## X2

Figure 1.19 Scatter plots showing the relationship between the dependent variable, Y, and concomitant variable, X, for the two treatment levels. The size of the error variance in ANOVA is determined by the dispersion of the marginal distributions. The size of the error variance in ANCOVA is determined by the dispersion of the conditional distributions. The higher the correlation between X and Y is, the greater the reduction in the error variance due to using analysis of covariance.

Figure 1.20 Analysis of covariance adjusts the concomitant-variable means, Xand X 2, so that they equal the concomitant-variable grand mean, X... When the concomitant-variable means differ, the absolute difference between adjusted means for the dependent variable, \Y adj.1 — Y adj.2\, can be less than that between unadjusted means, \Y — Y 2I, as in panels A and B, or larger, as in panel C.

Figure 1.20 Analysis of covariance adjusts the concomitant-variable means, Xand X 2, so that they equal the concomitant-variable grand mean, X... When the concomitant-variable means differ, the absolute difference between adjusted means for the dependent variable, \Y adj.1 — Y adj.2\, can be less than that between unadjusted means, \Y — Y 2I, as in panels A and B, or larger, as in panel C.

dispersion of the conditional distributions (see Figure 1.19). The higher the correlation between X and Y, in general, the narrower are the ellipses and the greater is the reduction in the error variance due to using analysis of covariance.

Figure 1.19 depicts the case in which the concomitant-variable means, X i and X 2, are equal. If participants are randomly assigned to treatment levels, in the long run the concomitant-variable means should be equal. However, if random assignment is not used, differences among the means can be sizable, as in Figure 1.20. This figure illustrates what happens to the dependent variable means when they are adjusted for differences in the concomitant-variable means. In panels A and B the absolute difference between adjusted dependent-variable means \Y adj.1 — Y adj.2\ is smaller than that between unadjusted means \ Y— Y 2 \. In panel C the absolute difference between adjusted means is larger than that between unadjusted means.

Analysis of covariance is often used in three kinds of research situations. One situation involves the use of intact groups with unequal concomitant-variable means and is common in educational and industrial research. Analysis of covariance statistically equates the intact groups so that their concomitant variable means are equal. Unfortunately, a researcher can never be sure that the concomitant variable used for the adjustment represents the only nuisance variable or the most important nuisance variable on which the intact groups differ. Random assignment is the best safeguard against unanticipated nuisance variables. In the long run, over many replications of an experiment, random assignment will result in groups that are, at the time of assignment, similar on all nuisance variables.

A second situation in which analysis of covariance is often used is when it becomes apparent that even though random assignment was used, the participants were not equivalent on some relevant variable at the beginning of the experiment. For example, in an experiment designed to evaluate the effects of different drugs on stimulus generalization in rats, the researcher might discover that the amount of stimulus generalization is related to the number of trials required to establish a stable bar-pressing response. Analysis of covariance can be used to adjust the generalization scores for differences among the groups in learning ability.

Analysis of covariance is useful in yet another research situation in which differences in a relevant nuisance variable occur during the course of an experiment. Consider the experiment to evaluate two approaches toward introducing long division that was described earlier. It is likely that the daily schedules of the eight classrooms provided more study periods for students in some classes than in others. It would be difficult to control experimentally the amount of time available for studying long division. However, each student could record the amount of time spent studying long division. If test scores on long division were related to amount of study time, analysis of covariance could be used to adjust the scores for differences in this nuisance variable.

Statistical control and experimental control are not mutually exclusive approaches for reducing error variance and minimizing the effects of nuisance variables. It may be convenient to control some variables by experimental control and others by statistical control. In general, experimental control involves fewer assumptions than does statistical control. However, experimental control requires more information about the participants before beginning an experiment. Once data collection has begun, it is too late to assign participants randomly to treatment levels or form blocks of dependent participants. The advantage of statistical control is that it can be used after data collection has begun. Its disadvantage is that it involves a number of assumptions such as a linear relationship between the dependent and concomitant variables and equal within-groups regression coefficients that may prove untenable in a particular experiment.

In this chapter I have given a short introduction to those experimental designs that are potentially the most useful in the behavioral and social sciences. For a full discussion of the designs, the reader is referred to the many excellent books on experimental design: Bogartz (1994), Cobb (1998), Harris (1994), Keppel (1991), Kirk (1995), Maxwell and Delaney (1990), and Winer, Brown, andMichels (1991). Experimental designs differ in a number of ways: (a) randomization procedures, (b) number of treatments, (c) use of independent samples or dependent samples with blocking, (d) use of crossed and nested treatments, (e) presence of confounding, and (f) use of covariates. Researchers have many design decisions to make. I have tried to make the researcher's task easier by emphasizing two related themes throughout the chapter. First, complex designs are constructed from three simple building block designs. Second, complex designs share similar layouts, randomization procedures, and assumptions with their building block designs.

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