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
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