13. Survival curves — drawing the staircase by hand · Dr. Seok's Tumour-Infiltrating Lymphocytes, First Steps

Now for the question you most want to ask. Do patients with more infiltration live longer?

First, something peculiar about survival data. Every patient carries two values —

OS_time — how long they were observed, in days
OS — what happened at the end of that. 1 means died, 0 means still alive

A patient with OS = 0 is censored. It means "they were alive at day 1,200 and after that we do not know". How you count those people is very nearly the whole of survival analysis.

The computation is startlingly simple. At every time t where a death occurred —

S(t) = S(previous) × (1 − deaths at t ÷ people still at risk at t)

A censored patient never steps the curve down; they simply drop quietly out of the denominator from then on.

One step per death time. This function is the same thing as til.km_curve().

Python Runs in your browser

import til
import numpy as np

def km(time, event):
    """Kaplan-Meier - ten lines is enough."""
    s, out = 1.0, [(0.0, 1.0)]
    for t in np.sort(np.unique(time[event == 1])):     # at each death time
        n = (time >= t).sum()                          # people still at risk
        d = ((time == t) & (event == 1)).sum()         # deaths at that time
        s *= 1 - d / n
        out.append((float(t), float(s)))
    return out

p = til.patients()
g = p[(p.tumor == "SKCM")].dropna(subset=["OS", "OS_time"])
curve = km(g.OS_time.values, g.OS.values)

print("melanoma: %d patients, %d deaths, %d censored"
      % (len(g), int(g.OS.sum()), int((g.OS == 0).sum())))
print("steps in the curve: %d" % (len(curve) - 1))
print()
for t, s in curve[::40][:8]:
    print("  %7.0f days  survival %5.1f%%" % (t, 100*s))

In melanoma the two curves part company visibly — 3,564 days against 1,618.

Python Runs in your browser

import til
import numpy as np
import matplotlib.pyplot as plt

p = til.patients()
g = p[p.tumor == "SKCM"].dropna(subset=["OS", "OS_time"])
hi, lo = til.split(g)          # split at the median infiltration rate

fig, ax = plt.subplots(figsize=(8, 4.4))
for grp, col, name in ((hi, "#ef4444", "high TIL"), (lo, "#3b82f6", "low TIL")):
    c = np.array(til.km_curve(grp.OS_time.values, grp.OS.values))
    ax.step(c[:, 0], c[:, 1], where="post", color=col, lw=2, label="%s (n=%d)" % (name, len(grp)))
ax.axhline(0.5, color="gray", ls="--", lw=1)
ax.set_xlim(0, 6000); ax.set_ylim(0, 1)
ax.set_xlabel("days"); ax.set_ylabel("survival"); ax.legend()
plt.tight_layout()

print("median survival, high TIL  %s days" % format(til.km_median(hi.OS_time.values, hi.OS.values), ",.0f"))
print("median survival, low TIL   %s days" % format(til.km_median(lo.OS_time.values, lo.OS.values), ",.0f"))

Switch cancer types. Some have curves that separate, others have curves that overlap.

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The melanoma result is striking. Median survival is 3,564 days (9.8 years) in the high-infiltration group and 1,618 days (4.4 years) in the low one — more than double.

But wait. Because a picture looks separated, is it? Human eyes are very good at manufacturing patterns that are not there. We need a number for how likely this difference is by chance. That is lesson 14.

🤔 Check yourself

Why must censored patients (OS = 0) not simply be dropped?

🤔 Check yourself

When does a Kaplan–Meier curve step down?