How do you measure risk and survival?

Start here

How many people are ill now, and how many became ill this year?

Prevalence counts everyone who has the disease at a given time. Incidence counts only new cases, among people who did not have the disease at the start.

Incidence proportion (risk) = new cases in a period ÷ people at risk at the start

Incidence rate = new cases ÷ total person-time at risk

Point prevalence = all current cases at one time ÷ population at that time

Person-time

the sum of the time each person was followed while still at risk. One person followed for one year contributes one person-year.

At risk

able to develop the disease: not already ill, and still being followed

Attack rate and secondary attack rate

In an outbreak, the incidence proportion over the short epidemic period is called the attack rate. The secondary attack rate counts new cases among the susceptible contacts of the first cases, within one incubation period. It shows how readily the infection spreads from person to person.

Attack rate = 36 ill ÷ 240 hostel residents = 15%

Secondary attack rate = 9 new cases ÷ 45 susceptible room-mates = 20%

Teaching example with made-up numbers. Despite its name, an attack rate is a proportion over a stated period, not a rate per unit of time. The incidence rate per person-time is also called incidence density.

Incidence, prevalence and duration

Why can a better treatment increase prevalence?

Prevalence depends on how many new cases appear and on how long each case lasts.

Prevalence ≈ incidence rate × average duration

2 per 1,000 person-years × 5 years = 0.010 → prevalence about 1%

This approximation holds only under two conditions.

1. A steady state. Incidence and average duration are not changing over time, and people do not move into or out of the population in a way linked to the disease.
2. An uncommon disease. The exact steady-state relation is P ÷ (1 − P) = I × D (Pearce, 2004). It reduces to P ≈ I × D only when 1 − P is close to 1.

I = 20 per 1,000 person-years, D = 10 years: I × D = 0.20, but exact P = 0.20 ÷ 1.20 = 16.7%, not 20%

The relation can also be used the other way round when only a survey and the average duration are available: incidence ≈ prevalence ÷ duration. Teaching example: a prevalence of 4 per 1,000 and an average duration of 3 months (0.25 years) give an incidence of about 16 per 1,000 person-years (16.1 with the exact form).

A treatment that keeps patients alive without curing them lengthens duration, so prevalence rises even when incidence does not change. Insulin in type 1 diabetes and antiretroviral therapy in HIV are examples. A faster cure, or a higher death rate, shortens duration and lowers prevalence.

Interactive

Change incidence and duration, and watch prevalence

Comparing death rates

Why can a town with higher death rates at every age have a lower crude death rate?

A crude death rate mixes the death rate at each age with the age structure of the population. Standardisation removes the effect of age structure so that populations can be compared.

The crude death rate is the number of deaths in a year ÷ the mid-year population, usually per 1,000. An age-specific death rate is the same calculation within one age group. Death rates are much higher in old age, so a population with more old people can have a higher crude death rate even when its death rate at every age is lower.

Teaching example with made-up numbers. Town A: 10 + 150 + 900 = 1,060 deaths. Town B: 50 + 160 + 400 = 610 deaths.

Town B has the higher death rate in every age group but the lower crude rate, because 10% of its people are aged 60 or more against 30% in town A. Age is linked both to which town people live in and to their chance of dying, so age confounds the comparison.

Direct standardisation

Apply each town's age-specific rates to the same standard population and count the deaths that would occur. With the standard population in thousands, each product below is a number of deaths.

Town A: 40 × 0.5 + 40 × 3.0 + 20 × 30.0 = 740 → 7.4 per 1,000

Town B: 40 × 1.0 + 40 × 4.0 + 20 × 40.0 = 1,000 → 10.0 per 1,000

After standardisation the order is reversed, and town B has the higher death rate. A standardised rate is useful only for comparison, and its value depends on the standard population chosen.

Indirect standardisation and the standardised mortality ratio

When the age-specific rates of a study population are not known, or rest on very few deaths, apply reference rates to the study population's own age structure. This gives the expected number of deaths. The standardised mortality ratio (SMR) is observed deaths ÷ expected deaths, often multiplied by 100. Teaching reference rates: 0.8, 3.5 and 35.0 per 1,000.

