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. So, if you know the answer, please help me! Thanks in advance . 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 Between Means 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
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