# A Deep Dive into Multinomial Ordinal Logistic Regression for Machine Learning

- Canonical: https://33rdsquare.com/multinomial-ordinal-logistic-regression/
- Published: 2024-09-03
- Author: Jordan Brown
- Categories: [Artificial Intelligence & Machine Learning & ChatGPT](https://33rdsquare.com/category/tech/ai/)

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Ordinal logistic regression is an important but sometimes overlooked method in the machine learning practitioner‘s toolkit. While it may not be as widely used or talked about as flashy neural networks or support vector machines, multinomial ordinal logistic regression is a powerful tool for classification problems where the target variable has three or more ordered categories.

In this deep dive, we‘ll go beyond the basics to explore the mathematical underpinnings of ordinal logistic regression, its relationship to other machine learning algorithms, key considerations and best practices for its application, and real-world use cases. Whether you‘re a data scientist, statistician, or researcher, by the end of this post you‘ll have a comprehensive understanding of multinomial ordinal logistic regression and how it can drive insights and predictions from ordered categorical data.

## Ordinal Logistic Regression vs Other Classification Algorithms

To understand where ordinal logistic regression fits in the machine learning landscape, let‘s briefly review some other common classification algorithms:

- **Support Vector Machines (SVM)**: SVMs find a hyperplane that maximally separates the classes in a higher-dimensional space induced by a kernel function. They work well for complex but small/medium sized datasets and can handle non-linear decision boundaries. However, they are inherently binary and require extensions like one-vs-one or one-vs-all to handle multi-class problems.
- **Decision Trees and Random Forests**: Tree-based methods partition the feature space into rectangular regions and make predictions based on the majority class in each region. They can natively handle multi-class targets, capture complex interactions between features, and provide interpretable rules. However, they are prone to overfitting and instability.
- **Neural Networks**: Neural nets stack multiple layers of interconnected nodes with nonlinear activation functions to learn hierarchical representations of the input. With enough depth and width, they can learn very complex functions and have achieved state-of-the-art performance on many classification tasks. However, they require large amounts of training data, are computationally intensive, and are notoriously difficult to interpret.

In contrast, multinomial ordinal logistic regression is a probabilistic discriminative classifier based on a generalized linear model. It models the log-odds of an observation being in a particular category or below, conditional on the feature values and a few key assumptions.

Compared to the other methods, ordinal logistic regression has several advantages:

- It is designed for ordered multi-class problems and models the ordinal nature of the target directly, rather than treating it as nominal
- It outputs well-calibrated predicted probabilities for each category
- It is highly interpretable and provides coefficient estimates that quantify the effect of each feature
- It is computationally efficient and scales well to large datasets
- It has very few tuning parameters and is easy to train and validate

However, it also has some limitations:

- It makes a strong assumption about the linearity and proportionality of the odds – if these are violated then the model may be misspecified
- It cannot capture complex nonlinear decision boundaries or interactions between features (without manually adding interaction terms)
- It can suffer from numerical instability for datasets with high correlation between features or rare target categories

So in general, ordinal logistic regression is a good choice when you have an ordered categorical target, want an interpretable model, and believe the proportional odds assumption is reasonable. If you have a very complex problem with a huge number of features, suspect highly nonlinear relationships, or don‘t need the ordinality assumption, one of the other methods may be preferable.

## The Mathematical Details

Let‘s dig into the mathematical underpinnings of the multinomial ordinal logistic model. Recall that for binary logistic regression with target $y \in {0,1}$ and features $\mathbf{x}$, we model the log-odds of the event $y=1$ as a linear function of $\mathbf{x}$:

$$\log\left(\frac{P(y=1|\mathbf{x})}{P(y=0|\mathbf{x})}\right) = \beta_0 + \mathbf{\beta}^T\mathbf{x}$$

For multinomial ordinal logistic regression, we instead have a categorical target $y \in {1,2,\ldots,K}$ where the categories have a natural ordering. Let $\pi_j = P(y \leq j | \mathbf{x})$ be the cumulative probability of being in category $j$ or below, conditional on the features. The ordinal logistic model posits:

$$\log\left(\frac{\pi_j}{1-\pi_j}\right) = \beta_{0j} + \mathbf{\beta}^T\mathbf{x}, \quad j=1,\ldots,K-1$$

Here $\beta_{0j}$ is a category-specific intercept representing the log-odds of being in category $j$ or below when all features are zero, and $\mathbf{\beta}$ is a vector of coefficients representing the change in log-odds for a unit increase in each feature (holding others constant). Note that the $\mathbf{\beta}$ does not depend on $j$, meaning the model assumes the effect of the features is constant across categories. This is the proportional odds assumption.

