X'(inverse of V)Xb=X'(inverse of V)y,
where V is replaced with its estimator, are solved to obtain the estimates of fixed effects parameters b.
It is assumed that the random effects u and the error vector e are normally distributed, uncorrelated and have expectations 0. Under the assumption that u and e are not correlated, V, the variance-covariance matrix of y, is equal to ZGZ’ + R, where G and R are the variance matrices of u and e, respectively.
Estimators of V, the variance-covariance matrix of y, can also be obtained in PROC MIXED by the MIVQUE0 method. For a short description of the method see reference (3), p.506. This method has two advantages over ML and REML; it does not require normality assumption (for computing the estimators) as do ML and REML and does not involve iterations. However simulation studies by Swallow and Monahan (1984) present evidence favoring ML and REML over MIVQUE0. PROC MIXED uses MIVQUE0 as starting values for the ML and RELM procedures.
For balanced data the REML method of PROC MIXED provides estimators and hypotheses test results that are identical to ANOVA (OLS method of GLM), provided that the ANOVA estimators of variance components are not negative. The estimators, as in GLM, are unbiased and have minimum variance properties. The ML estimators are biased in that case. In general case of unbalanced data neither the ML nor the REML estimators are unbiased and they do not have to be equal to those obtained from PROC GLM. There are many models involving forms of variance-covariance structure of observations that can not be analyzed using PROC GLM with TEST or PROC GLM with the REPEATED options. PROC MIXED can handle such cases. It also has to be mentioned that PROC GLM was design for analysis of fixed effects models and all computations are done under the assumption that there is only one variance component in the model, the error term. The RANDOM statement with the TEST option can be used to get the right tests in the case random effects are present in the model, but still some printed results, variances and standard errors, will be incorrect.
The PROC MIXED syntax is similar to the syntax
of PROC GLM. There are, however, a few important differences. The random
effects and repeated statements are used differently, random effects are
not listed in the model statement, GLM has MEANS and LSMEANS statements,
whereas MIXED has only the LSMEANS statement, GLM offers Type I, II, III
and IV tests for fixed effects, while MIXED offers TYPE I and TYPE III.
The following is a general form of PROC MIXED statement:
PROC MIXED options;
CLASS variable-list;
MODEL dependent=fixed effects/ options;
RANDOM random effects / options;
REPEATED repeated effects / options;
CONTRAST 'label' fixed-effect values | random-effect
values/ options;
ESTIMATE 'label' fixed-effect values | random-effect
values/ options;
LSMEANS fixed-effects / options;
MAKE 'table' OUT= SAS-data-set < options
>;
RUN;
The CONTRAST, ESTIMATE, LSMEANS, MAKE and RANDOM statements can appear multiple times, all other statements can appear only once.
The PROC MIXED and MODEL statements are required. The MODEL statement must appear after the CLASS statement if CLASS statement is used. The CONTRAST, ESTIMATE, LSMEANS, RANDOM and REPEATED statement must follow the MODEL statement. CONTRAST and ESTIMATE statements must follow RANDOM statement if the RANDOM is used.
A detailed description of all functions and options of each PROC MIXED
statement is given in SAS/STAT Software Changes and Enhancements through
Release 6.11 and SAS/STAT Software Changes and Enhancements for
Release 6.12, SAS Institute Inc. (1996). The following is a short summary
of selected, most often used, MIXED procedure statements.
Selected options:
DATA= SAS data set
Names SAS data set to be used by PROC MIXED. The default is the most
recently created data set.
METHOD=REML
METHOD=ML
METHOD=MIVQUE0
Specifies the estimation method. See Section 1 for a brief description of the methods and references. REML is the default method.
COVTEST
Prints asymptotic standard errors and Wald Z-test for variance-covariance
structure parameter estimates. For example, if a random effect A is included
in the model, then the estimator of the variance of A will be printed together
with the Wald test of the hypothesis that the variance of A is 0.
The COVTEST option is specified after Proc mixed and before semicolon;.
For example,
Proc mixed data=mydata method=reml covtest;
Lists classification variables (categorical independent variables
in the model). For example:
proc mixed data=mydata covtest;
Class group gender agecat;
The model statement names a single dependent variable and the fixed
effects, that is independent variables that are not random. An intercept
is included in the model by default. The NOINT option can be used to remove
the intercept.
NOTE: Even though PROC MIXED allows only for one dependent variable in the model statement, it is possible to use it to model, for example, multivariate repeated measures. In such case, the data set has to be properly prepared and should contain a variable indicating the measurement type. The correlation between observations on the same unit has to be modeled properly with the REPEATED statement. For example, suppose your observed data consist of heights and weights of children measured over several successive years. Your input data set should then contain variables similar to the following:
Y, all of the heights and weights, with a separate observation (line
in the data file) for each
VAR, indicating whether the measurement is a height or a weight
YEAR, indicating the year of measurement
CHILD, indicating the child on which the measurement was taken.
