Regression is a statistical staple of quantitative modeling. When business or academic researchers use the values of several variables as an explanation or prediction of a scale outcome, the method is called linear regression. But what happens when you want to know the median or an arbitrary quantile of the scale outcome? Enter quantile regression.
Quantile regression models the relationship between a set of independent variables and specific percentiles, or “quantiles,” of a dependent variable, most often the median. So, how does quantile regression work? What value does it add to business and research? This article explores the answer to those questions.

How quantile regression works

For quantile regression, a single numeric dependent variable is required. The target variable needs to be continuous. The predictors can be continuous variables or dummy variables for categorical predictors. Either the intercept term or at least one predictor is required to run an analysis.
It is a more flexible method than other linear and other regression methods. It can identify differing relationships at different parts of the distribution of the dependent variable. The result is regression coefficients that estimate an independent variable’s effect on a specified quantile of the dependent variable.
Consider the observation in healthcare, based on a linear regression model, that there is no relationship between the number of psychotherapy session hours and mental health at follow-up. But, what about estimating the median, or the 0.25 quantile, or the 0.90 quantile? That’s where quantile regression comes in. The math under the hood suggests that the relationship of hypothetical intervention and post-intervention mental health is positive for those with better post-intervention mental health, but a negative one for those with poorer post-intervention mental health.
Here’s another example. Suppose you want to know the relationship between total household income and the proportion of income that is spent on food. According to Engel’s law, the more the income increases, the smaller the proportion of income spent on food, even if absolute expenditure on food rises. But when you apply quantile regression to the data, you can determine which food expense can cover 90% of families (for 100 families with a given income).

The value of quantile regression

Quantile regression enables a more comprehensive analysis of the relationship between variables. Unlike Ordinary Least Squares regression, it makes no assumptions about the target variable, and it can resist the influence of outlying observations. For that reason, it is widely used for research in academia and industries such as retail, ecology, and finance.
Retailers, for example, can use quantile regression to research “close rate,” which is the percentage of shoppers who enter the store and make a purchase. Understanding how to predict a high close rate or the variables that differentiate between low and high close rates can help retail companies increase profitability and determine when to expand their business.
In ecology, complex interactions between organisms often cause unequal variations in statistical distributions of ecological data. Quantile regression enables a more complete picture. A granular example is the self-thinning of annual plants in the Chihuahuan desert of the southwestern U.S. To estimate the relationship in the reduction in density of mature plants with increasing seedling germination density, ecologists can use quantile regression to look at higher plant densities associated with upper quantiles, where competition for resources is greatest, and see gauge the effects. This knowledge is beneficial in the search for ways to offset the change in climate that is occurring globally.
In finance, researchers have used quantile regression analysis to identify the relationship between finance firm performance and gender diversity on their boards. Based on annual data from U.S. firms from 2007 to 2014, the application of quantile regression showed that the presence of women on the board has a positive effect on firm performance. This information can be of valuable assistance to firms seeking to grow their boardroom.
In academia, researchers have used quantile regression to predict student academic achievement.