Variance and Dissent
Absolute risk reductions, relative risks, relative risk reductions, and numbers needed to treat can be obtained from a logistic regression model

https://doi.org/10.1016/j.jclinepi.2008.11.004Get rights and content

Abstract

Objective

Logistic regression models are frequently used in cohort studies to determine the association between treatment and dichotomous outcomes in the presence of confounding variables. In a logistic regression model, the association between exposure and outcome is measured using the odds ratio (OR). The OR can be difficult to interpret and only approximates the relative risk (RR) in certain restrictive settings. Several authors have suggested that for dichotomous outcomes, RRs, RR reductions, absolute risk reductions, and the number needed to treat (NNT) are more clinically meaningful measures of treatment effect.

Study Design and Setting

We describe a method for deriving clinically meaningful measures of treatment effect from a logistic regression model. This method involves determining the probability of the outcome if each subject in the cohort was treated and if each subject was untreated. These probabilities are then averaged across the study cohort to determine the average probability of the outcome in the population if all subjects were treated and if they were untreated.

Results

Risk differences, RRs, and NNTs were derived using a logistic regression model.

Conclusions

Clinically meaningful measures of effect can be derived from a logistic regression model in a cohort study. These methods can also be used in randomized controlled trials when logistic regression is used to adjust for possible imbalance in prognostically important baseline covariates.

Introduction

In randomized controlled trials (RCTs), the effect of treatment on dichotomous outcomes can be reported using a variety of measures of treatment effect: absolute risk reduction, relative risk (RR), RR reduction, the number needed to treat (NNT), and the odds ratio (OR). Schechtman argues that both relative and absolute measures should be reported [1]. Cook and Sackett argue that for clinical decision making, the NNT is more meaningful than either the RR, the RR reduction, or the OR [2]. Jaeschke et al. suggest that the OR and the RR provide limited information [3]. Finally, Sinclair and Bracken argue that clinically important questions are best addressed using RRs, RR reductions, risk differences, and NNTs [4]. In the face of these proposals, some medical journals require that absolute risk reductions and the associated NNT be reported for any RCT with a dichotomous outcome [5].

Researchers are increasingly using observational studies to estimate the effect of treatment on outcomes. In nonrandomized studies, unlike in randomized trials, treated subjects often differ systematically from untreated subjects. Therefore, outcomes cannot be compared directly between treated and untreated subjects. Statistical methods must be used to adjust for systematic differences between treated and untreated subjects when estimating the effect of treatment on outcomes. A commonly used method for this purpose in the medical literature is the logistic regression model.

The use of the logistic regression model to determine the adjusted effect of treatment on dichotomous outcomes is ubiquitous in modern observational research. A limitation to the use of the logistic regression model in prospective or cohort studies is its use of the OR as the measure of treatment effectiveness. When the prevalence of the outcome is low (<10%), then the OR can be used to approximate the RR. However, when the outcome is common, the estimated OR will be farther away from 1 than the RR [6]. For instance, in the setting of a positive association between treatment and the outcome, the estimated OR can substantially overestimate the RR [6]. In general, the OR can be difficult to interpret, and has been criticized for a variety of reasons [3], [4], [7], [8].

The objective of this article is to describe how RRs, RR reductions, risk differences, and NNTs can be derived from a logistic regression model in a cohort or prospective study. The article is organized as follows: in Section 2, we describe how these more meaningful measures of treatment effect can be derived from a logistic regression model. In Section 3, we present a case study in which we demonstrate the utility of our proposed methods. Finally, in Section 4, we summarize our findings.

Section snippets

Estimating clinically meaningful measures of treatment effect using logistic regression models

The use of the logistic regression model is pervasive in modern medical research. Let us assume that in an observational study, a dichotomous outcome variable Y is observed for each subject (with Y = 1 denoting success and Y = 0 denoting failure). Furthermore, let Ti denote the treatment status of the ith subject (with T = 1 denoting treatment and T = 0 denoting no treatment), whereas X1i,X2i,,Xki denote the value of k covariates (or confounding variables) measured on this subject. The following

Case study

We illustrate the utility of our method by estimating the absolute and relative reduction in 3-year mortality associated with the receipt of a statin prescription at hospital discharge in a sample of patients discharged alive from hospital with a diagnosis of acute myocardial infarction (AMI). The data were collected as part of the Enhanced Feedback for Effective Cardiac Treatment (EFFECT) Study, a quality improvement initiative that is focused on improving the quality of care for

Discussion

In this article, we have described a method to derive RRs, RR reductions, risk differences, and NNTs using a logistic regression model in an observational cohort study. These methods allow for the estimation of measures of treatment effect that are more meaningful than the adjusted OR that is obtained directly from the logistic regression model.

As described in Section 3, Zhang and Yu have proposed a method for deriving a RR from an adjusted OR [15]. However, the approach proposed by Zhang and

Acknowledgments

The Institute for Clinical Evaluative Sciences (ICES) is supported in part by a grant from the Ontario Ministry of Health and Long Term Care. The opinions, results, and conclusions are those of the authors, and no endorsement by the Ministry of Health and Long-Term Care or by the ICES is intended or should be inferred. Dr. Austin is supported in part by a Career Investigator award from the Heart and Stroke Foundation of Ontario. The data used in this study were obtained from the EFFECT study.

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