This glossary collects the key concepts, methods and people you will meet in this lesson. Use it as a reference while you work through the material, or as a review before assessments. Type in the search box to filter entries.

Introduction and Overview

Many questions in public health research concern when an event happens as well as whether it happens. A trial of a smoking cessation program asks how long participants remain abstinent, a cohort study of young adults asks how long people who start without depressive symptoms remain free of them, and a tuberculosis program asks how long after diagnosis people complete treatment. In each case the outcome for every participant is a length of time from a defined starting point to a defined event. Data of this kind are called time-to-event data, and the methods for analyzing them are called survival analysis. The name reflects the origin of these methods in studies of mortality, although the event can be any well-defined change of state.

Time-to-event analysis extends Lesson 4. That lesson defined incidence risk as the proportion of an at-risk group that develops an outcome over a defined period, and incidence rate as new cases divided by person-time. Both measures make assumptions about follow-up. The risk requires that everyone is followed for the whole period, and the rate treats a person-month early in follow-up as equivalent to a person-month late in follow-up. Real cohorts rarely meet the first condition, because some participants leave before the period ends, and the second is often implausible. The methods in this lesson estimate the probability of remaining event-free over time while using all of the follow-up that each participant provides. Lesson 6 compared groups with the incidence rate ratio, and Section 4 of this lesson presents the hazard ratio as the time-to-event counterpart of that measure.

The lesson is conceptual and is written for people who design studies. This section sets out the design decisions that define a time-to-event outcome and the forms of incomplete follow-up called censoring. Section 2 computes an actuarial life table by hand, Section 3 computes a Kaplan-Meier estimate by hand and shows how to read a Kaplan-Meier curve, and Section 4 compares groups, explains the hazard ratio and the proportional hazards idea, and lists what a protocol with a time-to-event outcome must specify. These decisions belong in the methods of a study protocol, the written plan for a study as HSCI 207 Lesson 6 Section 3.1 defines it, and in a funding proposal they are summarized in the analysis plan.

The section begins with the two parts of a time-to-event outcome and the kinds of question that need them, and then introduces the running case that the whole lesson uses.

Questions About Whether and When

A time-to-event outcome has two parts for each participant. The first is a length of time, measured from a starting point that is defined in the same way for everyone. The second is a status that records how that time ended: either the event was observed, or observation stopped before any event was seen. Analysts usually code the status as 1 for an observed event and 0 for a participant whose observation stopped without the event, who is described as censored. Neither part is sufficient alone. Two participants who were each followed for eight months carry very different information if one had the event in month eight and the other moved away in month eight without having it.

Two features distinguish these data from other continuous outcomes. Follow-up times cannot be negative and are usually right-skewed, because a few participants remain event-free for a long time. More fundamentally, the event time of a censored participant is known only to exceed the time at which observation stopped. A participant who was followed for eight months without the event and then moved away tells the investigator that their event time is longer than eight months, and nothing more. Survival methods use this partial information. Methods that ignore it, or that treat the censoring time as if it were an event time, give biased answers, as the last part of this section shows.

The question being asked also matters. A question about the risk of the event by a fixed time (what proportion of participants screen positive for depressive symptoms within 24 months) can be answered with the methods of this lesson even when follow-up is incomplete. A question about timing (how quickly positive screens accumulate, and whether they accumulate faster in one group) can only be answered with them, because it concerns the whole course of follow-up.

Box 8.1 describes the hypothetical study that serves as the running case for this lesson. It extends the Campus Connection Study from Lesson 7 into a follow-up of young adults who screen negative for depressive symptoms at baseline, and its event is a first positive screen.

The running case fixes the setting, and the next part turns to the three design decisions that define its time-to-event outcome.

Three Design Decisions: Time Zero, the Event and the End of Follow-Up

Before any data are collected, a time-to-event protocol must define when the clock starts, what stops it, and how observation ends for participants who do not have the event. These three decisions determine what the time and status variables mean, and errors in any of them cannot be repaired at the analysis stage.

Box 8.2 recalls the thought experiment from HSCI 230 Lesson 3 that later work calls the target trial, which guides the first of these decisions, the choice of time zero.

Time zero

Time zero is the moment at which the clock starts for each participant. Common choices are the date of enrolment, the date of randomization in a trial, the date of diagnosis in a clinical cohort, and the date of a first prescription in a study of a medication. A good choice satisfies three conditions at the same moment: the participant meets the eligibility criteria, the participant's exposure group is determined, and follow-up for the event begins. Hernán et al. (2016) describe this alignment as part of specifying the target trial that an observational study tries to emulate, and they show that several avoidable biases arise when the three conditions are met at different times. In the Campus Connection Follow-up, time zero is the date on which the participant completes the baseline survey, because that is when eligibility (a PHQ-9 score below 10) is confirmed, when loneliness is measured, and when monthly follow-up begins.

Enrolment is usually spread over weeks or months, so time zero falls on different calendar dates for different participants. Survival analysis therefore measures time on a study time scale, defined as the time since each person's own time zero, and every participant starts at zero on that scale. Some cohort studies of chronic disease use age as the time scale instead, so that participants are compared with others of the same age. Figure 8.1 shows ten participants from the high-loneliness group of the Campus Connection Follow-up on the study time scale.

