Kaplan-Meier curve, median survival and the log-rank test

https://udgca1190.com.mx/en/herramientas/bioestadistica/kaplan-meier

Inputs

Follow-up time (one column)

30 values read

Copy the follow-up time column from your spreadsheet: one value per row, in whatever units you use (days, months, years). It is the time to the event for whoever had it and the time to the last observation for whoever did not.

Event (a column of 0s and 1s)

30 values read

A column with as many rows as the time column: 1 if the patient had the event and 0 if they were censored (follow-up closed or lost without the event occurring).

Group (one column, optional)

30 values read

Optional. A column of numeric codes with as many rows as the other two: without it a single curve is drawn; with it, one curve per group and the log-rank test. At most two groups are supported for now.

Confidence interval scale

Scale on which the Greenwood interval is built. The log-log scale (the default here) keeps the limits between 0 and 1 and behaves better in the tails; the logarithmic scale is R's own default.

First time of interest

Optional. The time at which you want to read survival, in the same units as the time column (7 days, for instance). Leave it empty or at 0 if you do not want to read any time. Beyond the last observed time of a group, S(t) is undefined.

Second time of interest

Optional. A second reading time, equal to or later than the first; empty or 0 if you do not need it.

Confidence level

Example loaded

Illustrative example: 30 patients with dengue followed to defervescence (the event), 15 with a negative NS1 antigen (group 0) and 15 with a positive one (group 1), with six censored losses to follow-up; times are in days (fictitious data, not real). Careful when reading the curve: here the event is a good outcome, so S(t) is the proportion of patients still febrile and a fast drop is good news.

Illustrative data, not real.

Results

Patients in the first curve

Events in the first curve

Median survival (first curve)

3 to 9

95% CI · quantile.survfit rule

Survival at the first time (first curve)

38.9%

15.3% to 62.2%

95% CI · Greenwood

Survival at the second time (first curve)

11.7%

0.9% to 37.6%

95% CI · Greenwood

Patients in the second curve

Events in the second curve

Median survival (second curve)

7 to 18

95% CI · quantile.survfit rule

Survival at the first time (second curve)

80.0%

50.0% to 93.1%

95% CI · Greenwood

Survival at the second time (second curve)

44.0%

18.5% to 67.1%

95% CI · Greenwood

Log-rank (χ²)

5.174

log-rank, rho = 0

Degrees of freedom

log-rank, rho = 0

Log-rank (p)

0.023

log-rank, rho = 0

Interpretation

Two curves were compared: in group 0 the number of patients was 15 and the number of events 12; in group 1, 15 and 12. At time 7, the estimated probability of still being event-free was 38.9% (95% CI 15.3% to 62.2%) in group 0 and 80.0% (95% CI 50.0% to 93.1%) in group 1. At time 14, the estimated probability of still being event-free was 11.7% (95% CI 0.9% to 37.6%) in group 0 and 44.0% (95% CI 18.5% to 67.1%) in group 1.

Median survival: 6 (95% CI 3 to 9) in group 0 and 14 (95% CI 7 to 18) in group 1. It is the time at which each curve crosses 50%; when a curve ends above 0.5 the median is not reached and must be reported that way, never as the last observed time.

Log-rank test: χ² = 5.174 with 1 degree of freedom, p = 0.023. At α = 0.05 the null hypothesis that the two curves are identical is rejected. The p value says the observed difference would be rare if there were none, but not how large it is: that is what the medians, the survival at the times of interest and the gap between the curves say.

Follow-up was censored in 6 of 30 patients (20.0%); the split by curve was group 0: 3/15 · group 1: 3/15. Censoring is informative only if it is unrelated to prognosis: loss to follow-up concentrated among the patients who were doing worse biases the curve upwards, and the manuscript should say why follow-up was lost.

The curves do not cross within the common follow-up, which is the situation in which the log-rank test and a single hazard ratio can be read without reservations.

Kaplan-Meier curves by group with their confidence band, censoring marks and the numbers-at-risk table

Explanation

Survival analysis answers a question a proportion cannot: how long something takes to happen. The "event" need not be death; it can be defervescence, discharge, relapse, seroconversion or catheter failure. What makes these data special is censoring: when the study closes, some patients have not had the event yet, and others were lost before having it. Those patients are neither failures nor successes: they contribute the information that they were event-free up to the day we stopped seeing them, and dropping them would bias the result.

