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HAZARD-BASED NONPARAMETRIC
SURVIVOR FUNCTION ESTIMATION
Ross L. Prentice
Seattle, WA, U.S.A.
Overview
- Joint work with Zoe Moodie and Jianrong Wu
- NP Estimation of F
where F (t 1
, t 2
) = P (T
1
t 1
, T
2
t 2
F = Φ(
F
1
F
2
where Λ(dt 1
, dt 2
) = F (dt 1
, dt 2
)/F (t
−
1
, t
−
2
- Hazard-based survivor function representation F = Φ(Λ) for
truncated T 1
and T 2
, (t 1
, t 2
)ε[0, τ 1
) × [0, τ 2
F = Φ(
Λ) readily obtained using a simple matrix calcu-
lation
E
and
V DL
special cases in simulation studies
- Comment on further research possibilities
Introduction
• T = (T
1
, T
2
) and let s = (s 1
, s 2
) < t = (t 1
, t 2
) ⇔ s 1
< t 1
and
s 2
< t 2
- Survivor function F defined by F (t) = P (T > t)
- Hazard function Λ defined by Λ(dt) = F (dt)/F (t
− )
• F = Φ(F
1
, F
2
F (ds) = F (s
−
)Λ(ds)
F (t) = Ψ(t) +
8 t
0
F (s
−
)Λ(ds) ∗
where Ψ(t) = F 1
(t 1
) + F
2
(t 2
) − 1 and F i
(t i
) = P (T
i
t i
i = 1, 2.
— ∗ is an inhomogeneous Volterra equation having a unique
Peano series solution
F (t) = Ψ(t) +
8 t
0
Ψ(s
−
)
P {(s, t]; Λ}Λ(ds)
where
P {(s, t]; Λ} = 1 +
∞ 3
m=
8
s<u 1
<···<um≤t
m
j=
Λ(du j
Representation F = Φ(Λ)
Since F (0) = 1 = F (τ
−
)
P {( 0 , τ ); α}
F (t
−
) = { 1 − Λ(∆t)}
− 1 ˜ P {(t, τ ); α}/
P {( 0 , τ ), α}
Notes:
lim
s↓t
F (s
−
)
− )Λ(dt) = α(dt)
P {(t, τ ), α}/
P {( 0 , τ ); α}
P is compact differentiable (Gill and Johansen, 1990, AS), as is
Φ, so that
F = Φ(
Λ) inherits such properties as strong consis-
tency, weak convergence to a Gaussian process, and bootstrap
applicability from
Calculation of ˆF = Φ(
u 11
< u 12
< · · · < u 1 I
= τ 1
uncensored T 1
values
u 21
< u 22
< · · · < u 2 J
= τ 2
uncensored T 2
values
Λ(∆t) estimators take positive values on a sub-
set of the grid formed by these uncensored failure times, zero
elsewhere
- Video order the points having positive hazard, starting from
bottom left and moving across rows. Denote these points by
t f
= (t 1 f
, t 2 f
), f = 1,... , s and let
λ f
denote the hazard rate
at t f
. Note that t s
= (τ 1
, τ 2
) and
λ s
= 1, whereas
λ f
< 1 for
f < s
F (∆t f
) the mass assignment at t f
, f = 1,... , s
λ f
= ˆp f
F
f
(t
−
f
) = ˆp f
3
{t 1 i
<t 1 f
or t 2 i
<t 2 f
}
p ˆ i
}, f = 1,... , s
Apˆ = 1
where ˆp
I
= (p 1
,... , p s
), 1 = (1 · · · 1) and
A =
A(
Λ) = (a fm
a fm
λ
− 1
f
, if f = m;
1 , if t 1 f
t 1 m
or t 2 f
t 2 m
0 , otherwise.
