Ordinal Logistic Regression

What Is Ordinal Logistic Regression?

Ordinal logistic regression (OLR) is a statistical technique used to predict a single ordered categorical variable using one or more other variables. It aims to model the relationship between independent variables and the probabilities of each category within the dependent variable.

OLR generalizes the multiple linear regression and binomial logistic regression models. Researchers use it to understand how changes in independent variables impact the odds of transitioning from one category to another. It is primarily applied in scenarios with more than three categories that exhibit natural ordering but unequal intervals, with common use in fields like medicine, social science, and education.

• OLR is a statistical technique to predict an ordered categorical variable by modeling the relationships between independent variables and the probabilities of falling into specific categories. It is particularly valuable when the dependent variable has ordered categories.
• OLR relies on several assumptions, including the proportional odds assumption, independence of observations, linearity in log odds, absence of multicollinearity, and the need for an adequate sample size.
• OLR is employed when the dependent variable has ordered categories, providing a meaningful order. In contrast, multinomial regression is chosen for analyzing dependent variables with numerous unordered categories, and logistic regression is used when there are only two ordered categories.

Ordinal Logistic Regression Explained

Ordinal logistic regression (OLR) can be defined as a mathematical method for modeling the relationship between multiple independent variables and an ordinal dependent variable. It builds upon the principles of logistic regression, with a key distinction being its consideration of the natural order of the dependent variable's categories. This consideration is achieved through the cumulative logit model, which determines the likelihood of an observation falling into a specific category of the dependent variable based on the values of the independent variables.

OLR involves several key steps in its functioning. Data preparation is the initial step, where data is cleaned and prepared. This includes scaling variables, addressing missing values, and dealing with outliers. Following data preparation, researchers move on to model specification, where they determine the appropriate ordinal level for the response variable and establish the hierarchy of categories.

Once the model is specified, maximum likelihood estimation is employed to find the model settings that maximize the likelihood of the observed data. This estimation process results in the prediction of the OLR model. Finally, to assess the model's performance, various metrics such as accuracy, precision, recall, and the F1 score are commonly used.

Assumptions

Assumptions for ordinal logistic regression:

• Proportional Odds: The odds of moving to a higher category remain consistent for different independent variable values.
• Independence: Observations are independent, with no systematic connections between them.
• Linearity: The relationship between variables and higher category probabilities is assumed to be linear.
• No Multicollinearity: Independent variables should not be highly correlated.
• Adequate Sample Size: A large sample size ensures reliable results.
• No Complete Separation: Avoid situations where variables perfectly predict outcomes.
• Proportional Odds Test: A statistical test checks the proportional odds assumption.

Examples

Let us use a few examples to understand the topic:

Example #1

Suppose Alex wants to understand how education level influences job satisfaction. He collects data from a group of employees and categorizes job satisfaction as "Low," "Medium," and "High." Education level is classified as "High School," "Bachelor's Degree," and "Master's Degree." He can use ordinal logistic regression to analyze the relationship between education level and job satisfaction, considering the ordinal nature of both variables. This analysis will reveal whether higher education levels are associated with higher job satisfaction while accounting for the ordinal structure of the data.

Example #2

Suppose in the real estate market, Megan wants to predict the price range of houses. She categorizes house prices into "Low," "Medium," and "High" based on their market values. She collects data on various factors, such as the number of bedrooms, square footage, and neighborhood. By applying ordinal logistic regression, she can assess how these factors are related to the likelihood of a house falling into different price ranges.

The analysis will help her understand the influence of each factor on housing price categories, considering the ordered nature of these categories, allowing her to make more informed pricing decisions in the real estate market.

When To Use?

Ordinal logistic regression is a versatile tool applicable in various fields such as psychology, social sciences, and other areas where ordinal outcomes are common. Below are situations and criteria for when to use OLR:

• Ordinal Variable with Natural Order: OLR is suitable when the dependent variable is ordinal and exhibits a natural order, where categories have a meaningful progression.
• Uneven Variance or Non-Normality: It is useful when dealing with non-normally distributed or unequally varying continuous dependent variables that do not meet the assumptions of linear regression.
• Meaningful Ordered Categories: Utilize OLR with ordered categorical variables with meaningful and interpretable order, such as assessing customer satisfaction levels.
• Understanding Connections: OLR is applicable when one wants to understand the relationships between ordinal outcomes and independent variables, taking into account the ordinal nature of the data.
• Predicting Multiple Categories: It is appropriate when the objective is to model or predict ordinal outcomes with more than two categories.
• More Than Three Ordered Categories: OLR is a valuable choice when the dependent variable comprises more than three naturally ordered categories.
• Consistent Connection Assumption: OLR works well when one can assume that the connections between independent factors and outcomes remain consistent across all levels of the dependent variable.

Ordinal Logistic Regression vs Multinomial Regression vs Logistic Regression

The differences between the three are as follows:

Basis	Ordinal Logistic	Multinomial	Logistic
When to Use	Ordered categories in the dependent variable	Unordered categories in the dependent variable	Only two ordered categories in the dependent variable
What it Models	Relationship between variables and cumulative probabilities of reaching a specific category or higher	Probabilities for each dependent category relative to independent variables	Probability of one category versus another category
Key Characteristics	Assumes consistent relationships, best for ≥3 ordered categories	No order assumption, suitable for >3 unordered categories	Applicable for binary outcomes

Frequently Asked Questions (FAQs)

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This article has been a guide to what is Ordinal Logistic Regression. We explain its assumptions, examples, when to use it, & comparison with multinomial regression. You may also find some useful articles here -

• Categorical Data
• Logistic Regression vs. Linear Regression
• Inferential Statistics