Survival Curve (Julia)

Authors

[Editor] Shixiang Wang;

[Contributors] Claude Code — AI-generated, independently AI-reviewed;

Modified

2026-10-19

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The Kaplan–Meier estimator reads out survival probability over time from right-censored follow-up data. There is no maintained Kaplan–Meier package in the Julia ecosystem (registry-checked 2026-10), so this tutorial implements the estimator and its Greenwood confidence band directly — twenty lines of base Julia — and renders the step curves with stairs!. The mechanics ARE the tutorial.

Example

Setup

  • System Requirements: Cross-platform (Linux/MacOS/Windows)
  • Programming Language: Julia
  • Dependencies: CairoMakie, DataFrames, Statistics 
using CairoMakie
using DataFrames
using Statistics
using Random

Data Preparation

We simulate 60 patients per arm (Control, Treatment) with exponential survival — median 12 vs 24 months — and uniform administrative censoring between 10 and 72 months.

function simulate_arm(rng, median; n = 60)
    true_time = -median / log(2) .* log.(rand(rng, n))
    censor = 10 .+ 62 .* rand(rng, n)
    DataFrame(Months = min.(true_time, censor),
              Event = true_time .<= censor)
end
rng = Xoshiro(141)
control = simulate_arm(rng, 12)
treat = simulate_arm(rng, 24)
control.Arm = fill("Control", nrow(control))
treat.Arm = fill("Treatment", nrow(treat))
surv = vcat(control, treat)
120×3 DataFrame
95 rows omitted
Row Months Event Arm
Float64 Bool String
1 30.2539 true Control
2 40.2043 false Control
3 10.2504 false Control
4 17.2625 true Control
5 14.8126 true Control
6 26.9555 true Control
7 4.53743 true Control
8 12.991 true Control
9 26.3117 true Control
10 23.9808 true Control
11 2.33873 true Control
12 9.1478 true Control
13 46.349 true Control
⋮ ⋮ ⋮ ⋮
109 9.95473 true Treatment
110 6.26144 true Treatment
111 18.4051 true Treatment
112 33.0882 true Treatment
113 35.1187 false Treatment
114 2.9565 true Treatment
115 14.5833 false Treatment
116 27.706 true Treatment
117 7.00252 true Treatment
118 10.8413 true Treatment
119 51.6202 false Treatment
120 26.747 true Treatment

Visualization

Kaplan–Meier Estimator, Step by Step

At each distinct event time the survival multiplies by the at-risk-adjusted survival fraction; stairs! renders the characteristic step curve.

function km_fit(times, events)
    order = sortperm(times)
    t = times[order]; e = events[order]
    n = length(t)
    s = 1.0
    ts = Float64[0.0]; ss = Float64[1.0]
    for tt in sort(unique(t))
        d = count(j -> t[j] == tt && e[j], 1:n)
        atrisk = count(j -> t[j] >= tt, 1:n)
        if d > 0
            s *= 1 - d / atrisk
            push!(ts, tt); push!(ss, s)
        end
    end
    (ts, ss)
end

arm_styles = Dict("Control" => :steelblue, "Treatment" => :tomato)
fig = Figure(size = (460, 360))
ax = Axis(fig[1, 1], xlabel = "Months since treatment",
          ylabel = "Survival probability")
for arm in ["Control", "Treatment"]
    sub = surv[surv.Arm .== arm, :]
    ts, ss = km_fit(sub.Months, sub.Event)
    stairs!(ax, ts, ss; step = :post, color = arm_styles[arm],
            linewidth = 1.8, label = "$arm (n=$(nrow(sub)))")
end
hlines!(ax, [0.5]; color = :gray60, linestyle = :dot, linewidth = 0.8)
axislegend(ax; framevisible = false, position = :lb)
ylims!(ax, 0, 1.02)
hidespines!(ax, :t, :r)
fig
Figure 1: Kaplan–Meier overall survival for the two arms

Greenwood Confidence Band

The Greenwood variance S² Σ dᵢ / (nᵢ(nᵢ − dᵢ)) gives pointwise 95% bands; plotting the band behind the step curve completes the publication layout.

function km_fit_ci(times, events)
    order = sortperm(times)
    t = times[order]; e = events[order]
    n = length(t)
    s = 1.0; cum = 0.0
    ts = Float64[0.0]; ss = Float64[1.0]; lo = Float64[1.0]; hi = Float64[1.0]
    for tt in sort(unique(t))
        d = count(j -> t[j] == tt && e[j], 1:n)
        atrisk = count(j -> t[j] >= tt, 1:n)
        if d > 0
            s *= 1 - d / atrisk
            cum += d / (atrisk * (atrisk - d))
            se = s * sqrt(cum)
            push!(ts, tt); push!(ss, s)
            push!(lo, max(s - 1.96 * se, 0.0)); push!(hi, min(s + 1.96 * se, 1.0))
        end
    end
    (ts, ss, lo, hi)
end

sub = surv[surv.Arm .== "Control", :]
ts, ss, lo, hi = km_fit_ci(sub.Months, sub.Event)
# Band has no step attribute: repeat points to build the step outline
xs_step = Float64[]; lo_step = Float64[]; hi_step = Float64[]
for i in eachindex(ts)
    if i > 1
        push!(xs_step, ts[i]); push!(lo_step, lo[i-1]); push!(hi_step, hi[i-1])
    end
    push!(xs_step, ts[i]); push!(lo_step, lo[i]); push!(hi_step, hi[i])
end
fig = Figure(size = (460, 360))
ax = Axis(fig[1, 1], xlabel = "Months since treatment",
          ylabel = "Survival probability")
band!(ax, xs_step, lo_step, hi_step; color = (:steelblue, 0.15))
stairs!(ax, ts, ss; step = :post, color = :steelblue, linewidth = 1.8)
hlines!(ax, [0.5]; color = :gray60, linestyle = :dot, linewidth = 0.8)
ylims!(ax, 0, 1.02)
hidespines!(ax, :t, :r)
fig
Figure 2: Kaplan–Meier curve with the Greenwood 95% confidence band

References

  1. Kaplan, E. L., & Meier, P. (1958). Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53(282), 457-481.
  2. Borgan, Ø., & Liestøl, K. (1990). A Note on Confidence Intervals and Bands for the Kaplan-Meier Estimator. Scandinavian Journal of Statistics, 17(1), 35-41.