VOLUME 31 : NUMBER 1 : February 2008
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To appreciate the significance of clinical trial results, clinicians need to understand the mathematical language used to describe treatment effects. When comparing intervention and control groups in a trial, results may be reported in terms of relative or absolute risk (or probability), or as more statistically sophisticated entities based on odds and hazard ratios. When events in the intervention group are significantly less frequent than in the control group, then relative risk, odds ratio and hazard ratio (and their confidence intervals) will be less than 1.0. If the converse holds true, these values will be greater than 1.0.
Key words: clinical trials, number needed to treat, odds, statistics.
Aust Prescr 2008;31:12-6
In randomised trials and systematic reviews of trials, the effects of new treatments on dichotomous outcomes (such as death vs survival) can be expressed in several ways including relative risk, absolute risk, odds ratio and hazard ratio. These figures help to determine if the new treatment has an advantage over other treatments or placebo.
Ways of expressing treatment effects
The absolute risk, number needed to treat, relative risk and odds ratio can be calculated by compiling a 2x2 table of study data. Values can then be derived using the equations shown in the box.
Absolute risk reduction, also termed risk difference, is the difference between the absolute risk of an event in the intervention group and the absolute risk in the control group.
In a trial of 441 patients at risk of developing pressure ulcers, patients were randomised to receive a sheepskin mattress overlay (intervention group) or usual treatment (control group) during their hospital stay.1The data from the trial can be represented in a 2x2 table (see Table 1).
The absolute risk reduction can then be calculated by subtracting the proportion of patients with ulcers in the sheepskin group from that in the control group.
Almost 17% of patients in the control group developed ulcers compared to 10% in the sheepskin group after 20 days of observation. This means that the absolute risk of developing ulcers in the sheepskin group was 7% less than in the control group.
If a treatment is effective and reduces the risk of an unwanted event, we see an absolute risk reduction. Conversely, if the treatment does not work and in fact increases the risk of the event, then we see an absolute risk increase.
It may be difficult to conceptualise the clinical relevance of the absolute risk reduction. The reciprocal of this value (1/absolute risk reduction) gives the number of patients who need to be treated for a certain period of time to prevent one event. This is termed the number needed to treat and can be useful for comparing the effectiveness of a number of different interventions. So in the ulcer trial, 14 patients need to have a sheepskin overlay for 20 days to prevent one of them from getting an ulcer.
It is important to appreciate that absolute risk will vary according to the event rates in both patient groups, whereas the relative risk usually remains unchanged across the spectrum of disease severity (see Table 2). Putting this another way, in 'low risk' patients (those with mild hypertension in Table 2) the absolute risk reduction will be small whereas in 'high risk' patients (those with moderate hypertension) absolute risk reduction will be larger. For both groups the relative risk (and relative risk reduction) is the same.2
Relative risk, also known as risk ratio, is the risk of an event in the experimental group divided by that in the control group. For the sheepskin trial, this can be calculated from the data in Table 1.
In the trial, 10% of patients in the sheepskin group developed ulcers compared to 17% in the control group. So the risk of getting ulcers with a sheepskin overlay was 0.58 of that in the control group.
In most trials where the treatment intends to prevent an undesirable outcome such as death or complication (prevention trials), efficacy will be denoted by a relative risk of less than 1.0. Treatment harm, reflecting an increased risk of an event (including adverse effect), will be denoted by a relative risk of more than 1.0. However, in trials where the treatment intends to reduce active disease (treatment trials) and promote a positive event, such as disease remission or symptom abatement, a relative risk of more than 1.0 confirms treatment efficacy. A relative risk of 1.0 indicates no difference between comparison groups. In all cases, statistical significance is assumed if the 95% confidence interval (CI) around the relative risk does not include 1.0.
The relative risk reduction equals the amount by which the relative risk has been reduced by treatment and is calculated as 1 – relative risk. For example in the sheepskin trial, sheepskin overlays reduced the risk of patients getting ulcers by 0.42 (1 – 0.58) or 42%.
Odds are the number of times an event happens divided by the number of times it does not within a group. Odds can also be expressed as the risk (or probability) of an event occurring over the risk of an event not occurring. To provide a numerical example: if 1/5 of the patients in a study suffer a stroke, the odds of their having a stroke is (1/5) ÷ (4/5) or 0.20/0.80, or 0.25. As the denominator is the same in both top and bottom expressions, it cancels out, leaving the number of patients with the event (1) divided by the number of patients without the event (4).
The odds ratio is the odds of an event occurring in one group divided by the odds of the same event in another group. In the sheepskin trial, the odds ratio can be calculated by dividing the odds of getting an ulcer in the sheepskin group by the odds in the control group.
The odds were about 0.11 in the sheepskin group and 0.20 in the control group. This means that the odds of developing an ulcer in the sheepskin group were 0.54 of that in the control group. Put another way, patients with a sheepskin overlay were half as likely to develop ulcers as patients given usual treatment.
Odds ratio is similar to relative risk. In the sheepskin trial the relative risk was 0.58 and the odds ratio was 0.54. For most clinical trials where the event rate is low, that is less than 10% of all participants have an event, the odds ratio and relative risk can be considered interchangeable. The relative risk and odds ratio will also be closer together when the treatment effect is small (that is, odds ratio and relative risk are close to 1) than when treatment effect is large. However, as the event rate increases above 15% or as the treatment effect becomes huge, the odds ratio will progressively diverge from the relative risk.
