Two and More Factors ANOVA - Agricultural Statistical Data Analysis Using Stata - page 127

In the first instance, an RCBD is estimated without a covariate. In
the second instance, the covariate cov is introduced into the model. In
a lima bean variety trial, it is difficult to harvest the crop where each
entry is at the same maturity and it is known that the more mature
the beans, the lower the ascorbic acid content. This covariate is then
percent dry matter content, which is an indicator of maturity. The c.
in front of this variable is telling the program to treat this variable as
a continuous variable and calculate it as a regression (see Chapter 10,
Correlation and Regression). In the second model, the residual mean
square is lower (148.682599 versus 56.4988289) and both the R 2 and
adjusted R 2 are higher. This indicates that the second model is more
precise in its estimate of the treatment effect.
With analysis of covariance, it is customary to present adjusted or
marginal means rather than the simple arithmetic means. The mar-
ginal means take into account the effect of the covariate and can have
a significant impact on the interpretation of results. The adjustment of
the means is calculated as follows:

ˆ

YYbX X =− ( − )

i • i • YX i • ••

he ˆ • Y i is the adjusted treatment mean. The Y i • represents the

observed mean and the b YX is the error regression coefficient. The X i •
and X •• represent the observed covariance mean and the overall mean
for all the covariance entries, respectively. To calculate the adjusted
mean for the first entry (lima bean variety), which has an observed
mean of 88.1, enter the following commands:

tabstat ascorbic, statistics(mean) by (var)
columns(variables)
tabstat cov, statistics(mean) by (var) columns(variables)
tabstat cov, statistics(mean) columns(variables)
anova ascorbic var rep c. cov
matrix list e(b)

The first two tabstat commands calculate the means for the
ascorbic (ascorbic acid content) and cov (dry weight percentage) vari-
ables by var (varieties). This gives us the mean for the first variety (i.e.,
88.1) and the mean for the corresponding covariate (i.e., 35.42). The
third tabstat command calculates the overall mean for cov (i.e.,