Logit vs. Probit

Attribute	Logit	Probit
Model Type	Linear probability model	Non-linear probability model
Link Function	Logit function	Standard normal cumulative distribution function
Interpretation of Coefficients	Log-odds ratios	Change in probability
Assumption	Errors follow logistic distribution	Errors follow standard normal distribution
Computational Complexity	Less computationally intensive	More computationally intensive

Introduction

Logit and Probit models are two commonly used statistical techniques for modeling binary outcomes. Both models are used in various fields such as economics, sociology, and epidemiology to analyze the relationship between a set of independent variables and a binary dependent variable. While both models are similar in many ways, they also have distinct differences in terms of their assumptions, interpretation, and estimation methods.

Assumptions

One key difference between Logit and Probit models lies in their underlying assumptions. The Logit model assumes that the errors follow a logistic distribution, while the Probit model assumes that the errors follow a standard normal distribution. This difference in assumptions can impact the interpretation of the coefficients in each model. In the Logit model, the coefficients represent the change in the log odds of the dependent variable for a one-unit change in the independent variable. In contrast, the coefficients in the Probit model represent the change in the probability of the dependent variable being equal to one for a one-unit change in the independent variable.

Interpretation

Another difference between Logit and Probit models is the way in which the results are interpreted. In the Logit model, the coefficients are typically interpreted in terms of odds ratios, which can be more intuitive for many researchers. For example, a coefficient of 0.5 in a Logit model would indicate that a one-unit increase in the independent variable is associated with a 50% increase in the odds of the dependent variable being equal to one. In contrast, the coefficients in a Probit model are not as easily interpretable, as they represent changes in probabilities rather than odds ratios.

Estimation

Both Logit and Probit models are estimated using maximum likelihood estimation, which is a common method for estimating parameters in statistical models. However, the estimation process for the two models differs slightly due to the different distributional assumptions. In the Logit model, the likelihood function is based on the logistic distribution, while in the Probit model, it is based on the standard normal distribution. This difference in likelihood functions can lead to slightly different parameter estimates for the two models, although the overall results are often quite similar.

Goodness of Fit

When comparing Logit and Probit models, it is important to consider the goodness of fit of each model. Goodness of fit measures how well the model fits the data, with higher values indicating a better fit. In general, both Logit and Probit models tend to have similar goodness of fit statistics, such as the likelihood ratio test or the Akaike Information Criterion (AIC). However, in some cases, one model may provide a better fit to the data than the other, depending on the specific characteristics of the dataset and the research question being addressed.

Robustness

Another important consideration when comparing Logit and Probit models is the robustness of the results. Robustness refers to the sensitivity of the results to changes in the model specification or assumptions. In general, both Logit and Probit models are considered to be robust to violations of their underlying assumptions, such as the independence of errors or the linearity of the relationship between the independent and dependent variables. However, in some cases, one model may be more robust than the other, depending on the specific characteristics of the data and the research question being addressed.

Conclusion

In conclusion, Logit and Probit models are two widely used statistical techniques for modeling binary outcomes. While both models have similar estimation methods and are commonly used in practice, they also have distinct differences in terms of their assumptions, interpretation, and estimation. Researchers should carefully consider the specific characteristics of their data and research question when choosing between Logit and Probit models, as each model has its own strengths and limitations. By understanding the differences between Logit and Probit models, researchers can make more informed decisions about which model is most appropriate for their analysis.