Fortunately, this is rarely a problem. Consider a meta-analysis of ligation versus sclerotherapy for oesophageal varices, which demonstrated a re-bleeding rate of 47% with sclerotherapy, as high an event rate as one is likely to find in most trials.3The odds ratio associated with treatment with ligation was 0.52, a large effect. Despite the high event rate and large effect, the relative risk was 0.60, not very different from the odds ratio. Thus choosing one measure or the other is unlikely to have an important influence on most treatment decisions.
The odds ratio is gradually losing favour as a measure of treatment effect4, particularly as data from which relative risk is derived can also be used to calculate absolute risk reduction and number needed to treat, which are more clinically useful.
Hazard ratio is a measure of relative risk over time in circumstances where we are interested not only in the total number of events, but in their timing as well. The event of interest may be death or it may be a non-fatal event such as readmission or symptom change.
Note that the relative risk and the incidence rate ratio were different, 0.58 versus 0.42, with the time-dependent relative risk suggesting a greater benefit from intervention than the overall relative risk, and which is also fairly close to the estimated hazard ratio of 0.39 (row j).
In contrast to the overall relative risk, both the time-dependent relative risk and hazard ratio take into account the timing of events which may not be evenly distributed throughout the study period.
The hazard ratio equals a weighted relative risk over the entire duration of a study and is derived from a time-to-event curve or Kaplan-Meier curve. This curve describes the status of both patient groups at different time points after a defined starting point. In the sheepskin study, events in the intervention group are not only less frequent overall than in the control group but they are delayed in time (Fig. 1). As some patients will be followed for a longer period of time than others (because they were recruited or randomised into the trial at an earlier time or because they remained in the study while others dropped out), the time-to-event curve usually extends beyond the mean follow-up duration.
As the trial progresses, at some point prediction of treatment effect becomes very imprecise (in our example at 20 days) because there are few patients available to estimate the probability of the outcome of interest. Confidence intervals around the survival curves would capture the precision of the estimate. Ideally then, we would estimate relative risk by applying an average, weighted for the number of patients available, over the entire study duration. Statistical methods allow just such an estimate which is the hazard ratio.
This derived (or 'crude') hazard ratio then needs to be 'adjusted' or corrected for differences in the two groups at baseline that might influence the outcome of interest. This issue is less of a concern if randomisation has rendered both groups similar in terms of their baseline characteristics. In our example, patients in the intervention group compared to control were older (mean age 63.2 years vs 61.1 years), more acutely ill (51% were emergency admissions vs 43%), and had greater prevalence of medical, as opposed to surgical, diagnoses (35% vs 27%). Applying the Cox proportional hazards regression model produces an adjusted hazard ratio which takes account of such imbalances.
In every other way the hazard ratio is similar to odds ratio and relative risk wherein treatment efficacy is denoted by a hazard ratio of less than 1.0 in prevention trials and a hazard ratio of more than 1.0 in treatment trials.
If there is a statistically significant difference in outcomes between treatment and control groups, the observed difference is very unlikely to have occurred due to the play of chance, even after accounting for imprecision in the difference related to the total number of events in both groups.
Statistical significance is defined arbitrarily in terms of a p value of less than 0.05. The p value however does not directly indicate the chance of an effect being present or not being present. Instead it tells us how often chance alone would give apparently favourable results. A p value of less than 0.05 tells us that there is less than 5% probability that chance alone would lead to such favourable results, but it says nothing directly about whether chance is the best explanation for the results.
Confidence intervals give us an estimate of the precision of the results. Conventionally 95% confidence intervals are used which, if the same trial were to be repeated many times over, define the range of values within which the true estimate would be found in 95% of occasions. The confidence interval represents the range of values within which we are 95% confident that the true population estimate lies. If the number of events such as death occurring over time is fairly small (as occurs with small samples and/or low case fatality rate), then the precision with which the true probability of the event can be estimated is relatively low, as reflected in wider confidence intervals. Narrower confidence intervals indicate more precise results. The 95% confidence intervals represent almost two standard deviations around the mean.
It is important to remember that the result is statistically significant if the confidence intervals do not cross the null value, such as 1.0 for relative risk and 0 for absolute risk reduction.
An understanding of the commonly used statistical measures of benefit is necessary if clinicians are to gain an appreciation of the efficacy of different therapies. For the majority of clinical trials, relative risk and odds ratio can be considered interchangeable as a measure of the relative change in the risk of a preventable event. The hazard ratio is a related measure that weights the risk change according to when events occur over time. Absolute risk reduction represents the absolute change in risk (expressed in percentage points) and its reciprocal represents the number of patients who would need to be treated over a given period of time to prevent one event.
1. Jolley DJ, Wright R, McGowan S, Hickey MB, Campbell DA, Sinclair RD, et al. Preventing pressure ulcers with the Australian medical sheepskin: an open-label randomised controlled trial. Med J Aust 2004;180:324-7.
2. Collins R, Peto R, MacMahon S, Hebert P, Fiebach NH, Eberlein KA, et al. Blood pressure, stroke, and coronary heart disease. Part 2, Short-term reductions in blood pressure: overview of randomised drug trials in their epidemiological context. Lancet 1990;335:827-38.
Conflict of interest: none declared