The Kaplan-Meier estimator (1958) uses that information. At each time with at least one event it computes the conditional probability of getting through it event-free, 1 − dᵢ/nᵢ, where nᵢ is the number still at risk just before that moment, and multiplies those probabilities one after another. The result is the step curve: it drops only when events occur and stays flat when there are only censorings, even though every censoring shrinks the risk set and makes later steps larger. That is why the numbers-at-risk table under the plot is not decoration: the tail of the curve, where two or three people are left, is the least reliable part and the most misleading at a glance.

The width of the confidence band comes from Greenwood's formula (1926), which accumulates the uncertainty of every step. It is computed on the log scale and then carried back to the survival scale: by default with the log-log transformation of Kalbfleisch and Prentice, which keeps the limits between 0 and 1 and behaves better in the tails, and alternatively on the log scale, which is R's own default and the one many textbooks show. It is the same difference that in R separates `conf.type = "log-log"` from `conf.type = "log"`, which is why the selector travels into the code.

Median survival is the time at which the curve crosses 50%. It is preferred over the mean because it does not depend on the censored tail, and it may not exist: if the curve ends above 0.5, the median is "not reached" and must be reported that way, never as the last observed time. Its confidence interval follows Brookmeyer and Crowley (1982): the same rule is applied to the lower and upper bands, so the lower limit comes from the band that falls fastest. When the curve sits at exactly 0.5 over a stretch, the median is the midpoint of that stretch; that is the rule of `quantile.survfit`, and this calculator reproduces it step by step.

The log-rank test (Mantel 1966; Peto and Peto 1972) compares two whole curves, not two percentages at one moment. At each event time it counts how many events would have been expected in one group if the two curves were identical, accumulates the difference between observed and expected, and turns it into a χ² with one degree of freedom. It comes with an important reading condition: it assumes the difference between groups points the same way throughout. If the curves cross, the log-rank loses power and a single hazard ratio stops summarising what happens; in that case compare survival at specific times and say so in the manuscript.

Equations

\hat S(t)=\prod_{t_i\le t}\left(1-\frac{d_i}{n_i}\right)

$d_i$: events occurring exactly at time t_i
$n_i$: patients at risk just before t_i (those with time greater than or equal to t_i)
$\hat S(t)$: estimated probability of still being event-free at t

The product-limit estimator of Kaplan and Meier (1958). When an event and a censoring tie, the event comes first: both are counted at the same time and the risk set includes both.

\var\left[\ln\hat S(t)\right]=g(t)=\sum_{t_i\le t}\frac{d_i}{n_i\,(n_i-d_i)},\qquad \se\left[\ln\hat S(t)\right]=\sqrt{g(t)}

$g(t)$: Greenwood sum accumulated up to t

Greenwood's formula (1926). The standard error is that of ln S(t), not of S(t): it is what sf$std.err returns in the R code on this page, and the interval is built on that scale before being carried back to the survival scale.

\CI_{1-\alpha}\left[\hat S(t)\right]=\left[\hat S(t)^{\exp(z\,\se_{\ell})},\;\hat S(t)^{\exp(-z\,\se_{\ell})}\right],\qquad \se_{\ell}=\frac{\sqrt{g(t)}}{\left|\ln\hat S(t)\right|}

$z$: normal quantile of the chosen confidence level
$\se_{\ell}$: standard error on the log-log scale

The log-log interval of Kalbfleisch and Prentice, the one behind conf.type = "log-log" and this calculator's default: the limits always stay within 0 and 1 without any clipping. It is undefined when S(t) = 0 or S(t) = 1.

\CI_{1-\alpha}\left[\hat S(t)\right]=\left[\hat S(t)\,e^{-z\sqrt{g(t)}},\;\min\left(1,\;\hat S(t)\,e^{z\sqrt{g(t)}}\right)\right]

$\sqrt{g(t)}$: standard error of ln S(t), the square root of the Greenwood sum

The interval on the logarithmic scale, the one behind conf.type = "log" and R's own default. Here the upper limit can exceed 1 and has to be clipped; when S(t) = 0 both limits are undefined.

m=\frac{t_{1}+t_{2}}{2},\qquad t_{1}=\min\left\{t:\hat S(t)\le\tfrac{1}{2}\right\},\qquad t_{2}=\min\left\{t:\hat S(t)<\tfrac{1}{2}\right\}

$m$: median survival
$t_{1}$: first time the curve reaches 50%
$t_{2}$: first time the curve drops below 50%

This is the rule of quantile.survfit, not simply "the first time below 0.5". When the curve crosses 50% in one step, t₁ and t₂ are the same time and the median is that time; when it sits at exactly 0.5 over a stretch, the median is the midpoint of that stretch. If the curve ends at 0.5 without dropping below it, the last observed time takes the place of t₂, and if it never reaches 50% the median is undefined. The interval applies the same rule to the lower and upper bands (Brookmeyer and Crowley 1982).