Selected Options of the model statement:
CHISQ, request c2 – tests (Wald tests) be performed for all fixed effects in addition to the F-tests.
DDFM=RESIDUAL
DDFM=CONTAIN
DDFM=BETWITHN
DDFM=SATTERTH,
The DDFM= options specifies the method for computing the denominator degrees of freedom for the tests of fixed effects. DDFM=SATTERTH will result in the Satterthwaite approximation for the denominator degrees of freedom. For balanced designs with random effects it will produce the same test results as RANDOM …/ TEST option in PROC GLM (if the default METHOD=REML is used in proc mixed).
P, requests that the predicted values be printed.
The RANDOM statement defines the random effects in the model. It
can be used to specify traditional variance components (independent random
effects with different variances) or to list correlated random effects
and specify a correlation structure for them with the TYPE=covariance-structure
option. A variety of structures are available (see references
5 and 6), most often used are either TYPE=VC, a variance components correlation
structure or TYPE=UN, an unstructured, that is, arbitrary covariance matrix.
TYPE=VC is the default structure. In the following example, the effect
of subject is random.
Proc mixed data=one method=reml covtest;
Class gender treat subject;
Model y=gender treat gender*treat /ddfm=satterth;
Random subject(gender);
Run;
In the next example there are two random effects specified (besides
the error term) and it is assumed that they are correlated.
Intercept and the slope coefficient in the regression equation
have fixed and random parts which are assumed to be correlated. The model
is:
yij = a0 +aj + b0*time +
bj*time + eij, where yij is observation i for person j.
The random effects, aj, bj and eij, are asumed to have normal distributions
with mean zero and different variances and it is also assumed that aj and
bj are correlated.
Proc mixed data=one method=reml covtest;
Class person;
Model y=time /solution;
Random intercept time /type=un subject=person;
Run;
The repeated statement is used in PROC MIXED to specify the covariance
structure of the error term. The repeated effect has to be categorical
and has to appear in the class statement and the data has to be sorted
accordingly. For example, suppose that for each subject a measurement was
taken at five equally spaced time points. The time is the repeated effect
and the data has to be sorted by subject and time within each subject.
If time is also used as a continuous independent variable in the model
then a new variable, say t, identical to time has to be defined and t should
be used in the class and repeated statements. For example:
Data one;
Set one;
T=time;
Run;
Proc sort data=one;
By group id t;
Run;
Proc mixed data=one covtest;
Class t group id;
Model y=group time group*time;
Repeated t /type=ar(1) subject=id;
Run;
The option TYPE in the REPEATED statement specifies the type of the error correlation structure. The one specified in the above example is the first-order autoregressive correlation. The subject option is needed to identify observations that are correlated. Observations within the same subject are correlated with the type of correlation specified in TYPE, observations from different subjects are independent.
The TYPE option allows for many types of correlation structures. Most
commonly used are autocorrelation, compound symmetry, Huynh-Feldt, Toeplitz,
variance components, unstructured and spatial. For the complete list and
examples, see references (7) and (8).
The CONTRAST statement is used when there is need for custom hypothesis
tests, the ESTIMATE statement, when there is need for custom estimates.
Although they were extended in PROC MIXED to include random effects, their
use is very similar to the CONTRAST and ESTIMATE statement in PROC GLM.
LABEL is required for every contrast or estimate statement. It identifies the contrast or estimated parameter on the output. It can not be longer than 20 characters.
FIXED-EFFECT is the name of an effect appearing in the MODEL statement.
RANDOM-EFFECT is the name of an effect appearing in the RANDOM statement.
VALUES are the coefficients of the contrast to be tested or the parameter to be estimated.
For example, suppose that we want to test if there is a significant effect of treat in group 2, where treat has three levels and group four levels. We also want to estimate the mean for treat 1 in group 2, the mean for treat 2 in group 2 and the difference between these two means. We will need the following CONTRAST and ESTIMATE statements to obtain these results.
Proc mixed data=one method=reml covtest;
Class group treat subject;
Model y=group treat group*treat /ddfm=satterth;
Random subject(group);
Contrast ‘treat in group 2’
Treat 1 –1 0 group*treat 0 0 0 1 –1 0 0 0 0 0 0 0,
Treat 0 1 –1 group*treat 0 0 0 0 1 –1 0 0 0 0 0 0;
Estimate ‘treat1 group2 mean’ intercept 1 group 0 1 0 0 treat 1
0 0
group*treat 0 0 0 1 0 0 0 0 0 0 0 0;
Estimate ‘treat2 group2 mean’ intercept 1 group 0 1 0 0 treat 0
1 0
Group*treat 0 0 0 0 1 0 0 0 0 0 0 0;
Estimate ‘mean diff t1g2-t2g2’ Treat 1 –1 0 group*treat 0 0 0 1
–1 0 0 0 0 0 0 0;
Run;
LSMEANS computes the least squares means of fixed effects. The ADJUST
option requests a multiple comparison adjustment to the p-values for pair-wise
comparisons of means. The following adjustments are available: BON (Bonferroni),
DUNNET, SCHEFFE, SIDAK, SIMULATE, SMM|GT2 and TUKEY. The ADJUST option
results in all possible pair-wise comparisons. If comparisons with a control
level are only needed then in addition to ADJUST option, PDIFF=control
should be used. The SLICE option allows to test the significance of one
effect at each level of another effect.