Town B expected deaths = 50 × 0.8 + 40 × 3.5 + 10 × 35.0 = 530

Town B: SMR = 610 ÷ 530 = 1.15, or 115

Town A expected deaths = 20 × 0.8 + 50 × 3.5 + 30 × 35.0 = 1,241

Town A: SMR = 1,060 ÷ 1,241 = 0.85, or 85

Town B had 15% more deaths than the reference rates predict, and town A about 15% fewer. Each SMR compares one town with the reference rates. Two SMRs are not strictly comparable with each other, because each is weighted by its own town's age structure.

Interactive

Change the age structure of town B and watch the crude and standardised death rates

Life tables

How is life expectancy calculated from death rates?

A current life table applies this year's age-specific death rates to an imaginary group of 100,000 newborns and follows them until all have died.

A cohort (generation) life table follows people actually born in the same period until all of them have died, which takes about a century. A current (period) life table assumes instead that people will meet today's death rates at every age. Life expectancy at birth from a current life table therefore summarises present mortality. It is not a forecast for babies born this year. A complete life table uses single years of age, and an abridged life table uses age groups.

Survivors at the next age: l × (1 − q) = 100,000 × 0.95 = 95,000

Person-years in the group: L = 20 × (100,000 + 95,000) ÷ 2 = 1,950,000

Last, open-ended group: L = l ÷ death rate = 49,761 ÷ 0.2 = 248,805

Life expectancy at 60: e = T ÷ l = 1,575,765 ÷ 82,935 = 19.0 years

q

probability of dying within the age group, for a person alive at its start

l

survivors at the start of the age group, out of 100,000 born

L

person-years lived within the age group, assuming deaths are spread evenly through it

T

person-years lived from that age onwards, the sum of L from that row to the last row

e

life expectancy, the average number of further years lived by people alive at that age

Teaching example with made-up death probabilities and a death rate of 0.2 per person-year at 80 years and over. Real life tables use narrower age groups, with special adjustments for the first year of life, where deaths are not spread evenly.

The same table gives other measures. Potential years of life lost adds up, for every death before a chosen age, the years between the age at death and that age. Health-adjusted life expectancy counts years lived in poor health as less than a full year of healthy life. The life table method is also used for survival after treatment, as the actuarial life table section below shows.

The formula

One 2 × 2 table, six measures

Teaching example with made-up numbers: a trial of DVT prophylaxis after surgery, 200 patients per group, outcome DVT within 30 days.

Relative risk (RR) = 0.15 ÷ 0.25 = 0.60

Relative risk reduction (RRR) = 1 − RR = 1 − 0.60 = 40%

Absolute risk reduction (ARR) = 0.25 − 0.15 = 0.10, or 10 percentage points

Number needed to treat (NNT) = 1 ÷ ARR = 1 ÷ 0.10 = 10

Odds ratio (OR) = (a × d) ÷ (b × c) = (30 × 150) ÷ (170 × 50) = 0.53

Round the NNT up to the next whole patient. When a treatment causes harm, the same calculation gives the number needed to harm (NNH).

Read the table the other way: attributable risk

In a cohort study the exposure usually causes the outcome. Treat "no prophylaxis" as the exposure. The risk difference is then called the attributable risk.

Attributable risk (AR) = risk in exposed − risk in unexposed = 0.25 − 0.15 = 10 per 100

Attributable risk percent = AR ÷ risk in exposed = 0.10 ÷ 0.25 = 40%

RR for the exposure = 0.25 ÷ 0.15 = 1.67. AR and ARR are the same subtraction. AR is the name used when an exposure causes the outcome, ARR when a treatment prevents it. Write "attributable risk" for the difference and "attributable risk percent" for the proportion, so the reader knows which you mean.

Comparison

Why does the same 40% reduction give a different NNT?

Relative measures stay similar when the baseline risk changes. Absolute measures do not.

Public health impact

How much disease in the whole population is due to the exposure?

The attributable fraction in the exposed depends only on the relative risk. The population attributable fraction also depends on how many people are exposed.

