The research design

The dataset has a dependent variable, number of words recalled from a memorized list. Three independent variables defined as factors 1) Grade, either fifth or twelfth graders 2) Feedback conditions: control (none), positive, or negative. WordType, that is the type of words used: A, B, C (we renamed the level for simplicity)

Running the model with the three factors and numcall as dependent variable, we obtain the following results.

Parameter Estimates table reports all tested coefficients for the contrasts used to cast the factor variables in the linear model. However, by default the factor are coded using simple coding, which implying comparing the last level with all other levels.

Custom contrast

We now wish to test the custom contrasts for WordType comparing group A against the pooled means of group B and C. We go to Factors Coding and we choose custom for the variable WordType.

Upon choosing custom for a variable, a new field appears and we can input the contrast weights we wish to test. Only one contrast per variable can be defined, but if more contrasts are required one can always run different analyses, one for each contrast. The coding weights are input with the simple syntax w1,w2,w3. The order of the weights follow the other of the factor levels in the datasheet. In our case, the following:

We also flag the option Focus on custom contrast, which produces a table singling out the results for the contrast (this options has other implications, see below).

The table reports the estimate of the contrast, the Cohen’s d, t-test and p-value of the contrast.

More results can be found in the Parameter Estimates table.

Here we see all model coefficients, including the ones associated with the custom contrast. For instance, row Feedback1 * WordType1 informs us on the interaction between our contrast and the comparison none vs negative of feedback, telling whether our contrast is different for Feedback=none as compared with Feedback=neg. The row wordType1 * Grade1 informs us on the interaction between our contrast and the comparison 5th vs 12th Grade, telling whether our contrast is different at grade 5th as compared with grade 12th.

In the table there is also a second contrast for WordType, named tech2. That is a technical contrast necessary to correctly estimate the model. Recall, in fact, that to cast a categorical variable with \(K\) levels into a linear model, we need \(K-1\) contrasts. However, tech2 is automatically coded to be orthogonal (uncorrelated) with our custom contrast, so custom contrast results are completely unaffected by the technical contrasts. We can ignore them.

Simple effects

We can further probe the results by asking for simple effects. Since we have a significant interaction between our contrast and Grade (see WordType1 * Grade1 above) we can probe it.

The results of both table refer exclusively to the custom contrast. So, the F-test in the ANOVA table tests the contrast at each level of grade, and so does the Parameter Estimates for simple effects table. However, if one wants the F-tests for the whole effect of the factor, so not only the custom contrasts, one can simply unflag the Focus on custom contrast.

Simple interactions

Although the 3-way interaction is not statistically significant, as an exercise we can ask for the Simple interactions

It is important to recall that in the presence of a custom contrast, and with he Focus on custom contrast flagged, the results pertain only to the custom contrast. Thus, the ANOVA table informs as whether the interaction contrast X feedback is significant at grade 5th and at grade 12th. The Parameter Estimates table breaks down these interactions into their single contrast interactions, namely custom * (none-neg) and custom * (pos-neg).

Unflagging Focus on custom contrast would return standard simple interactions, involving all contrast in the model.

More custom contrasts

We can set one custom contrast per variable in the same model. Assume, for instance, we also want to test the contrast for Feedback comparing group negative against group none. Because the Feedback levels are ordered neg,none, and pos, our contrast weights should be \(c=[1,-1,0]\).

We can see in the output the results for both contrasts defined in input.

The reaming of the tables can be interpreted as we did above. The only caveat to mention is for the simple interactions. When simple interactions involve factors with custom contrast, the results pertain the interactions between the target variable custom and the other factor. Thus, setting WordType as the target variable

The parameters table show the interaction custom contrast of WordType with the other contrast of Feedback, including the custom contrast of the latter variable. The ANOVA table tests that these interactions combined per level of Grade are statistical significant.