Thinking beyond the mean: a practical guide for using quantile regression methods for health services research

Health services and health economics research articles commonly use multivariate regression techniques to measure the relationship of health service utilization and health outcomes (the outcomes of interest) with clinical characteristics, sociodemographic factors, and policy changes (usually treated as explanatory variables). Common regression methods measure differences in outcome variables between populations at the mean (i.e., ordinary least squares regression), or a population average effect (i.e., logistic regression models), after adjustment for other explanatory variables of interest. These are often done assuming that the regression coefficients are constant across the population – in other words, the relationships between the outcomes of interest and the explanatory variables remain the same across different values of the variables. There are times, however, when researchers, policymakers, and clinicians may be interested in group differences across the distribution of a given dependent variable rather than only at the mean. Taking a more concrete example from the literature, research on individuals' consumption of alcohol consistently reported that higher alcohol prices were associated with lower alcohol consumption.[1] This led to a call for increases in taxes as a policy lever to reduce alcohol consumption and the subsequent social costs of alcoholism and alcohol abuse. However, these studies did not provide any information about whether increased price decreased alcohol use similarly for light drinkers, moderate drinkers, and heavy drinkers. Because there are positive social benefits for light drinkers and negative health and social consequences for heavy drinkers, analyzing the demand response of different types of drinkers was important to understanding who was most likely to modify their behavior due to increasing alcohol taxes. A subsequent study[2] found light and heavy drinkers were much less price elastic than moderate drinkers; that is, higher taxes did not reduce consumption nearly as much for light and heavy drinkers as it did for moderate drinkers. The policy implication is that increasing alcohol taxes might bring in revenue (and reduce alcohol-related accidents among moderate drinkers) but will have limited success in reducing the prevalence of heavy drinking and its sequelae. Another example is that associations of interest explaining health care and health outcomes may be very different among the highest utilizers of health care, compared to individuals at the bottom or middle of the distribution of health care utilization. As a simple illustration, Figure 1 plots the relationship between the number of hours attended of a hypothetical psychotherapy intervention (x-axis) and a fictitious scale of post-intervention mental health (higher score indicates better mental health on the y-axis) for a group of 400 individuals. In this example, the regression line from an ordinary least squares (OLS) regression model is essentially flat, suggesting that there is no relationship between number of psychotherapy session-hours and mental health at follow-up. To describe the association between number of session-hours and mental health for individuals with low and high post-treatment scores on the mental health scale using OLS, the analyst extends the line up or down to the 90th and 10th quantiles in a parallel fashion, as the OLS model assumes the association between hours of psychotherapy and mental health outcome remains the same at different levels of the mental health scale. Open in a separate window Figure 1. Prediction lines at 10th quantile, mean, and 90th quantile using ordinary least squares (OLS) regression In contrast, in Figure 2, we use quantile regression to allow slopes of the regression line to vary across quantiles of the mental health scale. Although the median line is flat as before, the 90th quantile prediction line is significantly increasing whereas the 10th quantile prediction line is significantly decreasing. This suggests that the association between the hypothetical intervention and post-intervention mental health is positive for those with better post-intervention mental health but there is a negative association among those with poorer post-intervention mental health. Quantile regression provides greater flexibility than other regression methods to identify differing relationships at different parts of the distribution of the dependent variable. Open in a separate window Figure 2. Prediction lines at 10th quantile, mean, and 90th quantile using quantile regression Health care expenditures are another area important to policy that is amenable to an analytical strategy that measures differences across the distribution. The average user of health care is obviously very different from the heavy user in terms of health status, but what about other factors such as race/ethnicity, gender, employment, insurance status and other factors of policy interest? Quantile regression allows for analysis of these other differences that exist among heavy health care users in a way that is not possible with commonly used regression methods. In previous applications, we have used quantile regression methods to assess racial and ethnic disparities in health care expenditures and in mental health care expenditures across different quantiles of expenditures, adjusting for covariates.[3] In the United States, disparities in the distribution of health care expenditures between Blacks and Whites, and between Hispanics and Whites diminish in the upper quantiles of expenditure, but remain significant throughout the distribution. This same pattern of persistent disparities was still evident in the highest education and income categories.