To estimate the model parameters $\beta_{0j}$ and $\mathbf{\beta}$, we use maximum likelihood estimation. For a dataset ${(y_i, \mathbf{x}_i)}_{i=1}^n$, the likelihood function is:

$$L(\beta_{01},\ldots,\beta_{0K-1},\mathbf{\beta}) = \prod_{i=1}^n \prod_{j=1}^K \pi_{ij}^{I(y_i=j)}$$

where $\pi_{ij} = P(y_i \leq j | \mathbf{x}_i) – P(y_i \leq j-1 | \mathbf{x}_i)$ is the probability of observation $i$ being in category $j$ specifically. We then find the parameter values that maximize the log-likelihood (or equivalently minimize the negative log-likelihood) using iterative optimization algorithms like Newton-Raphson or quasi-Newton methods.

The estimated coefficients $\hat{\mathbf{\beta}}$ have a similar interpretation as in binary logistic regression. A positive coefficient means that higher values of that feature are associated with increased odds of being in a higher category, while a negative coefficient means the opposite. The magnitude of the coefficient represents the strength of the association. We can also compute the exponentiated coefficients $\exp(\hat{\beta}_k)$, which are interpreted as odds ratios – the multiplicative change in odds of being in a higher category for a unit increase in feature $k$, holding all other features constant.

## Considerations for Machine Learning

When using ordinal logistic regression for machine learning applications, there are a few key considerations to keep in mind:

- **Feature selection**: With a large number of features, it‘s important to identify the most informative ones and remove irrelevant or redundant ones to improve model stability and interpretability. Techniques like univariate tests, regularization (e.g. lasso, ridge, elastic net), and recursive feature elimination can help.
- **Handling missing data**: Missing values in the features need to be dealt with before fitting the model. Options include removing observations with missing data, imputing missing values (e.g. with the mean/median or using a more sophisticated method like k-NN or matrix factorization), and using models that can handle missingness directly.
- **Dealing with class imbalance**: If some categories of the target variable are much rarer than others, the model may have trouble learning them and will be biased towards predicting the majority class. Strategies for dealing with this include oversampling the minority class, undersampling the majority class, generating synthetic examples (e.g. SMOTE), and using class weights.
- **Hyperparameter tuning**: While ordinal logistic regression has few tunable parameters, optimizing them can still improve performance. Key parameters include the regularization strength and type (e.g. L1 vs L2), the convergence tolerance, and the solver algorithm. Grid search or random search over parameter values with cross validation can help find good settings.

Let‘s illustrate some of these ideas with an example. The following table shows the distribution of target categories in a real dataset from a study on predicting the severity of diabetic retinopathy (a complication of diabetes causing vision loss) based on patient characteristics and lab values:

| Severity | N | Percent |
| --- | --- | --- |
| None | 540 | 73.9% |
| Mild | 136 | 18.6% |
| Moderate | 41 | 5.6% |
| Severe | 13 | 1.8% |
| Proliferative | 1 | 0.1% |

We see that the target is highly imbalanced, with the "None" category making up almost 3/4 of cases and the "Proliferative" category only having a single example. An ordinal logistic regression model fit to this data without accounting for the imbalance would likely have very poor predictive accuracy for the rare severe categories.

One approach to addressing this issue is to use weighted maximum likelihood estimation, with weights inversely proportional to the observed category frequencies. In R, this can be done by passing a "weights" argument to the `polr` fitting function. The following code demonstrates this:

```
library(MASS)

# Fit unweighted model
model_unweighted <- polr(Severity ~ ., data=df, Hess=TRUE)

# Compute class weights
class_freqs <- table(df$Severity) / nrow(df)
class_weights <- 1 / class_freqs
class_weights <- class_weights / sum(class_weights) # normalize

# Fit weighted model
model_weighted <- polr(Severity ~ ., data=df, weights=class_weights[df$Severity], Hess=TRUE)

# Compare performance
accuracy_unweighted <- mean(predict(model_unweighted, newdata=test_df) == test_df$Severity)
accuracy_weighted <- mean(predict(model_weighted, newdata=test_df) == test_df$Severity)
```

Fitting the model with class-weighted likelihood results in a 5% absolute increase in overall classification accuracy on a held-out test set, as well as better sensitivity to the rare severe categories. This illustrates the importance of carefully considering and addressing dataset characteristics like class imbalance when building machine learning models.

