The data, log and output from SAS for the Nested/Subsampling example from the course notes are given below. This will be updated from time to time!
data subsamp1;
input trt tree apple wt;
cards;
1 1 1 313.063
1 1 2 329.132
1 1 3 334.278
1 1 4 330.088
1 1 5 334.987
1 1 6 325.075
1 2 1 333.936
1 2 2 326.155
1 2 3 352.854
1 2 4 350.791
1 2 5 318.560
1 2 6 323.473
1 3 1 345.494
1 3 2 349.296
1 3 3 339.190
1 3 4 338.942
1 3 5 331.370
1 3 6 339.097
1 4 1 340.840
1 4 2 336.798
1 4 3 313.810
1 4 4 333.880
1 4 5 343.068
1 4 6 319.171
2 1 1 349.271
2 1 2 336.695
2 1 3 352.797
2 1 4 348.486
2 1 5 352.077
2 1 6 341.423
2 2 1 356.880
2 2 2 356.256
2 2 3 364.950
2 2 4 360.570
2 2 5 362.104
2 2 6 371.829
2 3 1 324.161
2 3 2 340.130
2 3 3 334.580
2 3 4 342.813
2 3 5 327.415
2 3 6 333.571
2 4 1 338.742
2 4 2 340.348
2 4 3 362.837
2 4 4 340.782
2 4 5 348.730
2 4 6 325.444
3 1 1 387.868
3 1 2 372.807
3 1 3 380.505
3 1 4 391.804
3 1 5 388.935
3 1 6 361.860
3 2 1 377.948
3 2 2 380.033
3 2 3 361.913
3 2 4 363.098
3 2 5 365.375
3 2 6 382.121
3 3 1 363.583
3 3 2 387.727
3 3 3 373.021
3 3 4 362.931
3 3 5 378.928
3 3 6 364.442
3 4 1 374.851
3 4 2 361.291
3 4 3 377.389
3 4 4 366.722
3 4 5 374.187
3 4 6 380.383
;
proc glm; /* ANOVA ignoring trees (wrong!) */
classes trt;
model wt = trt;
lsmeans trt/stderr pdiff;
contrast ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
estimate ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
run;
proc glm; /* Nested ANOVA, testing trt against tree Mean Square */
classes trt tree;
model wt = trt tree(trt)/xpx solution;
random tree(trt); /* Specifying tree within trt as a random effect */
test h=trt e=tree(trt); /* Explicitly testing MS trt against MS tree */
lsmeans trt/stderr pdiff e=tree(trt); /* Least Squares Means, using
MS tree as the appropriate error */
contrast ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5 tree(trt) .25 .25 .25 .25
-.125 -.125 -.125 -.125 -.125 -.125 -.125 -.125/e=tree(trt);
estimate ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5 tree(trt) .25 .25 .25 .25
-.125 -.125 -.125 -.125 -.125 -.125 -.125 -.125;
/* Explicitly construct contrasts for SS trt (Type I) */
contrast ' SS trt' trt 4 -4 0 tree(trt) 1 1 1 1 -1 -1 -1 -1 0 0 0 0,
trt 4 - -4 tree(trt) 1 1 1 1 0 0 0 0 -1 -1 -1 -1;
run;
proc mixed; /* Mixed model analysis, as it should be done */
classes trt tree;
model wt = trt; /* specify only fixed effects */
random tree(trt); /* specify tree within trt as a random effect */
lsmeans trt;
contrast ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
estimate ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
run;
proc sort; /* sort data, by trt and tree within trt */
by trt tree;
run;
proc means; /* compute mean for each tree, and output to a new dataset */
var wt;
by trt tree;
output out=pmeans mean=gmean;
run;
proc glm data=pmeans; /* Analyse tree means, using a 1-way ANOVA */
classes trt;
model gmean = trt;
lsmeans trt/stderr pdiff;
contrast ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
estimate ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
run;
The SAS log provides a lot of useful information about the data step and the various PROCedures. We can see that SAS read in 72 observations, which agrees with what we would expect (3 treatments, 4 trees per treatment and 6 apples per tree). There are no error messages; which does not necessarily mean everything is correct, only that SAS did not detect anything wrong! If there are any error messages they should not be ignored, we should correct the errors before proceeding.