For example, suppose that we want to compute the least squares means for group*treat and do pair-wise comparisons with the control being group 1 and treat 1. We also want to test for the significance of the treat effect within each group level using the SLICE option..
Proc mixed data=one method=reml covtest;
Class group treat subject;
Model y=group treat group*treat /ddfm=satterth;
Random subject(group);
lsmeans group*treat /adjust=bon pdiff=control('1' '1') slice=group;
Run;
The MAKE statement converts any table produced by PROC MIXED into
a sas data set. NOPRINT option can be used to prevent printing the requested
table. Only requested or default output can be converted into a sas data
set. Hence, in particular, the P option has to be used in the model statement
to produce a data set with predicted values, and the LSMEANS statement
has to be included to output least squares means. For example,
Proc mixed data=one method=reml covtest;
Class group treat subject;
Model y=group treat group*treat /ddfm=satterth p;
Random subject(group);
lsmeans group*treat /adjust=bon pdiff=control('1' '1') slice=group;
make ‘LSMeans’ out=gtmeans;
make ‘predicted’ out=pred noprint;
Run;
Proc print data=gtmeans;
Proc print data=pred;
Run;
Statistics Books:
1. Searle, Shayle R. (1987). Linear Models For Unbalanced Data, John Wiley & Sons.
2. Searle, Shayle R. (1971). Linear Models, John Wiley & Sons.
3. Searle, S.R., Casella, G., and McCulloch, C.E. (1992), Variance Components. John Wiley&Sons.
4. Verbeke, G., Molenberghs, G. (Editors) (1997), Linear Mixed Models in Practice. A SAS-Oriented Approach. Springer-Verlag
SAS Institute Books:
5. Littell, Ramon C., Milliken, George A., Stroup, Walter W., Wolfinger, Russell D. (1996). SAS System For Mixed Models, SAS Institute Inc.
6. SAS Institute Course Notes (1996). Advanced General Linear Models with an Emphasis on Mixed Models, SAS Institute Inc.
7. SAS/STAT Software Changes and Enhancements through Release 6.11, SAS Institute Inc. 1996.
8. SAS/STAT Software Changes and Enhancements for Release 6.12, SAS
Institute Inc. 1996.
In this example, 36 subjects are randomly assigned
to 12 group – treatment combinations, 3 to each combination. There are
three treatments and four groups. In the following program, factor treat
with 3 levels is the effect of the treatment and factor group with 4 levels
is the effect of the group.
As you can see below, the results from both procedures are identical.
options ls=76;
data one;
input y group treat subject;
cards;
22 1 1 1
23 1 1 2
25 1 1 3
17 1 2 4
18 1 2 5
23 1 2 6
12 1 3 7
16 1 3 8
14 1 3 9
8 2 1 10
9 2 1 11
10 2 1 12
16 2 2 13
17 2 2 14
20 2 2 15
29 2 3 16
30 2 3 17
36 2 3 18
3 3 1 19
7 3 1 20
5 3 1 21
1 3 2 22
2 3 2 23
1 3 2 24
4 3 3 25
7 3 3 26
8 3 3 27
11 4 1 28
15 4 1 29
8 4 1 30
34 4 2 31
37 4 2 32
33 4 2 33
27 4 3 34
28 4 3 35
24 4 3 36
;
run;
Proc mixed data=one method=reml;
Class group treat;
Model y=group treat group*treat;
lsmeans group*treat /adjust=bon pdiff=control('1' '1')
slice=group;
Contrast 'treat in group 2'
Treat 1 -1 0 group*treat 0 0 0 1 -1 0 0 0 0 0 0 0,
Treat 0 1 -1 group*treat 0 0 0 0 1 -1 0 0 0 0 0 0;
Estimate 'treat1 group2 mean' intercept 1 group 0 1
0 0 treat 1 0 0
group*treat 0 0 0 1 0 0 0 0 0 0 0 0;
Estimate 'treat2 group2 mean' intercept 1 group 0 1
0 0 treat 0 1 0
Group*treat 0 0 0 0 1 0 0 0 0 0 0 0;
Estimate 'mean diff t1g2-t2g2' Treat 1 -1 0 group*treat
0 0 0 1 -1 0 0 0 0 0 0 0;
Run;
proc GLM data=one;
class group treat;
Model y=group treat group*treat;
lsmeans group*treat /adjust=bon pdiff=control('1' '1')
slice=group;
Contrast 'treat in group 2'
