SAS proc mixed is a very powerful procedure for a wide variety of statistical analyses, including repeated measures analysis of variance. We will illustrate how you can perform a repeated measures ANOVA using a standard type of analysis using proc glm and then show how you can perform the same analysis using proc mixed. We use an example of from Design and Analysis by G. Keppel. Pages 414-416. This example contains eight subjects (sub) with one between subjects IV with two levels (group) and one within subjects IV with four levels (indicated by position dv1–dv4). These data are read into a temporary SAS data file called wide below.
DATA wide; INPUT sub group dv1 dv2 dv3 dv4; CARDS; 1 1 3 4 7 3 2 1 6 8 12 9 3 1 7 13 11 11 4 1 0 3 6 6 5 2 5 6 11 7 6 2 10 12 18 15 7 2 10 15 15 14 8 2 5 7 11 9 ; RUN; PROC PRINT DATA=wide ; RUN;OBS SUB GROUP DV1 DV2 DV3 DV4 1 1 1 3 4 7 3 2 2 1 6 8 12 9 3 3 1 7 13 11 11 4 4 1 0 3 6 6 5 5 2 5 6 11 7 6 6 2 10 12 18 15 7 7 2 10 15 15 14 8 8 2 5 7 11 9
We start by showing how to perform a standard 2 by 4 (between / within) ANOVA using proc glm.
PROC GLM DATA=wide; CLASS group; MODEL dv1-dv4 = group / NOUNI ; REPEATED trial 4; RUN;
The results of this analysis are shown below.
General Linear Models Procedure
Class Level Information
Class Levels Values
GROUP 2 1 2
Number of observations in data set = 8
General Linear Models Procedure
Repeated Measures Analysis of Variance
Repeated Measures Level Information
Dependent Variable DV1 DV2 DV3 DV4
Level of TRIAL 1 2 3 4
Manova Test Criteria and Exact F Statistics for
the Hypothesis of no TRIAL Effect
H = Type III SS&CP Matrix for TRIAL E = Error SS&CP Matrix
S=1 M=0.5 N=1
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.00829577 159.3911 3 4 0.0001
Pillai's Trace 0.99170423 159.3911 3 4 0.0001
Hotelling-Lawley Trace 119.54335260 159.3911 3 4 0.0001
Roy's Greatest Root 119.54335260 159.3911 3 4 0.0001
Manova Test Criteria and Exact F Statistics for
the Hypothesis of no TRIAL*GROUP Effect
H = Type III SS&CP Matrix for TRIAL*GROUP E = Error SS&CP Matrix
S=1 M=0.5 N=1
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.60915493 0.8555 3 4 0.5324
Pillai's Trace 0.39084507 0.8555 3 4 0.5324
Hotelling-Lawley Trace 0.64161850 0.8555 3 4 0.5324
Roy's Greatest Root 0.64161850 0.8555 3 4 0.5324
General Linear Models Procedure
Repeated Measures Analysis of Variance
Tests of Hypotheses for Between Subjects Effects
Source DF Type III SS Mean Square F Value Pr > F
GROUP 1 116.28125000 116.28125000 2.51 0.1645
Error 6 278.43750000 46.40625000
General Linear Models Procedure
Repeated Measures Analysis of Variance
Univariate Tests of Hypotheses for Within Subject Effects
Source: TRIAL
Adj Pr > F
DF Type III SS Mean Square F Value Pr > F G - G H - F
3 129.59375000 43.19791667 22.34 0.0001 0.0001 0.0001
Source: TRIAL*GROUP
Adj Pr > F
DF Type III SS Mean Square F Value Pr > F G - G H - F
3 3.34375000 1.11458333 0.58 0.6380 0.5693 0.6380
Source: Error(TRIAL)
DF Type III SS Mean Square
18 34.81250000 1.93402778
Greenhouse-Geisser Epsilon = 0.6337
Huynh-Feldt Epsilon = 1.0742
Now, we will illustrate how you can perform this same analysis in proc mixed. First, we need to reshape the data so it is in the shape expected by proc mixed. proc glm expects the data to be in a wide format, where each observation corresponds to a subject. By contrast, proc mixed expects the data to be in a long format where each observation corresponds to a trial. In this case, proc mixed expects that there would be four observations per subject and that each observation would correspond to the measurements on the four different trials. Below we show how you can reshape the data for analysis in proc mixed.
