Introduction to PROC MIXED - webpages.uidaho.edubrian/proc_mixed_documentation_uky.pdf · yij = a0...

Introduction to PROC MIXED

Table of Contents

1. Short description of methods of estimation used in PROC MIXED 2. Description of the syntax of PROC MIXED 3. References 4. Examples and comparisons of results from MIXED and GLM

- balanced data: fixed effect model and mixed effect model, - unbalanced data, mixed effect model

1. Short description of methods of estimation used in PROC MIXED.

The SAS procedures GLM and MIXED can be used to fit linear models. Proc GLM was designed to fit fixed effect models and later amended to fit some random effect models by including RANDOM statement with TEST option. The REPEATED statement in PROC GLM allows to estimate and test repeated measures models with an arbitrary correlation structure for repeated observations. The PROC MIXED was specifically designed to fit mixed effect models. It can model random and mixed effect data, repeated measures, spacial data, data with heterogeneous variances and autocorrelated observations.The MIXED procedure is more general than GLM in the sense that it gives a user more flexibility in specifying the correlation structures, particularly useful in repeated measures and random effect models. It has to be emphasized, however, that the PROC MIXED is not an extended, more general version of GLM. They are based on different statistical principles; GLM and MIXED use different estimation methods. GLM uses the ordinary least squares (OLS) estimation, that is, parameter estimates are such values of the parameters of the model that minimize the squared difference between observed and predicted values of the dependent variable. That approach leads to the familiar analysis of variance table in which the variability in the dependent variable (the total sum of squares) is divided into variabilities due to different sources (sum of squares for effects in the model). PROC MIXED does not produce an analysis of variance table, because it uses estimation methods based on different principles. PROC MIXED has three options for the method of estimation. They are: ML (Maximum Likelihood), REML (Restricted or Residual maximum likelihood, which is the default method) and MIVQUE0 (Minimum Variance Quadratic Unbiased Estimation). ML and REML are based on a maximum likelihood estimation approach. They require the assumption that the distribution of the dependent variable (error term and the random effects) is normal. ML is just the regular maximum likelihood method,that is, the parameter estimates that it produces are such values of the model parameters that maximize the likelihood function. REML method is a variant of maximum likelihood estimation; REML estimators are obtained not from maximizing the whole likelihood function, but only that part that is invariant to the fixed effects part of the linear model. In other words, if y = Xb + Zu + e, where Xb is the

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fixed effects part, Zu is the random effects part and e is the error term, then the REML estimates are obtained by maximizing the likelihood function of K'y, where K is a full rank matrix with columns orthogonal to the columns of the X matrix, that is, K'X = 0. It leads to REML estimator of the variance-covariance matrix of y, say V. It does not depend on the choice of matrix K. Then the generalized least squares equations, known also from the weighted least squares approach and the GLM procedure,

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.

2. Description of the syntax of PROC MIXED 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.

PROC MIXED <options>;

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;

CLASS variables;

Lists classification variables (categorical independent variables in the model). For example:

proc mixed data=mydata covtest;



Class group gender agecat;

MODEL dependent = fixed effects </options>;

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 χ2 – 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.

RANDOM random effects </options>;

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;

REPEATED repeated effects / options;

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).

CONTRAST ‘label’ fixed-effect values | random-effect values / options;

ESTIMATE ‘label’ fixed-effect values | random-effect values / options;

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 fixed-effects / options;

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;

MAKE 'table' OUT= SAS-data-set < options >;

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;

References 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.

3. Examples and comparisons of the results from PROC MIXED and PROC GLM.

Example1. Fixed effect model, balanced data.

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.

Program:

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;

Results:

The MIXED Procedure

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

Example 2. Mixed effect model, balanced data.

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.

Program:

options ls=76; data one; input y group time subject; cards; 22 1 1 1 23 1 1 2 25 1 1 3 17 1 2 1 18 1 2 2 23 1 2 3



12 1 3 1 16 1 3 2 14 1 3 3 8 2 1 4 9 2 1 5 10 2 1 6 16 2 2 4 17 2 2 5 20 2 2 6 29 2 3 4 30 2 3 5 36 2 3 6 3 3 1 7 7 3 1 8 5 3 1 9 1 3 2 7 2 3 2 8 1 3 2 9 4 3 3 7 7 3 3 8 8 3 3 9 11 4 1 10 15 4 1 11 8 4 1 12 34 4 2 10 37 4 2 11 33 4 2 12 27 4 3 10 28 4 3 11 24 4 3 12 ; run; proc sort data=one; by group subject time; run; Proc mixed data=one method=reml covtest; Class group time subject; Model y=group time group*time / DDFM=SATTERTH; RANDOM SUBJECT(group); lsmeans group*time /adjust=bon pdiff=control('1' '1') slice=group; Contrast 'time in group 2' time 1 -1 0 group*time 0 0 0 1 -1 0 0 0 0 0 0 0, time 0 1 -1 group*time 0 0 0 0 1 -1 0 0 0 0 0 0; Estimate 'time1 group2 mean' intercept 1 group 0 1 0 0 time 1 0 0 group*time 0 0 0 1 0 0 0 0 0 0 0 0; Estimate 'time2 group2 mean' intercept 1 group 0 1 0 0 time 0 1 0 Group*time 0 0 0 0 1 0 0 0 0 0 0 0; Estimate 'mean diff t1g2-t2g2' time 1 -1 0 group*time 0 0 0 1 -1 0 0 0 0 0 0 0; Run; proc GLM data=one; class group time subject;