Treat 1 -1 0 group*treat 0 0 0 1 -1 0 0 0 0 0 0 0,
Treat 0 1 -1 group*treat 0 0 0 0 1 -1 0 0 0 0 0 0;
Estimate 'treat1 group2 mean' intercept 1 group 0 1
0 0 treat 1 0 0
group*treat 0 0 0 1 0 0 0 0 0 0 0 0;
Estimate 'treat2 group2 mean' intercept 1 group 0 1
0 0 treat 0 1 0
Group*treat 0 0 0 0 1 0 0 0 0 0 0 0;
Estimate 'mean diff t1g2-t2g2' Treat 1 -1 0 group*treat
0 0 0 1 -1 0 0 0 0 0 0 0;
Run;
GROUP 4 1 2
3 4
TREAT 3 1 2
3
Tests of Fixed Effects
Source NDF DDF Type III F Pr > F
GROUP
3 24
121.60
0.0001
TREAT
2 24
34.11
0.0001
GROUP*TREAT 6
24 43.04
0.0001
ESTIMATE Statement Results
Parameter Estimate Std Error DF t Pr > |t|
treat1 group2 mean 9.00000000
1.35400640 24 6.65
0.0001
treat2 group2 mean 17.66666667
1.35400640 24 13.05
0.0001
mean diff t1g2-t2g2 -8.66666667
1.91485422 24 -4.53
0.0001
CONTRAST Statement Results
Source NDF DDF F Pr > F
treat in group 2 2
24 71.35
0.0001
Least Squares Means
Effect GROUP TREAT LSMEAN Std Error
GROUP*TREAT 1
1 23.33333333 1.35400640
GROUP*TREAT 1
2 19.33333333 1.35400640
GROUP*TREAT 1
3 14.00000000 1.35400640
GROUP*TREAT 2
1 9.00000000
1.35400640
GROUP*TREAT 2
2 17.66666667 1.35400640
GROUP*TREAT 2
3 31.66666667 1.35400640
GROUP*TREAT 3
1 5.00000000
1.35400640
GROUP*TREAT 3
2 1.33333333
1.35400640
GROUP*TREAT 3
3 6.33333333
1.35400640
GROUP*TREAT 4
1 11.33333333 1.35400640
GROUP*TREAT 4
2 34.66666667 1.35400640
GROUP*TREAT 4
3 26.33333333 1.35400640
Differences of Least Squares Means
Effect GROUP TREAT GROUP _TREAT Difference Std Error DF
GROUP*TREAT 1
2
1
1 -4.00000000
1.91485422 24
GROUP*TREAT 1
3
1
1 -9.33333333
1.91485422 24
GROUP*TREAT 2
1
1
1 -14.33333333 1.91485422
24
GROUP*TREAT 2
2
1
1 -5.66666667
1.91485422 24
GROUP*TREAT 2
3
1
1 8.33333333
1.91485422 24
GROUP*TREAT 3
1
1
1 -18.33333333 1.91485422
24
GROUP*TREAT 3
2
1
1 -22.00000000 1.91485422
24
GROUP*TREAT 3
3
1
1 -17.00000000 1.91485422
24
GROUP*TREAT 4
1
1
1 -12.00000000 1.91485422
24
GROUP*TREAT 4
2
1
1 11.33333333 1.91485422
24
GROUP*TREAT 4
3
1
1 3.00000000
1.91485422 24
Differences of Least Squares Means
t Pr > |t| Adjustment Adj P
-2.09 0.0475 Bonferroni
0.5224
-4.87 0.0001 Bonferroni
0.0006
-7.49 0.0001 Bonferroni
0.0000
-2.96 0.0068 Bonferroni
0.0752
4.35 0.0002 Bonferroni
0.0024
-9.57 0.0001 Bonferroni
0.0000
-11.49 0.0001 Bonferroni
0.0000
-8.88 0.0001 Bonferroni
0.0000
-6.27 0.0001 Bonferroni
0.0000
5.92 0.0001 Bonferroni
0.0000
1.57 0.1303 Bonferroni
1.0000
Tests of Effect Slices
Effect GROUP NDF DDF F Pr > F
GROUP*TREAT 1 2
24 11.96 0.0002
GROUP*TREAT 2 2
24 71.35 0.0001
GROUP*TREAT 3 2
24 3.66
0.0411
GROUP*TREAT 4 2
24 76.26 0.0001
General Linear Models Procedure
Class Level Information
GROUP 4 1 2 3 4
TREAT 3
1 2 3
General Linear Models Procedure
Dependent Variable: Y
Sum of Mean
Source
DF Squares
Square
F Value Pr > F
Model 11 3802.00000 345.63636 62.84 0.0001
Error 24 132.00000 5.50000
Corrected Total 35 3934.00000
R-Square C.V. Root MSE Y Mean
0.966446 14.07125
2.34521
16.6667
Source DF Type III SS Mean Square F Value Pr > F
GROUP
3 2006.44444 668.81481
121.60 0.0001
TREAT
2 375.16667
187.58333 34.11 0.0001
GROUP*TREAT 6 1420.38889
236.73148 43.04 0.0001
General Linear Models Procedure
Least Squares Means
Adjustment for multiple comparisons: Bonferroni
GROUP TREAT
Y Pr
> |T| H0:
LSMEAN LSMEAN=CONTROL
1
1 23.3333333
1
2 19.3333333
0.5224
1
3 14.0000000
0.0006
2
1 9.0000000
0.0001
2
2 17.6666667
0.0752
2
3 31.6666667
0.0024
3
1 5.0000000
0.0001
3
2 1.3333333
0.0001
3
3 6.3333333
0.0001
4
1 11.3333333
0.0001
4
2 34.6666667
0.0001
4
3 26.3333333
1.0000
GROUP*TREAT Effect Sliced by GROUP for Y
Sum of
Mean
GROUP DF
Squares
Square
