Example of an Ordinal Logistic Model

If the response variable has an ordinal modeling type, the platform fits the cumulative response probabilities to the logistic function of a linear model using maximum likelihood. Likelihood-ratio test statistics are provided for the whole model and lack of fit. Wald test statistics are provided for each effect.

If there is an ordinal response and a single continuous numeric effect, the ordinal logistic platform in Fit Y by X displays a cumulative logistic probability plot.

Details of modeling types are discussed in the Basic Analysis book. The details of fitting appear in the appendix Statistical Details. The method is discussed in Walker and Duncan (1967), Nelson (1976), Harrell (1986), and McCullagh and Nelder (1983).

Note: If there are many response levels, the ordinal model is much faster to fit and uses less memory than the nominal model.

As an example of ordinal logistic model fitting, McCullagh and Nelder (1983) report an experiment by Newell to test whether various cheese additives (A to D) had an effect on taste. Taste was measured by a tasting panel and recorded on an ordinal scale from 1 (strong dislike) to 9 (excellent taste). The data are in the Cheese.jmp sample data table.

1.	Select Help > Sample Data Library and open Cheese.jmp.

2.	Select Analyze > Fit Model.

3.	Select Response and click Y.

4.	Select Cheese and click Add.

5.	Select Count and click Freq.

6.	Click Run.

7.	From the red triangle next to Ordinal Logistic Fit, select Wald Tests.

Ordinal Logistic Fit

The model in this example reduces the –LogLikelihood of 429.9 to 355.67. This reduction yields a likelihood-ratio Chi-square for the whole model of 148.45 with 3 degrees of freedom, showing the difference in perceived cheese taste to be highly significant.

The Lack of Fit test happens to be testing the ordinal response model compared to the nominal model. This is because the model is saturated if the response is treated as nominal rather than ordinal, giving 21 additional parameters, which is the Lack of Fit degrees of freedom. The nonsignificance of Lack of Fit leads one to believe that the ordinal model is reasonable.

There are eight intercept parameters because there are nine response categories. There are only three structural parameters. As a nominal problem, there are

structural parameters.

When there is only one effect, its test is equivalent to the Likelihood-ratio test for the whole model. The Likelihood-ratio Chi-square is 148.45, different from the Wald Chi-square of 115.15, which illustrates the point that Wald tests are to be regarded with some skepticism.

To see whether a cheese additive is preferred, look for the most negative values of the parameters (Cheese D’s effect is the negative sum of the others, shown in Parameter Estimates and Preferences for Cheese Additives in Cheese.jmp.).

Parameter Estimates and Preferences for Cheese Additives in Cheese.jmp
Cheese	Estimate	Preference
A	–0.8622	2nd place
B	2.4896	least liked
C	0.8477	3rd place
D	–2.4750	most liked

You can also use the Fit Y by X platform for this model, which treats ordinal responses like nominal and shows a contingency table analysis. See Fit Y by X Platform Results for Cheese.jmp. The Fit Model platform can also be used, but you must change the ordinal response, Response, to nominal. See Fit Model Platform Results Setting Response to Nominal for Cheese.jmp. Nominal Fit Model results are shown in Fit Model Platform Results Setting Response to Nominal for Cheese.jmp. Note that the negative log-likelihood values (84.381) and the likelihood chi-square values (168.76) are the same.

Fit Y by X Platform Results for Cheese.jmp

Fit Model Platform Results Setting Response to Nominal for Cheese.jmp

If you want to see a graph of the response probabilities as a function of the parameter estimates for the four cheeses, add the Score variable as a response surface effect to the Fit Model dialog as shown. To create the model in Model Dialog For Ordinal Logistic Regression, select Score in the column selector list, and then select Response Surface from the Macros popup menu on the Fit Model dialog.

Model Dialog For Ordinal Logistic Regression

Validation is available only in JMP Pro.

Click Run to see the analysis report and the cumulative logistic probability plot in Cumulative Probability Plot for Ordinal Logistic Regression. The distance between each curve is the fitted response probability for the levels in the order for the levels on the right axis of the plot.

Cumulative Probability Plot for Ordinal Logistic Regression