O Life table is obtained by first dividing the period into a

series of time intervals.

O For example, [0,20),[20,40),[40,80).

O Flexible interval.

O Survival function:

𝑆 𝑡 = (𝑛′

𝑖− − 𝑑 𝑖

𝑛′𝑖−

)

𝑡(𝑖)≤𝑡

Assumption Censorings occur:

At the beginning of interval :

𝑛′(𝑖−) = 𝑛(𝑖−) − 𝑞(𝑖)

At the end of interval:

𝑛′(𝑖−) = 𝑛(𝑖−)

Uniform/Average throughout the interval:

𝑛′(𝑖−) = 𝑛(𝑖−) −

𝑞(𝑖)

2

Interval 𝒏(𝒊−) 𝒅(𝒊) 𝒒(𝒊) 𝒏′(𝒊−)

𝒏′(𝒊−) − 𝒅(𝒊)

𝒏′(𝒊−)

𝑺 (𝒕)

[0,10) 15 7 2 15 −2

2= 14

14 − 7

14= 0.5 0.5

[10,20) 6 3 3 6 −3

2= 4.5

4.5 − 3

4.5= 0.333 0.333

Given time: 1,1,2,3,3,5,6,7+,9+,12,14+,15+,17,19,19+

Interval 𝒏(𝒊−) 𝒅(𝒊) 𝒒(𝒊) 𝒏′(𝒊−)

𝒏′(𝒊−) − 𝒅(𝒊)

𝒏′(𝒊−)

𝑺 (𝒕)

[0,10) 15 7 2 15 − 2 = 13 13 − 7

13= 0.462 0.462

[10,20) 6 3 3 6 − 3 = 3 3 − 3

3= 0 0

Assumption: Uniform

Assumption: At the end of interval.

Hazard Function

O Assume death rate is constant during the

interval, and the average time survived in

that interval is (𝑛′(𝑖−)

−𝑑 𝑖

2)∆(𝑖).

O The life table estimate of the hazard

function is:

ℎ 𝑡 =𝑑(𝑖)

(𝑛′(𝑖−)

−𝑑 𝑖

2)∆(𝑖)

Comparison

Life Table

Estimating survival rate for each interval.

Survival curve changes in each interval

Correction of number at risk at the beginning of

interval

Kaplan-Meier

Estimating survival rate for each observation

outcome.

Survival curve changes in each observation outcome.

No correction of number at risk at the beginning of

inteval