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# Empirical Direction in Design and Analysis

By: Norman H. Anderson | Book details

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Page 395

NOTES

A good short treatment of analysis of covariance is given by Snedecor and Cochran (1980, Chapter 18). The book by Huitema (1980) is well-informed, thorough, and reasonably elementary. Among psychological texts, Maxwell and Delaney (1990) give an extensive discussion with a positive outlook on Ancova. Reichardt (1979) gives an illuminating discussion of Ancova with nonrandom groups, some of which is summarized in Section 15.5 on quasi-experimental design. I am indebted to Anthony Greenwald and Charles Reichardt for helpful comments on this chapter.

13.1.1a
Strictly speaking, Ancova is not simply Anova on the deviations from the regression line (Maxwell, Delaney, & Mannheimer, 1985). Intuitively, however, it is helpful to visualize Ancova in this way. Computer programs use a least squares analysis of Equation 2, which yields correct results.
13.1.1b
The adjustment of the means in Equation 4 assumes the regression is linear. With random groups, this linearity assumption is usually reasonable, even when the true regression is substantially nonlinear, because the X .j will be relatively close together.
13.1.1c
A covariate may yield a worthwhile reduction in MSerror even though its true correlation with Y is only ρ =.4, too small to be dependably statsig at α =.05. Hence a covariate may be retained for future use even though it is not statsig in the present application. One determinant in this decision is prior belief in the evidence value of the covariate, to which the evidence of the present experiment is subsidiary. In this situation, of course, a “false alarm” usually has low statistical cost and an α of.20 would seem often appropriate.
13.1.2a
Using blocks and Ancova together may be most effective. The historically first application of Ancova (Fisher, 1932) was used to illustrate a blocks-and-Ancova analysis by Snedecor and Cochran (1980, p. 371). Y was the current yield on 16 plots of tea bushes in Ceylon; X was yield in the previous year, prior to applying the four experimental treatments. In one set of these data, blocking reduced MSerror from 136 to 48; Ancova produced a further reduction to 27. Good design and analysis, at almost no cost, thus increased precision five-fold. Even 30% increases, which are more to be expected, will often be worthwhile (see similarly Maxwell and Delaney, 1990, pp. 395 ff).
13.1.2b
Significance tests are not too meaningful for assessing differences on the covariate between randomized groups. The difference in any actual sample is real in that sample (barring measurement unreliability). Whether the group differences on X are statsig depends strongly on N, which is irrelevant to the issue.
13.1.2c
Ancova with a correlated response measure also suffers a statistical bias. Unreliability in X causes the sample regression coefficient b1 to underestimate the true coefficient β1. Causal interpretation requires a structural regression, whereas standard Ancova incorporates a predictive regression. When Ancova with the structural regression would equalize the adjusted Y means, the standard Ancova will not, and vice versa. The outcome is thus ambiguous regardless of whether Ancova leaves the differences statsig or nonstatsig. Standard Ancova is thus conceptually invalid, although correction for the unreliability may be possible (Note 13.2.2b).

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