Interpret the key results for Ordinal Logistic Regression

Complete the following steps to interpret an ordinal logistic regression model. Key output includes the p-value, the coefficients, the log-likelihood, and the measures of association.

Step 1: Determine whether the association between the response and the terms is statistically significant

To determine whether the association between the response and each term in the model is statistically significant, compare the p-value for the term to your significance level to assess the null hypothesis. The null hypothesis is that there is no association between the term and the response. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that an association exists when there is no actual association.
P-value ≤ α: The association is statistically significant
If the p-value is less than or equal to the significance level, you can conclude that there is a statistically significant association between the response variable and the term.
P-value > α: The association is not statistically significant
If the p-value is greater than the significance level, you cannot conclude that there is a statistically significant association between the response variable and the term. You may want to refit the model without the term.
If there are multiple predictors without a statistically significant association with the response, you can reduce the model by removing terms one at a time. For more information on removing terms from the model, go to Model reduction.

For a categorical factor with more than 2 levels, the hypothesis for the coefficient is about whether that level of the factor is different from the reference level for the factor. To assess the statistical significance of the factor, use the test for terms with more than 1 degree of freedom. For more information on how to display this test, go to Select the results to display for Ordinal Logistic Regression.

Response Information Variable Value Count Return Appointment Very Likely 19 Somewhat Likely 43 Unlikely 11 Total 73
Logistic Regression Table Odds 95% CI Predictor Coef SE Coef Z P Ratio Lower Upper Const(1) -0.505898 0.938791 -0.54 0.590 Const(2) 2.27788 0.985924 2.31 0.021 Distance -0.0470551 0.0797374 -0.59 0.555 0.95 0.82 1.12
Key Results: P-value, Coefficients

An analysis of a patient satisfaction survey examines the relationship between the distance a patient came and how likely the patient is to return. In these results, the distance is statistically significant at the significance level of 0.05. You can conclude that changes in the distances are associated with changes in the probabilities that the different events occur.

Assess the coefficient to determine whether a change in the predictor variable makes any of the events more or less likely. The relationship between the coefficient and the probabilities depends on several aspects of the analysis, including the link function. Positive coefficients make the first event and the events that are closer to it more likely as the predictor increases. Negative coefficients make the last event and the events closer to it more likely as the predictor increases. For more information, go to Coef.

The coefficient for Distance is about −0.05, which suggests that longer distances are associated with higher probabilities of the response "Unlikely" and with lower probabilities of the response "Very Likely."

Step 2: Determine how well the model fits your data

To determine how well the model fits the data, examine the log-likelihood and the measures of association. Larger values of the log-likelihood indicate a better fit to the data. Because log-likelihood values are negative, the closer to 0, the larger the value. The log-likelihood depends on the sample data, so you cannot use the log-likelihood to compare models from different data sets.

The log-likelihood cannot decrease when you add terms to a model. For example, a model with 5 terms has higher log-likelihood than any of the 4-term models you can make with the same terms. Therefore, log-likelihood is most useful when you compare models of the same size. To make decisions about individual terms, you usually look at the p-values for the term in the different logits.

Larger values for Somers' D, Goodman-Kruskal gamma, and Kendall's tau-a indicate that the model has better predictive ability. Somers' D and Goodman-Kruskal gamma can be between -1 and 1. Kendall's tau-a can be between -2/3 and 2/3. Values close to the maximum indicate the model has good predictive ability. Values close to 0 indicate that the model does not have a predictive relationship with the response. Negative values are rare in practice because that performance is worse than when the model and the response are unrelated.

Ordinal Logistic Regression: Return Appointment versus Distance

Link Function: Logit

Response Information Variable Value Count Return Appointment Very Likely 19 Somewhat Likely 43 Unlikely 11 Total 73
Logistic Regression Table Odds 95% CI Predictor Coef SE Coef Z P Ratio Lower Upper Const(1) -0.505898 0.938791 -0.54 0.590 Const(2) 2.27788 0.985924 2.31 0.021 Distance -0.0470551 0.0797374 -0.59 0.555 0.95 0.82 1.12

Log-Likelihood = -68.987

Test of All Slopes Equal to Zero DF G P-Value 1 0.328 0.567
Goodness-of-Fit Tests Method Chi-Square DF P Pearson 97.419 101 0.582 Deviance 100.516 101 0.495
Measures of Association: (Between the Response Variable and Predicted Probabilities) Pairs Number Percent Summary Measures Concordant 832 55.5 Somers’ D 0.13 Discordant 637 42.5 Goodman-Kruskal Gamma 0.13 Ties 30 2.0 Kendall’s Tau-a 0.07 Total 1499 100.0
Key Results: Log-likelihood, Somers' D, Goodman-Kruskal gamma, Kendall's tau-a

For example, the manager of a physician's office studies factors that influence patient satisfaction. In this first set of results, the distance that a patient travels to a doctors office predicts how likely the patient is to say that they are to return. The log-likelihood is −68.987. Somers' D and Goodman-Kruskal gamma are 0.13. Kendall's tau-a is 0.07. These values, which are close to 0, suggest that the relationship between the distance and the response is weak. The p-value for the test that all slopes are zero is greater than 0.05, so the manager tries a different model.

In this second set of results, the distance and the square of the distance are both predictors. You cannot use the log-likelihood to compare these models because they have different numbers of terms. The measures of association are higher for the second model, which indicates that the second model performs better than the first model.

Ordinal Logistic Regression: Return Appointment versus Distance

Link Function: Logit

Response Information Variable Value Count Return Appointment Very Likely 19 Somewhat Likely 43 Unlikely 11 Total 73
Logistic Regression Table Odds 95% CI Predictor Coef SE Coef Z P Ratio Lower Upper Const(1) 6.38671 3.06110 2.09 0.037 Const(2) 9.31883 3.15929 2.95 0.003 Distance -1.25608 0.523879 -2.40 0.017 0.28 0.10 0.80 Distance*Distance 0.0495427 0.0214636 2.31 0.021 1.05 1.01 1.10

Log-Likelihood = -66.118

Test of All Slopes Equal to Zero DF G P-Value 2 6.066 0.048
Goodness-of-Fit Tests Method Chi-Square DF P Pearson 114.903 100 0.146 Deviance 94.779 100 0.629
Measures of Association: (Between the Response Variable and Predicted Probabilities) Pairs Number Percent Summary Measures Concordant 938 62.6 Somers’ D 0.29 Discordant 505 33.7 Goodman-Kruskal Gamma 0.30 Ties 56 3.7 Kendall’s Tau-a 0.16 Total 1499 100.0
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