Marginal Effects—Quantifying the Effect of Changes in Risk Factors in Logistic Regression Models

INTRODUCTION

This JAMA Guide to Statistics and Methods discusses the marginal effects approach to express the strength of the association between a risk factor and a binary outcome from a logistic regression.

Marginal effects can be used to express how the predicted probability of a binary outcome changes with a change in a risk factor. For example, how does 1-year mortality risk change with a 1-year increase in age or for a patient with diabetes compared with a patient without diabetes? This approach can make the results more easily understood. Marginal effects often are reported with logistic regression analyses to communicate and quantify the incremental risk associated with each factor.^1,2

In an article published in JAMA Psychiatry, Cummings et al³ studied factors that predicted access to US outpatient mental health facilities that accept Medicaid. Their main outcome had 3 categories, which were labeled “no access,” “some access,” and “good access.” An ordered logistic regression model was developed and results were presented as the change in the probability of each outcome for a change in certain demographic factors.

USE OF MARGINAL EFFECTS

Why Are Marginal Effects Used?

There are several ways to express the strength of the association between a risk factor and a binary outcome from a logistic regression. One popular approach is the odds ratio (OR).⁴ The odds are the ratio of the probability that an outcome occurs to the probability that the outcome does not occur. The ratio of the odds for 2 groups—the OR—is often used to quantify differences between 2 different groups; eg, treatment and control groups. Another approach is the risk ratio, which is the probability that the outcome occurs in the presence of the risk factor divided by the probability that the outcome occurs in the absence of the risk factor. Risk ratios are often easier to use in clinical practice than are ORs.^4,5

A third alternative is the marginal effect, which is the change in the probability that the outcome occurs as the risk factor changes by 1 unit while holding all the other explanatory variables constant. When the risk factor is continuous (eg, age), the change in the probability that the outcome occurs that is associated with a 1-unit change in the risk factor has been called a marginal effect. When the risk factor is discrete (eg, presence or absence of diabetes), the change has been called an incremental effect. In this chapter, the term marginal effect represents this strength of association measure in both instances.

What Are Marginal Effects?

Of the 3 approaches, marginal effects are the most intuitive because they are expressed as the change in the predicted probability that the outcome occurs that is associated with a 1-unit change in the risk factor. Unlike ORs, it ...