIV; Warren, White, & Fuller, 1974), Lord's (1960) method, Raaijmakers and Pieters's (1987) method (R&P), and structural equation modeling methods proposed by S?rbom (1978) and Hayduk (1996). Results show that EIV models are superior in terms of parameter accuracy, statistical power, and keeping Type I error close to the nominal value. Finally, we offer a program written in R that performs all needed computations for implementing EIV models so that ANCOVA can be used to obtain accurate results even when covariates are measured with error.     

Selecting a linear mixed model for longitudinal data: repeated measures analysis of variance, covariance pattern model, and growth curve approaches

With increasing popularity, growth curve modeling is more and more often considered as the 1st choice for analyzing longitudinal data. Although the growth curve approach is often a good choice, other modeling strategies may more directly answer questions of interest. It is common to see researchers fit growth curve models without considering alterative modeling strategies. In this article we compare 3 approaches for analyzing longitudinal data: repeated measures analysis of variance, covariance pattern models, and growth curve models. As all are members of the general linear mixed model family, they represent somewhat different assumptions about the way individuals change. These assumptions result in different patterns of covariation among the residuals around the fixed effects. In this article, we first indicate the kinds of data that are appropriately modeled by each and use real data examples to demonstrate possible problems associated with the blanket selection of the growth curve model. We then present a simulation that indicates the utility of Akaike information criterion and Bayesian information criterion in the selection of a proper residual covariance structure. The results cast doubt on the popular practice of automatically using growth curve modeling for longitudinal data without comparing the fit of different models. Finally, we provide some practical advice for assessing mean changes in the presence of correlated data.