How to Choose Between Logit and Probit Models

Deciding between Logit and Probit models can be crucial when conducting binary outcome analyses. Both are widely used in fields such as economics, finance, and political science. Key differences lie in the distribution assumptions and, consequently, the interpretation of the results. This guide will help you choose the right model by highlighting critical factors like data distribution, theoretical background, and practical considerations.

Understanding the Basics At the core of both Logit and Probit models is a binary dependent variable. This variable represents two outcomes (e.g., success/failure, 1/0, yes/no). Both models are designed to estimate the probability that a given event occurs. However, their primary distinction comes from the underlying assumptions about the error term distribution.

• Logit Model: Assumes a logistic distribution for the error terms.
• Probit Model: Assumes a normal distribution for the error terms.

These differing assumptions affect the shape of the curve that estimates the probabilities, especially at the tails. Logit models tend to have heavier tails, meaning they assume that outliers have a larger impact than in Probit models.

Choosing the Right Model: A Practical Guide

1. Data Characteristics: The first step in choosing between these models is understanding your data. Logit models are generally more forgiving with outliers and extreme values because of the heavier tails in the logistic distribution. If your data includes many outliers, Logit might be a better choice. However, if your data follows a more symmetrical, bell-curve-like distribution, Probit may be more appropriate.

2. Theoretical Considerations: In some cases, the choice may depend on theory or the field's conventions. Economists, for example, often prefer Probit models because of the assumption of normal distribution, which fits many economic models. On the other hand, Logit models are common in fields like political science, where the interpretation of odds ratios is more familiar.

3. Interpretation: One crucial distinction between the two models is in how we interpret the coefficients. In Logit models, the coefficients are interpreted in terms of odds ratios, which can be more intuitive for certain audiences. For instance, a coefficient in a Logit model can be explained as, "For a one-unit increase in the predictor variable, the odds of success increase by a certain percentage."

   In contrast, Probit models do not provide such a straightforward interpretation. The coefficients in a Probit model represent the change in the z-score of the latent variable (or the unobservable propensity) that is associated with the outcome. This makes Probit models less intuitive for non-statisticians, but it can be a better fit when dealing with underlying continuous distributions.

Illustrative Example: Consider the following example in the realm of credit risk analysis. A bank is assessing whether a loan applicant will default (1) or not default (0) on a loan. The dependent variable is binary, but the decision on which model to use will depend on the nature of the data.

• If the dataset contains outliers (e.g., very high or very low incomes that may influence default risk), a Logit model would be preferable because it is more robust to extreme values.
• If the data fits a bell-shaped distribution with fewer extreme values, a Probit model may offer more accurate estimates.

Model Fit and Selection Criteria Statistical software often provides goodness-of-fit measures like the Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC) to compare models. Generally, the model with the lower AIC or BIC is preferred. Both Logit and Probit models can be evaluated using these metrics, but it's essential to remember that these criteria should be complemented by theoretical considerations.

Computational Considerations In terms of computation, Logit models tend to be more straightforward and faster to estimate than Probit models. This is because the logistic function is easier to compute. For large datasets, this difference can be significant. If computational resources are a constraint, Logit models might be the better choice.

Table: Comparison Between Logit and Probit Models

Factor	Logit Model	Probit Model
Distribution	Logistic	Normal
Tail Behavior	Heavier tails, more robust to outliers	Lighter tails, sensitive to extreme values
Interpretation	Odds ratios	Z-scores of the latent variable
Computational Efficiency	Faster and easier to compute	Slightly more complex, slower
Common Application	Political science, social sciences	Economics, finance

Real-World Applications

• Logit models are often used in marketing for predicting customer behavior (e.g., whether a customer will buy a product).
• Probit models are more common in health economics and labor studies, where normally distributed errors are assumed.

Final Thoughts: Which Model Should You Choose? In summary, there is no definitive answer to whether a Logit or Probit model is superior. The choice depends on your data, theoretical assumptions, and the practical needs of your analysis. Here are a few final tips to guide your decision:

• Use Logit if:

  • You expect outliers or extreme values in your data.
  • You need a model that is computationally simpler and quicker to estimate.
  • You need to explain results in terms of odds ratios.

• Use Probit if:

  • Your data closely follows a normal distribution.
  • You are working in a field where the Probit model is the standard.
  • You are less concerned with intuitive interpretation and more focused on accurate modeling of underlying latent variables.

Test Both Models: Often, analysts will run both Logit and Probit models and compare the results. Since the estimates are usually very similar, the choice might boil down to ease of interpretation and field-specific preferences. The key is understanding the strengths and limitations of each model so that you can make an informed choice.