The data set analyzed in this task contains data from Littell, Freund, and Spector (1991). Subjects in the study participated in one of three different weightlifting programs, and their strength was measured once every other day for two weeks after they began the program. The first program increased the number of repetitions as the subject became stronger (RI), the second program increased the amount of weight as subjects became stronger (WI), and the subjects in the third program did not participate in weightlifting (CONT). The objective of this analysis is to investigate the effect each weightlifting program has on increasing strength over time. This section also illustrates how to prepare data in univariate form for this task.
Figure 16.2: Weightsmult Data
In order for you to perform the repeated measures analysis using the Analyst Application, your data must be in univariate form, which means that each response measurement is contained in a separate row. If your data are not in univariate form, you must create a new data table with this structure. This can be accomplished via the Stack Columns task in the Data menu.
The Stack Columns task creates a new table by stacking specified columns into a single column. The values in the other columns are preserved in the new table, and a source column in the new data set contains the names of the columns in the original data set that contained the stacked values.
You want to put the values for columns corresponding to the strength measurement variables s1 through s7 in individual rows, so you want to stack columns s1-s7. To stack the columns, follow these steps:
Figure 16.3: Stack Columns Dialog
The new data set is presented in the project tree under the Stack Columns folder. The Weightsmult with Stacked Columns folder contains the new data set with the Strength stacked column, and the Code node contains the SAS programming statements that generated the data set.
If a view of the Weightsmult with Stacked Columns data is displayed, close it. Then right-click on the data set node labeled Weightsmult with Stacked Columns, as displayed in Figure 16.4, and select Open to bring the new data set into the data table.
Figure 16.4: Stack Columns: Project Tree
The stacked columns data set contains two new variables. The Strength variable contains the strength measurements, and the _Source_ variable denotes the measurement times with seven distinct character values: s1, s2, s3, s4, s5, s6, and s7. However, in this analysis, time needs to be numeric. You can create a numeric variable called Time by using the Recode Values facility.
To create the Time variable, follow these steps:
Figure 16.5: Recode Values Information Dialog
Figure 16.6: Recode Values Dialog
The data set now includes a variable Time that contains numeric values for the time of strength measurement. Because the time values are contained in a new variable, you can delete the original variable from the data set by right-clicking on the _Source_ column in the data table and selecting Delete. Once you have deleted the column, the data set should contain four variables, Subject, Program, Strength, and Time, as displayed in Figure 16.7.
Figure 16.7: Weightsuni Data
Before proceeding with the analysis, you can save the new data set as Weightsuni by following these steps:
Note that the Weightsuni data are in univariate form and should be the same as the Weights data available in the Analyst Sample Library.
Figure 16.8 displays the dialog with Strength specified as the dependent variable and Subject, Program, and Time specified as classification variables.
Figure 16.8: Repeated Measures Dialog
Each experimental unit, a subject, needs to be uniquely identified in the Weightsuni data set. The value of the Subject variable for the first subject in each separate Program is 1, the value of the Subject variable for the second subject in each Program is 2, and so on. Because subjects participating in different programs have the same value from the Subject variable, you need to nest Subject within Program to uniquely define each subject.
To define the repeated measures model, follow these steps:
Figure 16.9: Repeated Measures: Model Dialog,Subjects Tab
Figure 16.10: Repeated Measures: Model Dialog,Model Tab
This identifies the repeated measurement effect.
Figure 16.11: Repeated Measures: Model Dialog,Repeated Tab
When analyzing repeated measures data, you must properly model the covariance structure within subjects to ensure that inferences about the mean are valid. Using the Repeated Measures task, you can select from a wide range of covariance types, where the most common types are compound symmetric, first-order autoregressive, and unstructured. To select the covariance structure for the analysis, follow these steps:
Figure 16.12: Repeated Measures: Model Dialog,Covariance Structure Tab
Close the Model dialog by clicking OK. When you have completed your selections, click OK in the main dialog to produce your analysis.
Figure 16.13: Repeated Measures: Project Tree
You can double-click on the Analysis for Compound Symmetric Covariances node in the project tree to view the results in a separate window.
Figure 16.14: Repeated Measures: Model Information
Figure 16.14 displays model information including the levels of each classification variable in the analysis. The Program variable has three levels while the Time variable has 7 levels. The "Dimensions" table displays information about the model and matrices used in the calculations. There are two covariance parameters estimated using the compound symmetry model: the variance of residual error and the covariance between two observations on the same subject. The 32 columns of the X matrix correspond to three columns for the Program variable, seven columns for the Time variable, 21 columns for the Program*Time interaction, and a single column for the intercept. You should always review this information to ensure that the model has been specified correctly.
Figure 16.15: Repeated Measures: Fitting Information
Figure 16.15 displays fitting information, including the iteration history, covariance parameter estimates, and likelihood statistics. The "Iteration History" table shows the sequence of evaluations to obtain the restricted maximum likelihood estimates of the variance components.
The "Covariance Parameter Estimates" table displays estimates of the variance component parameters. The covariance between two measurements on the same subject is 9.6. Based on an estimated residual variance parameter of 1.2, the overall variance of a measurement is estimated to be 9.6 + 1.2 = 10.8.
Figure 16.16: Repeated Measures: Tests for Fixed Effects
The "Type 3 Tests of Fixed Effects" table in Figure 16.16 contains hypothesis tests for the significance of each of the fixed effects, that is, those effects you specify on the Model tab. Based on a p-value of 0.0005 for the Program*Time interaction, there is significant evidence of a strong interaction between the weightlifting program and time of measurement at the level of significance.
The first-order autoregressive covariance structure has the property that observations on the same subject that are closer in time are more highly correlated than measurements at times that are farther apart. The first-order autoregressive covariance can be represented by , where w indicates the time between two measurements, stands for the correlation between adjacent observations on the same subject, and stands for the variance of an observation. For the first-order autoregressive covariance structure, the correlation between two measurements decreases exponentially as the length of time between the measurements increases.
To fit an additional repeated measures model with a first-order autoregressive covariance structure, follow these steps:
Note that the selections for the previous analysis are still specified.
Figure 16.17: Repeated Measures: Model Dialog,Covariance Structure tab
Although this analysis models only two different covariance structures, the Analyst Application provides a wide range of structures to choose from, including unstructured, Huynh-Feldt, Toeplitz, and variance components. To select other structures, click on the down arrow next to an Other check box and choose from the resulting drop-down list.
Double-click on the Analysis for First Order Autoregressive Covariances node in the project tree to view the results in a separate window.
Figure 16.18: Repeated Measures: Test for Fixed Effects for Autoregressive Covariance
Figure 16.18 displays the Type 3 tests for fixed effects based on the first-order autoregressive covariance model. Note that with a p-value greater than 0.30, the Program*Time interaction is not significant at the level of significance. The p-value is different from the p-value of the same test based on the compound symmetry covariance structure, and the two models lead to different conclusions. You can assess the model fit based on different covariance structures by comparing criteria that is provided in the Information Criteria Summary window in Figure 16.19.
Figure 16.19: Repeated Measures: Information Criteria Summary
The process of selecting the most appropriate covariance structure can be aided by comparing the Akaike's Information Criterion (AIC) and Schwarz's Bayesian Criterion (SBC) for each model. When you compare models with the same fixed effects but different variance structures, the models with the smallest AIC and SBC are deemed the best. In this example, the autoregressive model has lower values for both AIC and SBC, showing considerable improvement over the model with a compound symmetry structure. Based on the information criteria as well as the intuitively sensible property of the correlations being larger for nearby times than for far-apart times, the first-order autoregressive model is the more suitable fit for these data.