Logit vs Probit Models:

In statistical modelling, binary or dichotomous dependent variables are modelled using the logit and probit models. This implies that there are only two possible values for the outcome of interest.

Logit models are a form of a statistical model that is used to predict the probability of an event occurring. Logit models are also called logistic regression models. It is used to model the odds of success of an event as a function of independent variables.

Mathematically it can be written as below

logit(I) = log[P/(1-P)] = Z = b0 + b1X1 + b2X2 + ….. + bnXn

where P is the probability of an event occurring, and l is the odds of an event occurring. Z is the linear combination of independent variables with coefficients. 

From this we can also derive the probability of occurrence of the events.

P=σ(z)=1/(1+e^−Z)

The σ(Z) is also called a logistic or sigmoid function. As the value of Z approaches -infinity, the value of σ(Z) or P approaches 0. And, as the value of Z approaches +infinity, the value of σ(Z) or P approaches 1.

Probit models are similar to logit models, but they are based on the probit function instead of the logistic function. The Probit model determines the likelihood that an item or event will fall into one of a range of categories by estimating the probability that observation with specific features will belong to a particular category.

The Probit model can be represented using the following formula:

Pr(Y = 1|X) = Φ(Z) = Z = Φ(b0 + b1X1 + b2X2 + ….. + bnXn)

Where, Y is the dependent variable and represents the probability that the event will occur (hence, Y = 1) given the variables X. Φ is the cumulative standard normal distribution function. Z is the linear combination of independent variables (X) with coefficients (b0, b1, b2…bn). In the case of the logit model, we use logistic or sigmoid function instead of Φ which is cumulative standard normal distribution function.

Some of the distinguishing factors between the logit and probit models can be described as below:

• The logit model is used to model the odds of success of an event as a function of independent variables, while the probit model is used to determine the likelihood that an item or event will fall into one of a range of categories by estimating the probability that observation with specific features will belong to a particular category.
• In the case of the logit model, we use a logistic or sigmoid function instead of Φ which is a cumulative standard normal distribution function.
• Logistic regression models are also called logit models, while probit regression models are also called probit models.
• Logit models are used to model Logistic distribution while probit models are used to model the cumulative standard normal distribution.