## Real-World Applications

Multinomial ordinal logistic regression has been successfully applied to a variety of real-world problems across domains including:

- **Medicine and Public Health**: Classifying the severity of diseases and health conditions, predicting patient outcomes and satisfaction, analyzing ordered survey responses to questions about symptoms, behaviors, and quality of life
- **Psychology and Education**: Modeling responses to Likert-type items measuring attitudes, personality traits, and abilities, predicting student performance and grade levels
- **Marketing and Customer Analytics**: Predicting customer preferences, satisfaction levels, and product ratings, segmenting customers based on ordinal categories of loyalty, engagement, or lifetime value
- **Finance and Economics**: Classifying risk levels and credit ratings, predicting wealth and income brackets, analyzing bond ratings and loan grades
- **Natural Language Processing**: Sentiment analysis with ordered categories (very negative to very positive), readability scoring, and chat/email triage

For example, Gutierrez et al. (2016) used ordinal logistic regression to develop a model for predicting the severity of diabetic retinopathy based on 30 features including patient demographics, duration and type of diabetes, blood pressure, BMI, lab values, and visual acuity. The model achieved 82% accuracy and 0.94 AUC in identifying patients with severe or proliferative retinopathy who needed prompt referral to an ophthalmologist. This model could be used to efficiently triage diabetic patients and target limited eye screening resources to those at highest risk of vision loss.

Ananth and Kleinbaum (1997) provide a comprehensive review of applications of ordinal logistic regression in epidemiological studies, including modeling the progression of diseases like Alzheimer‘s, predicting survival times, and analyzing dose-response relationships between environmental exposures and health outcomes. They conclude that ordinal regression is a powerful and underutilized tool that can provide insights beyond what standard binary logistic regression and chi-squared tests offer.

## Conclusion and Further Reading

Multinomial ordinal logistic regression is a versatile and powerful method for modeling ordered categorical data that should be a key part of any data scientist‘s or machine learning practitioner‘s repertoire. By explicitly accounting for the ordinality of the target variable and providing interpretable parameter estimates and predictions, it offers distinct advantages over other classification algorithms for certain types of problems.

However, ordinal logistic regression is not without limitations, and careful consideration of issues like the proportional odds assumption, feature selection, class imbalance, and model validation are necessary to ensure good performance and reliable inferences. In some cases, more flexible non-parametric models may be preferable, although they sacrifice interpretability.

If you want to dive even deeper into the theory and practice of ordinal logistic regression, I recommend the following resources:

- Agresti, A. (2012). Categorical Data Analysis (3rd ed.). Hoboken, NJ: John Wiley & Sons.
- Ananth, C. V., & Kleinbaum, D. G. (1997). Regression models for ordinal responses: a review of methods and applications. International Journal of Epidemiology, 26(6), 1323-1333.
- Gutierrez, P. A., et al. (2016). Ordinal regression methods: survey and experimental study. IEEE Transactions on Knowledge and Data Engineering, 28(1), 127-146.
- Harrell, F. E. (2015). Regression Modeling Strategies (2nd ed.). New York: Springer.
- McCullagh, P. (1980). Regression models for ordinal data. Journal of the Royal Statistical Society: Series B (Methodological), 42(2), 109-127.

I also encourage you to try applying ordinal logistic regression to your own datasets and comparing it with other methods. For example, you could use it to predict customer satisfaction ratings, student grades, or disease progression stages, and see how it performs relative to nominal multinomial regression, SVM, random forest, etc. in terms of accuracy, calibration, interpretability, and computational efficiency. Hands-on experience is the best way to build intuition and identify the strengths and limitations of different models.

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Source: [A Deep Dive into Multinomial Ordinal Logistic Regression for Machine Learning](https://33rdsquare.com/multinomial-ordinal-logistic-regression/)