DATA long ; SET Wide; dv = dv1; trial = 1; OUTPUT; dv = dv2; trial = 2; OUTPUT; dv = dv3; trial = 3; OUTPUT; dv = dv4; trial = 4; OUTPUT; DROP dv1 - dv4 ; RUN; PROC PRINT DATA=long ; RUN;
You can compare the proc print for wide with the proc print for long to verify that the data were properly reshaped.
OBS SUB GROUP DV TRIAL 1 1 1 3 1 2 1 1 4 2 3 1 1 7 3 4 1 1 3 4 5 2 1 6 1 6 2 1 8 2 7 2 1 12 3 8 2 1 9 4 9 3 1 7 1 10 3 1 13 2 11 3 1 11 3 12 3 1 11 4 13 4 1 0 1 14 4 1 3 2 15 4 1 6 3 16 4 1 6 4 17 5 2 5 1 18 5 2 6 2 19 5 2 11 3 20 5 2 7 4 21 6 2 10 1 22 6 2 12 2 23 6 2 18 3 24 6 2 15 4 25 7 2 10 1 26 7 2 15 2 27 7 2 15 3 28 7 2 14 4 29 8 2 5 1 30 8 2 7 2 31 8 2 11 3 32 8 2 9 4
Now that the data are in the proper shape, we can analyze it with proc mixed.
The class and model statements are used much the same as with proc glm. However, the repeated statement is different. The repeated statement is used to indicate the within subjects (repeated) variables, but note that trial is on the class statement, unlike proc glm. This is because the data are in long format and that there indeed is a separate variable indicating the trials.
We also use the repeated statement to indicate which variable indicates the different subjects (via subject=sub) and we can specify the covariance structure among the repeated measures (in this case we choose compound symmetry via type=cs which is the same structure that proc glm uses. Unlike proc glm, proc mixed has a wide variety of covariance structures you can choose from so you can choose one that matches your data (see the proc mixed manual for more information on this).
PROC MIXED DATA=long; CLASS sub group trial; MODEL dv = group trial group*trial; REPEATED trial / SUBJECT=sub TYPE=CS; run;
As you see below, the results correspond to those produced by proc glm. Note that proc mixed does not produce Sums of Squares or Mean Squares. This is because proc mixed uses maximum likelihood estimation instead of a sums of squares style of computation.
The MIXED Procedure
Class Level Information
Class Levels Values
SUB 8 1 2 3 4 5 6 7 8
GROUP 2 1 2
TRIAL 4 1 2 3 4
REML Estimation Iteration History
Iteration Evaluations Objective Criterion
0 1 96.74510121
1 1 69.98784546 0.00000000
Convergence criteria met.
Covariance Parameter Estimates (REML)
Cov Parm Subject Estimate
CS SUB 11.11805556
Residual 1.93402778
Model Fitting Information for DV
Description Value
Observations 32.0000
Res Log Likelihood -57.0484
Akaike's Information Criterion -59.0484
Schwarz's Bayesian Criterion -60.2265
-2 Res Log Likelihood 114.0969
Null Model LRT Chi-Square 26.7573
Null Model LRT DF 1.0000
Null Model LRT P-Value 0.0000
Tests of Fixed Effects
Source NDF DDF Type III F Pr > F
GROUP 1 6 2.51 0.1645
TRIAL 3 18 22.34 0.0001
GROUP*TRIAL 3 18 0.58 0.6380
Proc mixed is much more powerful than proc glm. Because it is more powerful, it is more complex to use. This FAQ just scratches the surface in the use of proc mixed.