Consider the Cholesterol Stacked.jmp sample data table. A study was performed to test two new cholesterol drugs against a control drug. Five patients with high cholesterol were randomly assigned to each of four treatments (the two experimental drugs, the control, and a placebo). Each patient’s total cholesterol was measured at six times during the study: the first day in April, May, and June in the morning and afternoon. You are interested in whether either of the new drugs is effective at lowering cholesterol and in whether time and treatment interact.
Suppose that your model has J observation times. Then the number of covariance parameters in the covariance matrices for the three structures available are as follows:
See Repeated Measures for more information.
Open Cholesterol.jmp to see a format that is typically used for recording repeated measures data. For Mixed Model analysis of this data, each cholesterol measurement needs to be in its own row, as in Cholesterol Stacked.jmp. To construct Cholesterol Stacked.jmp, the data in Cholesterol.jmp was stacked using Tables > Stack.
The Days column in the stacked table was constructed using a formula. The Days column gives the number days of into the study when the cholesterol measurement was taken. Its modeling type is continuous. This is necessary because the AR(1) covariance structure requires the repeated effect be continuous.
1.
Open Cholesterol Stacked.jmp.
2.
Select Analyze > Fit Model.
3.
Select Keep dialog open so that you can return to the launch window in the next example.
4.
Select Y and click Y.
5.
Select Mixed Model from the Personality list.
6.
Select Treatment, Month, and AM/PM, and then select Macros > Full Factorial.
Fit Model Launch Window Showing Completed Fixed Effects Tab
7.
Select the Repeated Structure tab.
8.
Select Unstructured from the Structure list.
9.
Select Time and click Repeated. The Repeated column defines the repeated measures within a subject.
10.
Select Patient and click Subject.
Fit Model Launch Window Showing Completed Repeated Structure Tab
11.
Click Run.
The Fit Mixed report is shown in Fit Mixed Report - Unstructured Covariance Structure. Because you want to compare your three models using AICc or BIC, you are interested in the Fit Statistics report. The AICc for the unstructured model is 703.84.
Fit Mixed Report - Unstructured Covariance Structure
2.
On the Repeated Structure tab, select Residual from the Structure list.
3.
If you are continuing from the previous example, remove Time and Patient. Otherwise, a warning appears: “Repeated columns and subject columns are ignored when the Residual covariance structure is selected.” You are given the option to click OK to continue the analysis.
4.
Select the Random Effects tab.
5.
Select Patient and click Add.
6.
Select Patient in the Random Effects area, select the Treatment column, and then click Nest.
Fit Model Launch Window Showing Completed Random Effects Tab
7.
Click Run.
The Fit Mixed report is shown in Fit Mixed Report - Residual Error Covariance Structure. The Fit Statistics report shows that the AICc for the Residual model is 832.55, as compared to 703.84 for the Unstructured model.
Fit Mixed Report - Residual Error Covariance Structure
2.
If you are continuing from the previous example, select Patient[Treatment] on the Random Effects tab and then click Remove.
3.
Select the Repeated Structure tab.
4.
Select AR(1) from the Structure list.
5.
Select Days and click Repeated.
6.
Select Patient and click Subject.
Fit Model Launch Window Showing Completed Repeated Effects Tab
7.
Click Run.
The Fit Mixed report is shown in Fit Mixed Report - AR(1) Covariance Structure. The Fit Statistics report shows that the AICc for the AR(1) model is 652.63, compared with 832.55 for the Residual model and 703.84 for the Unstructured model. Based on the AICc criterion, the AR(1) model is the best of the three models.
Fit Mixed Report - AR(1) Covariance Structure
To explore these significant effects, select Marginal Model Inference > Profiler from the Fit Mixed report’s red triangle menu. The Marginal Model Profiler report (Marginal Profiler Plot for Treatment A) shows that, for Treatments A and B, cholesterol levels (Y) decrease steadily over the three months. However, when you set Treatment to Control or Placebo, you see virtually no change over the three months (Marginal Profiler Plot for Control).
The profiler also shows the main effect of AM/PM. To obtain the profiler, select Marginal Model Inference > Profiler. By setting Treatment and Month to all 12 combinations of their levels, you see that the predicted cholesterol level is consistently higher in the afternoon than in the morning.
Marginal Profiler Plot for Treatment A
Marginal Profiler Plot for Control
2.
Select Treatment*Month from the Choose an Effect list.
3.
Select All Pairwise Comparisons - Tukey HSD from the Choose Initial Comparisons list.
4.
In the All Pairwise Differences report (Portion of All Pairwise Differences Report for Treatment*Month Interaction Effect), find the row that corresponds to Treatment A in Month May and -Treatment B in -Month May. The p-value for the test of equal means is <.0001. The confidence interval for the difference is -39.62 to -11.38. You conclude that Treatment A reduces cholesterol levels by at least 11 points more than Treatment B in the first month of treatment.