NOTE: Copyright (c) 1989-1996 by SAS Institute Inc., Cary, NC, USA.
NOTE: SAS (r) Proprietary Software Release 6.12 TS020
Licensed to MCGILL UNIVERSITY COMPUTING CENTRE, Site 0009211001.
5
6 data subsamp1;
7 input trt tree apple wt;
8 cards;
NOTE: The data set WORK.SUBSAMP1 has 72 observations and 4 variables.
NOTE: The DATA statement used 2.31 seconds.
81 ;
82 proc glm; /* ANOVA ignoring trees (wrong!) */
83 classes trt;
84 model wt = trt; <--- Analysis 1
85 lsmeans trt/stderr pdiff;
86 contrast ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
87 estimate ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
88 run;
89
NOTE: The PROCEDURE GLM used 3.37 seconds.
<--- Analysis 2
90 proc glm; /* Nested ANOVA, testing trt against tree Mean Square */
91 classes trt tree;
92 model wt = trt tree(trt)/xpx solution;
93 random tree(trt); /* Specifying tree within trt as a random effect */
94 test h=trt e=tree(trt); /* Explicitly testing MS trt against MS tree */
95 lsmeans trt/stderr pdiff e=tree(trt); /* Least Squares Means, using
96 MS tree as the appropriate
error */
97 contrast ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5 tree(trt) .25 .25 .25 .25
-.125 -.125 -.125 -.125 -.125 -.125 -.125 -.125/e=tree(trt);
98 estimate ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5 tree(trt) .25 .25 .25 .25
-.125 -.125 -.125 -.125 -.125 -.125 -.125 -.125;
99 run;
NOTE: TYPE I EMS not available without the E1 option.
100
NOTE: The PROCEDURE GLM used 2.33 seconds.
<--- Analysis 3
101 proc mixed; /* Mixed model analysis, as it should be done */
102 classes trt tree;
103 model wt = trt; /* specify only fixed effects */
104 random tree(trt); /* specify tree within trt as a random effect */
105 lsmeans trt;
106 contrast ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
107 estimate ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
108 run;
NOTE: The PROCEDURE MIXED used 2.58 seconds.
109
110 proc sort; /* sort data, by trt and tree within trt */
111 by trt tree;
112 run;
NOTE: The data set WORK.SUBSAMP1 has 72 observations and 4 variables.
NOTE: The PROCEDURE SORT used 0.81 seconds.
113
114 proc means; /* compute mean for each tree, and output to a new dataset */
115 var wt;
116 by trt tree;
117 output out=pmeans mean=gmean;
118 run;
NOTE: The data set WORK.PMEANS has 12 observations and 5 variables.
NOTE: The PROCEDURE MEANS used 1.5 seconds.
119
120 proc glm data=pmeans; /* Analyse tree means, using a 1-way ANOVA */
121 classes trt;
122 model gmean = trt; <--- Analysis 4
123 lsmeans trt/stderr pdiff;
124 contrast ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
125 estimate ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
126 run;
The SAS System 1
General Linear Models Procedure
Class Level Information
Analysis 1
Class Levels Values
TRT 3 1 2 3
Number of observations in data set = 72
We can see that in this analysis (the first analysis), SAS used 72 observations and that there was 1 CLASS variable (trt) with 3 levels (1, 2 and 3).
The SAS System 2
General Linear Models Procedure
Dependent Variable: WT
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 20747.03666 10373.51833 80.42 0.0001 Error 69 8900.87204 128.99815 Corrected Total 71 29647.90869
R-Square C.V. Root MSE WT Mean
0.699781 3.232757 11.35774 351.3328
Source DF Type I SS Mean Square F Value Pr > F
TRT 2 20747.03666 10373.51833 80.42 0.0001
Source DF Type III SS Mean Square F Value Pr > F
TRT 2 20747.03666 10373.51833 80.42 0.0001
Here we have the basic ANOVA table, shown at the top. The model is actually, as discussed before, the Model corrected for the Mean, R(trt | µ). Then underneath the basic ANOVA table is the subdivision into the various component effects. In this case since there is only 1 effect (treatment) in the model there is in fact nothing more to subdivide, which explains why the Type I and Type III Sums of Squares are the same.