Calculation of ˆF = Φ(
F (t) =
3
{f|t 1 f
t 1
and t 2 f
t 2
}
p ˆ f
provides n.p. estimator of F for all tε[0, τ 1
) × [0, τ 2
Linkage to representation
F (∆t) = ˆα(∆t)
P {(t, τ ); ˆα}/
P {( 0 , τ ); ˆα}
λ f
λ f
} = ˆγ
− 1
f
, f = 1,... , s − 1
Then
P {( 0 , τ ); ˆα} = ( ˆα 1
· · · αˆ s− 1
) det
A
and
P {t k
, τ ); ˆα} = ( ˆα k+
· · · αˆ s− 1
)ˆp k
det A( ˆα 1
,... , αˆ k− 1
so
F (∆t k
) = ˆα k
(
fW=k
α ˆ f
) det
A
s− 1
1
αˆ f
det
A
p ˆ k
= ˆp k
, as desired
Generalization to Other Dimensions
• T = (T
1
,... , T
m
) for m = 1 or m > 2
- Same argument gives, for any m ≥ 1,
F (t
−
) = { 1 − Λ(∆t)}
− 1 ˜ P {(t, τ ); α}/
P ( 0 , τ ); α}
P {(t, τ ); α} = 1 +
∞ 3
n=
8
t<v 1
<···<τ
n
i=
α(dv i
t<u<τ
{1 + α(du)} (Gill and Johansen, 1990)
t<u<τ
{ 1 − Λ(du)}
− 1
So F (t
−
) = { 1 − Λ(∆t)}
− 1
t<u<τ
{ 1 − Λ(du)}
− 1
0 <u<τ
{ 1 − Λ(du)}
− 1
= { 1 − Λ(∆t)}
− 1
0 <u≤t
{ 1 − Λ(du)}
0 <u<t
{ 1 − Λ(du)} (K-M)
- For any m compact differentiability of transformation from Λ
to F allows asymptotic properties held by
Λ to be inherited
by
F = Φ(
Special Cases of ˆΛ
RE
V DL
with bivariate censoring
Λ requires iterative calculation, and provides no advantage over
direct calculation of
F via E-M algorithm
- Van der Laan’s (1996 AS) repaired NPMLE maximizes
L =
I
i=
J
j=
p
n
11
ij
ij
3
fεS 1 i
3
m>j
p fm
n
10
ij
3
f>i
3
mεS 2 j
p fm
n
01
ij
3
f>i
3
m>j
p fm
n
00
ij
after imposition of additional censoring, replacement of any
undistributed singly censored observation by an uncensored
observation, and coarsening of the uncensored component of
the singly censored observation to the pertinent partition ele-
ment
= 0 if n
11
ij
= 0. Otherwise p ij
0 and satisfies
n
11
ij
p ij
3
m<j
n ¯
10
im
3
uεS 1 i
3
v>m
p uv
3
f<i
n ¯
01
fj
3
u>f
3
vεS 2 j
p uv
3
f<i
3
m<j
n
00
fm
3
u>f
3
v>m
p uv
−n = 0
p ij
n
11
ij
n
3
m<j
n ¯
10
im
n
p ij
3
v>m
3
uεS 1 i
p uv
3
f<i
n ¯
01
fj
n
p ij
3
u>f
3
vεS 2 j
p uv
3
f<i
3
m<j
n
00
fm
n
p ij
3
u>f
3
v>m
p uv
- Simple iterative calculation using mass assignments from re-
duced data
F
E
for starting values
- Greenwood-like asymptotic variance estimator for
F
V DL
from
2
log L/∂p ij
∂p fm
Simulation Comparisons
(see also Van der Laan, 1997, Statistica Nederlandica)
(i) F : independence F (t 1
, t 2
) = F
1
(t 1
)F
2
(t 2
— Clayton (1978, Bmka)
F (t 1
, t 2
) = {F
1
(t 1
−θ
(t 2
−θ
− 1 }
−θ
− 1
with θ = 4.
— unit exponential margins
(ii) Censoring: C 1
— exponential, mean 0.5 (2/3 probability of cen-
soring)
C
2
— none; univariate (C 1
= C
2
); independent exponential
with mean 0.