\chi^{2}_{\mathrm{LR}}=\frac{(O_1-E_1)^{2}}{V},\qquad E_1=\sum_i\frac{d_i\,n_{1i}}{n_i},\qquad V=\sum_i\frac{n_{1i}\,n_{0i}\,d_i\,(n_i-d_i)}{n_i^{2}\,(n_i-1)}

$O_1$: events observed in the first group
$E_1$: events expected in the first group if the two curves were identical
$n_{1i}$: patients of the first group at risk just before t_i

The log-rank test with weight rho = 0 (Mantel-Haenszel), the one survdiff runs. With two groups it has one degree of freedom.

R code

# Kaplan-Meier survival curves and the log-rank test - Bioestadistica abierta, UDG-CA-1190
# Runs as is in R, RStudio or webR; prints the results as JSON at the end.
library(survival)
library(jsonlite)

# The three pasted columns, one row per patient:
tiempo <- c(2, 5, 3, 7, 3, 7, 4, 9, 4, 10, 5, 11, 5, 12, 6, 13, 6, 14, 7, 16, 8, 16,
  9, 18, 12, 19, 14, 20, 18, 21)
evento <- c(1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1,
  0, 1, 1, 0, 0, 0)
grupo <- c(0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1)
nivel <- 0.95
tipo_ic <- "log-log"
t1 <- 7   # first time of interest; 0 = none asked for
t2 <- 14   # second time of interest; 0 = none asked for

if (length(grupo) == 0) grupo <- rep(0, length(tiempo))   # no group column: a single curve
codigos <- sort(unique(grupo))                            # group 1 is the lowest numeric code
d <- data.frame(tiempo, evento, grupo = factor(grupo, levels = codigos))
etiquetas <- levels(d$grupo)
k <- length(etiquetas)

# conf.type is explicit: R defaults to "log", papers usually report "log-log"
sf <- survfit(Surv(tiempo, evento) ~ grupo, data = d, conf.type = tipo_ic, conf.int = nivel)

# survfit drops $strata when the grouping factor has a single level
filas <- if (is.null(sf$strata)) list(seq_along(sf$time)) else split(seq_along(sf$time), rep(1:k, sf$strata))

# Median survival with its CI (Brookmeyer-Crowley; the rule of quantile.survfit:
# first time with S < 0.5, midpoint of the plateau when S is exactly 0.5)
q <- quantile(sf, 0.5)
mediana <- function(g) {
  if (g > k) return(c(NA_real_, NA_real_, NA_real_))
  if (k == 1) c(q$quantile[[1]], q$lower[[1]], q$upper[[1]]) else c(q$quantile[g, 1], q$lower[g, 1], q$upper[g, 1])
}

# S(t*) with its CI. t = 0 means no time of interest was asked for (S(0) = 1
# is trivial), and past the last observed time of the group S(t) is not
# defined, although extend = TRUE would prolong the curve
s_en <- function(g, t) {
  if (g > k || t <= 0 || t > max(d$tiempo[d$grupo == etiquetas[g]])) return(c(NA_real_, NA_real_, NA_real_))
  s <- summary(sf, times = t, extend = TRUE)
  j <- if (is.null(s$strata)) 1 else which(as.integer(s$strata) == g)
  c(s$surv[j], s$lower[j], s$upper[j])
}