The event

The event must be defined precisely enough that two investigators reviewing the same record would agree on whether and when it occurred. The definition states the criterion (a PHQ-9 score of 10 or more), the method of ascertainment (the monthly online questionnaire) and the dating rule (the event is assigned to the month of the first qualifying score). The ascertainment schedule should be the same in every exposure group. If participants with high loneliness were contacted more often than others, their events would be detected sooner, and the comparison between groups would reflect the schedule as well as any effect of loneliness. Most analyses use the first occurrence of the event, so a participant leaves the group at risk once the event has happened. A composite event, such as hospitalization or death, counts whichever component occurs first, and each component needs its own definition.

Follow-up and how it ends

Each participant's follow-up ends at the earliest of three dates: the date of the event, the last date on which the participant was known to be event-free before leaving the study, and the administrative end of the study. The time variable is the interval from time zero to that date, and the status variable records whether the event ended it. Table 8.1 converts the follow-up records of ten participants from the high-loneliness group into times and statuses. Sections 2 to 4 use these ten participants, together with ten from the low-loneliness group, whenever a short dataset keeps the arithmetic manageable.

Table 8.1. Follow-up records of ten high-loneliness participants in the hypothetical Campus Connection Follow-up, reduced to a time and a status.

The protocol should also state what counts as loss to follow-up, because participants in an online cohort rarely announce that they are leaving. A common rule is that a participant who misses a stated number of consecutive questionnaires, for example three, is censored at the date of the last completed questionnaire. Writing the rule in advance prevents the analyst from choosing censoring dates after seeing the data, and it makes the rule the same for every exposure group.

Time zero, the event and the rules for ending follow-up together reduce each participant's record to a time and a status, as Table 8.1 does for ten participants. Five of those participants, numbers 2, 5, 7, 9 and 10, have a status of 0, and the next part of the section describes the forms that this incomplete information can take.

Censoring and Its Types

Censoring occurs when a participant's event time is only partly known. The form that standard survival methods handle is right censoring, in which the event, if it occurs at all, would occur after the participant's last observation. Right censoring has several sources, and a protocol should anticipate each of them. Click each card to see how the source arises in the Campus Connection Follow-up.

Two other forms of censoring arise less often in prospective cohorts. Left censoring occurs when the event happened before observation began at an unknown time. In a study of the age at which adolescents first use cannabis, a participant who reports at the first interview that they have already used it is left-censored, because the event is known only to have occurred before that interview. The Campus Connection Follow-up avoids left censoring by excluding participants who screen positive at baseline. Interval censoring occurs when the event is known only to have occurred between two observation times. Every event in the Campus Connection Follow-up is interval-censored within a month, because a score of 10 or more in month 5 means that symptoms reached the threshold at some point after the month 4 questionnaire. Treating the month of detection as the event time is acceptable when the gap between assessments is short relative to the length of follow-up. When assessments are a year apart, methods designed for interval-censored data are needed.

Late entry, also called left truncation, differs from censoring. It arises when participants join the group at risk some time after their time zero, for example when time zero is the date of diagnosis but people are recruited from a clinic months or years later. Anyone who had the event before they could be recruited is never seen. A late entrant is therefore counted as at risk only from the date of entry, and an analysis that counts the earlier time as observed time overstates early survival. Figure 8.2 places the four situations on one time axis.

Right censoring, the first situation in Figure 8.2, is the form of incomplete information that every method in the rest of the lesson handles. Those methods also make an assumption about why right-censored participants leave the study, and the next part examines that assumption.

Informative and Non-Informative Censoring

Every method in this lesson rests on the assumption that censoring is non-informative. The assumption states that, at any time during follow-up, participants who are censored have the same future risk of the event as participants in the same group who remain under observation. Censoring then removes people from the risk set without changing the risk of the people who remain, and the estimate for the remaining participants applies to the whole group.

Administrative censoring usually satisfies the assumption, because the closing date of a study has nothing to do with any participant's health. Loss to follow-up and withdrawal may not. In the Campus Connection Follow-up, a participant whose mood is worsening may stop answering monthly questionnaires before their score reaches 10. If participants like this are censored, the people who remain in the risk set are healthier than the group as a whole, fewer events are observed than actually occur, and the estimated probability of remaining event-free is too high. Censoring that depends on the participant's prognosis in this way is called informative censoring. If it is more common in one exposure group than in the other, it also biases the comparison between groups.

The assumption cannot be tested from the observed data, because the event times of censored participants are unknown. Design decisions make the assumption more plausible and its failure easier to assess. A protocol can include retention procedures that keep losses small (reminders by more than one channel, short questionnaires and modest incentives for completion), a record of the reason for every withdrawal, a plan to compare the baseline characteristics of participants who were censored early with those who were not, and a sensitivity analysis that recomputes the estimates under extreme assumptions, such as the assumption that every participant who was lost to follow-up had the event at the time of censoring. If the conclusions hold under those assumptions, informative censoring is an unlikely explanation for them.