F Value Pr > F
1
2 131.555556
65.777778 11.9596
0.0002
2
2 784.888889
392.444444 71.3535
0.0001
3
2 40.222222
20.111111 3.6566
0.0411
4
2 838.888889
419.444444 76.2626
0.0001
Dependent Variable: Y
Contrast DF Contrast SS Mean Square F Value Pr > F
treat in group 2 2 784.888889 392.444444 71.35 0.0001
T for H0:
Pr > |T| Std
Error of
Parameter
Estimate
Parameter=0
Estimate
treat1 group2 mean 9.0000000
6.65
0.0001
1.35400640
treat2 group2 mean 17.6666667
13.05
0.0001
1.35400640
mean diff t1g2-t2g2 -8.6666667
-4.53
0.0001
1.91485422
In this example, 12 subjects are randomly assigned to 4 groups,
3 to each group. There are three observations for each subject corresponding
to measurements taken at time 1, 2 and 3. In the following program, factor
time with 3 levels is the effect of the time and factor group with 4 levels
is the effect of the group.
A mixed effect model with fixed effect of group and time and random effect of subject will be used to analyze the data. It is assumed that the effect of the subject has a normal distribution with mean 0 and variance sigmaS squared (it measures between subject variability). It is also assumed that the error term has a normal distribution with mean 0 and variance sigmaE squared (it measures within subject error) and the error and subject effects are not correlated
As you can see below, the results of MIXED and GLM are not identical. The F and p-values for the tests are the same. Values from proc mixed have to be compared with the Tests of Hypotheses for Mixed Model Analysis from proc GLM, not with the main, General Linear Model Procedure, ANOVA table. The values in the main ANOVA table in proc GLM are incorrect for this example; they are computed under the assumption that subject is a fixed effect. However, the standard error of the lsmeans and requested estimates are not the same for proc MIXED and proc GLM. The ones printed by proc MIXED are correct. Again, proc GLM computed the standard error assuming that the subject effect is fixed. Note that the standard error for the third estimate, the mean difference between time 1 and time 2 in group 2 is the same for both. This is because when you compute that difference, the effect of the subject cancels out.
Also note that proc GLM results printed in the Test of Hypotheses table include the F-test for the significance of the subject effect. The test is not printed in proc Mixed. The corresponding table includes only the fixed effects. The estimates of the random effects, in this case sigmaS squared (variance of the subject effect) and sigmaE squared (variance of the error term) are printed in the table named Covariance Parameter Estimates. The test of significance is the Wald test. The estimates are consistent with the proc GLM results. The residual variance in proc MIXED is the same as MSS (mean sum of squares) for the error in proc GLM. The subject variance can be computed from the GLM Type III Expected Mean Square table.
Type III Expected Mean Square
GROUP Var(Error) + 3 Var(SUBJECT(GROUP)) + Q(GROUP,GROUP*TIME)
SUBJECT(GROUP) Var(Error) + 3 Var(SUBJECT(GROUP))
TIME Var(Error) + Q(TIME,GROUP*TIME)
GROUP*TIME Var(Error) + Q(GROUP*TIME)
According to that table, MSS(subject)=var(error)+3*var(subject). Hence var(subject)=(MSS(subject) – var(error))/3. Since the expected mean of MSS(error)=var(error), we can use MSS(error) as the estimate of var(error) and replace var(error) with MSS(error) in the above formula. Thus,
Var(subject)=(12.5278 – 1.9861)/3=3.5139,
which is the same as the value printed in the proc MIXED Covariance
Parameter Estimates table for the subject.
