## When random effects exists, why are the results same from proc mixed and proc glm procedure

Occasional Contributor
Posts: 7

# When random effects exists, why are the results same from proc mixed and proc glm procedure

When random effects exists, why are the results same from proc mixed and proc glm procedure ?

Say,This is a nonreplicated two-way cross-over study.

random effects is subject nested in sequence; fixed effects are treatment, sequence, period .

I mocked balanced data to try the proc mixed and proc glm , and find the same coefficient estimate there.

My main question is that as we know the

proc mixed treats random effect :subject within sequence as random effect

and

proc glm treats random effect as fixed effect.

Why for balanced data , they will produce the same result ?????

what is the rationale behind it. I have googled a lot of information, none of them provide me specific information.

The SAS DATASET, CODE AND RESULT are attached blow. (This is a mock).

DATA TRY;
INPUT SUBJECT\$ TREATMENT\$ PERIOD SEQUENCE CONC;

DATALINES;
001 A 1 1 2.90
002 A 1 1 3.14
003 A 1 1 3.49
004 A 1 1 5.28
005 B 1 2 2.39
006 B 1 2 3.7
007 A 1 1 3.68
008 B 1 2 1.8
009 B 1 2 2.28
010 B 1 2 2.44
001 B 2 1 2.65
002 B 2 1 1.96
003 B 2 1 3.18
004 B 2 1 3.66
005 A 2 2 3.83
006 A 2 2 4.62
007 B 2 1 2.22
008 A 2 2 3.5
009 A 2 2 1.76
010 A 2 2 4.88
;
RUN;

ods output lsmeans=result2;
ods output lsmeandiffcl=result1;
ods output overallanova=result3;
/*****/PROC GLM DATA=TRY;
CLASS TREATMENT PERIOD SEQUENCE SUBJECT;
MODEL CONC=TREATMENT PERIOD SEQUENCE SUBJECT(SEQUENCE)/SOLUTION;
RANDOM SUBJECT(SEQUENCE);
LSMEANS TREATMENT/STDERR PDIFF=control("A","B") CL ALPHA=0.1 ADJUST=T;
RUN;

/***/
ods output Estimates=result1;
ods output LSMeans=result2 ;
ods output Diffs=result3 ;
PROC MIXED DATA=TRY;
CLASS TREATMENT PERIOD SEQUENCE SUBJECT;
MODEL CONC=TREATMENT PERIOD SEQUENCE/SOLUTION;
RANDOM SUBJECT(SEQUENCE);
LSMEANS TREATMENT/PDIFF=control("A","B") CL ALPHA=0.1 ;
RUN;

 The SAS System

The GLM Procedure

 Class Level Information Class Levels Values TREATMENT 2 A B PERIOD 2 1 2 SEQUENCE 2 1 2 SUBJECT 10 001 002 003 004 005 006 007 008 009 010

 Number of Observations Read 20 Number of Observations Used 20

 The SAS System

The GLM Procedure

Dependent Variable: CONC

 Source DF Sum of Squares Mean Square F Value Pr > F Model 11 16.14470000 1.46770000 3.59 0.0402 Error 8 3.27122000 0.40890250 Corrected Total 19 19.41592000

 R-Square Coeff Var Root MSE CONC Mean 0.831519 20.18481 0.639455 3.168000

 Source DF Type I SS Mean Square F Value Pr > F TREATMENT 1 5.83200000 5.83200000 14.26 0.0054 PERIOD 1 0.06728000 0.06728000 0.16 0.6956 SEQUENCE 1 0.04608000 0.04608000 0.11 0.7457 SUBJECT(SEQUENCE) 8 10.19934000 1.27491750 3.12 0.0641

 Source DF Type III SS Mean Square F Value Pr > F TREATMENT 1 5.83200000 5.83200000 14.26 0.0054 PERIOD 1 0.06728000 0.06728000 0.16 0.6956 SEQUENCE 1 0.04608000 0.04608000 0.11 0.7457 SUBJECT(SEQUENCE) 8 10.19934000 1.27491750 3.12 0.0641

 Parameter Estimate Standard Error t Value Pr > |t| Intercept 3.178000000 B 0.49531959 6.42 0.0002 TREATMENT A 1.080000000 B 0.28597290 3.78 0.0054 TREATMENT B 0.000000000 B . . . PERIOD 1 -0.116000000 B 0.28597290 -0.41 0.6956 PERIOD 2 0.000000000 B . . . SEQUENCE 1 -0.710000000 B 0.63945485 -1.11 0.2991 SEQUENCE 2 0.000000000 B . . . SUBJECT(SEQUENCE) 001 1 -0.175000000 B 0.63945485 -0.27 0.7913 SUBJECT(SEQUENCE) 002 1 -0.400000000 B 0.63945485 -0.63 0.5490 SUBJECT(SEQUENCE) 003 1 0.385000000 B 0.63945485 0.60 0.5638 SUBJECT(SEQUENCE) 004 1 1.520000000 B 0.63945485 2.38 0.0448 SUBJECT(SEQUENCE) 007 1 0.000000000 B . . . SUBJECT(SEQUENCE) 005 2 -0.550000000 B 0.63945485 -0.86 0.4148 SUBJECT(SEQUENCE) 006 2 0.500000000 B 0.63945485 0.78 0.4568 SUBJECT(SEQUENCE) 008 2 -1.010000000 B 0.63945485 -1.58 0.1529 SUBJECT(SEQUENCE) 009 2 -1.640000000 B 0.63945485 -2.56 0.0334 SUBJECT(SEQUENCE) 010 2 0.000000000 B . . .