Box 8.3 describes competing events, which the methods in this lesson treat as censoring even though a person who has had one can no longer have the event of interest.

Non-informative censoring is an assumption that design can make more plausible and that the observed data cannot test. The next part shows what happens when the ten participants in Table 8.1 are summarized with ordinary methods that ignore censoring altogether.

Why Ordinary Summaries Fail

Consider the ten participants in Table 8.1 and the question of what proportion screen positive within 24 months. Five participants had the event (E = 5), three were censored before 24 months (C = 3), and two completed follow-up without the event. Two shortcuts are common, and both give the wrong answer.

The first shortcut treats every censored participant as event-free and divides the events by everyone enrolled, giving 5 ÷ 10 = 0.50. This understates the risk, because participants 2, 5 and 7 might have screened positive after they left the study, and the shortcut assumes that none did. The second shortcut drops the censored participants and divides the events by the participants who remain, giving 5 ÷ 7 = 0.71. This tends to overstate the risk, because the people removed are those known to have been event-free for part of the period, while every participant who had an event is kept. The data themselves only rule out values below 0.50 and above 0.80, which is the risk if all three censored participants had screened positive immediately after leaving. Section 3 shows that the Kaplan-Meier estimate of the 24-month risk for these ten participants is 0.64. It lies within those limits, and it uses the event-free months of each censored participant without inventing events for them.

Equation 8.1 sets out the two shortcuts and the limits that the data place on the risk. Calculator 8.1 computes the two shortcuts and the upper limit from the number enrolled, the number with an observed event and the number censored before the end of follow-up, and its presets load the ten participants in Table 8.1 and the 200 high-loneliness participants.

Averaging the observed times is also misleading. The mean of the ten follow-up times in Table 8.1 is 10.8 months, but it mixes event times with censoring times, and three of the censoring times are shorter than the event times they would eventually have become. The incidence rate from Lesson 4 does use all of the follow-up correctly: the ten participants contributed 2 + 3 + 5 + 5 + 8 + 10 + 12 + 15 + 24 + 24 = 108 person-months and had 5 events, a rate of 5 ÷ 108 = 0.046 per person-month, or 4.6 per 100 person-months. The rate summarizes the whole period in one number, however, and it assumes that the event rate stayed the same throughout. It cannot show whether positive screens clustered in the first months of follow-up or spread evenly across it. The life table in Section 2 and the Kaplan-Meier estimator in Section 3 describe how the probability of remaining event-free changes over time, and Section 4 returns to the rate as the basis of the hazard ratio.

Before those methods are introduced, the last part of this section returns to the choice of time zero.

Design Errors That Start at Time Zero

Two biases follow directly from the choice of time zero. HSCI 230 Lesson 10, Design-Specific and Temporal Biases, teaches both in detail. They are summarized here because each is prevented by a decision written into the protocol.

The accordion below describes immortal time bias and lead-time bias in turn, and its third panel states the feature that the two biases share.

This section has defined a time-to-event outcome through time zero, the event and the end of follow-up, described the forms of censoring and the non-informative censoring assumption, shown why ordinary summaries misstate the risk when follow-up is incomplete, and traced immortal time bias and lead-time bias to the choice of time zero. The exercise below applies these decisions to a study of readmission after a first heart attack, and the reflection, key takeaways and knowledge check that follow review the section. Section 2 then builds the actuarial life table, which uses the follow-up of every participant to describe how the probability of remaining event-free changes over time.

Introduction and Overview

The life table is the oldest method for describing survival. John Graunt's analysis of the London Bills of Mortality in 1662 estimated how many of 100 people born would still be alive at successive ages, and the same arithmetic later became the basis for pricing life insurance and annuities. Two kinds of life table are in use. A period life table, of the kind used to compute life expectancy in Lesson 4, applies the age-specific death rates of one calendar period to a hypothetical population. A cohort or clinical life table, the subject of this section, follows a real group of people from time zero and estimates the proportion who remain event-free at the end of successive intervals of follow-up. Cutler and Ederer (1958) set out the method for clinical follow-up studies, and it is often called the actuarial method.

The actuarial life table answers the question that defeated the shortcuts in Section 1. It uses the follow-up of censored participants up to the point at which they leave, and it never assumes anything about what happened to them afterwards. It does so by breaking follow-up into intervals and estimating, for each interval, the probability of getting through it without the event among those who started it event-free. Multiplying these conditional probabilities together gives the probability of remaining event-free to the end of any interval. The same logic, with intervals that end at each event time, becomes the Kaplan-Meier estimator in Section 3.

The section first explains why survival over several intervals can be written as a product of conditional probabilities. It then defines the columns of the table, works through a small example and the running case, sets out the assumptions behind the method, and closes with the step from grouped intervals to exact event times.

The Logic: Surviving One Interval at a Time

Suppose follow-up is divided into intervals of six months. To remain event-free for 18 months, a participant must get through the first interval without the event, then get through the second interval given that they reached it event-free, then get through the third interval given that they reached it event-free. Probability theory states that the probability of a sequence of events is the product of each event's probability conditional on the events before it. The probability of remaining event-free to the end of interval j is therefore the product of the conditional probabilities of surviving intervals 1 to j.