GROUP 4 1 2
3 4
TIME 3
1 2 3
SUBJECT 12 1 2 3 4 5 6 7 8 9
10 11 12
Covariance Parameter Estimates (REML)
Cov Parm Estimate Std Error Z Pr > |Z|
SUBJECT(GROUP)
3.51388889 2.10104164 1.67
0.0944
Residual
1.98611111 0.70219632 2.83
0.0047
Tests of Fixed Effects
Source NDF DDF Type III F Pr > F
GROUP
3 8
53.39 0.0001
TIME
2 16
94.45 0.0001
GROUP*TIME 6 16
119.19 0.0001
ESTIMATE Statement Results
Parameter Estimate Std Error DF t Pr > |t|
time1 group2 mean
9.00000000 1.35400640 13.2
6.65 0.0001
time2 group2 mean 17.66666667
1.35400640 13.2 13.05
0.0001
mean diff t1g2-t2g2 -8.66666667
1.15068418 16 -7.53
0.0001
CONTRAST Statement Results
Source NDF DDF F Pr > F
time in group 2 2
16 197.59
0.0001
Least Squares Means
Effect GROUP TIME LSMEAN Std Error DF t Pr > |t|
GROUP*TIME 1
1 23.33333333
1.35400640 13.2 17.23 0.0001
GROUP*TIME 1
2 19.33333333
1.35400640 13.2 14.28 0.0001
GROUP*TIME 1
3 14.00000000
1.35400640 13.2 10.34 0.0001
GROUP*TIME 2
1 9.00000000
1.35400640 13.2 6.65 0.0001
GROUP*TIME 2
2 17.66666667
1.35400640 13.2 13.05 0.0001
GROUP*TIME 2
3 31.66666667
1.35400640 13.2 23.39 0.0001
GROUP*TIME 3
1 5.00000000
1.35400640 13.2 3.69 0.0026
GROUP*TIME 3
2 1.33333333
1.35400640 13.2 0.98 0.3424
GROUP*TIME 3
3 6.33333333
1.35400640 13.2 4.68 0.0004
GROUP*TIME 4
1 11.33333333
1.35400640 13.2 8.37 0.0001
GROUP*TIME 4
2 34.66666667
1.35400640 13.2 25.60 0.0001
GROUP*TIME 4
3 26.33333333
1.35400640 13.2 19.45 0.0001
Tests of Effect Slices
Effect GROUP NDF DDF F Pr > F
GROUP*TIME 1
2 16
33.12 0.0001
GROUP*TIME 2
2 16
197.59 0.0001
GROUP*TIME 3
2 16
10.13 0.0014
GROUP*TIME 4
2 16
211.19 0.0001
General Linear Models Procedure
GROUP 4 1 2 3 4
TIME 3 1 2 3
SUBJECT 12 1 2 3 4 5 6 7 8 9
10 11 12
General Linear Models Procedure
Dependent Variable: Y
Sum of Mean
Source
DF Squares
Square
F Value Pr > F
Model 19 3902.22222 205.38012 103.41 0.0001
Error 16 31.77778 1.98611
Corrected Total 35 3934.00000
R-Square C.V. Root MSE Y Mean
0.991922 8.455767
1.40929
16.6667
Source DF Type III SS Mean Square F Value Pr > F
GROUP
3 2006.44444
668.81481 336.75
0.0001
SUBJECT(GROUP) 8
100.22222 12.52778
6.31
0.0009
TIME
2 375.16667
187.58333 94.45
0.0001
GROUP*TIME
6 1420.38889
236.73148 119.19
0.0001
Source Type III Expected Mean Square
GROUP Var(Error) + 3 Var(SUBJECT(GROUP)) + Q(GROUP,GROUP*TIME)
SUBJECT(GROUP) Var(Error) + 3 Var(SUBJECT(GROUP))
TIME Var(Error) + Q(TIME,GROUP*TIME)
GROUP*TIME
Var(Error) + Q(GROUP*TIME)
General Linear Models Procedure
Tests of Hypotheses
for Mixed Model Analysis of Variance
Dependent Variable: Y
Source: GROUP *
Error: MS(SUBJECT(GROUP))
Denominator Denominator
DF Type III
MS DF
MS
F Value Pr > F
3 668.81481481
8
12.527777778 53.3865 0.0001
* - This test assumes one or more other fixed effects are zero.
Source: SUBJECT(GROUP)
Error: MS(Error)
Denominator Denominator
DF Type III
MS DF
MS
F Value Pr > F
8 12.527777778
16
1.9861111111
6.3077 0.0009
Source: TIME *
Error: MS(Error)
Denominator Denominator
DF Type III
MS DF
MS
F Value Pr > F
2 187.58333333
16
1.9861111111
94.4476 0.0001
* - This test assumes one or more other fixed effects are zero.
