Differences Between Logit and Probit Models

The Logit and Probit models are two widely used techniques for analyzing binary outcome variables. Both are used to model the probability of a certain event occurring based on one or more predictor variables. Despite their similarities, they differ in their underlying assumptions and functional forms.

The Logit model, also known as logistic regression, assumes that the log-odds of the dependent variable being 1 (versus 0) is a linear function of the independent variables. Mathematically, the Logit model uses the logistic function, which produces probabilities bounded between 0 and 1. The logistic function is given by:

$P(Y=1\mid X)=\frac{1}{1+e^{-({\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+\cdots +{\beta }_k{X}_k)}}$P(Y=1∣X)=1+e−(β0​+β1​X1​+β2​X2​+⋯+βk​Xk​)1​

where $P(Y=1\mid X)$P(Y=1∣X) represents the probability of the dependent variable being 1, and $X_1,X_2,\dots ,X_k$X1​,X2​,…,Xk​ are the predictor variables. The coefficients ${\beta }_0,{\beta }_1,\dots ,{\beta }_k$β0​,β1​,…,βk​ are estimated from the data using maximum likelihood estimation.

In contrast, the Probit model assumes that the dependent variable is influenced by a latent (unobserved) variable which follows a standard normal distribution. The Probit model uses the cumulative distribution function (CDF) of the standard normal distribution to estimate probabilities. The Probit model can be expressed as:

$P(Y=1\mid X)=\Phi ({\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+\cdots +{\beta }_k{X}_k)$P(Y=1∣X)=Φ(β0​+β1​X1​+β2​X2​+⋯+βk​Xk​)

where $\Phi$Φ denotes the CDF of the standard normal distribution. Like the Logit model, the Probit model estimates the relationship between the independent variables and the probability of the dependent variable being 1.

Key Differences:

1. Functional Form: The Logit model uses the logistic function, while the Probit model uses the normal CDF. The logistic function has slightly heavier tails compared to the normal distribution, which can affect the estimated probabilities, especially for extreme values of the predictors.

2. Assumptions: The Logit model assumes a logistic distribution for the error term, whereas the Probit model assumes a normal distribution. This difference in distribution assumptions leads to different interpretations of the model coefficients.

3. Interpretation of Coefficients: In the Logit model, the coefficients represent changes in the log-odds of the dependent variable being 1 for a one-unit change in the predictor variables. In the Probit model, the coefficients represent changes in the z-score of the latent variable, which can be less intuitive to interpret directly.

4. Estimation: Both models use maximum likelihood estimation, but the computational methods and likelihood functions differ due to the different functional forms. Logit models tend to be computationally simpler, whereas Probit models may require more advanced numerical techniques.

5. Fit and Prediction: In practice, the choice between Logit and Probit models often depends on the specific application and the nature of the data. For most practical purposes, the predictions from Logit and Probit models are quite similar, but the choice of model can impact the interpretation of the results.

Applications and Examples:

To illustrate these differences, consider a dataset where we are predicting whether an individual will purchase a product (1 for purchase, 0 for no purchase) based on their income and age. Using both Logit and Probit models, we might find similar predicted probabilities, but the coefficients and their interpretations could differ. For instance, the Logit model might suggest that an increase in income leads to a higher log-odds of purchasing the product, while the Probit model might suggest a higher z-score of the latent variable.

In summary, while both Logit and Probit models are useful for binary outcome analysis, their differences in functional form, assumptions, and interpretation of coefficients can influence the choice of model depending on the specific research question and data characteristics.