6.2. Cumulative Logit Model: Example

The cumulative logit model (McCullagh 1980) could hardly be easier to implement in SAS. Whenever PROC LOGISTIC encounters more than two categories on the dependent variable, it automatically estimates a cumulative logit model. Let’s try it and see what we get. Then we’ll step back and discuss the underlying model.

In Chapter 5, we analyzed data in which the dependent variable was the response to the question, “If you found a wallet on the street, would you (1) keep the wallet and the money, (2) keep the money and return the wallet, or (3) return both the wallet and the money.” Although we treated these response categories as unordered, there’s also an obvious ordering: 1 is the most unethical response, 3 is the most ethical, and 2 is in the middle. To fit a cumulative logit model to this data, we submit the program:

PROC LOGISTIC DATA=my.wallet;
  MODEL wallet = male business punish explain;
RUN;

(In case you’ve forgotten, the explanatory variables are defined in Section 5.2.) Results are shown in Output 6.1.

The output looks virtually identical to output for a binary logit model, except for something new in the middle: the “Score Test for the Proportional Odds Assumption.” I’ll explain this in more detail later. For now, it’s sufficient to say that this statistic tests whether the ordinal restrictions are valid, and high p-values are desirable. In this case, we find no reason to reject the model. Another difference in the output is that there are two intercepts instead of just one. In general, the number of intercepts for all the ordinal models in this chapter is one less than the number of categories on the dependent variable. But like the usual intercept, these rarely have any substantive interest.

Output 6.1. LOGISTIC Output for Cumulative Logit Model, Wallet Data

  Data Set: MY.WALLET
  Response Variable: WALLET
  Response Levels: 3
  Number of Observations: 195
  Link Function: Logit

                              Response Profile

                         Ordered
                           Value  WALLET     Count

                               1       1        24
                               2       2        50
                               3       3       121


              Score Test for the Proportional Odds Assumption

                 Chi-Square = 5.1515 with 4 DF (p=0.2721)

    Model Fitting Information and Testing Global Null Hypothesis BETA=0

                             Intercept
               Intercept        and
 Criterion       Only       Covariates    Chi-Square for Covariates

 AIC             356.140       319.367         .
 SC              362.686       339.005         .
 -2 LOG L        352.140       307.367       44.773 with 4 DF (p=0.0001)
 Score              .             .          40.875 with 4 DF (p=0.0001)

                 Analysis of Maximum Likelihood Estimates

            Parameter Standard    Wald       Pr >    Standardized    Odds
Variable DF  Estimate   Error  Chi-Square Chi-Square   Estimate     Ratio

INTERCP1 1    -3.2691   0.5612    33.9325     0.0001            .    .
INTERCP2 1    -1.4913   0.5085     8.6008     0.0034            .    .
MALE     1     1.0636   0.3255    10.6768     0.0011     0.293955   2.897
BUSINESS 1     0.7370   0.3515     4.3971     0.0360     0.171635   2.090
PUNISH   1     0.6874   0.2246     9.3646     0.0022     0.259567   1.989
EXPLAIN  1    -1.0453   0.3392     9.4979     0.0021    -0.264113   0.352

Turning to the more familiar portions of the output, we see from the global tests that there is strong evidence that at least one of the coefficients is not 0. This is further confirmed by the lower portion of the output which shows that all the coefficients have p-values below .05. In the next section, we’ll see how the parameter estimates and odds ratios can be interpreted in a fashion that is nearly identical to interpretation in a binary logit model.

Comparing Output 6.1 with the multinomial logit results for the same data (Output 5.2), we find a few major differences. The most obvious is that the multinomial model has two coefficients for every explanatory variable while the cumulative model has only one. This makes the cumulative model much simpler to present and interpret. A less apparent difference is that the p-values for the tests of whether each variable has an effect on the dependent variable are all lower in the cumulative model. This is crucial for BUSINESS, which has a p-value of .10 in the multinomial model (with the 2 degrees of freedom ANOVA test) but .036 in the cumulative model. The differences in p-values illustrate the claim I made earlier, that hypothesis tests in the cumulative logit model are more powerful.