Two-Way Mixed-Effect Model

Auto Assembly Example

Assume that there are three methods of assembly of a transmission in an automobile assembly plant. A random sample of machine operators is taken from a large pool of operators and each operator is observed with each method of assembly twice in a completely crossed experiment. The response is the output of production.

Program Listing

OPTIONS LS=72 NODATE NONUMBER NOCENTER;

DATA auto (DROP=rep);
 DO assembly = 1 TO 3;   /* 3 assembly methods (fixed) */
  DO operator = 1 TO 4;  /* 4 operators (random) */
   DO rep = 1 TO 2;      /* 2 replicates */
    INPUT prod @@;
    OUTPUT;
   END;
  END;
 END;
 DATALINES;
 21 23   26 25   14 17   24 27
 30 27   27 26   19 24   33 31
 24 18   19 22   18 23   26 21
;

TITLE1 'Two-Way Mixed-Effect Model';
TITLE2 'Auto Assembly Example';
PROC GLM DATA=auto;
 TITLE3 'ANOVA Table';
 CLASS assembly operator;
 MODEL prod = assembly operator assembly*operator;
 RANDOM operator assembly*operator / TEST;
 RUN;
 TITLE3 'Hypothesis Test';
 TEST H=assembly    E=assembly*operator;
 RUN;
 TITLE3 'Least Square Means';
 LSMEANS assembly/PDIFF E=assembly*operator;
 RUN;
 TITLE3 'Treatment Means';
 MEANS assembly/LSD E=assembly*operator;
 RUN;
QUIT;

Output Listing

Two-Way Mixed-Effect Model
Auto Assembly Example
ANOVA Table

The GLM Procedure

    Class Level Information
 
Class         Levels    Values
assembly           3    1 2 3   
operator           4    1 2 3 4 


Number of Observations Read          24
Number of Observations Used          24
Two-Way Mixed-Effect Model
Auto Assembly Example
ANOVA Table

The GLM Procedure
 
Dependent Variable: prod   
                                      Sum of
Source                     DF        Squares    Mean Square   F Value
Model                      11    417.4583333     37.9507576      5.80
Error                      12     78.5000000      6.5416667          
Corrected Total            23    495.9583333                         

Source                 Pr > F
Model                  0.0026
Error                        

Corrected Total              


R-Square     Coeff Var      Root MSE     prod Mean
0.841721      10.86443      2.557668      23.54167

Source                     DF      Type I SS    Mean Square   F Value
assembly                    2    156.3333333     78.1666667     11.95
operator                    3    189.4583333     63.1527778      9.65
assembly*operator           6     71.6666667     11.9444444      1.83

Source                 Pr > F
assembly               0.0014
operator               0.0016
assembly*operator      0.1764


Source                     DF    Type III SS    Mean Square   F Value
assembly                    2    156.3333333     78.1666667     11.95
operator                    3    189.4583333     63.1527778      9.65
assembly*operator           6     71.6666667     11.9444444      1.83

Source                 Pr > F
assembly               0.0014
operator               0.0016
assembly*operator      0.1764


Two-Way Mixed-Effect Model
Auto Assembly Example
ANOVA Table

The GLM Procedure

Source                Type III Expected Mean Square
assembly              Var(Error) + 2 Var(assembly*operator) +           
                      Q(assembly)                                       
operator              Var(Error) + 2 Var(assembly*operator) + 6         
                      Var(operator)                                     
assembly*operator     Var(Error) + 2 Var(assembly*operator)             


Two-Way Mixed-Effect Model
Auto Assembly Example
ANOVA Table

The GLM Procedure
Tests of Hypotheses for Mixed Model Analysis of Variance
 
Dependent Variable: prod   
Source                     DF    Type III SS    Mean Square   F Value
assembly                    2     156.333333      78.166667      6.54
operator                    3     189.458333      63.152778      5.29
Error                       6      71.666667      11.944444          
Error: MS(assembly*operator)

Source                 Pr > F
assembly               0.0311
operator               0.0403
Error                        
Error: MS(assembly*operator)


Source                     DF    Type III SS    Mean Square   F Value
assembly*operator           6      71.666667      11.944444      1.83
Error: MS(Error)           12      78.500000       6.541667          
Source                 Pr > F
assembly*operator      0.1764
Error: MS(Error)             


Two-Way Mixed-Effect Model
Auto Assembly Example
Hypothesis Test

 
Dependent Variable: prod   

              Tests of Hypotheses Using the Type III MS
               for assembly*operator as an Error Term
 
Source                     DF    Type III SS    Mean Square   F Value
assembly                    2    156.3333333     78.1666667      6.54

Tests of Hypotheses Using the
       Type III MS for
      assembly*operator
      as an Error Term
 
Source                 Pr > F
assembly               0.0311


Two-Way Mixed-Effect Model
Auto Assembly Example
Least Square Means

 
Least Squares Means
Standard Errors and Probabilities Calculated Using the Type III MS for 
assembly*operator as an Error Term
                              LSMEAN
assembly     prod LSMEAN      Number
1             22.1250000           1
2             27.1250000           2
3             21.3750000           3


   Least Squares Means for effect assembly
     Pr > |t| for H0: LSMean(i)=LSMean(j)
                       
           Dependent Variable: prod
i/j              1             2             3
   1                      0.0276        0.6794
   2        0.0276                      0.0159
   3        0.6794        0.0159              


NOTE: To ensure overall protection level, only probabilities associated 
      with pre-planned comparisons should be used.


Two-Way Mixed-Effect Model
Auto Assembly Example
Treatment Means
 
 
t Tests (LSD) for prod

NOTE: This test controls the Type I comparisonwise error rate, not the 
experimentwise error rate.

Alpha                            0.05
Error Degrees of Freedom            6
Error Mean Square            11.94444
Critical Value of t           2.44691
Least Significant Difference   4.2284


Means with the same letter are not significantly different.
 
t
 
G
r
o
u
p
i
n
g          Mean      N    assembly

A        27.125      8    2       
                                  
B        22.125      8    1       
B                                 
B        21.375      8    3