# What is Ordinal Regression

##### What Is Ordinal Regression?

Ordinary linear regression is a powerful statistical technique used to model the relationship between a dependent variable and one or more independent variables. However, the assumption of linear relationships is often unrealistic in many real-world scenarios. For example, in social science research, we frequently work with categorical variables, such as various Likert scales or education levels. These variables do not have a natural ordering, and thus, linear regression is not appropriate.

This is where ordinal regression comes into play. It is a statistical technique used to model the relationship between a dependent variable and one or more independent variables when the dependent variable is ordinal, which is a type of categorical variable with a natural ordering.

The goal of ordinal regression is to predict the likelihood or probability that an observation belongs to a specific category or class, given the predictor variables. The prediction can be done for each level of the ordinal variable, creating a set of predicted probabilities that can be interpreted as the probability of the dependent variable being in a particular category or higher.

##### Types of Ordinal Regression

There are two main types of ordinal regression:

• Ordered Logistic Regression
• Proportional Odds Model
##### Ordered Logistic Regression

Ordered Logistic Regression, also known as a cumulative or proportional odds model, assumes that the coefficients of the predictor variables on the log-odds of the response variable are constant across the levels of the response variable. In other words, the effect of the predictors is the same across all levels of the response variable.

Ordered logistic regression is useful when the odds ratios between adjacent categories of the dependent variable are constant, which means that the odds of being in a specific category vs. all the categories below it are the same for every predictor variable. In other words, the odds of going from "poor" to "average" are the same as going from "average" to "good."

##### Proportional Odds Model

The proportional odds model is a special case of the ordered logistic regression model. It assumes that the relationship between the predictors and the dependent variable is the same across all levels of the outcome variable. However, it also assumes that the odds of being in a specific category of the dependent variable vs. all categories below it are proportional to the odds of being in that category vs. all categories above it.

Proportional odds models are convenient when the dependent variable has a natural ordering, and the relationship between the predictors and the dependent variable is linear. In other words, the coefficients of the predictor variables are the same across the levels of the response, but the odds ratios differ between adjacent categories of the dependent variable.

##### How Ordinal Regression Works?

Ordinal regression starts with the assumption that the categories of the ordinal variable have a natural ordering. The model then uses maximum likelihood estimates to determine the coefficients of the predictor variables. It estimates the likelihood of the outcome variable being in each category given the values of the predictor variables.

Unlike linear regression, which uses least squares to determine the coefficients, ordinal regression uses maximum likelihood estimation. This approach estimates the coefficients that maximize the likelihood of the observed data, given the predictor variables.

• Capturing Non-linear Relationships: Ordinal regression can capture non-linear relationships between the dependent variable and predictor variables, which is not possible with linear regression.
• Handling Categorical Variables: Ordinal regression can handle categorical variables with a natural ordering, such as education level, income, and social status, making it ideal for social science research.
• Interpretability: The predicted probabilities in ordinal regression can be easily interpreted in the context of the dependent variable, making it easy to communicate the results.
• Efficiency: Ordinal regression uses maximum likelihood estimation, which only requires a set of reasonable assumptions to estimate the coefficients. This saves computing resources and helps academics perform studies more efficiently.
##### Limitations of Ordinal Regression
• Assumptions: Ordinal regression relies on several assumptions, including linearity of the relationship between predictors and the response variable, independence of observations, and normality of residuals.
• Small Sample Size: When the sample size is small, it can affect the accuracy of the estimates and undermine the statistical power.
• Limited to Ordinal Data: Because ordinal regression can only handle data that have a natural ordering, researchers must find alternative techniques if their data falls outside of this category.
• Difficulties with Causal Inferences: Ordinal regression is not designed to make causal inferences as it does not account for other potential confounding variables in the model. This could be dangerous when drawing causal conclusions from the model.
##### Conclusion:

Ordinal regression is an essential statistical technique for researchers and data scientists who work with ordinal data. It is particularly useful for analyzing social science studies where variables have a natural order. Ordinal regression helps researchers capture nonlinear relationships between predictor variables and dependent variables and estimate the likelihood of the dependent variable being in specific categories or higher.

However, programmers should be aware of the assumptions and limitations of ordinal regression when applying this method to various analytical problems. Any research using this method must make a good effort to meet these conditions to get accurate results.