The value of this decomposition lies in the denominators. Each conditional probability is estimated only from the participants who were present and event-free at the start of that interval. A participant who was censored in month 8 contributes fully to the first interval, contributes partly to the second, and is simply absent from the third and fourth. Nobody has to guess what happened to that participant after month 8, and their eight event-free months are not wasted. Figure 8.3 shows the chain of conditional probabilities for the Campus Connection Follow-up life table computed later in this section.

Each link of the chain in Figure 8.3 is computed from counts recorded for its interval. The next part sets out those counts, and the quantities derived from them, as the columns of an actuarial life table.

The Columns of an Actuarial Life Table

An actuarial life table has one row per interval and seven columns. The subscript j numbers the intervals, so l₁ is the number entering the first interval and S₃ is the cumulative survival to the end of the third interval. Table 8.2 defines each column and shows how it is obtained.

Table 8.2. Columns of an actuarial life table, with the symbol for each quantity and how it is obtained.

The effective number at risk is the only column that needs explanation. Withdrawals leave at various points during the interval, so they were at risk for part of it. The actuarial method assumes that, on average, they were at risk for half of it, and it counts each withdrawal as half a person in the denominator. Equation 8.2 gives the formulas for one interval, and Calculator 8.2 applies them. The calculator starts from the first interval of the Campus Connection Follow-up, in which 200 participants entered, 12 withdrew and 14 screened positive.

Equation 8.2 deals with one interval at a time. A full table chains the intervals together, because the number entering each interval depends on the withdrawals and events in the interval before it, and the next part builds such a table for a small cohort followed in yearly intervals.

Worked Example: Three Yearly Intervals

The first worked example is deliberately small. One hundred people are followed in yearly intervals. In the first year, 10 withdraw and 8 have the event; in the second year, 6 withdraw and 10 have the event; and in the third year, 12 withdraw and 6 have the event. Each row's number entering is the previous row's number entering minus its withdrawals and events, so 100 − 10 − 8 = 82 people enter the second year and 82 − 6 − 10 = 66 enter the third.

Worked Example 8.1 applies Equation 8.2 to each of the three years and multiplies the conditional probabilities of surviving them to obtain the cumulative survival.

Worked Example 8.1 shows every step of the table on round numbers. The next part applies the same steps to the 200 high-loneliness participants of the running case.

The Running Case: 200 Participants Over 24 Months

The 200 high-loneliness participants in the Campus Connection Follow-up were followed in four six-month intervals. The counts in Worked Example 8.2 record, for each interval, how many participants screened positive and how many were censored before 24 months for any of the reasons listed in Section 1. The 114 participants who were still event-free and under observation at 24 months were censored administratively at the end of the study, and they do not appear as withdrawals because they completed the final interval.

Equation 8.3 states the rule behind the last column of Worked Examples 8.1 and 8.2: the cumulative survival to the end of an interval is the product of the conditional probabilities of surviving every interval up to and including that one.

Calculator 8.3 computes a full life table with up to five intervals. It opens with the running case; the second preset loads the three-interval example. Changing the number of events or withdrawals in one interval changes the number entering every later interval, which shows how each row depends on the rows before it. Figure 8.4, below the calculator, plots the four estimates for the running case.

The running case gives an estimated 24-month risk of 0.237 in the high-loneliness group. That estimate depends on how the 44 participants censored before 24 months were counted in the denominators, and the next part examines the rule that determined their contribution.

Why Withdrawals Are Counted as Half a Person

The half-interval assumption is a compromise between two extremes, and comparing the three versions for the first interval of the running case shows what is at stake. If the 12 withdrawals were counted as at risk for the whole interval, the denominator would be 200 and the conditional probability of the event would be 14 ÷ 200 = 0.070. This treats the withdrawals as if they had been observed event-free to month 6, which they were not, and it understates the probability of the event. If the withdrawals were removed entirely, the denominator would be 188 and the probability would be 14 ÷ 188 = 0.074. This discards the months that withdrawals spent at risk before they left, and it overstates the probability. The actuarial value, 14 ÷ 194 = 0.072, lies between them. The difference between the three versions grows with the number of withdrawals relative to the number entering and with the width of the interval.

The half-interval rule is one of three assumptions on which the actuarial method rests. The accordion below sets out each assumption and then the related choice of interval width.

Box 8.4 describes the kinds of data for which the actuarial method remains in use despite these assumptions.

The half-interval assumption and the choice of interval width both follow from grouping follow-up into fixed intervals. The final part of this section considers what changes when each interval is allowed to end at an observed event time.

From Grouped Intervals to Exact Event Times

The life table shows the probability of remaining event-free only at the interval boundaries: 0.928 at 6 months, 0.862 at 12 months, 0.808 at 18 months and 0.763 at 24 months. Between the boundaries it says nothing, and an analyst who chose three-month intervals would obtain somewhat different values. If the conditional probabilities are recomputed every time an event occurs, two things change. The intervals no longer need to be chosen, and withdrawals no longer need the half-interval assumption, because each censored participant is counted as fully at risk at every event time before their censoring time and as absent after it. That is the Kaplan-Meier estimator.

