Probit Model Regression: Understanding Its Use in Statistical Analysis

Why Probit?
Regression models come in many forms, and probit regression is one of the most fascinating due to its application in binary outcome modeling. But why should you care about it? Well, imagine you are trying to predict a yes-or-no decision, like whether someone will default on a loan or whether a voter supports a candidate. Ordinary linear regression won’t cut it here—probit regression will.

How Probit Differs from Other Models
Probit regression belongs to the family of generalized linear models (GLMs) and is specifically used when the dependent variable is binary. Unlike logistic regression, which uses the logistic function to estimate probabilities, probit regression uses the cumulative distribution function (CDF) of the standard normal distribution. Essentially, probit regression links the probability of an outcome occurring to the independent variables via a normal distribution.

Now you might ask: why go for probit over the more commonly used logistic regression? The answer lies in subtle differences in assumptions. While both logistic and probit models often produce similar results, they differ in their underlying distribution assumptions, with probit assuming a normal distribution of errors and logistic assuming a logistic distribution. In some fields like economics, the normality assumption of probit is preferred for theoretical reasons.

Practical Example: Loan Default Prediction
Let's get concrete. Imagine you're building a model to predict whether a customer will default on a loan. The dependent variable (Y) is binary, taking on a value of 1 if the customer defaults and 0 if they do not. The independent variables could include income, credit score, and employment status.

Using a probit model, you can estimate how changes in these independent variables influence the probability of default. The probit model would look something like this:

$P(Y=1\mid X)=\Phi ({\beta }_0+{\beta }_1\times \text{income}+{\beta }_2\times \text{credit score}+{\beta }_3\times \text{employment status})$P(Y=1∣X)=Φ(β0​+β1​×income+β2​×credit score+β3​×employment status)

Here, $\Phi$Φ is the CDF of the standard normal distribution, and the coefficients ${\beta }_1,{\beta }_2,{\beta }_3$β1​,β2​,β3​ tell you how much each variable influences the likelihood of default.

Step-by-Step Implementation in Python
Let’s take this theoretical knowledge and apply it in Python using the statsmodels library.

python
import statsmodels.api as sm
import numpy as np

# Simulating data
np.random.seed(0)
n = 1000
income = np.random.normal(50, 10, n)  # Simulated income
credit_score = np.random.normal(700, 50, n)  # Simulated credit score
employment_status = np.random.binomial(1, 0.8, n)  # 1 if employed, 0 if not

# Generate binary outcome
default = (0.3 * income + 0.5 * credit_score + 2 * employment_status + np.random.normal(0, 1, n)) > 800

# Define the dependent and independent variables
X = sm.add_constant(np.column_stack((income, credit_score, employment_status)))
y = default.astype(int)

# Probit model
probit_model = sm.Probit(y, X)
result = probit_model.fit()

print(result.summary())

Understanding the Output
The summary() method will show you various statistics such as the coefficients for each independent variable. These coefficients don’t directly tell you the change in probability, but you can interpret the sign and relative magnitude. For example, a positive coefficient for income means that as income increases, the likelihood of default decreases.

Marginal Effects
To get more meaningful insights, you might want to calculate the marginal effects, which will tell you how a unit change in an independent variable affects the probability of the outcome.

python
marginal_effects = result.get_margeff()
print(marginal_effects.summary())

The marginal effects output helps in understanding how, for instance, a $1 increase in income or a 1-point increase in credit score changes the probability of loan default.

When to Use Probit vs Logistic
Probit and logistic models are often used interchangeably in practice, but the choice between them can sometimes hinge on domain-specific preferences or the need for precision in modeling the probability distribution of errors. Logistic regression is more popular because of its interpretability—the logistic curve is easier for most to grasp intuitively. However, in certain fields like economics and finance, where the assumption of normality is more aligned with theoretical expectations, probit models are the go-to.

The following table provides a quick comparison between probit and logistic regression:

Model	Link Function	Assumed Distribution of Errors
Probit Regression	Cumulative normal distribution	Normal distribution
Logistic Regression	Logistic function	Logistic distribution

Interpreting Probit Coefficients
In probit regression, interpreting the coefficients directly can be tricky because they represent changes in the z-score (from the standard normal distribution) rather than changes in the probability itself. That's why we use marginal effects to interpret the relationship between independent variables and the probability of the outcome occurring.

For example, if the coefficient for credit score is 0.02, it doesn’t mean that a 1-point increase in credit score increases the probability of default by 0.02. Instead, it means that a 1-point increase in credit score increases the z-score of the cumulative normal distribution by 0.02, which in turn affects the probability.

Use Cases for Probit Regression
Probit models are especially useful in fields like:

1. Economics: Where normality of errors is a reasonable assumption, such as in modeling binary outcomes like labor participation, market entry, or consumer choice.
2. Marketing: Probit can be applied when analyzing customer responses to promotions, predicting binary outcomes like purchase decisions, subscription renewals, or opt-ins for services.
3. Medical Research: When studying outcomes like disease presence (yes/no), probit models are often used alongside logistic models for robustness.

Probit vs Other Binary Models
If you're still on the fence about using probit, consider this: it's not the only player in the binary regression world. Other models, like logit, and even more complex alternatives such as Bayesian hierarchical models or machine learning classifiers (e.g., Random Forest or Support Vector Machines), might outperform probit in specific cases, especially when your data isn’t perfectly suited to its assumptions.

Challenges and Limitations
While the probit model is powerful, it's not without limitations. One challenge is interpretability, particularly when trying to communicate results to a non-technical audience. The transformation from the linear predictor to probabilities is non-linear and can make intuitive explanations difficult. Additionally, probit models assume that the error terms follow a normal distribution, which might not always hold in real-world data.

Conclusion
Probit regression is a valuable tool for binary outcome prediction, especially in contexts where the normality of errors is a reasonable assumption. Though it can be slightly more complex to interpret than logistic regression, it offers benefits in certain fields where its theoretical foundations are preferred.

By focusing on practical examples and understanding its subtle differences from logistic regression, you’ll be better equipped to decide when and why to use probit in your statistical toolbox.