Logistic Regression and Marginal Effects

Intro

Logistic regression is a widely used statistical technique for modeling the relationship between a binary outcome and one or more predictor variables. It is commonly employed in various fields, such as healthcare, finance, and social sciences, to predict the probability of an event occurring. While the coefficients in logistic regression provide valuable insights, they are difficult to interpret. In this blog post, we will explore the use of marginal effects in logistic regression, unraveling their significance and practical applications.

The basics

In logistic regression, we model the log-odds of the event of interest occurring as a linear combination of predictor variables. The logistic function (sigmoid function) is then applied to this linear combination to convert it into a probability value between 0 and 1.

The logistic regression equation is given by:

\[\text{logit}(P(Y=1)) = \beta_0 + \beta_1X_1 + \beta_2X_2 + ... + \beta_pX_p\]

Where:

  • \(P(Y=1)\) is the probability of the event of interest (e.g., success) occurring.

  • \(logit (P(Y=1))\) is the log-odds (logit) of the event occurring.

  • \(\beta_0, \beta_1, \beta_2, ..., \beta_p\) are the coefficients of the model.

  • \(X_1, X_2, ..., X_p\) are the predictor variables.

Interpreting Coefficients in Logistic Regression

The coefficients \(\beta_1, \beta_2, ..., \beta_p\) in the logistic regression equation hold crucial information about the impact of each predictor variable on the log-odds of the event occurring. To interpret these coefficients, we need to consider their signs, magnitude, and statistical significance.

  1. Sign of Coefficients:

The sign of a coefficient reveals the direction of the relationship between the predictor variable and the log-odds of the event occurring. A positive coefficient (\(\beta > 0\)) indicates that an increase in the predictor variable leads to an increase in the log-odds (probability) of the event happening. Conversely, a negative coefficient (\(\beta < 0\)) suggests that an increase in the predictor variable is associated with a decrease in the log-odds (probability) of the event occurring.

  1. Magnitude of Coefficients:

The magnitude of a coefficient provides information about the strength of the relationship between the predictor variable and the log-odds of the event. Larger absolute values of coefficients indicate stronger influences on the probability.

  1. Statistical Significance:

Evaluating the statistical significance of coefficients is essential to determine if the relationship between a predictor variable and the event probability is meaningful or just due to random chance. Researchers often use hypothesis testing or confidence intervals to assess the statistical significance of coefficients.

Practical Example

Let’s consider an example to illustrate the interpretation of coefficients in logistic regression. Suppose we are studying the factors that influence whether a customer will make a purchase on an e-commerce website. Our logistic regression model includes two predictor variables: “Time Spent on Website” and “Number of Items Added to Cart.” The estimated coefficients are as follows:

  • \(\beta_{\text{Time Spent}} = 0.05\)

  • \(\beta_{\text{Number of Items}} = 0.2\)

Interpretation

  • \(\beta_{\text{Time Spent}} = 0.05\): For each additional minute a customer spends on the website, the log-odds (probability) of making a purchase increases by 0.05 units. This means that spending more time on the website is associated with a higher likelihood of making a purchase.

  • \(\beta_{\text{Number of Items}} = 0.2\): For each additional item a customer adds to the cart, the log-odds (probability) of making a purchase increases by 0.2 units. This suggests that adding more items to the cart is associated with a higher probability of making a purchase.

Interpretation Issues

Ideally, we want to understand what the model is telling us on the probability scale and not in the odds scale, much less in the estimation scale, the log-odds.

We might be tempted to convert log-odds to odds and then to probability and interpret everything as usual. But this would be very wrong!

Why?

The issue here is that on the probability scale the function is non-linear!

Marginal Effects

Marginal effects can be described as the change in outcome as a function of the change in the independent variable of interest holding all other variables in the model constant.

NOTE: In linear regression, the estimated regression coefficients are marginal effects and are more easily interpreted.*

One measure of change in a system is that the rate of change of the system is non-zero.

If we had a simple linear regression model for a trend, then the estimated rate of change would be the slope of the line. Technically, this is the instantaneous rate of change of the function that defines the line.

The idea can be extended to any function, even one as potentially complex non-linear function.

The problem we have is that in general we don’t have an equation for the non-linear from which we can derive the derivatives.

So how do we estimate the derivative of a non-linear function ?

One solution is to use the method of finite differences but to understand it properly we need to introduce first the basic concept of differential calculus.

Finite differences

Differential calculus is a branch of calculus that focuses on the study of rates of change and slopes of curves. It deals with the concept of derivatives, which represent the rate of change of a function at a specific point. Derivatives allow us to understand how a function is changing at a particular location and provide valuable insights into the behavior of functions.

The derivative of a function \(f(x)\) with respect to \(x\) is denoted as \(f'(x)\) or \(\frac{df(x)}{dx}\). It represents the slope of the tangent line to the graph of the function at a given point \(x\). The derivative can be interpreted as the instantaneous rate of change of \(f(x)\) with respect to \(x\).

For example, consider the function \(f(x) = x^2\). The derivative of \(f(x)\) is \(f'(x) = 2x\). This means that the slope of the tangent line to the graph of \(f(x)\) at any point \(x\) is twice the value of \(x\) at that point.

Differential calculus has wide-ranging applications in various fields, including physics, engineering, economics, and more. It is used to model and analyze phenomena that involve continuous changes, such as motion, growth, and optimization.

Finite differences are a numerical technique used to approximate derivatives of functions. The method involves computing the difference in function values at two nearby points and then dividing by the difference in their corresponding independent variable values. By using a small interval between the two points, we can get an approximation of the derivative at a specific point.

The finite difference formula for approximating the first derivative of a function \(f(x)\) at a point \(x\) is given by:

\[ f’(x) \approx \frac{f(x + h) - f(x)}{h} \]

Where:

  • \(h\) is a small value representing the step size or interval between the points.

Similarly, we can use finite differences to approximate higher-order derivatives, such as the second derivative:

\[ f’’(x) \approx \frac{f(x + h) - 2f(x) + f(x - h)}{h^2} \]

Finite differences are particularly useful when the analytical expression for the derivative is difficult to obtain or when dealing with discrete data points instead of continuous functions.

Differential calculus and finite differences are both essential tools in mathematics and scientific disciplines. Differential calculus provides a rigorous framework for studying rates of change and slopes of curves through derivatives. On the other hand, finite differences offer a practical and numerical approach to approximate derivatives when the analytical solution is not readily available or when dealing with discrete data. Together, these concepts enable us to better understand the behavior of functions and analyze real-world phenomena involving continuous and discrete changes.

Conclusion

In logistic regression, the estimated coefficients (log odds) indicate how the log odds of the outcome change with a one-unit increase in the predictor variable while holding other variables constant. However, interpreting these coefficients can be challenging, especially for non-statisticians. This is where marginal effects come to the rescue.

Marginal effects provide a more straightforward interpretation by measuring the impact of a one-unit change in the predictor variable on the probability of the event occurring (i.e., the probability of success). They express the change in the probability as a result of the predictor variable’s change, allowing us to understand the practical implications of the model.