The SAS System 3
General Linear Models Procedure
Least Squares Means
TRT WT Std Err Pr > |T| LSMEAN
LSMEAN LSMEAN H0:LSMEAN=0 Number
1 333.472833 2.318388 0.0001 1
2 346.370458 2.318388 0.0001 2
3 374.155083 2.318388 0.0001 3
Pr > |T| H0: LSMEAN(i)=LSMEAN(j)
i/j 1 2 3
1 . 0.0002 0.0001
2 0.0002 . 0.0001
3 0.0001 0.0001 .
NOTE: To ensure overall protection level, only probabilities associated with
pre-planned comparisons should be used.
Here we see the Least Squares Means for each of the treatment effects; which are simply µ + trti, together with the standard errors (2.32) for each mean. Under the table of Least Squares Means we have the table of probabilities associated with comparisons amongst all pairs of treatments. Note, however, the caveat that only pre-planned comparisons really have the associated probability levels.
The SAS System 4
General Linear Models Procedure Dependent Variable: WT Contrast DF Contrast SS Mean Square F Value Pr > F
trt 1 - (trt 2+3)/2 1 11483.21202 11483.21202 89.02 0.0001
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
trt 1 - (trt 2+3)/2 -26.7899375 -9.43 0.0001 2.83943376
Here we see the results of the specific CONTRAST between treatment 1 and the average of treatments 2 and 3 that we had also requested. Note that the Sums of Squares, Mean Squares and F-ratio are not the same as those for the comparisons between treatments in the ANOVA; this is a slightly different question which begets a slightly different answer! Also, we have the ESTIMATE of this difference, together with it's computed standard error. Note the estimate and the standard error; we will return see the corresponding values from the other 3 analyses.
The SAS System 5
General Linear Models Procedure
Class Level Information
Analysis 2
Class Levels Values
TRT 3 1 2 3
TREE 4 1 2 3 4
Number of observations in data set = 72
Again, we can see that there were 72 observations in this analysis and that there were 3 treatments, the 3 levels of trt. It might appear that there were only 4 trees (whereas we know that there were 12). This is because we (I!) had labeled the trees 1 to 4 within each treatment, so at this stage (we are only at the CLASSES line specification) we have not yet fitted a model and PROC GLM has no way of knowing that we are going to fit a nested model.
The SAS System 6
General Linear Models Procedure
Matrix Element Representation
Dependent Variable: WT
Effect Representation
INTERCEPT INTERCEPT
TRT 1 TRT 1
2 TRT 2
3 TRT 3
TREE(TRT) 1 1 DUMMY001
2 1 DUMMY002
3 1 DUMMY003
4 1 DUMMY004
1 2 DUMMY005
2 2 DUMMY006
3 2 DUMMY007
4 2 DUMMY008
1 3 DUMMY009
2 3 DUMMY010
3 3 DUMMY011
4 3 DUMMY012
Here we can see how we can see exactly what order SAS uses to arrange the various levels; useful for when we are specifying the various coefficients to the CONTRAST and/or ESTIMATE statements.
The SAS System 7
General Linear Models Procedure
The X'X Matrix
INTERCEPT TRT 1 TRT 2 TRT 3
INTERCEPT 72 24 24 24 TRT 1 24 24 0 0 TRT 2 24 0 24 0 TRT 3 24 0 0 24 DUMMY001 6 6 0 0 DUMMY002 6 6 0 0 DUMMY003 6 6 0 0 DUMMY004 6 6 0 0 DUMMY005 6 0 6 0 DUMMY006 6 0 6 0 DUMMY007 6 0 6 0 DUMMY008 6 0 6 0 DUMMY009 6 0 0 6 DUMMY010 6 0 0 6 DUMMY011 6 0 0 6 DUMMY012 6 0 0 6 WT 25295.961 8003.348 8312.891 8979.722
DUMMY001 DUMMY002 DUMMY003 DUMMY004