This section has built the actuarial life table from conditional probabilities, applied it to a small cohort and to the running case, and set out the assumptions on which it rests. The exercise below works through one row of a life table by hand, and the reflection, key takeaways and knowledge check that follow review the section. Section 3 then develops the Kaplan-Meier estimator in full.

Introduction and Overview

Edward Kaplan and Paul Meier published the product-limit estimator in 1958 (Kaplan & Meier, 1958), and it has become the standard way to describe survival in clinical and epidemiological research. The estimator applies the logic of the life table with one change. Instead of fixed intervals chosen in advance, each interval ends at an observed event time. At every event time the estimator computes the conditional probability of getting past that time among the participants still at risk, and it multiplies these conditional probabilities together. Censored participants are counted as fully at risk at every event time before they leave and are absent after it, so the half-interval assumption of the actuarial method is no longer needed.

This section computes a Kaplan-Meier estimate by hand for the ten high-loneliness participants introduced in Section 1, using a calculator that shows each step. It then explains how to read a published Kaplan-Meier curve: the survival probability at a chosen time, the median survival time, the censoring marks and the table of numbers at risk that should appear beneath every curve.

The section begins with the product-limit idea in symbols.

The Product-Limit Idea

Order the distinct event times as t₁ < t₂ < t₃ and so on. At each event time t_j, let n_j be the number of participants at risk just before that time, meaning those who have neither had the event nor been censored before t_j, and let d_j be the number who have the event at t_j. The conditional probability of getting past t_j is (n_j − d_j) / n_j. The Kaplan-Meier estimate of survival at time t, written Ŝ(t), is the product of these conditional probabilities over every event time up to and including t. The hat on S marks a quantity estimated from data.

Each step needs only the estimate at the previous event time, the number at risk and the number of events. Equation 8.4 writes one such step in symbols. Calculator 8.4 performs a single step, starting from the third event time in Worked Example 8.3: the estimate just before month 5 is 0.900, eight participants are at risk, and two screen positive.

Repeating this step at each event time in turn produces the whole estimate. The next part does so for the ten high-loneliness participants whose records appear in Table 8.1.

Worked Example: Ten Participants, Step by Step

Table 8.3 applies the estimator to the ten high-loneliness participants from Table 8.1. The rows are the distinct follow-up times in order. A row with an event changes the estimate; a row with only censoring leaves the estimate unchanged and reduces the number at risk at the next event time. Worked Example 8.3 sets out this calculation and interprets the resulting 24-month estimate.

Three conventions are worth stating explicitly, because published analyses and software follow them. Tied events at the same time are handled together in one step. When an event and a censoring are recorded at the same time, the event is assumed to come first, so the censored participant is still counted at risk at that time. The estimate is defined only up to the longest follow-up time; beyond month 24 the data say nothing about these participants.

Equation 8.5 gathers the steps of Table 8.3 into a single expression: the estimate at time t is the product of the conditional probabilities at every event time up to and including t.

The step-by-step Calculator 8.5 takes up to ten participants, each with a follow-up time and a status (1 for an event, 0 for censored). It sorts the times, forms the risk set at each time, and shows the working for all time points at once or for one time point at a time, together with the curve drawn so far. Try changing participant 5 from censored to an event at month 8 and watch every later step change. Then change participant 2's time from 3 to 20 months while keeping the status at 0 and notice that a censored participant who stays longer raises the number at risk at every later event time.

Worked Example 8.3 gives the estimate as a sequence of numbers. Published analyses present it as a curve, and the next part explains how to read one.

Reading a Kaplan-Meier Curve

A Kaplan-Meier curve plots the estimate against time as a step function. It starts at 1 at time zero, stays flat between event times, and drops at each event time by an amount that depends on the number of events and the number at risk. Figure 8.5 shows the curve for the ten participants in Table 8.3.

Four features of Figure 8.5 carry most of the information. The four tabs below explain how to read the survival at a chosen time, the median survival time, the censoring marks and the numbers at risk.

Uncertainty in the estimate is usually shown as a 95 percent confidence band around the curve. The standard error is computed with Greenwood's formula (Greenwood, 1926), which adds a contribution from each event time. The practical point for study design is that the band widens as the number at risk falls, so precision late in follow-up depends on retaining participants and on enrolling enough of them in the first place.

These reading rules apply to any Kaplan-Meier curve. The next part uses them to set the high-loneliness curve beside a curve for ten low-loneliness participants.

Comparing Two Curves by Eye

Figure 8.6 adds ten participants from the low-loneliness group to the ten from the high-loneliness group. In the low-loneliness group, participant 1 was censored at month 4, participants 2, 4 and 6 screened positive at months 6, 12 and 20, participants 3 and 5 were censored at months 9 and 18, and the remaining four completed 24 months without a positive screen.

