Logit and Probit models are both types of regression models commonly used in statistical analysis, particularly in the field of binary classification. This means that the outcome of interest can only take on two possible values / classes. In most cases, these models are used to predict whether or not something will happen in form of binary outcome. For example, a bank might want to know if a particular borrower might default on loan or otherwise. 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.
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 / binary possible outcomes or categorical 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 function is used to model the relationship between the predictors and the probability of the event occurring, and it produces an output on a continuous scale that ranges from 0 to 1.
Logit models generally take one of two forms: multinomial logits and binary logits. Multinomial logits predict a value from multiple mutually exclusive outcomes, while binary logits predict either a 1 or 0 outcome from a single variable. In both cases, the model takes into account independent variables that may influence the outcome, such as borrower’s credit score, income, debt-to-income ratio, loan amount, etc when predicting whether borrower would default on loan. The model then produces an estimated probability which is compared against a predetermined threshold to determine if the predicted outcome is correct or not.
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.
$$ P = \sigma(z) = \frac{1}{1 + e^{-Z}} $$
Logit models produces S-shaped curve such as the following:
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. Probit function is also called as probit link function. In probit model, the cumulative standard normal distribution function is used to model the relationship between the predictors and the probability of the event occurring. The output of the Probit model also ranges from 0 to 1, like that of the Logit model.
The Probit model can be represented using the following formula:
Pr(Y = 1|X) = Φ(Z)
Where,
Z = 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. 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 distribution function of standard normal distribution. The parameters (such as b0, b1, etc) are estimated using maximum likelihood estimation technique.
You may note that the key difference between logit and probit model is the sigmoid or logistic function and cumulative normal distribution function respectively.
Key Differences between Logit & Probit Models
The following are some of the key differences between the Logit and Probit models:
- Link function: The main difference between Logit and Probit models lies in the choice of the link function used to model the relationship between the predictor variables and the probability of the event occurring. In the case of the logit model, we use a logistic or sigmoid function while in case of probit models, the probit link function Φ used is a cumulative distribution function of the standard normal distribution.
- Model Assumptions: Logit and Probit models make different assumptions about the distribution of the error term. Logit models assume that the error term follows a logistic distribution, while Probit models assume that the error term follows a normal distribution.
- Usage: The logit model is more widely used than the probit models and has a more extensive literature.
- Outliers: Logit model is also more robust to outliers as it uses a logistic function but Probit model is more sensitive to outliers
The picture below represents the Logit & Probit models:
Thanks. This is quite informative. I feel confident that I can use these models in research now.
Thank you!!!!
What a refined concept and understanding!!
Thank you
Thank you
I have got important concept on the the two models.
Thank you for leaving your feedback.
Very helpful explanation. Thank you.