12. Survival curves — a staircase that only goes down · Dr. Seonggu Sohn's Data Visualization, First Steps

A survival curve is an odd chart. It is a staircase, not a line, and it only goes down.

The staircase part is clear enough. Events (deaths) happen on particular days. Nothing happens in between, so the curve must be flat there. Smooth it into a curve and you have claimed that events trickled in continuously.

The downward part is clearer still. The dead do not come back.

Real TCGA lung adenocarcinoma data, split at the median TIL infiltration. Notice how close the two staircases stay.

Python Runs in your browser

import matplotlib.pyplot as plt
import viz, numpy as np, json

F = json.load(open("viz_facts.json"))
K = F["km"]

fig, ax = plt.subplots(figsize=(6.4, 4))
for i, arm in enumerate(K["arms"]):
    c = np.array(arm["curve"])
    tag = "TIL high" if i == 0 else "TIL low"
    ax.step(c[:, 0], c[:, 1], where="post", lw=2, color=viz.PAL[i],
            label="%s (n=%d, events %d)" % (tag, arm["n"], arm["events"]))
ax.set_xlim(0, 3650); ax.set_ylim(0, 1.02)
ax.set_xlabel("days", fontsize=9); ax.set_ylabel("overall survival", fontsize=9)
ax.set_title("TCGA LUAD, split at median TIL %.2f%%" % K["cut"], fontsize=9)
ax.legend(fontsize=7, frameon=False, prop={"size": 7})
viz.bare(ax)
fig.tight_layout()
print("%d patients - longest follow-up %d days" % (K["n"], K["max_t"]))

Move the cut point. Median, top 25%, top 10% — where you cut changes the picture.

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Play with the widget and an uncomfortable fact emerges: moving the cut point changes how far the two curves separate.

This is the commonest trap in survival plots. It is common practice to try several thresholds and publish the one that separates best. A p-value chosen that way is no longer a p-value — it is the best of many attempts.

The honest way

cut at a pre-specified place such as the median. Or do not cut at all — put the continuous value into a Cox model

The common way

sweep the threshold for an optimum. If you do that, say so and correct for multiple testing

Cut at the top 10% and one side has 48 patients. Each step of the staircase then drops 2%.

Python Runs in your browser

import matplotlib.pyplot as plt
import viz, numpy as np

t = viz.til()
print("Moving the threshold changes the group sizes like this")
print()
for q in [25, 50, 75, 90]:
    cut = np.percentile(t.til_pct, q)
    hi = (t.til_pct >= cut).sum()
    print("  top %2d%% at %5.2f%%  ->  high %3d / low %3d" % (100-q, cut, hi, len(t)-hi))
print()
print("Smaller groups make jumpier curves - and make chance separation easier.")

🤔 Check yourself

Why is a Kaplan–Meier curve drawn as steps rather than a smooth line?

🤔 Check yourself

Why is the right-hand end of a survival curve especially untrustworthy?

🤔 Check yourself

Several thresholds were tried and the best-separating one was published. What is wrong?