Introduction to Ordinal Logistic Regression

Ordinal Logistic Regression (OLR), which is sometimes called Ordered Logit, is a way to describe ordinal dependent variables in statistics. A variable that is ordinal is one that has more than two levels or categories, but the categories naturally relate to each other in a certain order. This can be used for things like grades (bad, average, good, excellent), stages of a disease (early, middle, late), or levels of education (high school, bachelor’s, master’s, PhD).

Based on independent factors, the goal of Ordinal Logistic Regression is to guess the chance that an observation will fit into one of the ordered groups. It’s an addition to regular Logistic Regression that can deal with results that can be put into groups where the order of the groups is important.

This article will delve into the theory behind OLR, assumptions, how to implement it, and practical applications.

What is an Ordinal Logistic Regression?

Ordinal Logistic Regression (OLR) predicts ordinal dependent variables with natural ordering but uncertain spacing, such as customer satisfaction (low, medium, high). To handle ordered categories, logistic regression is extended. OLR uses the cumulative logit function to model result category probability. It assumes proportional odds, so thresholds are consistent. Multi-threshold intercepts are estimated by the model, but predictor coefficients are shared across categories.

Rating replies in surveys, medical research, and risk assessment is common. OLR is easier to interpret for ordered outcomes than multinomial logistic regression since it preserves ordinal relationships.

OLR has benefits, but it involves validating the proportionate odds assumption and is susceptible to violations, which may require generalized ordered logit models.

Ordinal Logistic Regression model assumptions

• Ordinal Response Variable: The dependent variable should have more than two groups that make sense in terms of their order.
• Proportional Odds: Each pair of outcome categories must have the same link with each other.
• Independence of Observations: The observations should not depend on each other.
• Linearity: Log-odds of the result are thought to be linearly related to the predictor variables.
• No multicollinearity: There should be no multicollinearity, which means that the predictor variables shouldn’t be strongly linked to each other.

Ordinal Logistic Regression learns from the data and tries to guess the chance that an observation will fit into one of these sorted groups.

Evaluating the Model

Once the model is fitted, we need to judge how well it works. Some common measures are:

• Pseudo R-Squared: This statistic shows how much of the variation in the outcome variable can be described by the model.
• Likelihood Ratio Tests: These tests look at how well the fitted model matches up with a base model.
• Classification Accuracy: Check how well the model can put observations in the right group.

Sometimes, seeing the total probability curves can also help you figure out how well the model works.

Applications of Ordinal Logistic Regression

In the real world, Ordinal Logistic Regression is used for many things, such as

• Medical Studies: guessing how bad a disease will be or how a situation will get worse over time, based on the order of the stages of the disease.
• Customer Satisfaction Surveys: Customer satisfaction surveys look at how customers rate goods or services using a scale of one to ten.
• Education: Looking at the results of education, like the amount of education gained, or guessing how likely it is that a student will graduate at different stages.

Advantages of Ordinal Logistic Regression

• Handles Ordinal Data Effectively: Ordinal Logistic Regression is developed for ordinal dependent variables, meaning categories have a meaningful order but not uniform distances. Traditional logistic regression, which handles binary outcomes, cannot handle this. OLR can model relationships with ordered outcome variables like Likert scale ratings (“Strongly Disagree” to “Strongly Agree”).
• Preserves the Ordinality of the Data: Ordinal Logistic Regression accounts for the ordered dependent variable, unlike nominal models. This is a huge benefit because interpreting ordinal variables as continuous or nominal can lose information and lead to inaccurate modeling assumptions. OLR preserves and uses natural ordering (e.g., low, medium, high) in the model.
• Interpretation of Coefficients: In Ordinal Logistic Regression, coefficient interpretation can reveal predictor-ordered response variable relationships. The model coefficients show the probability of advancing to a higher dependent variable category, making predictors easier to interpret. In a survey with ratings from 1 to 5, the coefficients show how a change in an independent variable (e.g., age, wealth) impacts the likelihood of rising from a lower to a higher rating.
• Efficient Use of Data: Ordinal Logistic Regression efficiently uses all data based on outcome variable structure and order. This is more illuminating than models that ignore category order or consider them as nominal. The model works effectively with smaller sample sizes, especially when the outcome variable has fewer categories.
• Assumptions are possible: Ordinal Logistic Regression assumes proportional odds, which states that categories have the same connection. In real-world situations like estimating customer satisfaction or education levels, where “good” to “very good” is likely similar, this assumption is valid. OLR is more accurate and dependable than multinomial logistic regression, which ignores data ordering, when this assumption holds.

Limitations of Ordinal Logistic Regression

• Proportional Odds Assumption: Odds that are relevant If this assumption is broken, the model might not work right and the results might not be what they seem to be.
• Interpretation of Coefficients: In ordinal logistic regression, it’s not as easy to read the coefficients as it is in simple logistic regression.
• Non-linearity: OLR assumes that the log-odds and the predictors have a straight line connection, but this may not always be the case. In these situations, factors may need to be transformed or nonlinear models may be needed.

Alternatives to Ordinal Logistic Regression

If the idea of proportional odds doesn’t hold, you can use other models, such as

• Partial Proportional Odds Model: The partial proportional odds model lets some violations of the proportional odds rule happen.
• Generalized Ordered Logit Model: This model doesn’t assume that the odds are proportional for each pair of groups that are next to each other.
• Multinomial Logistic Regression: This type of regression can be used if the answer variable is not in a certain order.

Ordinal Logistic Regression in python

To implement Ordinal Logistic Regression in Python, the statsmodels library provides the necessary tools. While statsmodels does not directly support ordinal logistic regression, we can use the OrderedModel class, which implements the proportional odds model.

Here is an example implementation:

import pandas as pd
import statsmodels.api as sm
from statsmodels.miscmodels.ordinal_model import OrderedModel

# Example data: assume a dataset with one ordinal dependent variable and two independent variables
data = pd.DataFrame({
    'education': [1, 2, 3, 4, 2, 1, 3, 4, 2, 3],
    'income': [30, 60, 40, 80, 55, 45, 70, 100, 65, 85],
    'age': [25, 45, 35, 50, 40, 30, 50, 60, 45, 55]
})

# Define the dependent variable (ordinal)
y = data['education']

# Define the independent variables
X = data[['income', 'age']]

# Add a constant for the intercept
X = sm.add_constant(X)

# Fit the ordinal logistic regression model
model = OrderedModel(y, X, distr='logit')
result = model.fit()

# Display the results
print(result.summary())

In conclusion

Ordinal Logistic Regression is a strong way to look at results that are ordered into categories. OLR can help make predictions when the dependent variable has a meaningful order but not a consistent space between categories. It does this by modeling the cumulative odds of being in a certain category or lower.

OLR is a strong and widely used tool in many fields, from social studies to healthcare, as long as certain assumptions are met. The proportional odds assumption is one of the most important of these.