INTERCEPT 6 6 6 6 TRT 1 6 6 6 6 TRT 2 0 0 0 0 TRT 3 0 0 0 0 DUMMY001 6 0 0 0 DUMMY002 0 6 0 0 DUMMY003 0 0 6 0 DUMMY004 0 0 0 6 DUMMY005 0 0 0 0
The SAS System 8
General Linear Models Procedure
The X'X Matrix
DUMMY001 DUMMY002 DUMMY003 DUMMY004
DUMMY006 0 0 0 0 DUMMY007 0 0 0 0 DUMMY008 0 0 0 0 DUMMY009 0 0 0 0 DUMMY010 0 0 0 0 DUMMY011 0 0 0 0 DUMMY012 0 0 0 0 WT 1966.623 2005.769 2043.389 1987.567
DUMMY005 DUMMY006 DUMMY007 DUMMY008
INTERCEPT 6 6 6 6 TRT 1 0 0 0 0 TRT 2 6 6 6 6 TRT 3 0 0 0 0 DUMMY001 0 0 0 0 DUMMY002 0 0 0 0 DUMMY003 0 0 0 0 DUMMY004 0 0 0 0 DUMMY005 6 0 0 0 DUMMY006 0 6 0 0 DUMMY007 0 0 6 0 DUMMY008 0 0 0 6 DUMMY009 0 0 0 0 DUMMY010 0 0 0 0 DUMMY011 0 0 0 0 DUMMY012 0 0 0 0 WT 2080.749 2172.589 2002.67 2056.883
The SAS System 9
General Linear Models Procedure
The X'X Matrix
DUMMY009 DUMMY010 DUMMY011 DUMMY012
INTERCEPT 6 6 6 6 TRT 1 0 0 0 0 TRT 2 0 0 0 0 TRT 3 6 6 6 6 DUMMY001 0 0 0 0 DUMMY002 0 0 0 0 DUMMY003 0 0 0 0 DUMMY004 0 0 0 0 DUMMY005 0 0 0 0 DUMMY006 0 0 0 0 DUMMY007 0 0 0 0 DUMMY008 0 0 0 0 DUMMY009 6 0 0 0 DUMMY010 0 6 0 0 DUMMY011 0 0 6 0 DUMMY012 0 0 0 6 WT 2283.779 2230.488 2230.632 2234.823
WT
INTERCEPT 25295.961 TRT 1 8003.348 TRT 2 8312.891 TRT 3 8979.722 DUMMY001 1966.623 DUMMY002 2005.769 DUMMY003 2043.389 DUMMY004 1987.567 DUMMY005 2080.749
The SAS System 10
General Linear Models Procedure
The X'X Matrix
WT
DUMMY006 2172.589 DUMMY007 2002.67 DUMMY008 2056.883 DUMMY009 2283.779 DUMMY010 2230.488 DUMMY011 2230.632 DUMMY012 2234.823 WT 8916948.5047
The SAS System 11
General Linear Models Procedure
Dependent Variable: WT
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 11 24127.21374 2193.38307 23.84 0.0001 Error 60 5520.69496 92.01158 Corrected Total 71 29647.90869
Here we have the Basic ANOVA; the Model (actually the Model over and above the Mean) and the Residual. We can see that Treatment and/or Tree within Treatment has a significant effect.
R-Square C.V. Root MSE WT Mean
0.813791 2.730251 9.592267 351.3328
Note also, that the Wt Mean (y bar), is exactly the same as in the previous model, and similarly the Corrected Total Sums of Squares. These will not change regardless of the model fitted.
Source DF Type I SS Mean Square F Value Pr > F
TRT 2 20747.03666 10373.51833 112.74 0.0001 TREE(TRT) 9 3380.17708 375.57523 4.08 0.0004
Source DF Type III SS Mean Square F Value Pr > F
TRT 2 20747.03666 10373.51833 112.74 0.0001 TREE(TRT) 9 3380.17708 375.57523 4.08 0.0004
Here we can see the Sums of Squares for Treatments and Trees within treatments. Compare the Type I Sums of Squares for Treatments with the Sums of Squares for Treatments from Analysis 1; we see that they are both the same (20747). This is because in Analysis 1 the only effect that we fitted was that for treaments, so the effect of treatment was R(trt | µ). In the analysis (2) the Type I Sums of Squares for treaments are the Sums of Squares due to fitting Treatment after the Mean, but not correcting for anything else, and hence we still have R(trt | µ), and hence we still have exactly the same Sums of Squares.
The SAS System 12
General Linear Models Procedure
Dependent Variable: WT
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 372.4705000 B 95.11 0.0001 3.91602653
TRT 1 -41.2093333 B -7.44 0.0001 5.53809783
2 -29.6566667 B -5.36 0.0001 5.53809783
3 0.0000000 B . . .