The curves can be compared at a fixed time, by their medians, or by the area beneath them. At 12 months the estimated probability of remaining event-free is 0.540 in the high-loneliness group and 0.762 in the low-loneliness group. The median is 15 months in the high-loneliness group and is not reached in the low-loneliness group, whose curve ends at 0.610. The area under each curve up to 24 months is the restricted mean survival time, the average number of event-free months within the first 24 (Royston & Parmar, 2013). It is 14.0 months for the high-loneliness participants and 19.9 months for the low-loneliness participants, a difference of about 5.9 event-free months. Each of these comparisons describes the two samples. Whether the difference is larger than chance would produce with ten participants per group is the question for the log-rank test in Section 4.

Before that comparison, the next part returns to the high-loneliness group alone and sets its Kaplan-Meier estimate beside the two shortcuts from Section 1.

Kaplan-Meier Compared with the Shortcuts

For the same ten high-loneliness participants, the three approaches give the following estimates of the probability of remaining event-free for 24 months. Table 8.4 sets out the three estimates and the assumption that each makes about the three participants censored early.

Table 8.4. Estimated probability of remaining event-free for 24 months among the ten high-loneliness participants under the two shortcuts and the Kaplan-Meier estimator, with the assumption each approach makes about the three early-censored participants.

The Kaplan-Meier estimate lies between the two shortcuts, and it is the only one of the three whose assumption is stated in terms that a design can make plausible. Efron (1967) showed that the estimator can be computed by redistributing each censored participant's share of the probability equally among the participants who remain at risk after the censoring time, which is a concrete way to see the non-informative censoring assumption at work.

The Kaplan-Meier estimate rests on assumptions of its own, and the final part of this section states them and lists what a report of a Kaplan-Meier analysis should include.

What the Estimator Assumes and What a Report Should Include

The Kaplan-Meier estimator makes no assumption about the shape of the survival curve, which is why it is called non-parametric. It does make three assumptions about the data. Censoring must be non-informative, as discussed in Section 1. Participants who enrolled early and late must have the same survival prospects at the same time since time zero, as for the life table. Event times must be recorded precisely enough that the order of events and censorings is correct, which is why monthly ascertainment works well in the Campus Connection Follow-up and annual ascertainment would not.

Box 8.5 lists the elements that a report of a Kaplan-Meier analysis should contain.

This section has computed the Kaplan-Meier estimate by hand, shown how to read and compare its curves, and stated its assumptions and the content of a report. The reflection, key takeaways and knowledge check that follow review the section, and Section 4 then turns from describing survival in each group to formal comparisons between groups.

Introduction and Overview

Sections 2 and 3 described survival in one group at a time. Most time-to-event studies compare groups: participants with and without an exposure, or participants randomized to an intervention or to control. This section introduces the two tools used for that comparison. The log-rank test asks whether two survival curves differ by more than chance would produce. The hazard ratio measures how much faster events occur in one group than in the other, and it is the time-to-event counterpart of the incidence rate ratio from Lesson 6. The section then explains the proportional hazards idea, which most hazard ratio analyses assume, and closes with the decisions that a protocol with a time-to-event outcome must state before data collection begins.

The section begins with the log-rank test.

Comparing Two Curves: The Log-Rank Test

Figure 8.6 in Section 3 showed that the high-loneliness participants had a lower Kaplan-Meier curve than the low-loneliness participants. The log-rank test asks whether a difference of that size could plausibly arise if the two groups had the same survival at every time. Its logic builds on the risk set. At each event time, the participants at risk in both groups are pooled, and the events at that time are shared out between the groups in proportion to their numbers at risk. If the null hypothesis is true, a group that makes up 40 percent of the pooled risk set should, on average, account for 40 percent of the events at that time. The expected number of events in group A at event time j is therefore the total number of events at that time multiplied by group A's share of the risk set.

Equation 8.6 adds these expected numbers over the event times and gives the test statistic that compares them with the observed number of events in group A.

Table 8.5 applies this logic to the twenty participants in Figure 8.6. Group A is the high-loneliness group. At month 12, one low-loneliness participant screened positive and high-loneliness participant 7 was censored; following the convention from Section 3, participant 7 is counted in the risk set at month 12. Worked Example 8.4 sets out this calculation and the resulting log-rank statistic.

The curves in Figure 8.6 look quite different, yet the test does not reject the null hypothesis. With ten participants per group and eight events in total, a large difference in survival is compatible with chance. This is a property of the design and says nothing in favour of equal survival. The information in a time-to-event comparison comes from the events, and eight events provide very little of it; the last part of this section shows how to plan for enough events. The ratio of observed to expected events also gives a rough estimate of the hazard ratio: (5 ÷ 3.24) ÷ (3 ÷ 4.76) = 2.45.

Calculator 8.6 performs the observed-and-expected arithmetic for up to eight event times. It opens with Table 8.5. Moving one event from group A to group B at the same time shows how sensitive a comparison with few events is.

The test is usually attributed to Mantel (1966) and to Peto and Peto (1972), and Bland and Altman (2004) give a short account for applied readers. Three of its properties matter for interpretation. It gives every event time the same weight, so late differences, where few participants remain at risk, count as much as early ones. It is most powerful when one group has a consistently higher hazard throughout follow-up, the situation described by the proportional hazards assumption below, and it can miss a real difference when curves cross. It extends directly to more than two groups and to comparisons within strata of a confounder, the stratified log-rank test, in the same way that the Mantel-Haenszel approach in Lesson 7 extends a single two-by-two table.

