Robert C. MacCallum

Power analysis and determination of sample size for covariance structure modeling.

A framework for hypothesis testing and power analysis in the assessment of fit of covariance structure models is presented. We emphasize the value of confidence intervals for fit indices, and we stress the relationship of confidence intervals to a framework for hypothesis testing. The approach allows for testing null hypotheses of not-good fit, reversing the role of the null hypothesis in conventional tests of model fit, so that a significant result provides strong support for good fit. The approach also allows for direct estimation of power, where effect size is defined in terms of a null and alternative value of the root-mean-square error of approximation fit index proposed by J. H. Steiger and J. M. Lind (1980). It is also feasible to determine minimum sample size required to achieve a given level of power for any test of fit in this framework. Computer programs and examples are provided for power analyses and calculation of minimum sample sizes.

Evaluating the use of exploratory factor analysis in psychological research.

Despite the widespread use of exploratory factor analysis in psychological research, researchers often make questionable decisions when conducting these analyses. This article reviews the major design and analytical decisions that must be made when conducting a factor analysis and notes that each of these decisions has important consequences for the obtained results. Recommendations that have been made in the methodological literature are discussed. Analyses of 3 existing empirical data sets are used to illustrate how questionable decisions in conducting factor analyses can yield problematic results. The article presents a survey of 2 prominent journals that suggests that researchers routinely conduct analyses using such questionable methods. The implications of these practices for psychological research are discussed, and the reasons for current practices are reviewed.

Sample size in factor analysis.

The factor analysis literature includes a range of recommendations regarding the minimum sample size necessary to obtain factor solutions that are adequately stable and that correspond closely to population factors. A fundamental misconception about this issue is that the minimum sample size, or the

The use of causal indicators in covariance structure models: some practical issues.

In conventional representations of covariance structure models, indicators are defined as linear functions of latent variables, plus error. In an alternative representation, constructs can be defined as linear functions of their indicators, called causal indicators, plus an error term. Such constructs are not latent variables but composite variables, and they have no indicators in the conventional sense. The presence of composite variables in a model can, in some situations, result in problems with identification of model parameters. Also, the use of causal indicators can produce models that imply zero correlation among many measured variables, a problem resolved only by the inclusion of a potentially large number of additional parameters. These phenomena are demonstrated with an example, and general principles underlying them are discussed. Remedies are described so as to allow for the evaluation of models that contain causal indicators.

The problem of equivalent models in applications of covariance structure analysis

For any given covariance structure model, there will often be alternative models that are indistinguishable from the original model in terms of goodness of fit to data. The existence of such equivalent models is almost universally ignored in empirical studies. A study of 53 published applications showed that equivalent models exist routinely, often in large numbers. Detailed study of three applications showed that equivalent models may often offer substantively meaningful alternative explanations of data. The importance of the equivalent model phenomenon and recommendations for managing and confronting the problem in practice are discussed.

Repairing Tom Swift's Electric Factor Analysis Machine

Proper use of exploratory factor analysis (EFA) requires the researcher to make a series of careful decisions. Despite attempts by Floyd and Widaman (1995), Fabrigar, Wegener, MacCallum, and Strahan (1999), and others to elucidate critical issues involved in these decisions, examples of questionable use of EFA are still common in the applied factor analysis literature. Poor decisions regarding the model to be used, the criteria used to decide how many factors to retain, and the rotation method can have drastic consequences for the quality and meaningfulness of factor analytic results. One commonly used approach-principal components analysis, retention of components with eigenvalues greater than 1.0, and varimax rotation of these components-is shown to have potentially serious negative consequences. In addition, choosing arbitrary thresholds for factor loadings to be considered large, using single indicators for factors, and violating the linearity assumptions underlying EFA can have negative consequences ...

Sample size in factor analysis: The role of model error

This article examines effects of sample size and other design features on correspondence between factors obtained from analysis of sample data and those present in the population from which the samples were drawn. We extend earlier work on this question by examining these phenomena in the situation in which the common factor model does not hold exactly in the population. We present a theoretical framework for representing such lack of fit and examine its implications in the population and sample. Based on this approach we hypothesize that lack of fit of the model in the population will not, on the average, influence recovery of population factors in analysis of sample data, regardless of degree of model error and regardless of sample size. Rather, such recovery will be affected only by phenomena related to sampling error which have been studied previously. These hypotheses are investigated and verified in two sampling studies, one using artificial data and one using empirical data.

Studying Multivariate Change Using Multilevel Models and Latent Curve Models

In longitudinal research investigators often measure multiple variables at multiple points in time and are interested in investigating individual differences in patterns of change on those variables. In the vast majority of applications, researchers focus on studying change in one variable at a time. In this article we consider methods for studying relations1.1ips between patterns of change on different variables. We show how the multilevel modeling framework, which is often used to study univariate change, can be extended to the multivariate case to yield estimates of covariances of parameters representing aspects of change on different variables. We illustrate this approach using data from a study of physiological response to marital conflict in older married couples, showing a substantial correlation between rate of linear change on different stress-related hormones during conflict. We also consider how similar issues can be studied using extensions of latent curve models to the multivariate case, and we show how such models are related to multivariate multilevel models.

Distinguishing Optimism From Pessimism in Older Adults: Is It More Important to Be Optimistic or Not to Be Pessimistic?

Confirmatory factor analysis revealed that the Life Orientation Test (LOT) consisted of separate Optimism and Pessimism factors among middle-aged and older adults. Although the two factors were significantly negatively correlated among individuals facing a profound life challenge (i.e., caregiving), they were only weakly correlated among noncaregivers. Caregivers also expressed less optimism than noncaregivers and showed a trend toward greater pessimism, suggesting that life stress may affect these dispositions. Pessimism, not optimism, uniquely predicted subsequent psychological and physical health; however, optimism and pessimism were equally predictive for stressed and nonstressed samples. By exploring optimism and pessimism separately, researchers may better determine whether the beneficial effects of optimism result from thinking optimistically, avoiding pessimistic thinking, or a combination of the two.

Chronic Stress Modulates the Immune Response to a Pneumococcal Pneumonia Vaccine

Objective Influenza and pneumonia account for significant morbidity and mortality, particularly in older individuals. Previous studies have shown that spousal caregivers of patients with dementia have poorer antibody and virus specific T cell responses to an influenza virus vaccine relative to noncaregiving control subjects. This study tested the hypothesis that stress can also significantly inhibit the IgG antibody response to a pneumococcal bacterial vaccine. Method We measured antibody titers of current caregivers, former caregivers, and control subjects after vaccination with a pneumococcal bacterial vaccine. Results Caregivers showed deficits relative to controls and former caregivers in their antibody responses to vaccination. Although the groups did not differ before vaccination or in the rise in antibody 2 weeks or 1 month after vaccination, current caregivers had lower antibody titers 3 and 6 months after vaccination than either former caregivers or controls. Conclusions These data, the first evidence that chronic stress can inhibit the stability of the IgG antibody response to a bacterial vaccine for pneumonia, provide additional evidence of health risks associated with dementia caregiving.