

More on Effect Size: Risk Versus Rate And Hazard Ratio
Correspondence Address: Source of Support: None, Conflict of Interest: None DOI: 10.4103/00283886.304082
In our previous series on measures of effect size, we have discussed odds ratio, risk ratio, risk difference, mean difference, and standardized difference.^{[1],[2]} In this final step towards the description of measures of effect size, we aim to cover the remaining measures of effect size and their interpretations. We will be discussing the difference between risk and rate. Finally, we will try to understand the concept of the hazard ratio (HR), which is commonly used in randomized controlled trials as well as longitudinal cohort studies.
A common confusion is to use the term “risk” and “rate” interchangeably. To illustrate the concept, read the results of the paper recently published,^{[3]} which was conducted in patients hospitalized with COVID19. The intervention (hydroxychloroquine) was received by 1561 patients and 3155 received the usual care. The primary outcome was 28day mortality. “Death within 28 days occurred in 421 patients (27.0%) in the hydroxychloroquine group and in 790 (25.0%) in the usualcare group (rate ratio, 1.09; 95% confidence interval [CI], 0.97 to 1.23; P = 0.15). The results suggest that patients in the hydroxychloroquine group were less likely to be discharged from the hospital alive within 28 days than those in the usualcare group (59.6% vs 62.9%; rate ratio, 0.90; 95% CI, 0.83 to 0.98).” To compute the rate ratio, the time to death and/or discharge was known for all the patients and the Kaplan–Meier curve was plotted for the groups. However, the same study also mentions the following: “Among the patients who were not undergoing mechanical ventilation at baseline, those in the hydroxychloroquine group had a higher frequency of invasive mechanical ventilation or death (30.7% vs 26.9%; risk ratio, 1.14; 95% CI, 1.03 to 1.27).” Note here that “rate ratio” has been replaced with “risk ratio.” Here the risk ratio has been calculated by the simple division of proportion of patients with the outcome in the experimental group by the control group. Only frequencies of events were compared without considering the time at which they occurred. This has already alluded in our previous paper in the series.^{[1]} The example illustrates important facts:
To calculate the rate, we need additional information, which is the time elapsed between the start of observation to the occurrence of an event, referred to in brief as “time to event.” Another example to illustrate the difference between risk and rate is cumulative incidence (CI) or risk and incidence rate (IR). Cumulative incidence (CI) or risk Risk is the probability of an outcome. In medical literature, it could be the probability of recovery, disease onset, disease relapse, death, etc. When the time factor is also considered, we might be interested in the outcome in relation to observation time. In [Figure 1], we have illustrated the example of eight people (N_{t0}) being followed up from time zero (t_{0}) to time 1 (t_{1}). The people may have entered at t_{0}without any history of stroke and developed stroke or remained well during the study. At t_{1}, there were two people who had a stroke. The CI of stroke would be the number of strokes divided by the number of people who had entered the study at t_{0}.
Therefore, cumulative incidence (CI) = New cases/N_{t0} In our example, it will be (2/8) or 0.25 or 25%. It defines the risk or probability of the disease which people have from time 0 to time 1. It can be represented as proportion, fraction, or percentage. Incidence rate (or incidence density) However, the IR is NOT the same as CI. In a population where people enter and leave the study at a variable time, we may find some people having a stroke during the study period [Figure 2]. The new events, per populationtime, is IR. Its unit is event per persontime.
Incidence Rate = #new events/population time All of us are familiar with the requirement of time in the denominator when we talk of “rate.” In physics, we all read that rate of change of distance (called “speed”) has a change of distance in the numerator and time in the denominator. Rate of change of velocity (called “acceleration”) has a change of velocity in the numerator and time in the denominator. The only difference is that in the medical world, time is not expressed in minutes, hours, or years; but as persondays, personyears, and so on. This is an important concept because some books use terms such as event rate but what they mean is probability or risk, because the denominator does not have persontime, it has a number of persons at risk.
Suppose you followed 100 patients for a fixed period of 3 months. The numbers of deaths in the first, second, and third months are 40, 20, and 10, respectively. The risk of death during the three periods is as follows: risk during the first month 40/100 = 0.4, risk during the second month 20/100 = 0.2 and that during the third month is 10/100 = 0.1, respectively. The risk of death in a time interval conditional upon a subject has survived to the beginning of that interval (conditional risk) is in the first month 40/100 = 0.4, in the second month 20/60 = 0.33 and in the third month 10/40 = 0.25. For a large sample size, as the measurement unit for a time becomes small, for example, onequarter of a month instead of one month, the conditional risk divided by the width of the time interval, for example, 0.25 month in the present case, will approach a smooth curve. This is known as the hazard rate (also called simply hazard, or hazard function). A hazard rate is the conditional risk of failure in an extremely smalltime interval divided by the width of the time interval. It is, in other words, the conditional, instantaneous risk. A formal definition may be as follows: The hazard rate is the limit of the number of events per unit time divided by the number at risk at the beginning of the time interval, as the time interval approaches 0. The concepts of hazard and proportional hazard are particularly important in survival analysis. Some authors use the term average hazard rate, which is similar to the IR; therefore, it may be wise to use the term “instantaneous hazard rate” to distinguish it from average hazard rate.
HR is the ratio of two hazard rates, conventionally one for the exposed group to that for the unexposed group. In a treatment trial, the HR is usually the hazard rate in the treated group divided by the hazard rate in the control group. When a study reports one HR, it is assumed that the difference between groups was proportional. HRs become meaningless when this assumption of proportionality is not met. Interpretation of hazard ratio In simple terms, you can interpret as a risk ratio averaged over time. Thus, in a treatment trial, HR of “1” means no difference in the effects of the two treatments. If it is less than “1” (and outcome in unfavorable and authors follow the convention of putting treatment hazard rate in the numerator), then treatment is associated with less rate of unfavorable outcome than the control. When it is more than “1,” then treatment is increasing the hazard. If we recall the earlier paper on survival analysis, we know that survival function is a probability. Further, if we assume that the effect of the covariate(s) in the risk for an event is constant over time (proportional hazard assumption), we can compare the groups with and without intervention to events overtime via HR. It is different from the rate ratio since it is conditional and instantaneous and adjusts for other covariates in a regression model. Financial support and sponsorship Nil. Conflicts of interest There are no conflicts of interest.
[Figure 1], [Figure 2]