The log-rank test addresses whether two curves differ by more than chance would produce. A measure of how much faster events occur in one group needs the hazard, which the next part defines.

The Hazard: An Event Rate Among Those Still at Risk

The hazard at time t is the rate at which events occur among participants who are still event-free at t, expressed per unit of time. It is the incidence rate from Lesson 4 computed over a very short interval around t. The survival probability can only stay level or fall as time passes, but the hazard can rise, fall or stay constant. The hazard of death after major surgery is highest in the days after the operation and then falls; the hazard of death from most chronic diseases rises with age; and the hazard of a sudden accidental event may be roughly constant.

The life table from Section 2 gives an estimate of the hazard within each interval. Dividing the events in an interval by the person-time at risk in that interval gives the average hazard over the interval. With the half-interval assumption applied to both withdrawals and events, the person-time in an interval of width h is approximately h × (r_j − d_j / 2).

Table 8.6 applies this approximation to the four six-month intervals of the high-loneliness life table in Worked Example 8.2.

Table 8.6. Estimated hazard of a first positive screen in each six-month interval for the 200 high-loneliness participants in the hypothetical Campus Connection Follow-up.

In this hypothetical cohort the hazard of a first positive screen declined slowly over follow-up, from about 1.25 to about 0.96 per 100 person-months. The overall rate in the last row, 42 events over 3,672 person-months, is the average of the interval hazards weighted by person-time.

The overall hazard of 1.14 per 100 person-months summarizes the high-loneliness group. The next part compares it with the hazard in the low-loneliness group.

The Hazard Ratio as a Ratio of Rates

The hazard ratio compares the hazard in one group with the hazard in another at the same time since time zero. If both hazards are constant over follow-up, the hazard ratio equals the incidence rate ratio from Lesson 6, and it can be computed from the events and person-time in each group. The 400 low-loneliness participants in the Campus Connection Follow-up had 40 first positive screens over 8,800 person-months, a rate of 0.45 per 100 person-months, compared with 1.14 per 100 person-months in the high-loneliness group.

Equation 8.7 gives the hazard ratio under constant hazards with its confidence interval, and Calculator 8.7 computes both from the events and person-time in each group, with a preset for the Campus Connection cohort.

The hazard ratio is (42 ÷ 3,672) ÷ (40 ÷ 8,800) = 2.52, with a 95 percent confidence interval from 1.63 to 3.88. In words, at any point during the 24 months, participants with high loneliness who had not yet screened positive did so at about two and a half times the rate of participants with low loneliness. The standard error depends only on the numbers of events, 42 and 40, which is another way of seeing that events carry the information in a time-to-event comparison.

In practice, hazard ratios are usually estimated with the Cox proportional hazards model (Cox, 1972). The Cox model does not assume that the hazards are constant. It allows the baseline hazard to take any shape over time and assumes only that the ratio of the hazards stays the same. Like logistic regression for the odds ratio, it can adjust a hazard ratio for confounders, and it is the time-to-event analogue of the regression adjustment described in Lesson 7. HSCI 341 interprets the hazard ratio from a Cox model without fitting one. For protocol writing, the point is that a hazard ratio from a Cox model has the same interpretation as the rate ratio above, provided its central assumption holds.

A single hazard ratio, whether computed from rates or from a Cox model, is a complete summary only if the ratio of the hazards stays the same over follow-up. The next part examines that assumption.

Proportional Hazards in Plain Language

The proportional hazards assumption states that the ratio of the hazards in the two groups is the same at every time during follow-up. Each hazard may rise or fall, but they rise and fall together. If participants with high loneliness screen positive at two and a half times the rate of participants with low loneliness in month 2, the assumption says they also do so at two and a half times the rate in month 20. A single hazard ratio is then a complete summary of how the groups differ over time.

Figure 8.7 shows schematic hazards and survival curves for a situation in which the assumption holds and for one in which it fails.

The assumption fails when the effect of the exposure changes over follow-up. A surgical treatment compared with medical management may carry a higher hazard of death in the weeks after the operation and a lower hazard afterwards, so the hazard ratio is above 1 early and below 1 later, and a single hazard ratio averages these into a number that describes neither period. Figure 8.7 contrasts the two situations. Kaplan-Meier curves that cross, or that separate early and then converge, suggest that the hazards are not proportional. More formal checks compare the log of the cumulative hazard in each group over time or test whether the hazard ratio drifts with time.

When the hazards are not proportional, the protocol can provide alternatives. The analyst can report separate hazard ratios for prespecified periods, such as the first six months and the remainder of follow-up; report the difference in survival at prespecified times; or report the difference in restricted mean survival time, which Royston and Parmar (2013) recommend as a summary that does not depend on proportional hazards. Hernán (2010) adds a further caution. Period-specific hazard ratios compare the participants who remain at risk at each time, and in later periods those participants are a selected group, because the most susceptible members of the higher-risk group have already had the event. Survival curves and risks at fixed times are less vulnerable to this problem, and he concludes that survival curves are more informative than hazard ratios and should be used more widely in observational studies.