Source: GROUP*TIME
Error: MS(Error)
Denominator Denominator
DF Type III
MS DF
MS
F Value Pr > F
6 236.73148148
16
1.9861111111
119.1935 0.0001
Least Squares Means
GROUP
TIME
Y
Std Err
Pr > |T|
LSMEAN LSMEAN
H0:LSMEAN=0
1
1 23.3333333
0.8136566 0.0001
1
2 19.3333333
0.8136566 0.0001
1
3 14.0000000
0.8136566 0.0001
2
1 9.0000000
0.8136566 0.0001
2
2 17.6666667
0.8136566 0.0001
2
3 31.6666667
0.8136566 0.0001
3
1 5.0000000
0.8136566 0.0001
3
2 1.3333333
0.8136566 0.1208
3
3 6.3333333
0.8136566 0.0001
4
1 11.3333333
0.8136566 0.0001
4
2 34.6666667
0.8136566 0.0001
4
3 26.3333333
0.8136566 0.0001
GROUP*TIME Effect Sliced by GROUP for Y
Sum of
Mean
GROUP DF
Squares
Square F Value
Pr > F
1
2 131.555556
65.777778 33.1189
0.0001
2
2 784.888889
392.444444 197.6000 0.0001
3
2 40.222222
20.111111 10.1259
0.0014
4
2 838.888889
419.444444 211.2000 0.0001
Contrast DF Contrast SS Mean Square F Value Pr > F
time in group 2 2 784.888889 392.444444 197.59 0.0001
T for H0:
Pr > |T| Std Error of
Parameter
Estimate Parameter=0
Estimate
time1 group2 mean 9.0000000
11.06
0.0001 0.81365658
time2 group2 mean 17.6666667
21.71
0.0001 0.81365658
mean diff t1g2-t2g2 -8.6666667
-7.53
0.0001 1.15068418
In this example, there are 2 subjects in group 1, 3 in group 2,
4 in group 3 and 3 in group 4. There are three observations for each subject
corresponding to measurements taken under three conditions, 1, 2 and 3
for subjects in groups 1 and 3 and two observations for each subject corresponding
to measurements taken at different conditions, 4 and 5 for subjects in
groups 2 and 4 . In the following program, factor cond with 5 levels is
the effect of the condition and factor group with 4 levels is the effect
of the group.
A mixed effect model with fixed effect of group and cond(group) and random effect of subject will be used to analyze the data. It is assumed that the effect of the subject has a normal distribution with mean 0 and variance sigmaS squared (it measures between subject variability). It is also assumed that the error term has a normal distribution with mean 0 and variance sigmaE squared (it measures within subject variability) and the error and subject effects are not correlated.
Note the use of the option E3 in the model statement. It makes proc mixed print the coefficients of the type 3 contrasts for the model effects hypotheses.
As can be seen below, the results of proc MIXED and proc GLM are
different in this case.
The MIXED Procedure
GROUP 4 1 2
3 4
COND 5
1 2 3 4 5
SUBJECT 12 1 2 3 4 5 6 7 8 9
10 11 12
Covariance Parameter Estimates (REML)
Cov Parm Estimate Std Error Z Pr > |Z|
SUBJECT(GROUP)
1.50219942 1.58123118 0.95
0.3421
Residual
2.98807617 1.27017905 2.35
0.0186
Type III Coefficients for COND(GROUP)
Effect GROUP COND Row 1 Row 2 Row 3 Row 4 Row 5 Row 6
INTERCEPT
0
0
0 0
0 0
GROUP
1
0
0
0 0
0 0
GROUP
2
0
0
0 0
0 0
GROUP
3
0
0
0 0
0 0
GROUP
4
0
0
0 0
0 0
COND(GROUP)
1 1
1
0
0 0
0 0
COND(GROUP)
1 2
0
1
0 0
0 0
COND(GROUP)
1 3
-1
-1
0 0
0 0
COND(GROUP)
2 4
0
0
1 0
0 0
COND(GROUP)
2 5
0
0
-1 0
0 0
COND(GROUP)
3 1
0
0
0
1 0
0
COND(GROUP)
3 2
0
0
0
0 1
0
COND(GROUP)
3 3
0
0
0 -1
-1 0
COND(GROUP)
4 4
0
0
0 0
0 1
COND(GROUP)
4 5
0
0
0 0
0 -1
Tests of Fixed Effects
Source NDF DDF Type III F Pr > F
GROUP
3
7.1 19.08
0.0009
COND(GROUP) 6
11.1 58.93
0.0001
ESTIMATE Statement Results
Parameter Estimate Std Error DF t Pr > |t|
