Evaluating the Proportional Odds Assumption in Ordinal Logistic Regression: Techniques and Python Implementation

The proportional odds model for ordinal logistic regression, introduced by McCullagh in 1980, extends binary logistic regression to scenarios involving ordinal dependent variables. This model is particularly useful when the outcome variable consists of ordered categories, such as ratings or rankings. The core assumptions of the proportional odds model include independence of observations, linearity of the log-odds, absence of multicollinearity among predictors, and the proportional odds assumption, which posits that the regression coefficients are constant across all thresholds of the ordinal dependent variable. 1. Introduction to the Proportional Odds Model In ordinal logistic regression, the dependent variable ( Y ) takes ordinal values from 1 to ( K ). The proportional odds model models the cumulative distribution probability ( \gamma_j = P(Y \leq j | X_i) ) for each threshold ( j ) (from 1 to ( K - 1 )) as a function of the explanatory variables ( X_i ): [ \log\left(\frac{\gamma_j}{1 - \gamma_j}\right) = \theta_j - \beta^T X_i ] Here, ( \theta_j ) are the intercepts for each category, and ( \beta ) represents the vector of regression coefficients that are assumed to be constant across all categories. The latent variable ( Y^* ) is derived from a linear regression model with error terms, and it relates to the observed ordinal variable ( Y ) through predefined thresholds. 2. Assessing the Proportional Odds Assumption: The Likelihood Ratio Test Brant (1990) proposed the likelihood ratio test to evaluate the proportional odds assumption. This test involves fitting two models: Unconstrained Model (Non-proportional odds): Allows each threshold to have its own set of regression coefficients. [ \log\left(\frac{\gamma_j}{1 - \gamma_j}\right) = \theta_j - \beta_j^T X_i ] Proportional Odds Model: Assumes a single set of regression coefficients for all thresholds. [ \log\left(\frac{\gamma_j}{1 - \gamma_j}\right) = \theta_j - \beta^T X_i ] The likelihood ratio test statistic ( \lambda ) is calculated by comparing the log-likelihoods of these two models: [ \lambda = 2 \times (\log L_{\text{full}} - \log L_{\text{reduced}}) ] The degrees of freedom for this test are ( (K-2) \times p ), where ( K ) is the number of categories and ( p ) is the number of predictors. The test statistic follows a chi-square distribution under the null hypothesis that the regression coefficients are equal across all cumulative logits. 3. Assessing the Proportional Odds Assumption: The Separate Fits Approach Another method proposed by Brant involves fitting separate binary logistic regression models for each threshold ( j ) (from 1 to ( K - 1 )). For each threshold, a binary variable ( Z_j ) is defined, taking the value 1 if the observation exceeds the threshold and 0 otherwise. The regression model for each binary variable is: [ \log\left(\frac{\pi_j}{1 - \pi_j}\right) = \theta_j - \beta_j^T X_i ] The proportional odds assumption can be tested by assessing whether the regression coefficients ( \beta_j ) are equal across all models. This is done by constructing a test statistic based on the differences between the coefficients, which is asymptotically chi-square distributed with ( (K-2) \times p ) degrees of freedom. Example: Application of the Two Proportional Odds Tests Data Preparation The "Wine Quality" dataset, containing 1,599 observations and 12 variables, was used for this example. The target variable, "quality," is ordinal and ranges from 3 to 8. To ensure sufficient observations in each category, categories 3 and 4 were combined into one category labeled 4, and categories 7 and 8 were combined into one category labeled 7, resulting in four levels. Outliers in the predictor variables were handled using the Interquartile Range (IQR) method, and three predictors—volatile acidity, free sulfur dioxide, and total sulfur dioxide—were selected and standardized. Model Fitting and Evaluation The dataset was loaded and preprocessed in Python using pandas and statsmodels. Binary logistic regression models were fit for each threshold, and the proportional odds model was implemented using the OrderedModel class. The results of these models showed significant discrepancies in the coefficients for volatile acidity, suggesting potential violation of the proportional odds assumption. Two tests were conducted to formally assess the assumption: Likelihood Ratio Test: Test statistic: ( \lambda = 53.207 ) Degrees of freedom: 6 P-value: ( 1.066 \times 10^{-9} ) Wald Test: Wald statistic: ( X^2 = 41.880 ) Degrees of freedom: 6 P-value: ( 1.232 \times 10^{-7} ) Both tests indicated that the proportional odds assumption was violated at the 5% significance level, suggesting that the proportional odds model may not be suitable for this dataset. Conclusion This paper aimed to illustrate how to test the proportional odds assumption in ordinal logistic regression and to encourage readers to explore Brant (1990)'s article for a deeper understanding. Brant's methods for assessing proportionality extend beyond the likelihood ratio and Wald tests, offering techniques to evaluate the overall adequacy of the model, such as testing the distribution of the latent variable ( Y^* ). The findings from this study highlight the importance of carefully validating model assumptions to ensure accurate and reliable results. Industry Insider Evaluation and Company Profiles Brant’s methods have been widely adopted in the field of statistical modeling due to their robustness and practicality. Industry experts emphasize the critical role of these tests in ensuring the reliability of ordinal regression models, particularly in fields like healthcare and social sciences where ordinal data are common. For instance, the statsmodels package, widely used for statistical modeling in Python, incorporates tools for conducting these tests, making it accessible for data scientists and researchers. The "Wine Quality" dataset, sourced from the UCI Machine Learning Repository, is a popular resource for machine learning and statistical analysis, licensed under CC BY 4.0. It showcases the practical application of ordinal logistic regression in real-world scenarios, emphasizing the need for rigorous model validation.