Attributable fraction in the exposed = (RR − 1) ÷ RR = (1.67 − 1) ÷ 1.67 = 40%

Population attributable fraction (PAF) = p(RR − 1) ÷ [p(RR − 1) + 1]

p

proportion of the population exposed

RR

relative risk of the disease in exposed compared with unexposed people

The first line is the attributable risk percent from the table above, written with the relative risk. Levin introduced the second formula in 1953 to estimate the population attributable fraction from the prevalence of a risk factor and its relative risk. It is biased when the association is confounded, even if an adjusted relative risk is used (Ferguson and colleagues, 2024).

Teaching examples with made-up numbers. A strong but rare exposure (RR 10, 2% exposed) accounts for fewer cases than a weak but common one (RR 1.5, 50% exposed). The relative risk measures the strength of an association. The population attributable fraction estimates how much of the disease in the population removing the exposure could prevent, if the association is causal.

Comparison

Risk or odds: what is the difference?

Risk divides the events by everyone in the group. Odds divide the events by the people who did not have the event.

Prophylaxis: risk = 30 ÷ 200 = 0.15, odds = 30 ÷ 170 = 0.176

No prophylaxis: risk = 50 ÷ 200 = 0.25, odds = 50 ÷ 150 = 0.333

OR = 0.176 ÷ 0.333 = 0.53, while RR = 0.60

Zhang and Yu (1998) pointed out that when the outcome occurs in more than 10% of the study population, the odds ratio no longer approximates the risk ratio, and they proposed a method to correct it. For a single 2 × 2 table the two are linked exactly:

RR = OR ÷ [(1 − P₀) + (P₀ × OR)] = 0.53 ÷ [0.75 + (0.25 × 0.53)] = 0.60

P₀ is the risk in the control (unexposed) group. The calculation uses the unrounded OR, 0.5294.

Interactive

Change the events in each group and watch RR, OR, ARR and NNT

Case-control studies

Why does a case-control study give an odds ratio and not a relative risk?

In a case-control study you decide how many controls to take. The proportion of cases in your table is set by you, so risk cannot be calculated.

The odds ratio still works because the odds of exposure among cases ÷ the odds of exposure among controls equals the odds ratio of disease in exposed versus unexposed people. Here (40 ÷ 40) ÷ (20 ÷ 80) = 4.0. When the disease is rare, the odds ratio is close to the relative risk: 4.0 against 3.94. Cornfield described estimating comparative rates from case-control data in this way in 1951. The designs are covered in the study designs chapter.

Matched case-control studies

How is the odds ratio calculated when each case has a matched control?

In a matched analysis the unit is the pair. Pairs in which the case and the control have the same exposure give no information about the odds ratio.

Teaching example with made-up numbers: 120 patients with a disease, each matched to one control of the same age, sex and area. Exposure is recorded for both members of every pair.

Matched OR = 40 ÷ 16 = 2.5

95% CI = exp[ln 2.5 ± 1.96 × √(1/40 + 1/16)] = 1.40 to 4.46

McNemar χ² = (|40 − 16| − 1)² ÷ (40 + 16) = 9.45, P = 0.002

The matched odds ratio divides the pairs in which only the case was exposed by the pairs in which only the control was exposed. The 30 pairs with both exposed and the 34 with neither exposed do not enter the calculation. Analysing the same 240 people as if they were unmatched, with 70 of 120 cases and 46 of 120 controls exposed, gives a different answer, OR 2.25. When controls have been matched, keep the pairs together in the analysis, with the matched odds ratio or with conditional logistic regression.

Precision

What do the confidence interval and P value add to an odds ratio?

The odds ratio comes from one sample. The 95% confidence interval gives the range of values compatible with the data. The P value tests whether the true ratio could be 1.

