Logit and probit models are statistical models that are used to model binary or dichotomous dependent variables. This means that the outcome of interest can only take on two possible values. In most cases, these models are used to predict whether or not something will happen. For example, a business might want to know if a particular advertising campaign will lead to an increase in sales. In this blog post, we will explain what logit and probit models are, and we will provide examples of how they can be used. As data scientists, it is important to understand the concepts of logit and probit models and when should they be used.

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## What are Logit models?

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. The logit model is based on the logistic function (also called the sigmoid function), which is used to model situations where there are two possible outcomes. The logistic function can be used to model a variety of situations, including binary dependent variables, dichotomous dependent variables, and categorical data.

The logit model is used to model the odds of success of an event as a function of independent variables. The following is the starting point of arriving at the logistic function which is used to model the probability of occurrence of an event.

A logit function can be written as follows:

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. The above equation can be solved further to arrive at the following function which can be used to determine the probability of occurrence of the events.

$$ P = \sigma(z) = \frac{1}{1 + e^{-Z}} $$

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.

## What are Probit models?

Probit models are a form of a statistical model that is used to predict the probability of an event occurring. 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. In the case of the Probit model, the dependent variable is categorical and can only take on one of the two values, such as yes or no, true or false.

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.

## What is the difference between the Logit and Probit models?

The following are some of the key differences between the Logit and Probit models:

- 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.

The picture below represents the Logit & Probit models:

Probit models as like the logit models are used to predict the probability of an event occurring. Probit models are similar to logit models, but they are based on probits instead logistic functions. 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 process for calculating probabilities in logit and probits differ from each other because logistic functions use linear combinations while probity uses cumulative standard normal distribution function.

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