Relative Risk Regression

Overview

The most common way to model associations with a dichotomous outcome variable is through logistic regression. Such associations can instead be estimated and communicated as relative risks (sometimes called risk ratios or prevalence ratios) under certain circumstances. In the case of a rare outcome (i.e. the prevalence of the outcome is ≤ 10%) the odds ratio will approximate the risk ratio and relative risk regression is not necessary. However, if the outcome is not rare and the study participants are sampled in such a way that the baseline risk in the unexposed can be estimated (e.g. cohort study, cross-sectional study or case-cohort study) then relative risk regression can be performed and may in fact be the preferred analysis method. These recommendations assume that the association itself is the quantity of interest (rather than optimal prediction) and that a dichotomous outcome is meaningful. Some analyses ignore important information by treating a continuous quantity as dichotomous or by discarding time to event data in order to use logistic regression, and relative risk regression does not fix this problem. Relative risk regression can be implemented quite easily with most standard software packages.

Description

The relative risk (also called the risk ratio or prevalence ratio or relative prevalence) is

• Easy to interpret and explain

• Often the quantity of interest (although additive risk should also be considered)

• Estimable via relative risk regression using standard statistical software

Using logistic regression and the corresponding odds ratios may be necessary

• Logistic regression is still used for case-control studies

• Logistic regression is to similar relative risk regression for rare outcomes

• Logistic regression is fine to estimate direction and significance for main effects

Relative risks can be estimated from odds ratios (see Zhang & Yu, JAMA, 1999)
RR = OR/[(1-probability in reference group) + (probability in reference group x OR)]

• Why not just do this, and skip the relative risk regression?

-Adjustment for confounding is still on the OR scale
-Confidence intervals are too narrow

Relative risk regression is preferred as it allows the direct estimation of relative risks

• Log link

log(Y) = constant + β*X + error

• Working variance assumption: binomial or Poisson

• Robust standard errors to relax variance assumptions

Readings

Methodological Articles

Describing the problem

Dangers of dichotomization
Cohen J. “The cost of dichotomization.” Applied Psychological Measurement. 1983;7(3):249.
This Psychology article highlights the ways that dichotomization can distort associations and reduce statistical power.

MacCallum RC, Zhang S, Preacher KJ, Rucker DD. “On the practice of dichotomization of quantitative variables.” Psychol Methods. Mar 2002;7(1):19-40.
The authors present the case that dichotomization is rarely defensible and often will yield misleading results.

DeCoster J, Iselin AM, Gallucci M. “A conceptual and empirical examination of justifications for dichotomization.” Psychol Methods. Dec 2009;14(4):349-366.
This paper offers a more positive view of dichotomization, along with simulation analyses examining when dichotomization leads to worse statistical performance and when it does not.

OR interpretation
Sinclair JC, Bracken MB. Clinically useful measures of effect in binary analyses of randomized trials. Journal of clinical epidemiology. Aug 1994;47(8):881-889.
The authors critically evaluate the clinical utility of the odds ratio versus a risk ratio or risk difference; they examine the potential for error if the odds ratio is used to represent a risk ratio.

Liberman AM. How much more likely? The implications of odds ratios for probabilities. American Journal of Evaluation. 2005;26(2):253.
This paper considers the challenge of fitting odds ratios into policy discussions where the contrasting probabilities themselves are of greater interest.

Lovasi GS, Underhill LJ, Jack D, Richards C, Weiss CC, and Rundle A, At odds: concerns raised by odds ratios for common outcomes in research on physical activity and obesity. The Open Epidemiology Journal, 2012. 5: 13-17.
This narrative review discusses the problem of interpreting odds ratios within a particular outcome context (research addressing physical activity and obesity), with solutions suggested.

OR interactions
Morabia A, Ten Have T, Landis JR. Interaction fallacy. Journal of clinical epidemiology. Jul 1997;50(7):809-812.
This foundational paper highlights conditions in which an interaction apparent on an odds ratio scale would represent a statistical artifact rather than addressing true effect modification.

Campbell UB, Gatto NM, Schwartz S. Distributional interaction: Interpretational problems when using incidence odds ratios to assess interaction. Epidemiol Perspect Innov. Mar 3 2005;2(1):1.
Graphics and equations are used to provide a guided tour of why interactions on an odds ratio scale may not represent interactions on a relative risk scale, and vice versa.

Krieger N, Chen JT, Ware JH, Kaddour A. Race/ethnicity and breast cancer estrogen receptor status: impact of class, missing data, and modeling assumptions. Cancer Causes Control. Dec 2008;19(10):1305-1318.
The authors consider the possibility that using odds ratios may yield inflated estimates of racial/ethnic disparities.

Implementation recommendations

Spiegelman D, Hertzmark E. Easy SAS calculations for risk or prevalence ratios and differences.American journal of epidemiology. Aug 1 2005;162(3):199-200.
An invited editorial note with SAS code declares there is no longer sufficient justification for working with odds ratios if they are not the parameter of interest.

Lumley T, Kronmal RA, Ma Y. Relative risk regression in medical research: models, contrasts, estimators, and algorithms. Biostatistics Working Paper Series. Seattle, WA: University of Washington; 2006:26.
This working paper discusses the motivation for relative risk regression and notes on how to implement relative risk regression across statistical software packages.

Zhang J, Yu KF. What’s the relative risk? A method of correcting the odds ratio in cohort studies of common outcomes. JAMA. Nov 18 1998;280(19):1690-1691.
The authors offer a “back of the envelope” way to estimate relative risks from published odds ratios if the outcome prevalence in the reference group is known.

McNutt LA, Wu C, Xue X, Hafner JP. Estimating the relative risk in cohort studies and clinical trials of common outcomes. American journal of epidemiology. May 15 2003;157(10):940-943.
The potential for bias from using odds ratios in prospective studies is discussed, and simulation studies are used to provide guidance on implementation of relative risk regression.

Barros AJ, Hirakata VN. Alternatives for logistic regression in cross-sectional studies: an empirical comparison of models that directly estimate the prevalence ratio. BMC Med Res Methodol. Oct 20 2003;3:21.
Cox, Poisson, and Log-Binomial regression approaches for cross-sectional studies are discussed and compared with logistic regression.

Selected examples

Nijem K, Kristensen P, Al-Khatib A, Bjertness E. Application of different statistical methods to estimate relative risk for self-reported health complaints among shoe factory workers exposed to organic solvents and plastic compounds. Norsk epidemiologi. 2005;15(1).
This example provides estimates from logistic regression alongside those from log-Binomial and Cox regression; convergence problems and robust variance estimates are also discussed.

Hauksdottir A, Steineck G, Furst CJ, Valdimarsdottir U. Long-term harm of low preparedness for a wife’s death from cancer–a population-based study of widowers 4-5 years after the loss. Am J Epidemiol 2010;172(4):389-96.
This example uses log-Binomial regression to estimate relative risks as the primary analysis.

Black JL, Macinko J. The Changing Distribution and Determinants of Obesity in the Neighborhoods of New York City, 2003-2007. Am J Epidemiol 2010;171:765-775.
The authors use multi-level Poisson models to estimate relative risks for the common outcome of obesity in the NYC-based Community Health Survey.

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