(iii) Truncation: τ 1
= τ 2
= 0. 5978 , F
i
(τ i
) = 0. 55 , i = 1, 2
Exclude sample from summary statistics if risk set empty at τ
(iv) Sample size: n = 30, 60 , 120 , 240; 1000 repetitions
(v) Partitions for
F
V DL
: Vertical, and horizontal strips
[0, − log(.85)), [− log(.85), − log(.70)), [− log(.70), − log(.55)),
[− log(.55), − log(.55)]
Also considered wider and narrow strips
Table 2. Sample means and standard deviations (in parentheses) for various estimators of the marginal
survivor functions F 1 and F 2. Each row is based on 1000 simulations at sample size n = 120.
T 1 Survival Probability T 2 Survival Probability
Failure Model .85 .70 .55 .85 .70.
Censoring on T 1 Only
Independence FˆE .850 (.038) .700 (.055) .552 (.070) .850 (.054) .702 (.072) .551 (.081)
Fˆ KM .850 (.036)^ .701 (.049)^ .552 (.064)^ .851 (.033)^ .700 (.042)^ .550 (.046)
ˆ FRE .851 (.035) .701 (.050) .553 (.066) .850 (.039) .700 (.050) .550 (.047)
Clayton
ˆ FE .850 (.036) .700 (.049) .551 (.063) .850 (.042) .702 (.055) .555 (.066)
ˆ FKM .850 (.036) .701 (.049) .552 (.064) .851 (.033) .701 (.041) .553 (.044)
Fˆ RE .851 (.036)^ .702 (.048)^ .559 (.060)^ .851 (.037)^ .701 (.045)^ .553 (.046)
Univariate Censoring
Independence FˆE .852 (.053) .703 (.067) .554 (.078) .850 (.053) .702 (.070) .552 (.079)
ˆ FKM .850 (.036) .701 (.049) .552 (.064) .851 (.035) .702 (.050) .553 (.065)
ˆ FRE .851 (.044) .701 (.056) .552 (.071) .850 (.042) .702 (.056) .553 (.070)
Clayton FˆE .851 (.043) .701 (.057) .552 (.071) .850 (.042) .701 (.056) .555 (.069)
Fˆ KM .850 (.036)^ .701 (.049)^ .552 (.064)^ .851 (.036)^ .702 (.049)^ .555 (.064)
ˆ FRE .851 (.041) .702 (.054) .556 (.069) .850 (.041) .702 (.055) .560 (.067)
Bivariate Censoring
Independence
ˆ FE .847 (.063) .701 (.092) .552 (.117) .853 (.061) .700 (.095) .550 (.122)
ˆ FKM .850 (.035) .702 (.051) .552 (.064) .852 (.033) .700 (.051) .551 (.065)
Fˆ RE .849 (.052)^ .702 (.070)^ .558 (.072)^ .852 (.049)^ .702 (.069)^ .557 (.075)
(Greenwood)
◦ (.042) (.057) (.068) (.042) (.057) (.069)
Clayton FˆE .849 (.048) .701 (.072) .548 (.100) .851 (.046) .703 (.071) .552 (.097)
ˆ FKM .850 (.035) .702 (.051) .552 (.064) .850 (.034) .700 (.050) .548 (.063)
ˆ FRE .850 (.047) .704 (.065) .563 (.069) .849 (.047) .704 (.063) .561 (.068)
(Greenwood) (.040) (.054) (.065) (.040) (.054) (.065)
◦ Entries are the square root of the average across samples of Greenwood-like variance estimators.
Discussion/Further Research
- Compact differentiable transformation
F (t
−
) = { 1 − Λ(∆t)}
− 1 ˜ P {(t, τ ); α}/
P {( 0 , t); α}
developed, where α(dt) = Λ(dt)/{ 1 − Λ(∆t)}, for truncated
failure time data
- Nonparametric plug-in estimators
F = Φ(
Λ) calculated using
a simple matrix calculation
F
E
is too inefficient; and asymptotically efficient
F
V DL
may
require very large sample sizes to out-perform simple plug-in
estimators
F
P C
and
F
D
- Further research possibilities:
— Obtain
F
1
F
2
estimators via partitioning [0, τ 1
], [0, τ 2
], fol-
lowed by Prentice-Cai or Dabrowska procedures
— Impose hazard rate form of Prentice-Cai or Dabrowska es-
timators (with restriction to avoid negative mass) in
F =
Λ) calculation.
