Survival analysis

A time-to-event analysis measures the time from a fixed starting point to an event, and the follow-up of participants whose outcome is unknown is censored. A Kaplan–Meier curve estimates survival over that time, a hazard is a rate of events per unit of time, and a hazard ratio compares two such rates and is not strictly equivalent to a relative risk. Reporting the analysis means stating those definitions, the estimates with their uncertainty and the numbers at risk, and the test and model behind any comparison.

Starting point, event and censoring

Time to event is the period in a clinical trial or other health study from recruitment to the occurrence, usually the first occurrence, of a disease end point, the resolution of a sign or symptom, or the censoring of a participant's follow-up, for example through loss to follow-up, death from another cause or withdrawal. ¹

The outcome of a time-to-event analysis is not the event itself but the time from a given starting point to the event: for example, what is of interest is the time between hospitalisation and death, not the death itself. ² Survival is not the endpoint in these analyses, death is, so time-to-event analysis is the most accurate and preferred term. ²

A Kaplan–Meier plot gives the probability of surviving from a fixed starting point over time, and the event whose likelihood is plotted may be, for example, death, survival or disease-free survival. ³ Participants who become uncooperative or lose contact with study personnel, or who are lost to follow-up or have other events such as protocol violations, are censored, in part because it is not known whether they went on to die or to have the event. ^{3, 4}

Kaplan–Meier estimation

Kaplan–Meier estimation, also called the product limit estimate, gives a cumulative estimate of survival over time. ³ The time axis is divided into segments, often defined by clinic visits, and the method does not determine exactly when an event such as death occurs, only that it happened between two visits or other prespecified intervals. ³ In each interval the probability of survival is conditional: it is the probability of surviving to the beginning of the interval multiplied by the probability of surviving the interval, given survival to its beginning. ⁴ The probability at each observation point is multiplied by that at the next, then by the one after, and so on. ³ Survival at a given time is related to the number of events divided by the number of individuals known to have survived. ³ A censored observation is not included in the denominator for interval survival calculations. ³ Kaplan–Meier plots are useful for comparing cumulative survival between two treatments, given the vicissitudes of patient participation in a study. ³

In a worked example, an active-treatment group of 350 and a placebo group of 370 are followed in three-month intervals to 18 months. ⁴ In the six-month interval for active treatment, 340 begin, 21 die and 1 is censored, leaving 318, so survival over the interval is 318/340 (0.9353), and multiplied by survival to the start of the interval it gives a cumulative survival of 0.9714 × 0.9353 = 0.9085. ⁴ The survival probabilities can be computed in Microsoft Excel, although the Kaplan–Meier function may not be in standard versions and an add-in may be needed, and the curves can also be graphed in SigmaPlot. ⁴

The probability of survival, or of not experiencing a disease event, is plotted on the ordinate and time on the abscissa. ⁴ A grid below the time axis typically lists the numbers of participants still available at each visit, those who have not died, had another event or been censored. ³ The method requires that participants enrolled early and late in the study have identical probabilities of survival, and that a participant whose data are censored at any observation point has the same likelihood of survival as those who continue in the study. ³

Kaplan–Meier curves give a simple means of visualising the cumulative proportion of patients surviving or experiencing an event at multiple time points, and they can give useful information about temporal trends, essentially a picture of the event rate over time. ¹ They show disease incidences at each of several observation points during a trial. ³

Reading a survival estimate

A Kaplan–Meier percentage such as 95% survival or 5% events is already an estimate, not a proportion of the kind descriptive statistics reports, and it cannot be used as though it were one. ^{5, 6} Many people have a problem with this, and one statistician has seen the two treated alike several times. ^{7, 6}

Where 100 patients are enrolled, one death is observed and only 80 patients reach the one-year visit, some divide the event by the 100 enrolled at baseline, which is not correct because the 20 missing patients may all have died, and some divide it by the 80, so the answer turns on the denominator. ^{8, 9, 10} Survival analysis instead censors each patient when their last information is available, so a patient missing at the one-year visit but known to be alive at day 300 contributes at least that information to the calculation. ¹⁰

Where the censoring falls relative to the events moves the estimate, and this is the part people find hard to understand. ¹¹ With the same number of events, survival is much lower when many patients are censored before the events than when the censoring comes after them. ¹¹ So it helps to look at the Kaplan–Meier curves and not only the estimate, because the curves give an idea of when patients were censored. ^{11, 12} One statistician trusts Kaplan–Meier estimates, having seen them hold up time after time in interim analyses and in registries that run for a long time, although they are really sensitive at the beginning, especially in interim analyses while patients are still entering the trial. ¹²

Hazard

A hazard is the rate at which events happen. ¹³ A hazard rate is an incidence rate: the number of new events per population at risk per unit of time. ¹⁴ More precisely, a hazard is the instantaneous event rate, the probability that if an event has not occurred in one period it will occur in the next. ¹⁴ A hazard differs from incidence: a hazard is a rate, counted per unit of time, here a year, whereas incidence is the proportion of new cases occurring over a given period of many units of time, so over several years and not per year. ¹⁴ A hazard rate is reported as, for example: the hazard rate for death after radical prostatectomy was 0.4% at 5 years, 0.7% at 10 years and 1% at 15 years. ¹⁴

Hazard rates are often indicated on Kaplan–Meier or survival curves, which show the incidence rate, of death, at any given time in a study. ¹⁴