95% CI = exp[ln(OR) ± 1.96 × SE], where SE of ln(OR) = √(1/a + 1/b + 1/c + 1/d)

OR 0.53, 95% CI 0.32 to 0.88 · χ² = 6.25, P = 0.012

• The interval does not include 1, so the result is statistically significant at the 5% level. This matches P below 0.05.
• For ratios (OR, RR, HR), 1 means no difference. For differences (ARR), 0 means no difference.
• In the same table: RR 0.60 (95% CI 0.40 to 0.90). ARR 10 points (95% CI 2.2 to 17.8). NNT 10 (95% CI 5.6 to 45.0). The NNT interval is found by inverting the limits of the ARR interval (Cook and Sackett, 1995).
• When the ARR is not statistically significant, its interval includes 0 and the NNT interval passes through infinity. Teaching example: 50 and 40 events in 200 patients per group, ARR 5 points (95% CI −3.2 to 13.2), NNT 20. The interval runs from 31.5 patients treated to harm one (NNTH), through infinity, to 7.6 patients treated to benefit one (NNTB). Altman (1998) proposed the labels NNTB and NNTH and showed how to report such an interval.
• The P value does not show how large the effect is. A wide interval shows an imprecise estimate, even when P is below 0.05.

P values in general are covered in the hypothesis testing chapter.

Confounding

How do you remove the effect of a confounder from an odds ratio?

Split the data into strata of the confounder, calculate the odds ratio in each stratum, and combine them with the Mantel-Haenszel method.

Teaching example with made-up numbers: a case-control study of an exposure and a disease. Older people are more often exposed and more often have the disease, so age can distort the crude odds ratio.

OR(MH) = Σ(a × d ÷ n) ÷ Σ(b × c ÷ n)

= (3,800 ÷ 500 + 15,300 ÷ 500) ÷ (1,800 ÷ 500 + 6,300 ÷ 500)

= (7.6 + 30.6) ÷ (3.6 + 12.6) = 38.2 ÷ 16.2 = 2.36

The crude odds ratio (3.67) is larger than the odds ratio in either age group (2.11 and 2.43), because age is linked to both the exposure and the disease. The Mantel-Haenszel odds ratio, 2.36, is the age-adjusted estimate. Pooling is sensible only when the stratum odds ratios are similar. If they differ greatly, report them separately, because the effect differs between the strata (effect modification).

Stratification works for a small number of categorical confounders. With several confounders, use logistic regression, described in the Practical Statistics chapter on logistic regression, or a propensity score, described in the chapter on propensity scores. Confounding itself is explained in the bias and validity chapter.

Survival analysis

What do you do with patients whose follow-up ends before the event?

Survival analysis studies the time from a starting point to an event, such as death or relapse. Patients enter on different dates, and some leave before the event. Their time is still used.

Removing censored patients, as in the opening case, keeps every death but removes people who were alive, so survival is underestimated. Counting them as alive for the whole period overestimates it. The Kaplan-Meier method uses each patient's time up to censoring. It assumes that patients who are censored have the same survival prospects as those who continue to be followed (Goel et al., 2010).

• Right censoring: the event had not happened by the last contact. This is the usual type, and the one the Kaplan-Meier method handles.
• Left censoring: the event is known to have happened before observation began, but not when.
• Interval censoring: the event is known only to have happened between two visits, as when a complication is found at a scheduled review.

The Kaplan-Meier method

How is a Kaplan-Meier curve calculated?

At each death, work out the fraction of patients at risk who survived it. Multiply these fractions in order. The product is the probability of surviving to that time.

S(t) = S(previous death time) × (n − d) ÷ n

n

patients still at risk just before that time: not dead and not yet censored

d

deaths at that time

Teaching data, standard treatment, 15 patients. Months to death, with + for censored: 1, 2, 3+, 4, 5, 7, 8, 9+, 10, 12, 14, 16+, 18, 22+, 24+.

At month 4 only 12 patients are at risk: 2 have died and 1 was censored at month 3. A censored patient leaves the denominator but does not cause a drop in the curve.

The standard error of S(t) comes from the numbers at risk and the deaths up to that time. For standard treatment at 12 months:

SE[S(t)] = S(t) × √Σ d ÷ [n(n − d)], summed over death times up to t

SE = 0.413 × √0.1091 = 0.136 → 95% CI 0.146 to 0.680

With only 6 patients still at risk, the interval is wide. This simple interval can fall below 0 near the end of a curve: at 18 months here it would start at −0.026.

Interactive

Switch patients between died and censored, and watch the Kaplan-Meier curve

The life table method

What if you only know the interval in which each death happened?

