Kaplan-Meier Estimation &Log-Rank Test

Survival of Ventilated and Control Flies (Old Falmouth Line 107)

R.Pearl and S.L. Parker (1922). “Experimental Studies on the Duration of Life. V. On the Influence of Certain Environmental Factors on Duration of Life in Drosophila,” The American Naturalist, Vol. 56, #646, pp. 385-405

Data Description

• 946 Flies are Venilated on Day 1• 931 Flies are not Ventilated on Day 1 (Controls)• Counts of Survivors and Non-Survivors

Reported every 6 Days Subsequently (Days 7,13,…,85)

• Goal 1: Estimate the Survival Function for the two groups: S(t) = P(T > t) where T is the (random) Survival time of a fly

• Goal 2: Test whether the (population) Survival Functions differ for the 2 Groups

DataDays VentLive VentDie CntlLive CntlDie

1 946 0 931 07 933 13 916 15

13 918 15 902 1419 895 23 865 3725 851 44 788 7731 761 90 680 10837 659 102 573 10743 557 102 452 12149 454 103 370 8255 351 103 272 9861 263 88 182 9067 175 88 111 7173 97 78 41 7079 11 86 2 3985 0 11 0 2

VentDie and CntlDie represent numbers dying in period just prior to this day

Kaplan-Meier Estimation - Notation

• For each fly we observe the pair (ti,i) where ti is the ith fly’s time to death or censoring and i is a censoring indicator (1=censored, 0=not censored (actual death))

• Observed failure times: t(1) < … <t(k)

– Number of failures at t(i) di

– Subjects censored at t(i) treated as if censored between t(i) and t(i+1). Number censored in [t(i),t(i+1)) mi

– Subjects at risk Prior to t(i) ni = (di+mi)+…+(dk+mk) (this removes all who have died or censor prior to t(i))

• Note: For this dataset, there is no censoring (all flies were observed to die)

Kaplan-Meier Estimates

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Kaplan-Meier Estimates Days VentLive VentDie CntlLive CntlDie Vlambda Clambda Vsurv Csurv

1 946 0 931 0 0.0000 0.0000 1.0000 1.00007 933 13 916 15 0.0137 0.0161 0.9863 0.9839

13 918 15 902 14 0.0161 0.0153 0.9704 0.968919 895 23 865 37 0.0251 0.0410 0.9461 0.929125 851 44 788 77 0.0492 0.0890 0.8996 0.846431 761 90 680 108 0.1058 0.1371 0.8044 0.730437 659 102 573 107 0.1340 0.1574 0.6966 0.615543 557 102 452 121 0.1548 0.2112 0.5888 0.485549 454 103 370 82 0.1849 0.1814 0.4799 0.397455 351 103 272 98 0.2269 0.2649 0.3710 0.292261 263 88 182 90 0.2507 0.3309 0.2780 0.195567 175 88 111 71 0.3346 0.3901 0.1850 0.119273 97 78 41 70 0.4457 0.6306 0.1025 0.044079 11 86 2 39 0.8866 0.9512 0.0116 0.002185 0 11 0 2 1.0000 1.0000 0.0000 0.0000

Note:

For the ventilated flies, 13 died at day 7 (actually between days 1 and 7), out of 946 that were at risk prior to that period: 13/946=.0137. At day 13, 15/933=.0161 was the proportion dying

The Survival function at day 7 is obtained as (1-.0137)=.9863. At day 13, the survival function is (1-.0137)(1-.0161) = .9704

Kaplan Meier Survival Estimates - Ventilated and Control Flies

0

0.1

0.2

0.3

0.4

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0.8

0.9

1

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85

Days

P(S

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Vsurv

Csurv

Log-Rank Test• Used to test whether two (or more) survival

functions are equal• Involves obtaining the “expected” number of

deaths in (say) the treatment group at time t(i) if the hazard functions for the two groups were equal: – Let n1i and n2i be the numbers at risk just prior to t(i) for the 2

groups, with total at risk ni = n1i + n2i

– Let d1i and d2i be the numbers dying at t(i) for the 2 groups, with total deaths di = d1i + d2i

– Then the expected deaths for group 1 is: e1i=di(n1i/ni) which represents the total deaths at t(i) times the fraction of the total at risk that are in group 1

Log-Rank Test

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Log-Rank Test (Fly Data)

17.23

45.827

45.827931

55.1049

55.1049946

47.31)61.5(61.586.340

55.1049946

45.82755.1049931946931

86.34055.1049946

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Days VentLive VentDie CntlLive CntlDie e1 v1 d1-e11 946 0 931 0 0.00 0.00 0.007 933 13 916 15 14.11 6.90 -1.11

13 918 15 902 14 14.63 7.14 0.3719 895 23 865 37 30.26 14.51 -7.2625 851 44 788 77 61.53 28.18 -17.5331 761 90 680 108 102.81 43.48 -12.8137 659 102 573 107 110.37 44.56 -8.3743 557 102 452 121 119.28 45.47 -17.2849 454 103 370 82 102.13 37.40 0.8755 351 103 272 98 110.75 37.64 -7.7561 263 88 182 90 100.29 31.32 -12.2967 175 88 111 71 93.97 24.76 -5.9773 97 78 41 70 90.56 17.02 -12.5679 11 86 2 39 87.86 2.48 -1.8685 0 11 0 2 11.00 0.00 0.00

Sum #N/A 946 #N/A 931 1049.55 340.86 -103.55

Both tests are highly significant