A hazard ratio is not a risk ratio

Under proportional hazards, the survival curve of the exposed group is the survival curve of the unexposed group raised to the power of the hazard ratio: S₁(t) = S₀(t)^HR. This relationship shows how far the hazard ratio can differ from the risk ratio at a fixed time. If 90 percent of low-loneliness participants remain event-free at 24 months and the hazard ratio is 2.5, then 0.90^2.5 = 0.768 of high-loneliness participants remain event-free. The 24-month risks are 0.100 and 0.232, a risk ratio of 2.32. If the event were common, with only half of the unexposed group event-free at 24 months, the same hazard ratio of 2.5 would give risks of 0.500 and 0.823, a risk ratio of 1.65. The hazard ratio and the risk ratio are close when the event is rare and diverge as it becomes common, in the same way that the odds ratio and the risk ratio diverged in Lesson 6.

Equation 8.8 states the relationship between the two survival curves. Calculator 8.8 applies it to translate a hazard ratio into the probability of remaining event-free, the cumulative risk, the risk ratio and the risk difference at a chosen time, and its presets include the uncommon and common events described above.

The check of the proportional hazards assumption, the summary to report if it fails and the choice of effect measure are decisions for the analysis plan. The final part of this section gathers them, with the other decisions in this lesson, into the content of a protocol.

What a Protocol With a Time-to-Event Outcome Must Specify

The decisions in this lesson must be written into the protocol before data collection begins. Most of them cannot be corrected later, and stating them in advance prevents analytic choices from being shaped by the results. Table 8.7 lists the elements with the Campus Connection Follow-up as the example.

Table 8.7. Elements that a protocol with a time-to-event outcome must specify, with the Campus Connection Follow-up as the example.

Planning for enough events

The log-rank example showed that a time-to-event comparison with few events has little power, whatever the number of participants. Schoenfeld (1983) derived the number of events needed to detect a given hazard ratio with a two-sided test. The number of participants follows by dividing the required events by the proportion of participants expected to have the event during follow-up.

Equation 8.9 gives both quantities. Calculator 8.9 evaluates them and adds an allowance for loss to follow-up, with a preset for the Campus Connection protocol.

For a new cohort designed to detect a hazard ratio of 1.8 with a two-sided significance level of 0.05 and 80 percent power, with one third of participants in the high-loneliness group, the formula gives (1.960 + 0.842)² ÷ [(1/3) × (2/3) × (ln 1.8)²] = 102.2, which rounds up to 103 events. If about 15 percent of participants are expected to screen positive within 24 months, the study needs 103 ÷ 0.15 = 687 participants (rounded up) before any allowance for loss to follow-up. The protocol targets a hazard ratio of 1.8, smaller than the 2.52 observed in the hypothetical pilot, because pilot estimates from small samples are imprecise and tend to be too large when a pilot is continued only if its results look promising, and because the target should be the smallest effect that would matter for campus mental health planning.

Worked Example 8.5 shows how the elements of Table 8.7 and this sample size calculation are written as the time-to-event subsection of the Campus Connection Follow-up protocol.

The inflation for loss to follow-up in the last paragraph of Worked Example 8.5 divides 687 by 0.80 and rounds up, the same approach used to inflate a survey sample for expected non-response. Losses reduce the number of observed events, so a protocol that expects heavy loss should either plan a larger cohort or a longer follow-up.

This section has compared groups with the log-rank test and the hazard ratio, set out the proportional hazards assumption and its alternatives, and gathered the design decisions of the whole lesson into the content of a protocol. The note below describes how the lesson applies to writing the time-to-event subsection of a protocol, and the reflection, key takeaways and knowledge check that follow review the section before the final review of the lesson.

Bringing It All Together

This lesson treated time-to-event analysis as a way of estimating incidence when follow-up is incomplete. Section 1 set out the design decisions that give a time-to-event outcome its meaning (time zero, the event and the end of follow-up) and the forms of censoring that make follow-up incomplete. It showed that the shortcuts of treating censored participants as event-free or dropping them misstate the risk, and that the methods of the lesson depend on censoring being non-informative. Sections 2 and 3 estimated the probability of remaining event-free over time, first with the actuarial life table and then with the Kaplan-Meier estimator, and showed how to read survival at a time, the median, censoring marks and numbers at risk from a curve.

Section 4 compared groups. The log-rank test compares observed with expected events, the hazard ratio compares event rates among participants still at risk, and the proportional hazards assumption states that this ratio is the same throughout follow-up. Because the information in these comparisons comes from events, the number of events needed is the starting point of a sample size calculation. Each of these decisions belongs in the protocol, written before any data are collected.

The final reflection asks you to bring these decisions together for a new study.

Final Knowledge Assessment

Complete all 15 questions below with 100% accuracy to finish this lesson. You must also pass every section knowledge check and complete every reflection, including the final reflection above, before submitting.