The actuarial (life table) method groups follow-up into intervals. A patient censored during an interval is counted as at risk for half of it.

q = d ÷ (n − c ÷ 2), and S = product of (1 − q) over the intervals

n

patients alive and followed at the start of the interval

d

deaths during the interval

c

patients censored during the interval

The same 15 standard-treatment patients, in 6-month intervals:

First interval: q = 4 ÷ 14.5 = 0.276, S = 1 − 0.276 = 0.724

The two methods agree closely while many patients remain, and differ at the right-hand end, where few are at risk. The Kaplan-Meier method uses the exact times, so it is preferred when they are known. The life table method suits large studies with follow-up at fixed visits, and it is the basis of population life tables.

Comparing curves

Are the two survival curves different?

The log-rank test compares the deaths observed in each group with the deaths expected if both groups had the same survival.

At each death time, expected deaths in a group = all deaths at that time × (that group's number at risk ÷ total number at risk). Add these across every death time.

χ² = Σ (O − E)² ÷ E = 4.30² ÷ 5.70 + 4.30² ÷ 11.30 = 4.87, 1 degree of freedom, P = 0.027

• Statistical software uses a version with the exact variance: χ² = 5.12, P = 0.024. Both lead to the same conclusion here.
• The test compares the whole curves, not survival at one time point.
• It gives a P value only. It does not tell you how large the difference is. For that you need the hazard ratio.
• It assumes the hazard ratio is roughly constant over time (Goel et al., 2010). Curves that cross are a warning sign.

Survival at one fixed time, such as 1 year, can be compared between groups, but the result depends on the time chosen. The log-rank test uses every death time and gives each the same weight. The generalised Wilcoxon (Gehan) test weights each death time by the number at risk, so early deaths count for more, and the Tarone-Ware test uses the square root of the number at risk, a weight between the two.

Hazard ratio and Cox regression

What does a hazard ratio of 0.33 mean?

Hazard is the event rate at a given moment among patients who have not yet had the event. The hazard ratio compares that rate between two groups.

h(t) = h₀(t) × exp(b₁x₁ + b₂x₂ + …)

h₀(t)

baseline hazard: the hazard when every x is 0. Its shape over time is left unspecified.

x

explanatory variables: treatment group, age, stage

exp(b)

the hazard ratio for a one-unit change in x, with the other variables held constant

• HR 0.33: at any time during follow-up, patients on the new drug died at about one-third the rate of patients on standard treatment. The hazard was 67% lower.
• For a harmful event, HR below 1 favours the new treatment and HR above 1 means a higher event rate.
• Cox regression adjusts the hazard ratio for other variables, in the same way that multiple regression adjusts a difference in means. See the correlation and regression chapter.
• The model assumes proportional hazards: the hazard ratio is constant over time. Check this before relying on one hazard ratio.
• A hazard is a rate, not a probability, so it can be greater than 1. The standard-treatment group had 10 deaths in 155 person-months, 0.065 per person-month or 0.77 per person-year. A risk can never be greater than 1.
• Proportional hazards can be checked by plotting log(−log S(t)) against log time for each group, where the lines should be roughly parallel, or by testing an interaction between the variable and time in the Cox model. When the hazards are not proportional, see the Practical Statistics chapter on Cox regression.
• Rates (events ÷ person-time) are compared with Poisson regression, covered in the Practical Statistics chapter on count models.
• A real trial: in START, immediate versus deferred antiretroviral therapy, the primary end point occurred at 0.60 vs 1.38 events per 100 person-years, hazard ratio 0.43 (95% CI 0.30 to 0.62) (Lundgren et al., 2015).

Survival distributions

What shape can the hazard take over time, and why does it matter?

A constant hazard gives an exponential survival curve. A rising or a falling hazard bends the curve in different ways, even with the same median.

With a constant hazard λ, the chance of the event in the next month is the same however long a patient has already survived. Survival times then follow an exponential distribution.

