3.5 - SAS Output for ANOVA

The first set of output includes descriptions of the ANOVA run. These are somewhat trivial here, but will become important as we work with more complicated situations.

ANOVA of Greenhouse Data

The Mixed Procedure

Model Information
Data Set WORK.GREENHOUSE
Dependent Variable Height
Covariance Structure Diagonal
Estimation Method Type 3
Residual Variance Method Factor
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Residual
Class Level Information
Class Levels Values
fert 4 Control F1 F2 F3
Dimensions
Covariance Parameters 1
Columns in X 5
Columns in Z 0
Subjects 0
Max Obs Per Subject 24

The main output we are interested in is the Type 3 Analysis of Variance. It does not include the Total SS, but other than that we get the same ANOVA table that we developed earlier.

Type 3 Analysis of Variance
Sources DF Sum of Squares Mean Square Expected Mean Square Error Term Error DF F Value Pr > F
fert 3 251.440000 83.813333 Var(Residual)+Q(fert) MS(Residual) 20 27.46 <.0001
Residual 20 61.033333 3.051667 Var(Residual)        
Covariance Parameter Estimates
Cov Parm Estimate
Residual 3.0517
Fit Statistics
-2 Res Log Likelihood 86.2
AIC (smaller is better) 88.2
AICC (smaller is better) 88.5
BIC (smaller is better) 89.2
Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
fert 3 20 27.46 <.0001

The next output that we should discuss is that labeled as “Type 3 Tests of Fixed Effects”. This is the standard output we would get if we weren’t specifying “method = type 3”. This is the method SAS uses to calculate the ANOVA (if this is confusing don't worry, we haven't talked about this yet). These are the Fcalculated and p-value for the test of any variables that are specified in the model statement. Note that the F and p-values are identical to that which we see in the full ANOVA table.

Diagnostic Plots

The Mixed procedure produces the following diagnostic plots:

sas output
sas output

We included the LSmeans statement in the PLM procedure and so we get these:

fert Least Squares Means
fert Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper
Control 21.0000 0.7132 20 29.45 <.0001 0.05 19.5124 22.4876
F1 28.6000 0.7132 20 40.10 <.0001 0.05 27.1124 30.0876
F2 25.8667 0.7132 20 36.27 <.0001 0.05 24.3790

27.3543

F3 29.2000 0.7132 20 40.94 <.001 0.05 27.7124 30.6876

In the "Least Squares Means" table note that the t-value and P > t are testing Null hypotheses that the estimate = 0. (They aren't usually very important). The Lower and Upper values are the 95% confidence limits for the estimates. You may also note that there is a standard error listed after the estimates of the Least Squares Means. Here in this example we see only one value here for all the treatment levels. The reason is that these standard error are based on the residual variance estimate (MSE). The LS means will be the same as the original arithmetic means that we got in the Summary procedure because we have equal sample sizes.

With unequal sample sizes or if there is a covariate present, the LSmeans can differ from the original sample means. Plots of final results should be made with the LSmeans as shown below.. When we make plots of these, we are constrained by the software to only use error bars showing the confidence limits.

sas output

The results of the Tukey test appear in the "Difference of Least Squares Means". Notice this is different than the previous table because this table is testing each pairwise comparison. For example, the first row compares the control to the F1. We interpret this output as we would any other confidence interval for two means. The confidence interval does not contain zero, so we reject the null and conclude the difference is not zero.

Differences of fert Least Squares Means

Adjustment for Multiple Comparisons: Tukey

fert _fert Estimate Standard Error DF t Value  Pr > |t| Adj P Alpha Lower Upper Adj Lower Adj Upper
Control F1 -7.6000 1.0086 20 -7.54 <.0001 <.0001 0.05 -9.7038 -5.4962 -10.4229 -4.7771
Control F2 -4.8667 1.0086 20 -4.83 0.0001 0.0006 0.05 -6.9705 -2.7628 -7.6896 -2.0438
Control F3 -8.2000 1.0086 20 -8.13 <.0001 <.0001 0.05 -10.3038 -6.0962 -11.0229 -5.3771
F1 F2 2.7333 1.0086 20 2.71 0.0135 0.0599 0.05 0.6295 4.8372 -0.08957 5.5562
F1 F3 -0.6000 1.0086 20 -0.59 0.5586 0.9324 0.05 -2.7038 1.5038 -3.4229 2.2229
F2 F3 -3.3333 1.0086 20 -3.30 0.0035 .0171 0.05 -5.4372 -1.2295 -6.1562 -0.5104

Tukey Grouping for fert

Least Squares Means (Alpha=0.05)

LS means with the same letter are not significantly different
fert Estimate    
F3 29.2000   A
      A
F1 28.6000 B A
    B  
F2 25.8667 B  
       
Control 21.000   C

The lettering here can be used now to label the LSmeans, using the Graphics Editor. Although this may seem tedious, Minitab will not produce the means plot using the LSmeans. Minitab makes a means plot, but but only from the raw data (using the original sample means).