Understanding Ordered Logit and Probit Regression

When dealing with ordinal data, researchers often face the challenge of choosing between ordered logit and probit regression. These models are both statistical tools designed to handle the ordinal nature of the dependent variable. While they share some conceptual similarities, they differ in their approach and assumptions. This article aims to clarify the differences between ordered logit and probit regression, and to provide insights into when and how to use them effectively.

Introduction to Ordered Regression Models

Ordered regression models are used when the dependent variable is ordinal. An ordinal variable is one where the categories have a natural, meaningful order, such as a Likert scale (1 Low, 5 High), or educational levels (1 No High School, 2 High School, 3 Bachelor's, 4 Master's, 5 Doctorate).

Ordered Logit Model

The ordered logit model is based on the logistic distribution. It assumes that there is a latent continuous variable y underlying the observed ordinal variable x. The idea is that the observed ordinal response is a result of the latent variable exceeding certain threshold values.

Mathematically, the ordered logit model can be described as follows:

Model: P(x j | y) P(y cj 1) - P(y cj)

where cj 1 are the cutoff points (or thresholds).

The logistic function used in the model implies that the odds ratio increases in a specific way across the categories. The parameters of the model are estimated using maximum likelihood estimation techniques.

Ordered Probit Model

The ordered probit model, on the other hand, is based on the normal (Gaussian) distribution. Similar to the logit model, it also assumes a latent continuous variable y but uses the cumulative normal distribution function instead of the logistic function.

Model: P(x j | y) Φ(cj 1 - μ) - Φ(cj - μ)

where Φ() is the cumulative distribution function of the standard normal distribution, and μ is the mean of the latent variable.

Key Differences Between Ordered Logit and Probit

Despite some differences in the underlying distribution, both models are quite similar in their practical applications. The logistic and normal distributions are relatively close to each other in behavior, and thus, the choice between logit and probit models often boils down to preference and familiarity rather than significant differences in results.

The primary differences lie in the link functions used:

Logit Model: Uses the logistic function, which has an S-shaped curve and is symmetric around the origin.

Probit Model: Uses the cumulative normal distribution function, which also has an S-shaped curve but is defined by the standard normal distribution.

Applications and Interpretation

Both models are powerful tools for analyzing ordinal data. They allow researchers to make inferences about the relationship between the dependent variable and independent variables while respecting the ordinal nature of the data.

The key differences in their results are often negligible in practical applications. Therefore, applied researchers usually focus on choosing the model that fits their data best rather than making a theoretical distinction between logit and probit.

Interpretation of the coefficients in both models is similar to that of regular regression analysis. The coefficients represent the change in the log-odds (logit) or latent variable (probit) associated with a one-unit change in the independent variable.

Conclusion

While the ordered logit and probit models share many similarities, the choice between them should be based on practical considerations rather than theoretical differences. Both models are valuable for analyzing ordinal data, and the results are often very similar. As an applied researcher, focus on using the model that best suits your data and research context. The key takeaway is to understand the underlying assumptions and the practical implications of using these models in your analysis.