S(t) = e^−λt, median = 0.693 ÷ λ, mean = 1 ÷ λ

Standard treatment: λ = 10 ÷ 155 = 0.0645 per month

Median = 0.693 ÷ 0.0645 = 10.7 months, S(12) = e^{−0.0645 × 12} = 0.461

• The Kaplan-Meier estimates for the same patients are a median of 10 months and S(12) = 0.413. The exponential model is a smooth approximation that assumes the hazard does not change.
• The exponential distribution has no memory. A patient alive at 12 months has the same 67.9% chance of surviving another 6 months as a new patient has of surviving the first 6.
• With a constant hazard, 63.2% of patients have the event before the mean survival time, so the mean (15.5 months here) is longer than the median (10.7 months).
• The Weibull distribution adds a shape parameter. The hazard rises over time when the shape is above 1, as with ageing or a progressive disease, falls when it is below 1, as when the risk is highest soon after an operation, and is constant when it equals 1, which is the exponential case (Clark et al., 2003).
• Cox regression does not assume any of these shapes. It assumes only that the ratio of the hazards stays constant. Parametric models such as exponential and Weibull regression are an option when the shape is known.

Decide

Which measure fits your study design?

Answer the questions. Each answer links to the section that explains it.

By specialty

Risk and survival problems by specialty

Common myths

Common mistaken beliefs

• ✗Myth: An odds ratio is the same as a relative risk.

  ✓They are close only when the outcome is rare. With 25% and 15% risks, OR 0.53 and RR 0.60. With 5 and 3 per 1,000, OR 0.599 and RR 0.60.

• ✗Myth: A 40% relative risk reduction means the same benefit for every patient.

  ✓The absolute benefit depends on baseline risk. At 25% baseline risk the NNT is 10. At 5% it is 50.

• ✗Myth: Patients lost to follow-up should be removed from a survival analysis.

  ✓Censor them at their last contact. The time they were observed alive still counts.

• ✗Myth: Median survival is the average survival time.

  ✓It is the time at which the Kaplan-Meier curve reaches 50%. If the curve stays above 50%, the median is "not reached".

• ✗Myth: A hazard ratio of 0.5 means patients live twice as long.

  ✓It means the event rate is halved at any moment during follow-up. How many extra months that gives depends on the baseline hazard. Read the medians and the curves for time.

• ✗Myth: The right-hand end of a Kaplan-Meier curve is as reliable as the start.

  ✓Few patients remain at risk, so each death causes a large drop. In the teaching data, 1 standard-treatment patient was at risk at 24 months.

Why this matters to you

Five consequences of choosing the wrong measure

1. A protocol that measures the wrong thing

   A one-visit survey cannot give incidence. A title that promises incidence from a cross-sectional design will be questioned by the research committee.

2. A wrong survival figure in your thesis

   Removing patients lost to follow-up, as in the opening case, underestimates survival.

3. A common viva error

   Reporting "risk" or "relative risk" from a case-control study.

4. Misreading papers

   The same 40% relative reduction means 10 fewer events per 100 patients at 25% baseline risk, and only 2 fewer at 5%.

5. Unclear advice to patients

   A patient cannot weigh a treatment from "40% fewer DVTs" alone. Give the absolute change: 25 in 100 becomes 15 in 100.

History

History of risk and survival measures

1. 1951

   Cornfield

   Published "A method of estimating comparative rates from clinical data", using case-control data on cancer of the lung, breast and cervix. The odds ratio from such data approximates the relative risk when the disease is rare.

2. 1953

   Levin

   Introduced a method of estimating the population attributable fraction from the prevalence of a risk factor and its relative risk.

3. 1958

   Edward Kaplan and Paul Meier

   Published "Nonparametric estimation from incomplete observations" in the Journal of the American Statistical Association: the method for estimating survival when some follow-up is incomplete. It is also called the product-limit estimate.

4. 1959

   Mantel and Haenszel

   Published "Statistical aspects of the analysis of data from retrospective studies of disease", with a method of combining 2 × 2 tables from separate strata. The Mantel-Haenszel method is still used for odds ratios.

5. 1966

   Mantel

   Applied the combination of 2 × 2 tables, one at each death time, to the comparison of survival data.

6. 1972

   Richard Peto and Julian Peto

   Described the logrank test as particularly appropriate for comparing life tables, including in clinical trials.

7. 1972

   D.R. Cox

   Published "Regression models and life-tables": the hazard is a function of explanatory variables multiplied by an unspecified function of time. This is the Cox proportional hazards model.