Portion of All Pairwise Differences Report for Treatment*Month Interaction Effect
1.
Select Multiple Comparisons from the Fit Mixed red triangle menu.
2.
Select AM/PM from the Choose an Effect list.
3.
Select All Pairwise Comparisons - Tukey HSD from the Choose Initial Comparisons list.
4.
5.
Select Equivalence Tests from the Tukey HSD All Pairwise Comparisons red triangle menu.
7.
Click OK.
Equivalent TOST Test and Equivalence Test Scatterplot for AM/PM Effect
1.
After following steps in Covariance Structure: AR(1) Example, return to the Fit Model launch window.
3.
Select Treatment and Days then select Macros > Full Factorial.
Fit Model Launch Window Showing Fixed Effects Tab
4.
Click Run.
The Fit Mixed report is shown in Fit Mixed Report - AR(1) Covariance Structure with Continuous Fixed Effect. You see that the interaction of Treatment and Days is highly significant indicating different regressions for the drugs.
Fit Mixed Report - AR(1) Covariance Structure with Continuous Fixed Effect
The data in the Split Plot.jmp sample data table come from a study of the effects of tenderizer and length of cooking time on meat. Six beef carcasses were randomly selected from carcasses at a meat packaging plant. From the right rib-eye muscle of each carcass, three rolled roasts were prepared under uniform conditions. Each of these three roasts was assigned a tenderizing treatment at random. After treatment, a coring device was used to mark four cores of meat near the center of each.
1.
Open the Split Plot.jmp sample data table.
2.
Select Analyze > Fit Model.
3.
Select Y and click Y.
4.
Select Mixed Model from the Personality list.
5.
Select Tenderizer and Roasting Time, and then select Macros > Full Factorial.
Fit Model Launch Window Showing Completed Fixed Effects Tab
6.
Select the Random Effects tab.
7.
Select Carcass and click Add to create the random carcass effect.
8.
Select Carcass and Tenderizer and click Cross.
Fit Model Launch Window Showing Completed Random Effects Tab
9.
Click Run.
Fit Mixed Report
1.
Select Marginal Model Inference > Profiler from the Fit Mixed report’s red triangle menu.
Marginal Model Profiler with Roasting Time Set to 30 Minutes
In Multiple Comparisons, Partial View, notice that both the papain and vinegar tenderizers result in significantly lower tenderness scores than the control when roasting time is either 30 or 36 minutes. However, at 42 minutes, there are no significant differences. At 48 minutes, papain gives a value lower than the control, but vinegar does not. Papain gives lower tenderness scores than does vinegar at all times except 42 minutes.
3.
Select Multiple Comparisons from the Fit Mixed red triangle menu.
4.
Select Tenderizer*Roasting Time.
5.
Select All Pairwise Comparisons - Tukey HSD and then click OK.
Multiple Comparisons, Partial View shows a partial list of pairwise comparisons. Most of the differences between papain and vinegar that you observed in the profiler are statistically significant. Therefore, it appears that papain is the better tenderizer.
Multiple Comparisons, Partial View
Consider the Uniformity Trial.jmp sample data table. An agronomic uniformity trial was conducted on an 8x8 grid of plots. In a uniformity trial, a test crop is grown on a field with no experimental treatments applied. The response variable, often yield, is measured. The idea is to characterize variability in the field as background for planning a designed experiment to be conducted on that field. (See SAS for Mixed Models, 2nd Edition, 2006, pp. 447.)
1.
Open the Uniformity Trial.jmp sample data table.
2.
Select Analyze > Fit Model.
3.
Select Keep dialog open so that you can return to the launch window in the next example.
4.
Select Yield and click Y.
5.
Select Mixed Model from the Personality list.
6.
Select the Repeated Structure tab.
7.
Choose Spatial from the list next to Structure.
8.
Choose Spherical from the list next to Type.
9.
Select Row and Column and click Repeated.
Completed Fit Model Launch Window Showing Repeated Structure Tab
10.
Click Run.
The Fit Mixed report is shown in Fit Mixed Report - Spatial Spherical. The Actual by Predicted Plot shows that the predicted yield is a single value. This is because only spatial covariance was fit. The Fit Statistics report shows that -2 Log Likelihood is 227.68, and the AICc is 234.08.
Fit Mixed Report - Spatial Spherical
2.
Select Repeated Structure tab.
3.
Select Residual from the Structure list.
4.
Remove Row and Column from the Repeated effects list. Otherwise, a warning appears: “Repeated columns and subject columns are ignored when the Residual covariance structure is selected.”
5.
Click Run.
2.
Select the Repeated Structure tab.
