MODELLING SURVIVAL DATA

• A S I N G L E B I N A R Y C OVA R I AT E

• A S I N G L E C AT E G O R I C A L

C OVA R I AT E O R FAC TO R

COMPARISON OF TOPICS

BINARY COVARIATE CATEGORICAL COVARIATE

x ∈ {0,1} x ∈ {c0, c1, …,cK-1}

Ci0 = 1 –

β∈ R βk∈ R, for k=1,…,K−1

BINARY COVARIATE

CATEGORICAL COVARIATE

h(t,0) = h0(t)

h(t,1) = h0(t)exp(β)

h(t,c0) = h0(t)

h(t,c1) = h0(t)exp(β1)

….

h(t,cK-1) = h0(t)exp(βK-1)

Hazard Ratio = exp(β)

*group 1 relative to group 0

Hazard Ratio = exp(βk)

*when ck ≠ 0, relative to c0

Hazard Ratio = exp(βk –βj)

*when ck ≠ 0 and cj ≠ 0

EXAMPLE OF SINGLE BINARY COVARIATE

The effect of “RACE” on the effectiveness of the drug

treatment. Individuals have been classified as “white”

and “other”.

𝒇 = 𝒘𝒉𝒊𝒕𝒆 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑎𝑠 𝟎𝒐𝒕𝒉𝒆𝒓 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑎𝑠 𝟏

The fitted hazard rate is:

ℎ 𝑡, 𝑓 = ℎ0 𝑡 exp −0.29𝑓

The hazard ratio for “other” relative to “white” is:

𝑒𝛽 = 0.75

EXAMPLE OF CATEGORICAL COVARIATE OR FACTOR

The effect of drug used on reversion to drug use. Each

individual has been categorized according to heroin or

coccaine use (hard drugs), where

𝑥 = 𝑐0, 𝑐1, 𝑐2, 𝑐3

𝑐0 = 𝟎 ; 𝒅𝒐𝒆𝒔 𝒏𝒐𝒕 𝒖𝒔𝒆 𝒄𝒐𝒄𝒄𝒂𝒊𝒏𝒆 𝒐𝒓 𝒉𝒆𝒓𝒐𝒊𝒏

𝟏 ; 𝒖𝒔𝒆 𝒄𝒐𝒄𝒄𝒂𝒊𝒏𝒆 𝒂𝒏𝒅 𝒉𝒆𝒓𝒐𝒊𝒏

𝑐1 = 𝟎 ; 𝒅𝒐𝒆𝒔 𝒏𝒐𝒕 𝒖𝒔𝒆 𝒉𝒆𝒓𝒐𝒊𝒏 𝟏 ; 𝒖𝒔𝒆 𝒉𝒆𝒓𝒐𝒊𝒏

𝑐2 = 𝟎 ; 𝒅𝒐𝒆𝒔 𝒏𝒐𝒕 𝒖𝒔𝒆 𝒄𝒐𝒄𝒄𝒂𝒊𝒏𝒆 𝟏 ; 𝒖𝒔𝒆 𝒄𝒐𝒄𝒄𝒂𝒊𝒏𝒆

𝑐3 = 1 − 𝑐0 − 𝑐1 − 𝑐2

EXAMPLE OF CATEGORICAL COVARIATE OR FACTOR

The fitted hazard rate function is :

ℎ(𝑡, 𝑓) = ℎ0 𝑡 exp (0.078𝑐1 − 0.25𝑐2 − 0.16𝑐3)

that is,

ℎ 𝑡, 𝑐0 = ℎ0 𝑡 ℎ 𝑡, 𝑐1 = ℎ0 𝑡 x 1.08

ℎ 𝑡, 𝑐2 = ℎ0 𝑡 x 0.78

ℎ 𝑡, 𝑐3 = ℎ0 𝑡 x 0.85