Recall the maineffects model fit to the Neuralgia data set in Example 60.2. The Treatment*Sex
interaction, which was previously shown to be nonsignificant, is added back into the model for this discussion.
In the following statements, the ODDSRATIO
statement is specified to produce odds ratios of pairwise differences of the Treatment
parameters in the presence of the Sex
interaction. The LSMEANS
statement is specified with several options: the E
option displays the coefficients that are used to compute the LSmeans for each Treatment
level, the DIFF
option takes all pairwise differences of the LSmeans for the levels of the Treatment
variable, the ODDSRATIO
option computes odds ratios of these differences, the CL
option produces confidence intervals for the differences and odds ratios, and the ADJUST=BON
option performs a very conservative adjustment of the pvalues and confidence intervals.
proc logistic data=Neuralgia; class Treatment Sex / param=glm; model Pain= TreatmentSex Age; oddsratio Treatment; lsmeans Treatment / e diff oddsratio cl adjust=bon; run;
The results from the ODDSRATIO
statement are displayed in Output 60.17.1. All pairwise differences of levels of the Treatment
effect are compared. However, because of the interaction between the Treatment
and Sex
variables, each difference is computed at each of the two levels of the Sex
variable. These results show that the difference between Treatment
levels A and B is insignificant for both genders.
To compute these odds ratios, you must first construct a linear combination of the parameters, , for each level that is compared with all other levels fixed at some value. For example, to compare Treatment
=A with B for Sex
=F, you fix the Age
variable at its mean, 70.05, and construct the following vectors:
Treatment 
Sex 
Treatment*Sex 


Intercept 
A 
B 
P 
F 
M 
AF 
AM 
BF 
BM 
PF 
PM 
Age 


1 
1 
0 
0 
1 
0 
1 
0 
0 
0 
0 
0 
70.05 


1 
0 
1 
0 
1 
0 
0 
0 
1 
0 
0 
0 
70.05 


0 
1 
–1 
0 
0 
0 
1 
0 
–1 
0 
0 
0 
0 
Then the odds ratio for Treatment
A versus B at Sex
=F is computed as . Different vectors must be similarly constructed when Sex
=M because the resulting odds ratio will be different due to the interaction.
Output 60.17.1: Odds Ratios from the ODDSRATIO Statement
Odds Ratio Estimates and Wald Confidence Intervals  

Odds Ratio  Estimate  95% Confidence Limits  
Treatment A vs B at Sex=F  0.398  0.016  9.722 
Treatment A vs P at Sex=F  16.892  1.269  224.838 
Treatment B vs P at Sex=F  42.492  2.276  793.254 
Treatment A vs B at Sex=M  0.663  0.078  5.623 
Treatment A vs P at Sex=M  34.766  1.807  668.724 
Treatment B vs P at Sex=M  52.458  2.258  >999.999 
The results from the LSMEANS statement are displayed in Output 60.17.2 through Output 60.17.4.
The LSmeans are computed by constructing each of the coefficient vectors shown in Output 60.17.2, and then computing . The LSmeans are not estimates of the event probabilities; they are estimates of the linear predictors on the logit scale
and therefore are estimated log odds. In order to obtain event probabilities, you need to apply the inverselink transformation
by specifying the ILINK option in the LSMEANS
statement. Notice in Output 60.17.2 that the Sex rows of the coefficient vectors do not select either Sex
=F or Sex
=M. Instead, the LSmeans are computed at an average of these two levels, so only one result needs to be reported. For more
information about the construction of LSmeans, see the section Construction of Least Squares Means in Chapter 45: The GLM Procedure.
Output 60.17.2: Treatment LSMeans Coefficients
Coefficients for Treatment Least Squares Means  