 Note: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable.

 The SAS System

The GLM Procedure

 Source Type III Expected Mean Square TREATMENT Var(Error) + Q(TREATMENT) PERIOD Var(Error) + Q(PERIOD) SEQUENCE Var(Error) + 2 Var(SUBJECT(SEQUENCE)) + Q(SEQUENCE) SUBJECT(SEQUENCE) Var(Error) + 2 Var(SUBJECT(SEQUENCE))

 The SAS System

The GLM Procedure

Least Squares Means

 TREATMENT CONC LSMEAN Standard Error H0:LSMEAN=0 H0:LSMean1=LSMean2 Pr > |t| Pr > |t| A 3.70800000 0.20221338 <.0001 0.0054 B 2.62800000 0.20221338 <.0001

 TREATMENT CONC LSMEAN 90% Confidence Limits A 3.708000 3.331975 4.084025 B 2.628000 2.251975 3.004025

 Least Squares Means for Effect TREATMENT i j Difference BetweenMeans 90% Confidence Limits for LSMean(i)-LSMean(j) 2 1 -1.080000 -1.611780 -0.548220

Proc mixed

 The SAS System

The Mixed Procedure

 Model Information Data Set WORK.TRY Dependent Variable CONC Covariance Structure Variance Components Estimation Method REML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Containment

 Class Level Information Class Levels Values TREATMENT 2 A B PERIOD 2 1 2 SEQUENCE 2 1 2 SUBJECT 10 001 002 003 004 005 006 007 008 009 010

 Dimensions Covariance Parameters 2 Columns in X 7 Columns in Z 10 Subjects 1 Max Obs Per Subject 20

 Number of Observations Number of Observations Read 20 Number of Observations Used 20 Number of Observations Not Used 0

 Iteration History Iteration Evaluations -2 Res Log Like Criterion 0 1 50.47676453 1 1 48.01890254 0.00000000

 Convergence criteria met.

 Covariance Parameter Estimates Cov Parm Estimate SUBJECT(SEQUENCE) 0.4330 Residual 0.4089

 Fit Statistics -2 Res Log Likelihood 48.0 AIC (smaller is better) 52.0 AICC (smaller is better) 52.9 BIC (smaller is better) 52.6

 Solution for Fixed Effects Effect TREATMENT PERIOD SEQUENCE Estimate Standard Error DF t Value Pr > |t| Intercept 2.6380 0.4103 8 6.43 0.0002 TREATMENT A 1.0800 0.2860 8 3.78 0.0054 TREATMENT B 0 . . . . PERIOD 1 -0.1160 0.2860 8 -0.41 0.6956 PERIOD 2 0 . . . . SEQUENCE 1 0.09600 0.5050 8 0.19 0.8540 SEQUENCE 2 0 . . . .

 Type 3 Tests of Fixed Effects Effect Num DF Den DF F Value Pr > F TREATMENT 1 8 14.26 0.0054 PERIOD 1 8 0.16 0.6956 SEQUENCE 1 8 0.04 0.8540

 Least Squares Means Effect TREATMENT Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper TREATMENT A 3.7080 0.2902 8 12.78 <.0001 0.1 3.1684 4.2476 TREATMENT B 2.6280 0.2902 8 9.06 <.0001 0.1 2.0884 3.1676

 Differences of Least Squares Means Effect TREATMENT _TREATMENT Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper TREATMENT B A -1.0800 0.2860 8 -3.78 0.0054 0.1 -1.6118 -0.5482

Occasional Contributor
Posts: 7

## Re: When random effects exists, why are the results same from proc mixed and proc glm procedure

You can find the coefficient estimate for treatment A and B are almost the same values from prc mixed and proc glm .

Occasional Contributor
Posts: 7

## Re: When random effects exists, why are the results same from proc mixed and proc glm procedure

Yes, Absolutely. My question is why it is the same reslut from prc glm and proc mixed.

Posts: 2,655

## Re: When random effects exists, why are the results same from proc mixed and proc glm procedure

Because your data and model are balanced, the point estimates should be the same (at least to a few decimal places).  However, variance estimates are quite different.  Look at the size of the standard errors for the least squares means.  For GLM (a narrow inference approach), they are 0.2022..., while for MIXED (a broad inference approach), they are 0.2902, an increase of almost 45%.  Note also that the F test for sequence differs for the two approaches, due to the nesting of subjects within treatment*period.

That is the difference.  Take a look at Littell et al.'s SAS for Mixed Models, 2nd ed. for additional material that compares GLM to MIXED.

Steve Denham

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