Loss of subjects in a repeated measures ANOVA due to
missing data can be a serious problem. If you use proc glm to perform you
analysis, it will omit observations listwise, meaning that if any of the
observations for a subject are missing, the entire subject will be omitted from the
analysis. Consider the data file below based on an example of from Design and
Analysis by G. Keppel. Pages 414-416. This example contains 8 subjects (sub) with
one between subjects IV with 2 levels (group) and 1 within subjects IV with 4 levels. We
have inserted 4 missing values to illustrate the impact of missing data in this kind of
design.
DATA wide; INPUT sub group dv1 dv2 dv3 dv4; CARDS; 1 1 3 4 7 3 2 1 6 . 12 9 3 1 7 13 11 11 4 1 0 3 . 6 5 2 5 6 11 7 6 2 10 12 18 . 7 2 10 15 15 14 8 2 5 . 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 . 12 9 3 3 1 7 13 11 11 4 4 1 0 3 . 6 5 5 2 5 6 11 7 6 6 2 10 12 18 . 7 7 2 10 15 15 14 8 8 2 5 . 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;
Note the number of observations available for analysis
is only four, and that four have been omitted due to missing data. 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
NOTE: Observations with missing values will not be included in this analysis.
Thus, only 4 observations can be used in this analysis.
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
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 36.00000000 36.00000000 0.46 0.5673
Error 2 156.25000000 78.12500000
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 47.25000000 15.75000000 5.32 0.0397 0.1430 0.0629
Source: TRIAL*GROUP
Adj Pr > F
DF Type III SS Mean Square F Value Pr > F G - G H - F
3 2.50000000 0.83333333 0.28 0.8371 0.6556 0.7898
Source: Error(TRIAL)
DF Type III SS Mean Square
6 17.75000000 2.95833333
Greenhouse-Geisser Epsilon = 0.3474
Huynh-Feldt Epsilon = 0.7547
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 . 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 . 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 . 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 . 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. Proc mixed does not delete missing data listwise.
It analyzes all of the data that are present. For the analysis to be valid, it
is assumed that the data are missing at random. Rarely, however, are data truly missing at random. To the
extent that there are systematic factors that led to the data being missing, the analysis
will not be valid. In using this kind of analysis, we recommend that you assess and
present information regarding the reasons for missing data and an assessment of the extent
to which it was non-random.
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, proc mixed analyzed
all eight of the subjects and had far less missing data than the analysis with
proc glm.
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 81.93159646
1 3 63.43970119 0.00138808
2 1 63.39025490 0.00006552
3 1 63.38810898 0.00000018
4 1 63.38810333 0.00000000
Convergence criteria met.
Covariance Parameter Estimates (REML)
Cov Parm Subject Estimate
CS SUB 10.83244625
Residual 2.29522110
Model Fitting Information for DV
Description Value
Observations 28.0000
Res Log Likelihood -50.0728
Akaike's Information Criterion -52.0728
Schwarz's Bayesian Criterion -53.0686
-2 Res Log Likelihood 100.1456
Null Model LRT Chi-Square 18.5435
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 2.37 0.1748
TRIAL 3 14 17.04 0.0001
GROUP*TRIAL 3 14 0.40 0.7556
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.