diff c1g1-c1g3
6.75000000 1.83513125
16.5 3.68 0.0020
CONTRAST Statement Results
Source NDF DDF F Pr > F
cond 1 vs 2 in group
1 11.1
8.37 0.0146
cond 1 vs 2 in group
1 11.1
9.41 0.0106
Least Squares Means
Effect GROUP COND LSMEAN Std Error DF t Pr > |t|
COND(GROUP) 1
1 22.50000000 1.49837839
16.5 15.02 0.0001
COND(GROUP) 1
2 17.50000000 1.49837839
16.5 11.68 0.0001
COND(GROUP) 1
3 14.00000000 1.49837839
16.5 9.34 0.0001
COND(GROUP) 2
4 9.00000000
1.22342083 16.5 7.36 0.0001
COND(GROUP) 2
5 17.66666667 1.22342083
16.5 14.44 0.0001
COND(GROUP) 3
1 15.75000000 1.05951352
16.5 14.87 0.0001
COND(GROUP) 3
2 12.00000000 1.05951352
16.5 11.33 0.0001
COND(GROUP) 3
3 17.00000000 1.05951352
16.5 16.05 0.0001
COND(GROUP) 4
4 11.33333333 1.22342083
16.5 9.26 0.0001
COND(GROUP) 4 5
34.66666667 1.22342083 16.5 28.34
0.0001
Tests of Effect Slices
Effect GROUP NDF DDF F Pr > F
COND(GROUP) 1
2 11.1
12.22 0.0016
COND(GROUP) 2
1 11.1
37.71 0.0001
COND(GROUP) 3
2 11.1
9.06 0.0047
COND(GROUP) 4
1 11.1
273.31 0.0001
General Linear Models Procedure
GROUP 4
1 2 3 4
COND 5
1 2 3 4 5
SUBJECT 12 1 2 3 4 5 6
7 8 9 10 11 12
General Linear Models Procedure
Dependent Variable: Y
Sum of
Mean
Source
DF Squares
Square F Value
Pr > F
Model 17 1463.66667 86.09804 29.95 0.0001
Error 12 34.50000 2.87500
Corrected Total 29 1498.16667
R-Square C.V. Root MSE Y Mean
0.976972 10.07277
1.69558
16.8333
Source DF Type III SS Mean Square F Value Pr > F
GROUP
3 353.91667
117.97222 41.03
0.0001
SUBJECT(GROUP) 8
53.25000
6.65625 2.32
0.0919
COND(GROUP)
6 1056.50000
176.08333 61.25
0.0001
General Linear Models Procedure
Source Type III Expected Mean Square
GROUP
Var(Error) + 2.4667 Var(SUBJECT(GROUP))
+ Q(GROUP,COND(GROUP))
SUBJECT(GROUP) Var(Error) + 2.5 Var(SUBJECT(GROUP))
COND(GROUP)
Var(Error) + Q(COND(GROUP))
General Linear Models Procedure
Tests of Hypotheses
for Mixed Model Analysis of Variance
Source: GROUP *
Error: 0.9867*MS(SUBJECT(GROUP)) + 0.0133*MS(Error)
Denominator Denominator
DF Type III
MS DF
MS
F Value Pr > F
3 117.97222222
8.09
6.6058333333 17.8588
0.0006
* - This test assumes one or more other fixed effects are zero.
Source: SUBJECT(GROUP)
Error: MS(Error)
Denominator Denominator
DF Type III
MS
DF
MS
F Value Pr > F
8
6.65625
12
2.875
2.3152 0.0919
Source: COND(GROUP)
Error: MS(Error)
Denominator Denominator
DF Type III
MS DF
MS
F Value Pr > F
6 176.08333333
12
2.875
61.2464 0.0001
Least Squares Means
COND GROUP
Y
Std Err Pr > |T|
LSMEAN LSMEAN
H0:LSMEAN=0
1
1 22.5000000
1.1989579 0.0001
2
1 17.5000000
1.1989579 0.0001
3
1 14.0000000
1.1989579 0.0001
4
2
9.0000000 0.9789450
0.0001
5
2 17.6666667
0.9789450 0.0001
1
3 15.7500000
0.8477912 0.0001
2
3 12.0000000
0.8477912 0.0001
3
3 17.0000000
0.8477912 0.0001
4
4 11.3333333
0.9789450 0.0001
5
4 34.6666667
0.9789450 0.0001
Least Squares Means
COND(GROUP) Effect Sliced by GROUP for Y
Sum of
Mean
GROUP DF
Squares Square
F Value Pr > F
1
2 73.000000
36.500000 12.6957
0.0011
2
1 112.666667
112.666667 39.1884
0.0001
3
2 54.166667
27.083333 9.4203
0.0035
4
1 816.666667
816.666667 284.1000 0.0001
Dependent Variable: Y
Contrast DF Contrast SS Mean Square F Value Pr > F
cond 1 vs 2 in group 1
25.0000000 25.0000000 8.70
0.0122
cond 1 vs 2 in group 1
28.1250000 28.1250000 9.78
0.0087
T for H0:
Pr > |T|
Std Error of
Parameter
Estimate
Parameter=0
Estimate
diff c1g1-c1g3
6.75000000 4.60
0.0006 1.46841752