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Chapter 11 11.1 Introduction to One-way ANOVA In the previous two chapters we distinguished between two distinct applications of the t-test - the independent samples t-test and the correlated samples t-test. This same distinction applies to the new procedure introduced in this chapter, One-way ANOVA. The present chapter will introduce this procedure for independent samples, and the ANOVA procedure for correlated samples will be presented in Chapter 12. As with the t-test, ANOVA also tests for significant differences between groups. But while the t-test is limited to the comparison of only two groups, one-way ANOVA can be used to test differences in three or more groups. Several hypotheses worth investigating in our project involve the comparison of more than two groups. For example, EZ execs would certainly be interested in any differences in leader performance in relation to employees' sex role type or leadership style. However, as you will see, there are four levels of both of these quasi-independent variables (and so four groups to compare). Since the t-test only permits comparison of two groups, it is necessary to use analysis of variance (ANOVA) procedures for these comparisons. The ANOVA procedure produces an F statistic, a value whose probability enables the researcher to reject or retain the null hypothesis, i.e., to conclude whether or not the differences in the scores on the dependent variable are statistically significant or due to chance. As with the t-test test, ANOVA is appropriate when the data are interval or ratio level, when the groups show similar variances, and when the data are normally distributed. ANOVA is based upon a comparison of variance attributable to the independent variable (variability between groups or conditions) relative to the variance within groups resulting from random chance. In fact, the formula for computing the F statistic involves dividing the between-group variance estimate by the within-group variance estimate. While different procedures are used to compute the ANOVA for independent vs. correlated samples designs, each procedure yields an F statistic which is evaluated in essentially the same way. Thus, regardless of which design is employed, the end result is an F-value, and when the probability of occurrence of the F value is less than .05, we conclude that there are significant differences groups, i.e., variation which cannot be attributed to chance.
When three or more groups are being analyzed in the ANOVA, there frequently arises the need to carry out more specific two-group comparisons in order to determine where the major treatment effect is occurring. A significant F-value only indicates that there is a significant difference somewhere between the groups - it does not indicate which groups are different. To determine this, secondary comparisons among all group means are needed subsequent to the ANOVA. These two-group comparisons are commonly referred to as individual comparisons, follow-up tests, or post-hoc tests. Let's assume, for example, that you are a clinical psychologist who has designed an experiment to assess the effectiveness of different types of therapy on the treatment of phobias in a sample of eighty men and women. You assign each participant to one of four groups (either a control group or one of three different types of therapy). After an adequate treatment period, you rate the amount of improvement each subject has shown (higher score = more improvement). The mean improvement scores (hypothetical, of course!) for the four groups are as follows:
You perform an ANOVA on the four groups, and find that there is significant overall variation (or differences) between the treatment groups. From this analysis, you can conclude that the independent variable (therapy) is having a significant effect, but you are unable to state where the effect is occurring. Is psychoanalytic therapy more effective than receiving no treatment at all? Is cognitive-behavioral therapy more effective than either psychoanalytic or client-centered therapies? Might there also be a significant difference between psychoanalytic and client-centered approaches? These questions require that we carry out more specific individual comparisons between the pairs of groups in which we're interested. There are a number of guidelines for performing these two-group comparisons. For example, most researchers agree that it is appropriate to perform individual comparisons only if the result of the overall ANOVA is significant. Furthermore, one must decide at some point in the experimental process whether these comparisons are to be a priori (planned before the data are collected), or post hoc (decided upon after collecting and studying the data). Last, one must decide which specific a priori or post hoc technique (since there are many) best suits the situation (or researcher). While we are not able to review all the relevant guidelines for performing individual comparisons within this text, suffice it to say that these comparisons are generally considered an important and necessary part of the analysis of an experimental design in which there are three groups or more |