3.
Select Row and Column and click Repeated.
4.
Select Spatial with Nugget from the Structure list.
5.
Select Spherical from the Type list.
6.
Click Run.
The Fit Mixed report is shown in Fit Mixed Report - Spatial Spherical with Nugget. Notice that the log likelihoods are essentially equal to those of the spherical with no nugget model, and the AICc is slightly higher (236.36 compared to 234.08). The Repeated Effects Covariance Parameter Estimates report shows that the Nugget covariance parameter has an estimate of zero. There does not appear to be any evidence for a nugget effect.
Fit Mixed Report - Spatial Spherical with Nugget
8.
Select Repeated Structure tab.
9.
To test anisotropicity, select Spatial Anisotropic from the Structure list.
10.
Select Spherical from the Type list.
11.
Click Run.
The Fit Mixed report is shown in Fit Mixed Report - Spatial Anisotropic Spherical . The fit statistics indicate not as good a fit as the isotropic (spatial structure) spherical model (AICc 262.62 compared to 234.08). The Repeated Effects Covariance Parameter Estimates report shows that the estimates for the Row (Spatial Spherical Row) and Column (Spatial Spherical Column) covariances are very close. There is no evidence to suggest that spatial correlations within rows and columns of the grid differ.
Fit Mixed Report - Spatial Anisotropic Spherical
Repeat step 1 through step 4 in Select the Type of Spatial Covariance. Change Spatial with Nugget to Spatial and then change Spherical to the other available types: Power, Exponential, Gaussian. The observed AICc values for these types and the other fits that you performed are summarized in Fit Statistics for Spatial Models Fit.
Run the Graph Builder script from the Uniformity Trial.jmp sample data table. Graph Builder Plot of Proposed Complete and Incomplete Block Designs shows the plot of the proposed complete and incomplete block designs for the field. The color indicates the quarter fields that would serve as complete blocks. The numbered points represent the subquarter fields that would serve as incomplete blocks.
Graph Builder Plot of Proposed Complete and Incomplete Block Designs
2.
Select Repeated Structure tab.
3.
Select Residual from the Structure list.
4.
Remove Row and Col from the effect. Otherwise, a pop-up dialog appear stating, “Repeated columns and subject columns are ignored when the Residual covariance structure is selected.”
5.
Select the Random Effects tab.
6.
Select Quarter and click Add.
7.
Click Run.
Both AICc’s and BICs for the competing models are shown in Fit Statistics for Spatial and Block Models. The spherical covariance structure results in the best model fit. This indicates that, for future studies using this field, a completely randomized design with spatially correlated errors is preferred.
Consider the Tiretread Stacked.jmp sample data table. The effect of three factors (Silica, Silane, and Sulfur) is studied for four responses (Abrasion, Modulus, Elong, and Hardness).
Open Tiretread.jmp to see a format that is typically used for recording repeated measures data. For Mixed Model analysis of this data, each repeated measurement needs to be in its own row, as in Tiretread Stacked.jmp. To construct Tiretread Stacked.jmp, the data in Tiretread.jmp was stacked using Tables > Stack.
1.
Open the Tiretread Stacked.jmp sample data table.
2.
Select Analyze > Fit Model.
3.
Select Y and click Y.
4.
Select Mixed Model from the Personality list.
5.
Select Characteristic and click Add.
6.
Select Silica, Silane, and Sulfur, and select Response Surface from the Macros list.
8.
Select Characteristic from the Select Columns list.
9.
Click Cross.
Fit Model Launch Window Showing Completed Fixed Effects Tab shows the completed fixed effects tab.
Fit Model Launch Window Showing Completed Fixed Effects Tab
10.
Select the Repeated Structure tab.
11.
Select Unstructured from the Structure list.
12.
Select Characteristic and click Repeated.
13.
Select Tire and click Subject.
Fit Model Launch Window Showing Repeated Structure Tab
14.
Click Run.
The Fit Mixed report is shown in Fit Mixed Report. The Repeated Effects Covariance Parameter Estimates report gives the estimated variances and covariances for the four responses. All of the confidence intervals for the covariance estimates include zero. This suggests that the analysis of the responses that treats them as independent is not necessarily incorrect. Although, there might not be enough data to detect nonzero covariances.
Fit Mixed Report
1.
Select Marginal Model Inference > Surface Profiler from the Fit Mixed red triangle menu.
Note that Surface Profiler Showing the Response Surface for MODULUS and Silica = 1.2 shows a cross-section of the response surface for the Silica value 1.2.
Surface Profiler Showing the Response Surface for MODULUS and Silica = 1.2
1.
Select Marginal Model Inference > Profiler from the Fit Mixed red triangle menu.
Profiler for the Correlated Tire Tread Analysis