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The analysis of variance, introduced by Sir Ronald Fisher near the beginning of the twentieth century, is widely used by behavioral and social scientists. As a class of statistical models, “ANOVA” provides a means for analyzing data that is both rigorous logically and mathematically, and sufficiently broad to address questions posed in a wide spectrum of investigations. This entry describes the range of different analysis of variance models, the questions they address, the types of data for which they are appropriate, and the logic by which they operate. Several newer developments and recent thinking about ANOVA procedures are described and demonstrated in an investigation of students' motivation.
The main function performed by ANOVA is to compare systematically the mean response levels of two or more independent groups of observations, or of a set of observations measured at two or more points in time. Analysis of variance techniques are validly applied in each of three types of investigations (Bock, 1975): (a) experiments, in which subjects are assigned at random to treatments determined by the investigator: (b) comparative studies, whose purpose is to describe differences among naturally occurring populations; and (c) surveys, in which the responses of subgroups of a single population are to be compared.
Consider several examples. First, an experiment might be conducted in which one randomly assigned group of students receives no formal spelling instruction, a second group receives 15 minutes of spelling instruction per day, and a third group 30 minutes per day. At the end of the experiment, the effectiveness of spelling instruction is assessed by comparing the mean scores of the three groups on a common spelling test. Second, a recent large-scale comparative study examined differences among the achievement levels of students attending American public schools, private schools with religious affiliation, and other private schools. Although many different analyses were employed, ANOVA provides the most direct comparison of mean achievement across the three types of institutions. Third, the International Association for the Evaluation of Educational Achievement (IEA) international surveys sampled children in such a way that the final sample represented the entire age cohort attending school within each participating country. Subjects within a country may be subclassified by their responses even after the data are collected, to examine differences in mean response le vel among subgroups of interest. In an example described below, Swedish 13-year olds are classified according to their fathers' occupations, and average levels of achievement motivation are compared using ANOVA. In a survey, unlike other types of investigations, removing the researcher-defined subclassifications still yields an intact population (all Swedish 13-year olds). Nevertheless, whenever mean comparisons are of concern, ANOVA remains an appropriate and powerful analytic tool.
The examples above are described as having one measured response variable. Studies that yield two or more interrelated response measures (i.e., multiple dependent variables) require multivariate analysis of variance (“MANOVA”) tests and estimates. For example, the spelling experiment might have included a subtest covering words that were explicitly part of the curriculum and another covering words of similar difficulty that were not taught. Achievement in three types of American schools may have been measured in terms of multiple subject areas (mathematics, reading, social studies, etc.) or in terms of both cognitive and affective outcomes.
In each case, summing the measures will obscure important differences among the subscales, while separate analyses of each scale may give contradictory or confusing results and limit the replicability of the investigation. Appropriate MANOVA procedures maintain the integrity of the original measures while providing tests and estimates that pertain to the set of responses jointly. Repeated measures analysis is employed in investigations in which the same subjects are observed at two or more points in time or under two or more experimental conditions. These include pretest-post-test studies or educational or social interventions, longitudinal studies in which data are collected on the same scale repeatedly, and within-subject experiments in which subjects are measured on the same response variable under several experimental conditions. Both univariate and multivariate analysis of variance models may be used with repeated measures data, depending on the distribution of the multiple measurements. The two types of analyses are described and compared in Bock (1975 Chap. 7). (J. D. Finn) Full Reading ... PPT Download ...