The Hazards of Hazard Ratios

Abstract

The hazard ratio (HR) is the main, and often the only, effect measure reported in many epidemiologic studies. For dichotomous, non–time-varying exposures, the HR is defined as the hazard in the exposed groups divided by the hazard in the unexposed groups. For all practical purposes, hazards can be thought of as incidence rates and thus the HR can be roughly interpreted as the incidence rate ratio. The HR is commonly and conveniently estimated via a Cox proportional hazards model, which can include potential confounders as covariates. Unfortunately, the use of the HR for causal inference is not straightforward even in the absence of unmeasured confounding, measurement error, and model misspecification. Endowing a HR with a causal interpretation is risky for 2 key reasons: the HR may change over time, and the HR has a built-in selection bias. Here I review these 2 problems and some proposed solutions. As an example, I will use the findings from a Women's Health Initiative randomized experiment that compared the risk of coronary heart disease of women assigned to combined (estrogen plus progestin) hormone therapy with that of women assigned to placebo.1 By using a randomized experiment as an example, the discussion can focus on the shortcomings of the HR, setting aside issues of confounding and other serious problems that arise in observational studies. The Women's Health Initiative followed over 16,000 women for an average of 5.2 years before the study was halted due to safety concerns. The primary result from the trial was a HR. As stated in the abstract1 and shown in Table 1 of the article, “Combined hormone therapy was associated with a hazard ratio of 1.24.”1 In addition, Table 2 provided the HRs during each year of follow-up: 1.81, 1.34, 1.27, 1.25, 1.45, and 0.70 for years 1, 2, 3, 4, 5, and 6+, respectively. Thus, the HR reported in the abstract and Table 1 can be viewed as some sort of weighted average of the period-specific HRs reported in Table 2. This bring us to Problem 1: although the HR may change over time, some studies report only a single HR averaged over the duration of the study's follow-up. As a result, the conclusions from the study may critically depend on the duration of the follow-up. For example, the average HR in the WHI would have been 1.8 if the study had been halted after 1 year of follow-up, 1.7 after 2 years,2 1.2 after 5 years, and—who knows—perhaps 1.0 after 10 years. The 24% increase in the rate of coronary heart disease that many researchers and journalists consider as the effect of combined hormone therapy is the result of the arbitrary choice of an average follow-up period of 5.2 years. A trial with a shorter follow-up could have reported an 80% increase, whereas a longer trial might have found little or no increase at all. The magnitude of the average HR depends on the length of follow-up because the average HR ignores the distribution of events during the follow-up. The average HR can take the value 1.0 if the hazard in the exposed is identical to the hazard in the unexposed during the entire follow-up, or if the hazard in the exposed is higher during, say, the first 5 years and lower afterward. Incidentally, the same problem arises whether the average HR is directly estimated in a cohort study, as discussed here, or estimated via the odds ratio of a properly designed case-control study with incidence density sampling. One might then conclude that we should forget about the average HR and restrict our attention to the period-specific HRs, which seem to capture the potentially time-varying magnitude of the effect. This brings us to Problem 2: the period-specific HRs have a built-in selection bias. To describe the bias, consider that the (discrete-time) hazard during period t is defined as the risk of the outcome during period t among those who reached period t free of the outcome. In the Women's Health Initiative, the calculation of the HR during year t was restricted to women who did not develop coronary heart disease—the “survivors”—between baseline and the beginning of year t. The HR after year 5 was 0.7, which means that the disease rate after year 5 was lower in the treatment arm (the hazard in the numerator of the HR) than in the placebo arm (the hazard in the denominator). However, this apparently protective effect of hormone therapy after year 5 is hardly surprising if one bears in mind that women vary in their susceptibility to heart disease. A certain proportion of all women enrolled in the trial were particularly prone to develop heart disease if they were exposed to hormone therapy or other factors (for simplicity, let's refer to them as the “susceptible women”). The proportion of susceptible women in the trial was of course unknown but, because of randomization, it was expected to be the same in both the treatment and placebo arms at baseline. However, these susceptible women were preferentially excluded from the treatment arm as they developed heart disease