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The subjects are divided into two groups of four subjects using the the variable trt. Here are the basic descriptive statistics at each of the four time points combined and broken out by treatment group.
Next, we will graph the eight cell means with the user written command, profileplot. You can download this command by typing search profileplot is the Stata command window. Repeated measure anova assumes the within-subject covariance structure is compound symmetric. The covariance matrix above does not appear to have compound symmetry. We will discuss within-subject covariance in greater detail later in the presentation. Now that we have looked at some of the descriptive statistics we can reshape the data into long form using the reshape command.
The i option gives the variable that identifies the subject while the j option creates a new variable that indicates the time period. In anova parlance this design has both between-subject and within-subject effects, i.
In particular this design is sometimes referred to as a split-plot factorial analysis of variance. In Stata, with the data in long form, we need to specify the error terms for both the between-subject and within-subject effects. In general, the rule is that there is one single error term for all of the between-subject effects and a separate error term for each of the within-subject factors and for the interaction of within-subject factors.
Our model is relatively straight forward with only two error terms. The between-subject effect is treatment trt and its error term is subject nested in treatment id trt. The within-subject factor time time. Its error term is the residual error for the model. Repeated measures anova have an assumption that the within-subject covariance structure is compound symmetric, also known as, exchangeable. With compound symmetry the variances at each time are expected to be equal and all of the covariances are expected to be equal to one another.
Stata lets you take the lack of compound symmetry into account by including the repeated option in the anova command which computes p-values for conservative F-tests.
The treatment-by-time interaction is significant as are the two main-effects for treatment and time. These values are indicators of the p-value is even if the data do not meet the compound symmetry assumption.
We can view the pooled within-subject covariance matrix by listing the Srep matrix. Inspection of the pooled within-subject covariance matrix casts doubt on the validity of the compound symmetry assumption. Fortunately, the p-values for the conservative F-test still indicate significant effects for the trt time interaction and the time main effect.
Since the treatment-by-time interaction is significant we should try to explain the interaction. One way to do this is through the use of test of simple effects. We will begin by looking at the effect of time at each treatment level. The simple effect of time has three degrees of freedom for each level of the treatment for a total of six degrees of freedom. This test of simple effects will use the residual error for the model as its error term. We will use the contrast command to do the test of simple effects.
Since each of the tests of simple effects involves four time points will follow up with pairwise comparisons using the margins command with the pwcompare option. The tests of treatment at each tie point require the use of the pooled error. That is, pooling id trt and the residual error. This is easily accomplished by removing id trt from the anova command.
Note that the residual degrees of freedom is now Now we can run the simple effects of treatment at each time, again using the contrast command. Since there are two levels of treatment at each time point there are a total of four degrees of freedom. Since each test is one degree of freedom, we do not have to do any follow up tests. A graph of the interaction is always useful. We will use the margins command and marginsplot to produce the plot. An alternative to repeated measures anova is to run the analysis as a repeated measures mixed model.
We will do this using the xtmixed command. Note that we do not have to specify the error terms, we only need to specify the name of the variable on which the data are repeated, in this case id. Here is what the xtmixed command looks like.
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