14. The log-rank test — the gap between two curves as one number · Dr. Seok's Tumour-Infiltrating Lymphocytes, First Steps

How do you decide whether two survival curves differ? You want to measure something like "the area between the curves", but censoring makes that awkward.

The log-rank test takes a different route. At every time a death happened it asks —

"A hundred people were alive at this moment and 40 of them were in group A. Then one died. If the two groups were no different, the expected number of deaths in group A would be 0.4. Did the death actually happen in A, or in B?"

Pile up that observed − expected over every death time. The further the total is from zero, the more the two groups differ.

73 people died in the high-infiltration group, where 102.5 would have been expected. About 30 fewer deaths than chance.

Python Runs in your browser

import til
import numpy as np
from scipy import stats

p = til.patients()
g = p[p.tumor == "SKCM"].dropna(subset=["OS", "OS_time"])
hi, lo = til.split(g)

t = np.concatenate([hi.OS_time.values, lo.OS_time.values])
e = np.concatenate([hi.OS.values, lo.OS.values])
grp = np.concatenate([np.ones(len(hi)), np.zeros(len(lo))])    # 1 = high TIL

O = E = V = 0.0
for ti in np.sort(np.unique(t[e == 1])):
    n  = (t >= ti).sum()                       # risk set at that moment
    n1 = ((t >= ti) & (grp == 1)).sum()        # of which, group A
    d  = ((t == ti) & (e == 1)).sum()          # deaths at that time
    d1 = ((t == ti) & (e == 1) & (grp == 1)).sum()
    if n < 2:
        continue
    O += d1                                    # observed
    E += d * n1 / n                            # expected
    V += d * (n1/n) * (1 - n1/n) * (n - d) / (n - 1)   # variance

z = (O - E) / np.sqrt(V)
print("observed deaths (high TIL)   %6.1f" % O)
print("expected if no difference    %6.1f" % E)
print("difference                   %6.1f" % (O - E))
print("z = (O-E)/sqrt(V) = %.3f" % z)
print("two-sided p = %.2e" % (2*stats.norm.sf(abs(z))))

Three of the 12 cancer types come in under p < 0.05. Most z values are negative, but two are positive.

Python Runs in your browser

import til

p = til.patients()
rows = []
for t, g in p.dropna(subset=["OS", "OS_time"]).groupby("tumor"):
    r = til.surv_test(g)                  # split at the median, then log-rank
    if r:
        rows.append((t, r["n_hi"] + r["n_lo"], r["z"], r["p"]))

rows.sort(key=lambda x: x[3])
print("%-6s%7s%8s%12s   direction" % ("type", "n", "z", "p"))
for t, n, z, pv in rows:
    direction = "more TIL, longer" if z < 0 else "more TIL, shorter"
    star = " *" if pv < 0.05 else ""
    print("%-6s%7s%8.2f%12.2e   %s%s" % (t, format(n, ","), z, pv, direction, star))

Bar length is |z|. Green means "more infiltration, longer life"; red is the opposite.

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Read the result honestly.

SKCM (melanoma) — z = −4.38, p = 1.2×10⁻⁵. Strong. That TIL is a prognostic factor in melanoma has been known for a long time, and this data says so too.
CESC and BRCA — p of 0.004 and 0.017, in the same direction.
The other nine — p between 0.2 and 0.87. They say nothing.
PRAD, PAAD and UCEC — z is positive. The direction is reversed.

UVM (uveal melanoma) is not in the table at all. More than half its infiltration rates are zero, so it cannot be split in two at the median. Some data cannot be cut.

We must not stop here. We just ran 12 tests. What "three are significant" actually means has to be settled in lesson 16.

🤔 Check yourself

Under what assumption is the 'expected number of deaths' computed in a log-rank test?

🤔 Check yourself

How do you correctly read z = −4.38 (with high infiltration as group 1)?