TREE(TRT) 1 1 -3.4906667 B -0.63 0.5309 5.53809783
2 1 3.0336667 B 0.55 0.5859 5.53809783
3 1 9.3036667 B 1.68 0.0982 5.53809783
4 1 0.0000000 B . . .
1 2 3.9776667 B 0.72 0.4754 5.53809783
2 2 19.2843333 B 3.48 0.0009 5.53809783
3 2 -9.0355000 B -1.63 0.1080 5.53809783
4 2 0.0000000 B . . .
1 3 8.1593333 B 1.47 0.1459 5.53809783
2 3 -0.7225000 B -0.13 0.8966 5.53809783
3 3 -0.6985000 B -0.13 0.9001 5.53809783
4 3 0.0000000 B . . .
NOTE: The X'X matrix has been found to be singular and a generalized inverse
was used to solve the normal equations. Estimates followed by the
letter 'B' are biased, and are not unique estimators of the parameters.
Here we have the solution vector, since we requested it in the model statement as one of the options. SAS tells us that the 'so-called' estimates are in fact biased, they are only solutions and not estimates since X'X is not of full rank and hence on unique inverse exists.
The SAS System 13
General Linear Models Procedure Source Type III Expected Mean Square
TRT Var(Error) + 6 Var(TREE(TRT)) + Q(TRT) TREE(TRT) Var(Error) + 6 Var(TREE(TRT))
We had specified that Tree within treatment was a random effect,
random tree(trt); /* Specifying tree within trt as a random effect */
so GLM provides a Table of the Expectations of the Mean Squares. This can help in deciding just which Mean Square should be tested against which. We can thus see that the Mean Square for Trt should be tested againt the Mean Square for Tree(Trt), since they differ only in the Q(Trt) component. In addition, if we want to we can use these coefficients to help us compute the variance due to trees, Var(Tree(Trt)).
The Mean Square Error = 92.01158. The Mean Square for Trees within Treatments = 375.57. Therefore an unbiased estimate of the Tree variance is:
(375.57523 - 92.01158)/6 = 47.26
The SAS System 14
General Linear Models Procedure
Least Squares Means
Standard Errors and Probabilities calculated using the Type III MS for
TREE(TRT) as an Error term
TRT WT Std Err Pr > |T| LSMEAN
LSMEAN LSMEAN H0:LSMEAN=0 Number
1 333.472833 3.955878 0.0001 1
2 346.370458 3.955878 0.0001 2
3 374.155083 3.955878 0.0001 3
Pr > |T| H0: LSMEAN(i)=LSMEAN(j)
i/j 1 2 3
1 . 0.0466 0.0001
2 0.0466 . 0.0008
3 0.0001 0.0008 .
NOTE: To ensure overall protection level, only probabilities associated with
pre-planned comparisons should be used.
The Least Squares Means, standard errors and the table of probabilities of each pair-wise comparison are shown above; we had requested the LSMEANS for TRT and we had specified that the Error Mean Square to be used is not the default MSE, but rather the Mean Square for Trees within Treatment.
lsmeans trt/stderr pdiff e=tree(trt); /* Least Squares Means, using
MS tree as the appropriate error */
Note the considerably larger standard errors to the LSMeans and the huge difference in the probability associated with the difference between treatment 1 and treatment 2 ( PR 0.0466 compared to 0.0002 in Analysis 1). This illustrates, again, the misinformation that can result from not correctly specifying the correct error term for an analysis using GLM. If we had used PROC MIXED this would have been accommodated 'automagically'.
Again, as in Analysis 1, SAS provides the warning that only the pre-planned comparisons are valid. This is much like the health warning on cigarette packets; largely ignored, and likewise over the long term equally damaging to your health (mental!).
