## Example 53.16 Using the LSMEANS Statement

Recall the main-effects model fit to the Neuralgia data set in Example 53.2. 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 LS-means for each Treatment level, the DIFF option takes all pairwise differences of the LS-means 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 p-values and confidence intervals.

```proc logistic data=Neuralgia;
class Treatment Sex / param=glm;
model Pain= Treatment|Sex Age;
oddsratio Treatment;
lsmeans Treatment / e diff oddsratio cl adjust=bon;
run;
```

The results from the ODDSRATIO statement are displayed in Output 53.16.1. 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 53.16.1 Odds Ratios from the ODDSRATIO Statement
Odds Ratio Estimates and Wald Confidence Intervals
Label 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 53.16.2 through Output 53.16.4.

The LS-means are computed by constructing each of the coefficient vectors shown in Output 53.16.2, and then computing . The LS-means are not estimates of the event probabilities; they are estimates of the linear predictors on the logit scale. In order to obtain event probabilities, you need to apply the inverse-link transformation by specifying the ILINK option in the LSMEANS statement. Notice in Output 53.16.2 that the Sex rows do not indicate either Sex=F or Sex=M. Instead, the LS-means are computed at an average of these two levels, so only one result needs to be reported. For more information about the construction of LS-means, see the section Construction of Least Squares Means of Chapter 41, The GLM Procedure.

Output 53.16.2 Treatment LS-Means 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 LS-means shown in Output 53.16.3 are all significantly nonzero at the 0.05 level. These LS-means are predicted population margins of the logits; that is, they estimate the marginal means over a balanced population, and they are effectively the within-Treatment means appropriately adjusted for the other effects in the model. The LS-means are not event probabilities; in order to obtain event probabilities, you need to apply the inverse-link transformation by specifying the ILINK option in the LSMEANS statement. For more information about LS-means, see the section LSMEANS Statement of Chapter 19, Shared Concepts and Topics.

Output 53.16.3 Treatment LS-Means
Treatment Least Squares Means
Treatment Estimate Standard Error z Value Pr > |z| Alpha Lower Upper
A 1.3195 0.6664 1.98 0.0477 0.05 0.01331 2.6257
B 1.9864 0.7874 2.52 0.0116 0.05 0.4431 3.5297
P -1.8682 0.7620 -2.45 0.0142 0.05 -3.3618 -0.3747

Pairwise differences between the Treatment LS-means, requested with the DIFF option, are displayed in Output 53.16.4. The LS-mean for the level that is displayed in the _Treatment column is subtracted from the LS-mean for the level in the Treatment column, so the first row displays the LS-mean for Treatment level A minus the LS-mean for Treatment level B. The Pr > |z| column indicates that the A and B levels are not significantly different; however, both of these levels are different from level P. If the inverse-link transformation is specified with 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 53.16.1; these are constructed at each level of the interacting covariate Sex. In contrast, there is only one LS-means odds ratio for Treatment level A versus B in Output 53.16.4. This odds ratio is computed at an average of the interacting effects by creating the vectors shown in Output 53.16.2 (the Row1 column corresponds to and the Row2 column corresponds to ) and computing .

Output 53.16.4 Differences and Odds Ratios for the Treatment LS-Means
Differences of Treatment Least Squares Means
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
Odds Ratio
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 with the LSMESTIMATE statement. In the following statements, the LS-means for the two treatments are contrasted against the LS-mean 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= Treatment|Sex Age;
lsmestimate treatment 1 0 -1, 0 1 -1 / joint;
run;
```

Output 53.16.5 displays the results from the LSMESTIMATE statement. The "Least Squares Means Estimate" 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 53.16.3. The "Chi-Square 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.

Output 53.16.5 Custom LS-Mean Tests
Least Squares Means Estimates
Effect Label Estimate Standard Error z Value Pr > |z|
Treatment Row 1 3.1877 1.0376 3.07 0.0021
Treatment Row 2 3.8547 1.2126 3.18 0.0015

Chi-Square Test for Least Squares Means
Estimates
Effect Num DF Chi-Square Pr > ChiSq
Treatment 2 12.13 0.0023

If you want to work with LS-means 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 LS-means analysis partitioned into sets that are defined by the Sex variable:

```proc logistic data=Neuralgia;
class Treatment Sex / param=glm;
model Pain= Treatment|Sex Age;
slice Treatment*Sex / sliceby=Sex diff oddsratio cl adjust=bon;
run;
```

The results for Sex=F are displayed in Output 53.16.6 and Output 53.16.7. The joint test in Output 53.16.6 tests the equality of the LS-means of the levels of Treatment for Sex=F, and rejects equality at level 0.05. In Output 53.16.7, the odds ratios and confidence intervals match those reported for Sex=F in Output 53.16.1, and multiplicity adjustments are performed.

Output 53.16.6 Joint Test of Treatment Equality for Females
Chi-Square Test for Treatment*Sex
Least Squares Means Slice
Slice Num DF Chi-Square Pr > ChiSq
Sex F 2 8.22 0.0164

Output 53.16.7 Differences of the Treatment LS-Means for Females
Simple Differences of Treatment*Sex Least Squares Means
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
Odds Ratio
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 53.16.8 and Output 53.16.9.

Output 53.16.8 Joint Test of Treatment Equality for Males
Chi-Square Test for Treatment*Sex
Least Squares Means Slice
Slice Num DF Chi-Square Pr > ChiSq
Sex M 2 6.64 0.0361

Output 53.16.9 Differences of the Treatment LS-Means for Males
Simple Differences of Treatment*Sex Least Squares Means