![]()
Chapter 1
Absolute Risk Reduction
Robert Newcomb
1.1 Introduction
Many response variables in clinical trials are binary: the treatment was successful or unsuccessful; the adverse effect did or did not occur. Binary variables are summarized by proportions, which may be compared between different arms of a study by calculating either an absolute difference of proportions or a relative measure, the relative risk or the odds ratio. In this article we consider several point and interval estimates for the absolute difference between two proportions, for both unpaired and paired study designs. The simplest methods encounter problems when numerators or denominators are small; accordingly, better methods are introduced. Because confidence interval methods for differences of proportions are derived from related methods for the simpler case of the single proportion, which itself can also be of interest in a clinical trial, this case is also considered in some depth. Illustrative examples relating to data from two clinical trials are shown.
1.2 Preliminary Issues
In most clinical trials, the unit of data is the individual, and statistical analyses for efficacy and safety outcomes compare responses between the two (or more) treatment groups. When subjects are randomized between these groups, responses of subjects in one group are independent of those in the other group. This leads to unpaired analyses. Crossover and split-unit designs require paired analyses. These have many features in common with the unpaired analyses and will be described in the final section.
Thus, we study n1 individuals in group 1 and n2 individuals in group 2. Usually, all analyses are conditional on n1 and n2. Analyses conditional on n1 and n2 would also be appropriate in other types of prospective studies or in cross-sectional designs. (Some hypothesis testing procedures such as the Fisher exact test are conditional also on the total number of “successes” in the two groups combined. This alternative conditioning is inappropriate for confidence intervals for a difference of proportions; in particular in the event that no successes are observed in either group, this approach fails to produce an interval.) The outcome variable is binary: 1 if the event of interest occurs, 0 if it does not. (We do not consider here the case of an integer-valued outcome variable; typically, this involves the number of episodes of relapse or hospitalization, number of accidents, or similar events occurring within a defined follow-up period. Such an outcome would instead be modeled by the Poisson distribution.) We observe that r1 subjects in group 1 and r2 subjects in group 2 experience the event of interest. Then the proportions having the event in the two groups are given by p1 = r1/n1 and p2 = r2/n2. If responses in different individuals in each group are independent, then the distribution of the number of events in each group is binomial.
Several effect size measures are widely used for comparison of two independent proportions:
Difference of proportions p1 − p2
Ratio of proportions (risk ratio or relative risk) p1/p2
Odds ratio (p1/(1 − p1))/(p2/(1 − p2))
In this article we consider in particular the difference between two proportions, p1 − p2, as a measure of effect size. This is variously referred to as the absolute risk reduction, risk difference, or success rate difference. Other articles in this work describe the risk ratio or relative risk and the odds ratio. We consider both point and interval estimates, in recognition that “confidence intervals convey information about magnitude and precision of effect simultaneously, keeping these two aspects of measurement closely linked” [1]. In the clinical trial context, a difference between two proportions is often referred to as an absolute risk reduction. However, it should be borne in mind that any term that includes the word “reduction” really presupposes that the direction of the difference will be a reduction in risk—such terminology becomes awkward when the anticipated benefit does not materialize, including the nonsignificant case when the confidence interval for the difference extends beyond the null hypothesis value of zero. The same applies to the relative risk reduction, 1 − p1/p2. Whenever results are presented, it is vitally important that the direction of the observed difference should be made unequivocally clear. Moreover, sometimes confusing labels are used, which might be interpreted to mean something other than p1 − p2; for example, Hashemi et al. [2] refer to p1 − p2 as attributable risk. It is also vital to distinguish between relative and absolute risk reduction.
In clinical trials, as in other prospective and cross-sectional designs already described, each of the three quantities we have discussed may validly be used as a measure of effect size. The risk difference and risk ratio compare two proportions from different perspectives. A halving of risk will have much greater population impact for a common outcome than for an infrequent one. Schechtman [3] recommends that both a relative and an absolute measure should always be reported, with appropriate confidence intervals.
The odds ratio is discussed at length by Agresti [4]. It is widely regarded as having a special preferred status on account of its role in retrospective case-control studies and in logistic regression and meta-analysis. Nevertheless, it should not be regarded as having gold standard status as a measure of effect size for the 2 × 2 table [3,5].
1.3 Point and Interval Estimates for a Single Proportion
Before considering the difference between two independent proportions in detail, we first consider some of the issues that arise in relation to the fundamental task of estimating a single proportion. These issues have repercussions for the comparison of proportions because confidence interval methods for p1 − p2 are generally based closely on those for proportions. The single proportion is also relevant to clinical trials in its own right. For example, in a clinical trial comparing surgical versus conservative management, we would be concerned with estimating the incidence of a particular complication of surgery such as postoperative bleeding, even though there is no question of obtaining a contrasting value in the conservative group or of formally comparing these.
The most commonly used estimator for the population proportion π is the familiar empirical estimate, namely, the observed proportion p = r/n. Given n, the random variable R denoting the number of subjects i...