Model y=group subject(group) time group*time; RANDOM SUBJECT(GROUP) /TEST; lsmeans group*time /stderr; lsmeans group*time /adjust=bon pdiff=control('1' '1') slice=group; Contrast 'time in group 2' time 1 -1 0 group*time 0 0 0 1 -1 0 0 0 0 0 0 0, time 0 1 -1 group*time 0 0 0 0 1 -1 0 0 0 0 0 0; Estimate 'time1 group2 mean' intercept 1 group 0 1 0 0 time 1 0 0 group*time 0 0 0 1 0 0 0 0 0 0 0 0; Estimate 'time2 group2 mean' intercept 1 group 0 1 0 0 time 0 1 0 Group*time 0 0 0 0 1 0 0 0 0 0 0 0; Estimate 'mean diff t1g2-t2g2' time 1 -1 0 group*time 0 0 0 1 -1 0 0 0 0 0 0 0; Run;

Results:

The MIXED 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

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



GROUP 3 8 53.39 0.0001 TIME 2 16 94.45 0.0001 GROUP*TIME 6 16 119.19 0.0001



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





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



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


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





Model 19 3902.22222 205.38012 103.41 0.0001

Error 16 31.77778 1.98611



0.991922 8.455767 1.40929 16.6667


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


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


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


time in group 2 2 784.888889 392.444444 197.59 0.0001




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

Example 3. Mixed effect model, unbalanced data.

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.

Program:

options ls=76; data one; input y group cond subject; cards; 22 1 1 1 23 1 1 2 17 1 2 1 18 1 2 2 12 1 3 1 16 1 3 2 8 2 4 3 9 2 4 4 10 2 4 5 16 2 5 3 17 2 5 4 20 2 5 5



13 3 1 6 17 3 1 7 15 3 1 8 18 3 1 9 11 3 2 6 12 3 2 7 11 3 2 8 14 3 2 9 17 3 3 6 18 3 3 7 19 3 3 8 14 3 3 9 11 4 4 10 15 4 4 11 8 4 4 12 34 4 5 10 37 4 5 11 33 4 5 12 ; run; proc sort data=one; by group subject cond; run; Proc mixed data=one method=reml covtest; Class group cond subject; Model y=group cond(group) / DDFM=SATTERTH e3; RANDOM SUBJECT(group); lsmeans cond(group) /adjust=bon pdiff=control('1' '1') slice=group; Contrast 'cond 1 vs 2 in group 1' cond(group) 1 -1 0 0 0 0 0 0 0 0; contrast 'cond 1 vs 2 in group 3' cond(group) 0 0 0 0 0 1 -1 0 0 0; Estimate 'diff c1g1-c1g3' group 1 0 -1 0 cond(group) 1 0 0 0 0 -1 0 0 0 0; Run; proc GLM data=one; class group cond subject; Model y=group subject(group) cond(group); RANDOM SUBJECT(GROUP) /TEST; lsmeans cond(group) /stderr; lsmeans cond(group) /adjust=bon pdiff=control('1' '1') slice=group; Contrast 'cond 1 vs 2 in group 1' cond(group) 1 -1 0 0 0 0 0 0 0 0; contrast 'cond 1 vs 2 in group 3' cond(group) 0 0 0 0 0 1 -1 0 0 0; Estimate 'diff c1g1-c1g3' group 1 0 -1 0 cond(group) 1 0 0 0 0 -1 0 0 0 0; Run;

Results:



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



GROUP 3 7.1 19.08 0.0009 COND(GROUP) 6 11.1 58.93 0.0001





diff c1g1-c1g3 6.75000000 1.83513125 16.5 3.68 0.0020



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



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


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





Model 17 1463.66667 86.09804 29.95 0.0001

Error 12 34.50000 2.87500



0.976972 10.07277 1.69558 16.8333


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


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


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



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




diff c1g1-c1g3 6.75000000 4.60 0.0006 1.46841752



Introduction to PROC MIXED - webpages.uidaho.edubrian/proc_mixed_documentation_uky.pdf · yij = a0...

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