Logistic Regression vs. Probit Regression

Attribute	Logistic Regression	Probit Regression
Model Type	Generalized linear model	Generalized linear model
Link Function	Logit function	Probit function
Assumption	Assumes errors follow a logistic distribution	Assumes errors follow a standard normal distribution
Interpretation of Coefficients	Interpreted as log odds ratios	Interpreted as changes in z-scores
Computational Complexity	Computationally simpler	Computationally more complex

Introduction

Logistic regression and probit regression are two popular statistical methods used for binary classification problems. Both models are used to predict the probability of a binary outcome based on one or more predictor variables. While they have similar goals, there are some key differences between logistic regression and probit regression in terms of their assumptions, interpretation, and implementation.

Assumptions

One of the main differences between logistic regression and probit regression lies in the assumptions they make about the underlying distribution of the error term. Logistic regression assumes that the error term follows a logistic distribution, while probit regression assumes that the error term follows a standard normal distribution. This difference in assumptions can impact the interpretation of the coefficients in each model.

Interpretation of Coefficients

In logistic regression, the coefficients represent the log odds of the outcome variable being in a particular category. This means that a one-unit increase in a predictor variable results in a multiplicative change in the odds of the outcome variable being in a certain category. In probit regression, the coefficients represent the change in the z-score of the outcome variable for a one-unit increase in the predictor variable. This difference in interpretation can affect how the coefficients are used to make predictions.

Model Fitting

When it comes to model fitting, logistic regression and probit regression use different link functions to relate the predictor variables to the outcome variable. Logistic regression uses the logistic function, while probit regression uses the cumulative distribution function of the standard normal distribution. This difference in link functions can impact the computational complexity of fitting the models and the ease of interpretation of the results.

Goodness of Fit

Another important consideration when comparing logistic regression and probit regression is the goodness of fit of the models. Goodness of fit measures how well the model fits the data, and can be assessed using various metrics such as the likelihood ratio test, AIC, and BIC. While both logistic regression and probit regression can be used to assess goodness of fit, the choice of model may depend on the specific characteristics of the data and the research question.

Implementation

From an implementation perspective, logistic regression is more commonly used in practice compared to probit regression. This is partly due to the fact that logistic regression is easier to interpret and has a more straightforward link function. Additionally, logistic regression is supported by many statistical software packages, making it more accessible to researchers and practitioners. However, probit regression may be preferred in certain cases where the assumptions of logistic regression are not met.

Conclusion

In conclusion, logistic regression and probit regression are both valuable tools for binary classification problems. While they share some similarities in terms of their goals, there are important differences in their assumptions, interpretation, and implementation. Researchers and practitioners should carefully consider these differences when choosing between logistic regression and probit regression for their specific research questions and data characteristics.