This article explains the basics of Probit regression as an alternative method to logistic regression for analyzing binary outcomes. Probit regression utilizes the cumulative distribution function of the normal distribution to model the relationship between a binary outcome variable and independent variables. It provides a step-by-step example of calculating probabilities and estimating the model parameters. Probit regression is compared to logistic regression, highlighting their similarities and differences. Overall, logistic regression is typically preferred due to its simplicity and ease of interpretation.
A Crucial Method in Binary Outcome Analysis
When analyzing binary outcomes, many of us automatically turn to logistic regression. While logistic regression is a popular method, it’s not the only one available. There are other methods, such as the Linear Probability Model (LPM), Probit regression, and Complementary Log-Log (Cloglog) regression, which can also be effective. Unfortunately, there is a lack of information on these alternative methods.
The Linear Probability Model (LPM)
The LPM is not commonly used because it struggles to capture the curvilinear relationship between a binary outcome and independent variables.
Probit Regression
Probit regression uses the cumulative distribution function (CDF) of the normal distribution to model the relationship between a binary outcome and independent variables. It can be seen as a variation of logistic regression, which uses the logistic CDF. While there are technical articles on Probit regression available, they can be difficult for non-technical readers to understand.
In this article, we will explain the basic principles of Probit regression, its applications, and compare it to logistic regression.
Background
When examining the relationship between a binary outcome variable and an independent variable, we often see an S-shaped curve or sigmoid curve. This curve resembles a cumulative distribution function (CDF) of a random variable. Therefore, it makes sense to use the CDF to model the relationship. The two most commonly used CDFs are the logistic and the normal distributions. Logistic regression utilizes the logistic CDF, while Probit regression uses the cumulative distribution function (CDF) of the normal distribution.
The Basic Concept behind Probit Regression
Probit regression works by assuming that there is an unobservable latent variable, represented as Ai, which influences whether an individual will experience the binary outcome (e.g., depression). In our example, the weight of an individual determines the value of the latent variable, and the probability of experiencing the outcome increases with an increase in the latent variable.
To estimate the parameters of the Probit regression equation, we need to determine the probability of the outcome for different values of the independent variable. We can do this by calculating the probabilities using the inverse cumulative distribution function (CDF) of the normal distribution.
Practical Calculations
In our example, we have data on the weight and depression status of a sample of individuals. We can calculate the probability of depression for each weight by grouping the data and using the inverse CDF of the normal distribution.
Once we have the estimated latent variable Ai, we can estimate the model parameters by running a simple linear regression between Ai and the independent variable.
Mathematical Structure
The mathematical equations behind Probit regression involve the use of the standardized normal distribution and the inverse of the cumulative distribution function (CDF) of the normal distribution. These equations allow us to estimate the probabilities and the model parameters.
Probit vs Logit
Probit and logistic regression yield comparable results in terms of predicted probabilities. However, there is a minor distinction in their sensitivity to extreme values. Probit regression is less sensitive to extreme values compared to logistic regression. If you want your model to be sensitive at extreme values, logistic regression might be preferred.
A Practical AI Solution for Your Company
If you’re looking to evolve your company with AI and stay competitive, consider using the Back to the Basics: Probit Regression as a guide. AI can redefine the way you work, and it’s essential to identify automation opportunities, define measurable KPIs, select the right AI solution, and implement gradually. For AI KPI management advice and continuous insights, connect with us at hello@itinai.com.
Spotlight on a Practical AI Solution – AI Sales Bot
The AI Sales Bot from itinai.com/aisalesbot is designed to automate customer engagement and manage interactions throughout the entire customer journey. Discover how AI can redefine your sales processes and customer engagement by exploring our solutions at itinai.com.
