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Applications for Quantile Regression in Epidemiology

  • Epidemiologic Methods (P Howards Section Editor)
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Abstract

Purpose of Review

To illustrate the utility of quantile regression in epidemiology for outcomes that are continuous and when exposure effects may differ across the distribution of the outcome. Linear regression methods estimate only the effects at the mean level which may be an incomplete and biased summary of the effect of exposures for some continuous health outcomes.

Recent Findings

There are several variations of the quantile regression method including classical linear quantile regression, nonparametric quantile regression for growth trajectories, and the modified quantile regression for case–control designs. Such methods offer several applications including (1) the use of quantile regression to test whether the effects of exposure are similar across quantiles, (2) the use of quantile regression for risk prediction, and (3) the use of quantile regression to examine the effects of growth trajectories over time.

Summary

Quantile regression is an important tool for understanding continuous health outcomes, especially outcomes that are not normally distributed, as it offers insight into the relation of exposures with respect to the distribution of the outcome. Quantile regression methods have the potential to deepen and expand the existing quantitative evidence from more common mean-based analyses.

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Author information

Authors and Affiliations

  1. Department of Biostatistics, Columbia University Mailman School of Public Health, New York, NY, USA

    Ying Wei

  2. Department of Epidemiology, Columbia University Mailman School of Public Health, 722 West 168th St, New York, NY, 10032, USA

    Rebecca D. Kehm, Mandy Goldberg & Mary Beth Terry

  3. Herbert Irving Comprehensive Cancer Center, Columbia University Medical Center, New York, NY, USA

    Mary Beth Terry

Authors
  1. Ying Wei
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  2. Rebecca D. Kehm
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  3. Mandy Goldberg
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  4. Mary Beth Terry
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Corresponding author

Correspondence to Mary Beth Terry.

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Appendix

Appendix

Table 2. Computational packages and basic syntax of quantile regression
Full size table

Basic Quantile Regression Syntax in R

> install.packages(“quantreg”).

> library(quantreg).

> fit = rq(y~×1 + ×2, tau = .5, data = data).

Note: tau is the quantile level(s) of interest. It could a single value for a fixed quantile level, or a vector of quantile levels, tau = c(0.25, 0.5, 0.75). The function rq() will return regression quantiles from multiple quantiles. If tau is smaller than 0 or larger than 1, the function will return the entire quantile process.

Basic Quantile Regression Syntax in SAS

PROC QUANTREG.

DATA = sas-data-set;

CLASS X1;

MODEL Y = X1 X2 / QUANTILE = 0.25 0.5 0.75;

RUN;

Note: if the option QUANTILE = ALL, it returns the entire quantile process. Same as in R, the default value is 0.5, corresponding to the median.

Statistical Inference of Quantile Regression in R and SAS

To obtain statistical inference of quantile regression in R, we need to use the function summary.rq(object, se = “nid”, ...), where object is the returned object from the function rq(), and the parameter se specify the inference methods. In SAS, the inference options are specified at the PROC QUANTREG Statement following the syntax “PROC QUANTREG CI= <NONE|RANK|...> ALPHA = value ;” where ALPHA is the significance level, and CI specifies the choice of inference. The table below lists the available methods in R and SAS.

  

Options

Inference method

Subcategories

R

SAS

Direct

i.i.d. model

n.i.d. model

se =” iid”

se = “nid”

CI = SPARCITY/IID

CI = SPARCITY

Rank Score

 

se =” rank”

CI = RANK

resampling

Pairwise

se =” boot”, bsmethod = “xy”

Not available

 

Parzen, Wei and Ying

se =” boot”, bsmethod = “pxy”

Not available

 

MCMB

se =” boot”, bsmethod = “mcmb”

CI = RESAMPLING

 

Wild

se =” boot”, bsmethod = “wild”

Not available

R script for the nonparametric quantile regression for growth trajectories with B-spline approximation

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Wei, Y., Kehm, R.D., Goldberg, M. et al. Applications for Quantile Regression in Epidemiology. Curr Epidemiol Rep 6, 191–199 (2019). https://doi.org/10.1007/s40471-019-00204-6

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