Statistical Methods for Psychology

appropriate to adjust for random differences among groups. (An analysis of variance on the

covariate [Distract] showed no significant effects.)

Notice that Exhibit 16.5 presents partialeta-squared for the effects. These effect-size

measures can be calculated as the difference between two values, divided by ( ).

For example, the model without the dummy variables for Task has an 5 .2762. This leaves

1 2 .2762 5 72.38% of the variation unexplained. When we add in the Task variables (going

to the full model) we have 5 .8027. This is an increase of .8027 2 .2762 5 .5265, which

accounts for .5265/.7238 5 72.74% of the variation that had been left unexplained. This is

the value given in Exhibit 16.5 for Task. Similar calculations will reproduce the other values.

Adjusted Means

The method of obtaining adjusted means is simply an extension of the method employed in

the Conti and Musty example. We want to know what the cell means would have been if

the treatment combinations had not differed on the covariate.

From the full model we have

which equals

Since we want to know what the Ymeans would be if the treatments did not differ on the

covariate, we will set for all treatments.

For all observations in the appropriate row of the design matrix, with

, is

10101000 112.518

Applying the regression coefficients and taking the intercept into account, we have

=9.804

2 1.404(0) 2 6.661(0) 1 1.668(0) 1 0.292512(112.518)

Y 11 = 2 14.654 2 8.561(1) 1 19.071(0) 2 2.900(1) 1 1.437(0) 1 3.006(1)

C replaced by C

Cell 11

C=C=112.518

2 6.661TG 211 1.668TG 221 0.292512Distract

YN = 2 14.654 2 8.561T 11 19.071T 22 2.900G 11 1.437G 21 3.006TG 112 1.404TG 12

YN =b 01 b 1 C 1 b 2 T 11 b 3 T 21 b 4 G 11 b 5 G 21 b 6 TG 111 b 7 TG 121 b 8 TG 211 b 9 TG 22

R^2

R^2

R^212 R^2 reduced

618 Chapter 16 Analyses of Variance and Covariance as General Linear Models

3. Smoking Group
   Dependent Variable: ERRORS

Smoking Group

NonSmokers

Delayed smokers

Active Smokers

Mean

15.360a

19.696a

19.721a

Std. Error

1.264

1.266

1.261

Lower

Bound

12.859

17.191

17.225

Upper

Bound

17.862

22.202

22.217

95% Confidence

Interval

aEvaluated at covariates appeared in the model: DISTRACT 5 112.52.

Exhibit 16.5 (continued)