This example uses data from Cole and Grizzle (1966) to illustrate a commonly occurring repeated measures ANOVA design. Sixteen dogs are randomly assigned to four groups. (One
animal is removed from the analysis due to a missing value for one dependent variable.) Dogs in each group receive either
morphine or trimethaphan (variable Drug
) and have either depleted or intact histamine levels (variable Depleted
) before receiving the drugs. The dependent variable is the blood concentration of histamine at 0, 1, 3, and 5 minutes after
injection of the drug. Logarithms are applied to these concentrations to minimize correlation between the mean and the variance
of the data.
The following SAS statements perform both univariate and multivariate repeated measures analyses and produce Output 45.7.1 through Output 45.7.7.
data dogs; input Drug $12. Depleted $ Histamine0 Histamine1 Histamine3 Histamine5; LogHistamine0=log(Histamine0); LogHistamine1=log(Histamine1); LogHistamine3=log(Histamine3); LogHistamine5=log(Histamine5); datalines; Morphine N .04 .20 .10 .08 Morphine N .02 .06 .02 .02 Morphine N .07 1.40 .48 .24 Morphine N .17 .57 .35 .24 Morphine Y .10 .09 .13 .14 Morphine Y .12 .11 .10 . Morphine Y .07 .07 .06 .07 Morphine Y .05 .07 .06 .07 Trimethaphan N .03 .62 .31 .22 Trimethaphan N .03 1.05 .73 .60 Trimethaphan N .07 .83 1.07 .80 Trimethaphan N .09 3.13 2.06 1.23 Trimethaphan Y .10 .09 .09 .08 Trimethaphan Y .08 .09 .09 .10 Trimethaphan Y .13 .10 .12 .12 Trimethaphan Y .06 .05 .05 .05 ;
proc glm; class Drug Depleted; model LogHistamine0--LogHistamine5 = Drug Depleted Drug*Depleted / nouni; repeated Time 4 (0 1 3 5) polynomial / summary printe; run;
The NOUNI option in the MODEL statement suppresses the individual ANOVA tables for the original dependent variables. These analyses are usually of no interest in a repeated measures analysis. The POLYNOMIAL option in the REPEATED statement indicates that the transformation used to implement the repeated measures analysis is an orthogonal polynomial transformation, and the SUMMARY option requests that the univariate analyses for the orthogonal polynomial contrast variables be displayed. The parenthetical numbers (0 1 3 5) determine the spacing of the orthogonal polynomials used in the analysis.
The "Repeated Measures Level Information" table gives information about the repeated measures effect; it is displayed in Output 45.7.2. In this example, the within-subject (within-dog) effect is Time
, which has the levels 0, 1, 3, and 5.
The multivariate analyses for within-subject effects and related interactions are displayed in Output 45.7.3. For the example, the first table displayed shows that the TIME effect is significant. In addition, the Time
*Drug
*Depleted
interaction is significant, as shown in the fourth table. This means that the effect of Time
on the blood concentration of histamine is different for the four Drug
*Depleted
combinations studied.
Output 45.7.3: Multivariate Tests of Within-Subject Effects
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no Time Effect H = Type III SSCP Matrix for Time E = Error SSCP Matrix S=1 M=0.5 N=3.5 |
|||||
---|---|---|---|---|---|
Statistic | Value | F Value | Num DF | Den DF | Pr > F |
Wilks' Lambda | 0.11097706 | 24.03 | 3 | 9 | 0.0001 |
Pillai's Trace | 0.88902294 | 24.03 | 3 | 9 | 0.0001 |
Hotelling-Lawley Trace | 8.01087137 | 24.03 | 3 | 9 | 0.0001 |
Roy's Greatest Root | 8.01087137 | 24.03 | 3 | 9 | 0.0001 |
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no Time*Drug Effect H = Type III SSCP Matrix for Time*Drug E = Error SSCP Matrix S=1 M=0.5 N=3.5 |
|||||
---|---|---|---|---|---|
Statistic | Value | F Value | Num DF | Den DF | Pr > F |
Wilks' Lambda | 0.34155984 | 5.78 | 3 | 9 | 0.0175 |
Pillai's Trace | 0.65844016 | 5.78 | 3 | 9 | 0.0175 |
Hotelling-Lawley Trace | 1.92774470 | 5.78 | 3 | 9 | 0.0175 |