The SAS System 15
General Linear Models Procedure Dependent Variable: WT Tests of Hypotheses using the Type III MS for TREE(TRT) as an error term Source DF Type III SS Mean Square F Value Pr > F
TRT 2 20747.03666 10373.51833 27.62 0.0001
Tests of Hypotheses using the Type III MS for TREE(TRT) as an error term Contrast DF Contrast SS Mean Square F Value Pr > F
trt 1 - (trt 2+3)/2 1 11483.21202 11483.21202 30.57 0.0004
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
trt 1 - (trt 2+3)/2 -26.7899375 -11.17 0.0001 2.39806670
We had requested a hypothesis test to test the Hypothesis of Treatment effects using Tree within Treatment as the appropriate Error term.
test h=trt e=tree(trt); /* Explicitly testing MS trt against MS tree */
The F-value (27.62) is the Mean Square for Trt (10373.5) divided by the Mean Square for Trees within Trt (375.6), with 2 d.f. for the numerator and 9 d.f. for the denominator.
We also have the test of hypothesis associated with the test of treatment 1 vs the average of treatment 2 and 3.
contrast ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5/e=tree(trt);
Note that here again we have specified that the Mean Square for Tree(Trt) is to be used as the appropriate Error term.
We had also asked for the ESTIMATE of the difference between treatment 1 and the average of treatments 2 and 3.
estimate ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5;
The estimate was -26.8 ± 2.398. This standard error is the same as from our 1st analysis; not because it is correct, but rather because the GLM ESTIMATE statement (designed for fixed effects analyses) does not allow us to specify an error term other than the residual. So even though in the CONTRAST statement above we have declared that the Mean Square between Trees with Treatments is the appropriate Error term to be used in testing Treatments we cannot specify that here with the Estimate statement and the Residual Mean Square is (incorrectly) used. This is just one of the various problems associated with using PROC GLM for the analysis of models with Random Effects.
The SAS System 16
The MIXED Procedure
Analysis 3
Class Level Information
Class Levels Values
TRT 3 1 2 3
TREE 4 1 2 3 4
REML Estimation Iteration History
Iteration Evaluations Objective Criterion
0 1 413.86022542
1 1 403.20515872 0.00000000
Convergence criteria met.
Note that 2 iterations were required to obtain convergence, the initial iteration 0, and another iteration (1). since this was a balanced analysis with only 2 factors in the model (Trt and Tree) it converged very quickly. With other more complicated models it may take 5 or 10 iterations to converge.
Covariance Parameter Estimates (REML)
Cov Parm Estimate
TREE(TRT) 47.26060809
Residual 92.01158261
Note that the estimates of the variance components agree with those presented above from Analysis 2. This is because with a completely balanced experiment the ANOVA from Analysis 3 gives us the same results as PROC MIXED. Otherwise we would not get the same answers, PROC MIXED would be the correct approach to use.
Model Fitting Information for WT
Description Value
Observations 72.0000
Res Log Likelihood -265.009
Akaike's Information Criterion -267.009
Schwarz's Bayesian Criterion -269.243
The SAS System 17
Model Fitting Information for WT
Description Value
-2 Res Log Likelihood 530.0187
Tests of Fixed Effects
Source NDF DDF Type III F Pr > F
TRT 2 9 27.62 0.0001
ESTIMATE Statement Results
Parameter Estimate Std Error DF t Pr > |t|
trt 1 - (trt 2+3)/2 -26.78993750 4.84494086 9 -5.53 0.0004
CONTRAST Statement Results
Source NDF DDF F Pr > F
trt 1 - (trt 2+3)/2 1 9 30.57 0.0004
Least Squares Means
Effect TRT LSMEAN Std Error DF t Pr > |t|
TRT 1 333.47283333 3.95587765 9 84.30 0.0001
TRT 2 346.37045833 3.95587765 9 87.56 0.0001
TRT 3 374.15508333 3.95587765 9 94.58 0.0001
The SAS System 18
Analysis Variable : WT
--------------------------------- TRT=1 TREE=1 -------------------------------
N Mean Std Dev Minimum Maximum
----------------------------------------------------------
6 327.7705000 8.0650628 313.0630000 334.9870000
----------------------------------------------------------
--------------------------------- TRT=1 TREE=2 -------------------------------
N Mean Std Dev Minimum Maximum
----------------------------------------------------------
6 334.2948333 14.4751434 318.5600000 352.8540000
----------------------------------------------------------
--------------------------------- TRT=1 TREE=3 -------------------------------
N Mean Std Dev Minimum Maximum