Parameter  Treatment  Sex  Row1  Row2  Row3 
Intercept: Pain=No  1  1  1  
Treatment A  A  1  
Treatment B  B  1  
Treatment P  P  1  
Sex F  F  0.5  0.5  0.5  
Sex M  M  0.5  0.5  0.5  
Treatment A * Sex F  A  F  0.5  
Treatment A * Sex M  A  M  0.5  
Treatment B * Sex F  B  F  0.5  
Treatment B * Sex M  B  M  0.5  
Treatment P * Sex F  P  F  0.5  
Treatment P * Sex M  P  M  0.5  
Age  70.05  70.05  70.05 
The Treatment
LSmeans shown in Output 60.17.3 are all significantly nonzero at the 0.05 level. These LSmeans are predicted population margins of the logits; that is, they estimate the marginal means over a balanced population, and they are effectively the withinTreatment
means appropriately adjusted for the other effects in the model. The LSmeans are not event probabilities; in order to obtain
event probabilities, you need to apply the inverselink transformation by specifying the ILINK option in the LSMEANS
statement. For more information about LSmeans, see the section LSMEANS Statement in Chapter 19: Shared Concepts and Topics.
Pairwise differences between the Treatment
LSmeans, requested with the DIFF
option, are displayed in Output 60.17.4. The LSmean for the level that is displayed in the _Treatment column is subtracted from the LSmean for the level in the
Treatment column, so the first row displays the LSmean for Treatment
level A minus the LSmean for Treatment
level B. The difference (–0.6669) is the estimated difference in log odds, or equivalently the log odds ratio of the two
treatments. The Pr > z column indicates that the A and B levels are not significantly different; however, both of these
levels are significantly different from level P. If the inverselink transformation is specified by the ILINK option, then
these differences do not transform back to differences in probabilities.
There are two odds ratios for Treatment
level A versus B in Output 60.17.1; these are constructed at each level of the interacting covariate Sex
. In contrast, there is only one LSmeans odds ratio for Treatment
level A versus B in Output 60.17.4. This odds ratio is computed at an average of the interacting effects by creating the vectors shown in Output 60.17.2 (the Row1 column corresponds to , and the Row2 column corresponds to ) and computing .
Because multiple tests are performed, you can protect yourself from falsely significant results by adjusting your pvalues for multiplicity. The ADJUST=BON option performs the very conservative Bonferroni adjustment, and adds the columns labeled with 'Adj' to Output 60.17.4. By comparing the Pr > z column to the Adj P column, you can see that the pvalues are adjusted upwards; in this case, there is no change in your conclusions. The confidence intervals are also adjusted for multiplicity—all adjusted intervals are wider than the unadjusted intervals, but again your conclusions in this example are unchanged. You can specify other adjustment methods by using the ADJUST= option.
Output 60.17.4: Differences and Odds Ratios for the Treatment LSMeans
Differences of Treatment Least Squares Means Adjustment for Multiple Comparisons: Bonferroni 