8. 1980

   Freeman and Hutchison

   Set out how prevalence, incidence and duration are related in a steady-state population, so that two of them can be used to obtain the third.

9. 1988

   Laupacis, Sackett and Roberts

   Introduced the number of patients a clinician needs to treat to prevent one adverse event: the number needed to treat.

10. 1998

    D.G. Altman

    Suggested the labels NNTB and NNTH, and showed how to report a confidence interval for the number needed to treat when it includes infinity.

11. 1998

    Zhang and Yu

    Showed that when an outcome occurs in more than 10% of a study population, the odds ratio no longer approximates the risk ratio, and proposed a correction.

Guidelines

Risk and survival in reporting guidelines

Steps

Kaplan-Meier analysis, step by step

1. Define time zero (randomisation, diagnosis or surgery) and the event (death from any cause, relapse).
2. For each patient, record the time from time zero to the event or to censoring, and whether the event happened.
3. Sort the times. At each event time, count the patients at risk and the events.
4. Calculate (n − d) ÷ n at each event time and multiply in order to get S(t).
5. Draw a step curve for each group with censoring ticks and the numbers at risk below the time axis.
6. Read the median survival and the survival at fixed times, such as 1 year, with 95% CIs.
7. Compare the groups with the log-rank test.
8. Estimate the size of the difference with a hazard ratio from Cox regression, and check that the hazards are proportional.
9. Report the number censored in each group and the length of follow-up.

Reading a paper

Five checks on any paper that reports risk or survival

• Does the measure fit the design: OR from a case-control study, RR or HR from a cohort study or trial?
• Is the risk in each group, or the ARR or NNT, given next to every relative measure?
• Are the numbers at risk shown under the Kaplan-Meier curves, and how many patients were censored in each group?
• Was follow-up long enough, and was loss to follow-up similar in the groups?
• Is the hazard ratio given with a 95% CI, and did the authors check proportional hazards (do the curves cross)?

Why is it called that?

Word surgery

Exam answers

Model answer outlines

Past papers

Every TNMGRMU question on risk and survival

Each one is answered in this chapter. Tap a sitting to open the university's own question paper.

1. ×3Odds ratioDM Critical Care Medicine · Jan 2024MCh Gynaecological Oncology · Aug 2022MCh Gynaecological Oncology · Aug 2019
2. ×3Survival analysisDM Neonatology · Aug 2023MD Community Medicine · Nov 2020MD Community Medicine · Oct 2018
3. ×2Kaplan-meier curvesMCh Surgical Oncology · Aug 2025DM Cardiology · Aug 2022
4. ×2Incidence and prevalencePh.D. RM · Jan 2018Ph.D. RM · Jul 2015
5. ×2Kaplan Meir AnalysisPh.D. RM · Jan 2017Ph.D. RM · Jan 2014
6. ×1NNT (number needed to treat)MCh Surgical Oncology · Aug 2025
7. ×1Kaplan Meier curve and Funnel plotDM Neonatology · Jan 2024
8. ×1Compare and Contrast Relative Risk and Hazards RatioMD Community Medicine · May 2023
9. ×1Describe Kaplan-Meier and Water fall plotDM Medical Oncology · Feb 2023
10. ×1Write a note on Odds Ratio and Confidence IntervalDM Neonatology · Aug 2022
11. ×1Survival analysis and Kaplan Meier curveDM Neonatology · Nov 2020
12. ×1How do you assess the Effectiveness of Anti-Retroviral Therapy for HIV Clinical trial using Kaplan - Meir Analysis?Ph.D. RM · Dec 2020
13. ×1Compare and contrast ‘Risk’ presented as ‘Incidence’ in terms of person-years and ‘Hazard rate’ through survival analysisMD Community Medicine · Oct 2019
14. ×1P Value and Odds ratioDM Cardiology · Feb 2019
15. ×1Attributable riskMD Community Medicine · Oct 2018
16. ×1Kaplan Meier AnalysisMCh Cardio Vascular and Thoracic Surgery · Aug 2015
17. ×1Odds Ratio (OR)MD Community Medicine · Feb 2005
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