Comparing survival between groups

The curves of two treatments can be compared preliminarily by visual inspection, and the time at which they diverge is called separation. ³ One way to evaluate the difference is the log-rank test statistic, which sets the total numbers of events observed in each group against the numbers expected. ³ For example, if 2 of 100 participants alive at the first visit in one group die by the second visit, the observed risk is 0.02, and a second group with 50 alive at the first visit has 50 × 0.02 = 1 expected death at the second; that group's total expected deaths is the sum of the expected events at all observation points. ³

Another way to analyse survival data is a Cox proportional hazards regression model, which relates the hazard of an event at a given time to a baseline hazard, a set of exposure variables and a regression coefficient for each. ³ Hazard rates and hazard ratios are found in time-to-event studies with binary outcomes, often alive or dead, and they are the output of Cox proportional hazards regression, which can also identify which factors are associated with living or dying. ^{14, 2}

Hazard ratio

The hazard ratio is commonly calculated to analyse statistical differences in survival or time-to-event data between treatment groups. ¹ A hazard ratio is a ratio of incidence rates, whereas an odds ratio and a risk ratio are ratios of proportions. ² It is the ratio of event incidence in one group to that in another, and, unlike an odds ratio, it gives a time-specific risk and not a cumulative risk over time. ³ How it is read depends on the event: where absence of the event is beneficial, as with absence of cancer, a hazard ratio below 1 indicates a beneficial outcome of therapy compared with control, and where presence of the event is beneficial, as when the disease is cured, a hazard ratio above 1 indicates a desirable outcome. ¹

In a placebo-controlled trial of amoxicillin–clavulanate for acute otitis media, 161 children received the drug or placebo over seven days, and the Kaplan–Meier curves for time to treatment failure separated early and remained separate. ¹ The hazard ratio of 0.38 showed a 62% attenuation in progression to treatment failure, calculated as 100 × (1 − 0.38). ¹ Treatment failure was based on the child's overall condition, adverse events included, together with otoscopic signs of acute otitis media. ¹

An overall hazard ratio gives no information about relative speed, that is, how much faster an end point occurs in one group than in another. ¹ Instead, it indicates the chance that a treated person who has not had the event at one time point will have it at a subsequent time point, compared with people in a control group or in another treatment group. ¹ The larger the hazard ratio, the greater the chance that the end point will occur sooner in a person in the treatment group than in a person in the control group. ¹ A hazard ratio of 2.0 does not mean that an event occurs twice as quickly in one group, but it does mean that the person has twice the risk of experiencing the outcome compared with a person in the other group. ¹

Hazard ratio and relative risk

A hazard ratio is a ratio of event rates over time and is not strictly equivalent to relative risk, which is a ratio of event numbers, although the two concepts have sometimes been used interchangeably. ¹ In descriptions of survival analyses, relative risk cannot be used instead of hazard ratio, because although the two can be interpreted in more or less the same way, they are calculated differently and relative risks do not account for the timing of the events of interest. ¹⁵

A hazard ratio can show differences that a relative risk or an odds ratio cannot. ¹ In a hypothetical trial, ten patients in each of two groups are followed for ten years and three fractures occur in each group, but in the placebo group all three occur in year 1 and in the treatment group all three in year 9. ¹ Person-years at risk are (7 × 10) + (3 × 1) = 73 on placebo, giving a rate of 3 ÷ 73 = 0.04, and (7 × 10) + (3 × 9) = 97 on treatment, giving 3 ÷ 97 = 0.03, so the hazard ratio is 0.03 ÷ 0.04 = 0.75, a 25% risk reduction in fractures over ten years compared with placebo. ¹ The ten-year odds ratio for the same data is (3/7) ÷ (3/7) = 1 and the ten-year relative risk is 30% ÷ 30% = 1, both indicating no difference over the whole study period. ¹ Clinically relevant outcomes between groups can easily be missed when only one time point is used, such as the end of a trial. ¹ So single point-in-time comparisons, such as relative risk or odds ratio analyses, can miss temporal patterns such as early worsening of diabetic complications, latency periods for cancers and acute effects of hormone replacement therapy on cardiovascular disease. ¹

Reporting a time-to-event analysis

The SAMPL guidelines, Statistical Analyses and Methods in the Published Literature, tell authors, journal editors and reviewers how to report basic statistical methods and results, and for a survival or time-to-event analysis they give these items. ^{16, 17}

• Describe the purpose of the analysis.
• Identify the dates or events that mark the beginning and the end of the time period analysed.
• Specify the circumstances under which data were censored.
• Specify the statistical methods used to estimate the survival rate.
• Confirm that the assumptions of survival analysis were met.
• For each group, give the estimated survival probability at appropriate follow-up times with confidence intervals and the number of participants at risk for death at each time. Especially where events are not common, it is often more helpful to plot the cumulative probability of not surviving.
• Reporting median survival times with confidence intervals is often useful because it lets the results be compared with those of other studies.
• Consider presenting the full results in a graph, such as a Kaplan–Meier plot, or in a table.
• Specify the statistical methods used to compare two or more survival curves.
• When comparing two or more survival curves with hypothesis tests, report the P value of the comparison.
• Report the regression model used to assess the associations between the explanatory variables and survival or time to event.
• Report a measure of risk, such as a hazard ratio, for each explanatory variable, with a confidence interval.

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