# Log-rank test (rho = 0, Mantel-Haenszel). It needs two groups, and it is NOT
# defined when the variance of the statistic is 0: survdiff then either stops
# with a singular system (every event tied at one time, or no group at risk
# when the events happen) or reports "on 0 degrees of freedom" with chisq = 0.
# In both situations chi2, the degrees of freedom and p are NA, not 0 and 1.
chi2 <- NA_real_; gl <- NA_real_; p <- NA_real_
if (k == 2) {
  lr <- tryCatch(survdiff(Surv(tiempo, evento) ~ grupo, data = d, rho = 0), error = function(err) NULL)
  if (!is.null(lr) && lr$var[1, 1] > 0) {
    chi2 <- lr$chisq
    gl <- length(lr$n) - 1
    p <- pchisq(chi2, gl, lower.tail = FALSE)
  }
}

n_de <- function(g) if (g > k) NA_real_ else sum(d$grupo == etiquetas[g])
ev_de <- function(g) if (g > k) NA_real_ else sum(d$evento[d$grupo == etiquetas[g]])
# Life table of one group, as survfit reports it (std.err is the SE of log S)
col <- function(g, campo) if (g > k) numeric(0) else sf[[campo]][filas[[g]]]

res <- list(n_1 = n_de(1), eventos_1 = ev_de(1), mediana_1 = mediana(1),
            s_t1_1 = s_en(1, t1), s_t2_1 = s_en(1, t2),
            n_2 = n_de(2), eventos_2 = ev_de(2), mediana_2 = mediana(2),
            s_t1_2 = s_en(2, t1), s_t2_2 = s_en(2, t2),
            chi2_logrank = chi2, gl = gl, p_logrank = p,
            t_1 = I(col(1, "time")), riesgo_1 = I(col(1, "n.risk")), ev_1 = I(col(1, "n.event")),
            cens_1 = I(col(1, "n.censor")), s_1 = I(col(1, "surv")), ee_1 = I(col(1, "std.err")),
            lo_1 = I(col(1, "lower")), hi_1 = I(col(1, "upper")),
            t_2 = I(col(2, "time")), riesgo_2 = I(col(2, "n.risk")), ev_2 = I(col(2, "n.event")),
            cens_2 = I(col(2, "n.censor")), s_2 = I(col(2, "surv")), ee_2 = I(col(2, "std.err")),
            lo_2 = I(col(2, "lower")), hi_2 = I(col(2, "upper")))
cat(toJSON(res, auto_unbox = TRUE, digits = NA))

# Equivalent in RStudio: survminer::ggsurvplot(sf, risk.table = TRUE)

This is the very code that validates the calculator: copy it and run it in R or RStudio to reproduce the result.

In-browser verification with R will arrive in a forthcoming version; meanwhile, copy the code and run it in RStudio.

Methods for a manuscript

Survival functions were estimated with the Kaplan-Meier method [1] with 95% CIs based on Greenwood's formula [2] on the log-log of Kalbfleisch and Prentice [6] scale; the median and its CI follow Brookmeyer and Crowley [5] and the curves were compared with the log-rank test [3,4]. Calculations used the "Kaplan-Meier and log-rank" calculator of Bioestadística abierta (Research Group UDG-CA-1190, https://udgca1190.com.mx/en/herramientas/bioestadistica/kaplan-meier), verified against R's survival package [7] (survfit, quantile.survfit and survdiff).

A paragraph ready for the Methods section; bracketed numbers refer to the reference list.

References

01 Kaplan EL, Meier P. Nonparametric estimation from incomplete observations. Journal of the American Statistical Association. 1958;53(282):457–481. doi:10.1080/01621459.1958.10501452 Original source
02 Greenwood M. The natural duration of cancer. Reports on Public Health and Medical Subjects No. 33. London: His Majesty's Stationery Office; 1926. Original source
03 Mantel N. Evaluation of survival data and two new rank order statistics arising in its consideration. Cancer Chemotherapy Reports. 1966;50(3):163–170. PMID: 5910392 Original source
04 Peto R, Peto J. Asymptotically efficient rank invariant test procedures. Journal of the Royal Statistical Society. Series A (General). 1972;135(2):185–207. doi:10.2307/2344317 Original source
05 Brookmeyer R, Crowley J. A confidence interval for the median survival time. Biometrics. 1982;38(1):29–41. doi:10.2307/2530286 Original source
06 Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data. 2nd ed. Hoboken, NJ: John Wiley & Sons; 2002. doi:10.1002/9781118032985 Complementary
07 Therneau TM. A Package for Survival Analysis in R. R package survival version 3.8-6. CRAN; 2026. Complementary
08 Bland JM, Altman DG. Survival probabilities (the Kaplan-Meier method). BMJ. 1998;317(7172):1572–1580. doi:10.1136/bmj.317.7172.1572 PMID: 9836663 Didactic reading
09 Bland JM, Altman DG. The logrank test. BMJ. 2004;328(7447):1073. doi:10.1136/bmj.328.7447.1073 PMID: 15117797 Didactic reading