Treatment  _Treatment  Estimate  Standard Error  z Value  Pr > z  Adj P  Alpha  Lower  Upper  Adj Lower  Adj Upper  Odds Ratio  Lower Confidence Limit for Odds Ratio 
Upper Confidence Limit for Odds Ratio 
Adj Lower Odds Ratio 
Adj Upper Odds Ratio 
A  B  0.6669  1.0026  0.67  0.5059  1.0000  0.05  2.6321  1.2982  3.0672  1.7334  0.513  0.072  3.663  0.047  5.660 
A  P  3.1877  1.0376  3.07  0.0021  0.0064  0.05  1.1541  5.2214  0.7037  5.6717  24.234  3.171  185.195  2.021  290.542 
B  P  3.8547  1.2126  3.18  0.0015  0.0044  0.05  1.4780  6.2313  0.9517  6.7576  47.213  4.384  508.441  2.590  860.612 
If you want to jointly test whether the active treatments are different from the placebo, you can specify a custom hypothesis test among LSmeans by using the LSMESTIMATE statement. In the following statements, the LSmeans for the two treatments are contrasted against the LSmean of the placebo, and the JOINT option performs a joint test that the two treatments are not different from placebo.
proc logistic data=Neuralgia; class Treatment Sex / param=glm; model Pain= TreatmentSex Age; lsmestimate treatment 1 0 1, 0 1 1 / joint; run;
Output 60.17.5 displays the results from the LSMESTIMATE statement. The "Least Squares Means Estimates" table displays the differences of the two active treatments against the placebo, and the results are identical to the second and third rows of Output 60.17.4. The "ChiSquare Test for Least Squares Means Estimates" table displays the joint test. In all of these tests, you reject the null hypothesis that the treatment has the same effect as the placebo.
If you want to work with LSmeans but you prefer to compute the Treatment
odds ratios within the Sex
levels in the same fashion as the ODDSRATIO
statement does, you can specify the SLICE
statement. In the following statements, you specify the same options in the SLICE
statement as you do in the LSMEANS
statement, except that you also specify the SLICEBY=
option to perform an LSmeans analysis partitioned into sets that are defined by the Sex
variable:
proc logistic data=Neuralgia; class Treatment Sex / param=glm; model Pain= TreatmentSex Age; slice Treatment*Sex / sliceby=Sex diff oddsratio cl adjust=bon; run;
The results for Sex=F are displayed in Output 60.17.6 and Output 60.17.7. The joint test in Output 60.17.6 tests the equality of the LSmeans of the levels of Treatment
for Sex=F, and rejects equality at level 0.05. In Output 60.17.7, the odds ratios and confidence intervals match those reported for Sex=F in Output 60.17.1, and multiplicity adjustments are performed.
Output 60.17.7: Differences of the Treatment LSMeans for Females
Simple Differences of Treatment*Sex Least Squares Means Adjustment for Multiple Comparisons: Bonferroni 


Slice  Treatment  _Treatment  Estimate  Standard Error  z Value  Pr > z  Adj P  Alpha  Lower  Upper  Adj Lower  Adj Upper  Odds Ratio  Lower Confidence Limit for Odds Ratio 
Upper Confidence Limit for Odds Ratio 
Adj Lower Odds Ratio 
Adj Upper Odds Ratio 
Sex F  A  B  0.9224  1.6311  0.57  0.5717  1.0000  0.05  4.1193  2.2744  4.8272  2.9824  0.398  0.016  9.722  0.008  19.734 
Sex F  A  P  2.8269  1.3207  2.14  0.0323  0.0970  0.05  0.2384  5.4154  0.3348  5.9886  16.892  1.269  224.838  0.715  398.848 
Sex F  B  P  3.7493  1.4933  2.51  0.0120  0.0361  0.05  0.8225  6.6761  0.1744  7.3243  42.492  2.276  793.254  1.190  >999.999 
Similarly, the results for Sex=M are shown in Output 60.17.8 and Output 60.17.9.
Output 60.17.9: Differences of the Treatment LSMeans for Males
Simple Differences of Treatment*Sex Least Squares Means Adjustment for Multiple Comparisons: Bonferroni 


Slice  Treatment  _Treatment  Estimate  Standard Error  z Value  Pr > z  Adj P  Alpha  Lower  Upper  Adj Lower  Adj Upper  Odds Ratio  Lower Confidence Limit for Odds Ratio 
Upper Confidence Limit for Odds Ratio 
Adj Lower Odds Ratio 
Adj Upper Odds Ratio 
Sex M  A  B  0.4114  1.0910  0.38  0.7061  1.0000  0.05  2.5496  1.7268  3.0231  2.2003  0.663  0.078  5.623  0.049  9.028 
Sex M  A  P  3.5486  1.5086  2.35  0.0187  0.0560  0.05  0.5919  6.5054  0.06286  7.1601  34.766  1.807  668.724  0.939  >999.999 
Sex M  B  P  3.9600  1.6049  2.47  0.0136  0.0408  0.05  0.8145  7.1055  0.1180  7.8021  52.458  2.258  >999.999  1.125  >999.999 