Roy's Greatest Root | 1.92774470 | 5.78 | 3 | 9 | 0.0175 |
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no Time*Depleted Effect H = Type III SSCP Matrix for Time*Depleted E = Error SSCP Matrix S=1 M=0.5 N=3.5 |
|||||
---|---|---|---|---|---|
Statistic | Value | F Value | Num DF | Den DF | Pr > F |
Wilks' Lambda | 0.12339988 | 21.31 | 3 | 9 | 0.0002 |
Pillai's Trace | 0.87660012 | 21.31 | 3 | 9 | 0.0002 |
Hotelling-Lawley Trace | 7.10373567 | 21.31 | 3 | 9 | 0.0002 |
Roy's Greatest Root | 7.10373567 | 21.31 | 3 | 9 | 0.0002 |
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no Time*Drug*Depleted Effect H = Type III SSCP Matrix for Time*Drug*Depleted E = Error SSCP Matrix S=1 M=0.5 N=3.5 |
|||||
---|---|---|---|---|---|
Statistic | Value | F Value | Num DF | Den DF | Pr > F |
Wilks' Lambda | 0.19383010 | 12.48 | 3 | 9 | 0.0015 |
Pillai's Trace | 0.80616990 | 12.48 | 3 | 9 | 0.0015 |
Hotelling-Lawley Trace | 4.15915732 | 12.48 | 3 | 9 | 0.0015 |
Roy's Greatest Root | 4.15915732 | 12.48 | 3 | 9 | 0.0015 |
Output 45.7.4 displays tests of hypotheses for between-subject (between-dog) effects. This section tests the hypotheses that the different
Drug
s, Depleted
s, and their interactions have no effects on the dependent variables, while ignoring the within-dog effects. From this analysis,
there is a significant between-dog effect for Depleted
(p-value=0.0229). The interaction and the main effect for Drug
are not significant (p-values=0.1734 and 0.1281, respectively).
Output 45.7.4: Tests of Between-Subject Effects
Source | DF | Type III SS | Mean Square | F Value | Pr > F |
---|---|---|---|---|---|
Drug | 1 | 5.99336243 | 5.99336243 | 2.71 | 0.1281 |
Depleted | 1 | 15.44840703 | 15.44840703 | 6.98 | 0.0229 |
Drug*Depleted | 1 | 4.69087508 | 4.69087508 | 2.12 | 0.1734 |
Error | 11 | 24.34683348 | 2.21334850 |
Univariate analyses for within-subject (within-dog) effects and related interactions are displayed in Output 45.7.6. The results for this example are the same as for the multivariate analyses; this is not always the case. In addition, before the univariate analyses are used to make conclusions about the data, the result of the sphericity test (requested with the PRINTE option in the REPEATED statement and displayed in Output 45.7.5) should be examined. If the sphericity test is rejected, consider using the adjusted G-G or H-F-L probabilities. See the section Repeated Measures Analysis of Variance for more information.
Output 45.7.6: Univariate Tests of Within-Subject Effects
Source | DF | Type III SS | Mean Square | F Value | Pr > F | Adj Pr > F | |
---|---|---|---|---|---|---|---|
G - G | H-F-L | ||||||
Time | 3 | 12.05898677 | 4.01966226 | 53.44 | <.0001 | <.0001 | <.0001 |
Time*Drug | 3 | 1.84429514 | 0.61476505 | 8.17 | 0.0003 | 0.0039 | 0.0023 |
Time*Depleted | 3 | 12.08978557 | 4.02992852 | 53.57 | <.0001 | <.0001 | <.0001 |
Time*Drug*Depleted | 3 | 2.93077939 | 0.97692646 | 12.99 | <.0001 | 0.0005 | 0.0002 |
Error(Time) | 33 | 2.48238887 | 0.07522391 |
Output 45.7.7 is produced by the SUMMARY
option in the REPEATED
statement. If the POLYNOMIAL option is not used, a similar table is displayed using the default CONTRAST
transformation. The linear, quadratic, and cubic trends for Time
, labeled as ‘Time_1’, ‘Time_2’, and ‘Time_3’, are displayed, and in each case, the Source labeled ‘Mean’ gives a test for
the respective trend.
Output 45.7.7: Tests of Between-Subject Effects for Transformed Variables
Contrast Variable: Time_1 |
Source | DF | Type III SS | Mean Square | F Value | Pr > F |
---|---|---|---|---|---|
Mean | 1 | 2.00963483 | 2.00963483 | 34.99 | 0.0001 |
Drug | 1 | 1.18069076 | 1.18069076 | 20.56 | 0.0009 |
Depleted | 1 | 1.36172504 | 1.36172504 | 23.71 | 0.0005 |
Drug*Depleted | 1 | 2.04346848 | 2.04346848 | 35.58 | <.0001 |
Error | 11 | 0.63171161 | 0.05742833 |