Factorial Design

An Example of a 2 × 2 Design

Table 1 presents hypothetical data and means in which the data are presented for scores under two drugs and two dosage levels. Each of the four cell means is based on three observations. That is, three different [Page 349]individuals were given either Drug 1 or Drug 2 and either 10 mg or 20 mg of the drug. The 12 individuals were randomly assigned to one of the four combinations of drug and dosage level. Higher mean adjustment scores indicate better adjustment.

As in a simple, independent-sample t test or one-way analysis of variance (ANOVA), the variability within each treatment condition must be determined. The variability within the four groups is defined as SSWG where

With three observations in each group, there are 3 − 1 = 2 degrees of freedom (df) within each group. The degrees of freedom within all groups are

The within-groups variability is known as the mean square (MS) within groups, defined as

The overall difference between drugs is Factor A, Drug Type. The null and alternative hypotheses are

The variability among the drug means is

With a levels of Factor A, the degrees of freedom will be

The MS for Factor A is

The F test for Factor A is

To test the null hypothesis at α= .01, we need the critical value CV = F.99(1,8) = 11.26. The null hypothesis is rejected because FA = 75.0 > 11.26. The overall mean, MA = 8.0, for Drug 1 is significantly greater than the overall mean, MA = 3.0, for Drug 2. Drug 1 produces significantly greater adjustment that does Drug 2.

Applying similar calculations to Factor B, Dosage Level, we have the hypotheses

The variability among the drug means

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