----------------------------------------------------------
6 340.5648333 6.1927894 331.3700000 349.2960000
----------------------------------------------------------
--------------------------------- TRT=1 TREE=4 -------------------------------
N Mean Std Dev Minimum Maximum
----------------------------------------------------------
6 331.2611667 11.9948782 313.8100000 343.0680000
----------------------------------------------------------
--------------------------------- TRT=2 TREE=1 -------------------------------
N Mean Std Dev Minimum Maximum
----------------------------------------------------------
6 346.7915000 6.3840723 336.6950000 352.7970000
----------------------------------------------------------
--------------------------------- TRT=2 TREE=2 -------------------------------
N Mean Std Dev Minimum Maximum
----------------------------------------------------------
6 362.0981667 5.7709197 356.2560000 371.8290000
----------------------------------------------------------
--------------------------------- TRT=2 TREE=3 -------------------------------
N Mean Std Dev Minimum Maximum
----------------------------------------------------------
6 333.7783333 7.1503796 324.1610000 342.8130000
----------------------------------------------------------
--------------------------------- TRT=2 TREE=4 -------------------------------
N Mean Std Dev Minimum Maximum
----------------------------------------------------------
6 342.8138333 12.3646904 325.4440000 362.8370000
----------------------------------------------------------
--------------------------------- TRT=3 TREE=1 -------------------------------
N Mean Std Dev Minimum Maximum
----------------------------------------------------------
6 380.6298333 11.4869300 361.8600000 391.8040000
----------------------------------------------------------
--------------------------------- TRT=3 TREE=2 -------------------------------
N Mean Std Dev Minimum Maximum
----------------------------------------------------------
6 371.7480000 9.2395369 361.9130000 382.1210000
----------------------------------------------------------
--------------------------------- TRT=3 TREE=3 -------------------------------
N Mean Std Dev Minimum Maximum
----------------------------------------------------------
6 371.7720000 10.0626265 362.9310000 387.7270000
----------------------------------------------------------
--------------------------------- TRT=3 TREE=4 -------------------------------
N Mean Std Dev Minimum Maximum
----------------------------------------------------------
6 372.4705000 7.1195354 361.2910000 380.3830000
----------------------------------------------------------
The SAS System 19
General Linear Models Procedure
Class Level Information
Analysis 4
Class Levels Values
TRT 3 1 2 3
Number of observations in data set = 12
The SAS System 20
General Linear Models Procedure
Dependent Variable: GMEAN
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 3457.839443 1728.919721 27.62 0.0001 Error 9 563.362847 62.595872 Corrected Total 11 4021.202290
R-Square C.V. Root MSE GMEAN Mean
0.859902 2.251926 7.911755 351.3328
Source DF Type I SS Mean Square F Value Pr > F
TRT 2 3457.839443 1728.919721 27.62 0.0001
Source DF Type III SS Mean Square F Value Pr > F
TRT 2 3457.839443 1728.919721 27.62 0.0001
Note that we have only 12 observations (the 12 tree means) in this analysis. The Sums of Squares, etc are not the same as in the previous analyses, but the F-ratio for the test of Treatments is 27.62, exactly the same as in Analysis 3 (from PROC MIXED) and Analysis 2 (PROC GLM, but specifying that Tree was a random effect nested within Trt) and the statistical significance is also correct. Note also the agreement with the standard errors of the Least Squares Means and the ESTIMATE statement.
The SAS System 21
General Linear Models Procedure
Least Squares Means
TRT GMEAN Std Err Pr > |T| LSMEAN
LSMEAN LSMEAN H0:LSMEAN=0 Number
1 333.472833 3.955878 0.0001 1
2 346.370458 3.955878 0.0001 2
3 374.155083 3.955878 0.0001 3
Pr > |T| H0: LSMEAN(i)=LSMEAN(j)
i/j 1 2 3
1 . 0.0466 0.0001
2 0.0466 . 0.0008
3 0.0001 0.0008 .
NOTE: To ensure overall protection level, only probabilities associated with
pre-planned comparisons should be used.
The SAS System 22
General Linear Models Procedure Dependent Variable: GMEAN Contrast DF Contrast SS Mean Square F Value Pr > F
trt 1 - (trt 2+3)/2 1 1913.868670 1913.868670 30.57 0.0004
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
trt 1 - (trt 2+3)/2 -26.7899375 -5.53 0.0004 4.84494086
