Abidemi - ICSA-Final

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Estimation of Discrete Survival Function Through
the Modeling of Diagnostic Accuracy for
Mismeasured Outcome Data
Abidemi K. Adeniji, PhD
The 10th ICSA International Conference
Shanghai Jiao Tong University
Shanghai, P. R. China
December 21, 2016

Research Team
Acknowledgements
..
. .
Hee-Koung Joeng, PhD, Merck
..
. .
Ming-Hui Chen, PhD, University of Connecticut
..
. .
Naitee Ting, PhD, Boehringer Ingelheim

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Results and Discussion
Overview:
.
..1 Introduction
.
..2 Methods
.
..3 Results and Discussion
3 / 33

Outline
.
..1 Introduction
.
..2 Methods
.
..3 Results and Discussion

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Misclassiﬁed outcomes can occur from false diagnostic test results.
● Alzheimer’s Disease:
Imaging vs. Autopsy
Dubois B, [2014]; Raman MR, [2014]; Johnson KA, [2012]; Fearing MA, [2007].
4 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Imaging vs. Autopsy
● Hepatitis:
Quantitative assays vs. Qualitative assays
Khuroo MS, [2014]; Franzeck FC, [2013]; Kamili S, [2012]; Conjeevaram HS, [2006].
4 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Imaging vs. Autopsy
● Hepatitis:
● Oncology:
Investigator reads vs. Central reads
Floquet A, [2015]; Dodd LE, [2011]; Amit O, [2010]; FDA, [2007].
4 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Imaging vs. Autopsy
● Hepatitis:
● Oncology:
● Ebola:
Antigen rapid test vs. RT-PCR
Schieﬀelin J, [2016]; Iwen PC, [2014]; WHO Ebola Response Team, [2014].
4 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Imaging vs. Autopsy
● Hepatitis:
● Oncology:
● Ebola:
Antigen rapid test vs. RT-PCR
Schieﬀelin J, [2016]; Iwen PC, [2014]; WHO Ebola Response Team, [2014].
● Helicobacter pylori:
Urea breathe test vs. Biopsy
Kalali B, [2015]; Queiroz DMM, [2013]; Choi J, [2011]; Gatta L, [2006]; Gisbert JP, [2006].
4 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
In time to event analysis with a binary outcome, event
misclassiﬁcation is common.
Standard methodology in survival analysis ignores such errors,
leading to incorrect inferences.
5 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
In time to event analysis with a binary outcome, event
misclassiﬁcation is common.
Standard methodology in survival analysis ignores such errors,
leading to incorrect inferences.
Racine-Poon and Hoel [1984]; Magder and Hughes [1997];
Snapinn [1998]; Richardson and Hughes [2000]; and McKeown
and Jewell [2010].
5 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Discrete Time Survival Data
Two Types of Discrete Time
...1 Derived Discrete
...2 Intrinsically Discrete
6 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Discrete time survival data are common in social science,
behavior science, economics, and biomedical science.
Derived Discrete:
7 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Derived Discrete:
◇ Example Data 1: Obeysekara, 2013
7 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Derived Discrete:
● Host selection of Tiphia
● Observing grubs once a week for three weeks
● Endpoint: Time to death of a grub
. .Time
.0

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Derived Discrete:
. .Time
.0 .t1

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Derived Discrete:
. .Time
.0 .t1 .t2

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Derived Discrete:
. .Time
.0 .t1 .t2 .t3

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Derived Discrete:
. .Time
.0 .t1 .t2 .t3
.
.
.Event

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Derived Discrete:
. .Time
.0 .t1 .t2 .t3
.
.
.Event
7 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Derived Discrete:
. .Time
.0 .t1 .t2 .t3
.
.
.Event
● By grouping, recorded into a discrete time
7 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
◇ Example Data 2: SEER (Surveillance, Epidemiology, and End
Results) breast cancer data in Joeng et al., 2015.
8 / 33
AGE DX DX YR GRADE STime ER PR STAGE STAT REC
(YYMM)
050 2002 3 0705 2 2 3 1
053 2001 3 0508 1 1 3 4

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
● Endpoint: Time to death
● Unit of survival time: Month - which is rounded due to patient
conﬁdentiality
8 / 33
(YYMM)
050 2002 3 0705 2 2 3 1
053 2001 3 0508 1 1 3 4

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
● Endpoint: Time to death
● Unit of survival time: Month - which is rounded due to patient
conﬁdentiality
● By rounding, recorded into a discrete time
8 / 33
(YYMM)
050 2002 3 0705 2 2 3 1
053 2001 3 0508 1 1 3 4

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Intrinsically Discrete:
◇ Example Data 3: PACO (Policy-Academic-Career Outcome)
data from a research assistant (RA) project
● How training programs (F31) funded by NIH aﬀect the career
development of biomedical scientists
● Endpoint: Time to get a degree
9 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Intrinsically Discrete:
◇ Example Data 3: PACO (Policy-Academic-Career Outcome)
data from a research assistant (RA) project
● How training programs (F31) funded by NIH aﬀect the career
development of biomedical scientists
● Endpoint: Time to get a degree
● The event can not happen between the units of survival time
9 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
◇ Example Data 4: VIRAHEP-C study in Adeniji et al., 2014
● Viral resistance to antiviral therapy of chronic Hepatitis C
● Endpoint: Time to viral negativity
● The viral loads of HCV (Hepatitis C virus) are measured only at
clinical visits and the event happens only over these visits
● None of the information is available between the unit of
survival time.
10 / 33
Time
ViralLoad(IU/ml)
0 t1 t2 t3 t4 t5 t6 t7 t8

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Notations
True outcomes
T∗
: discrete survival time
h∗
(j) = P(T∗
= tj ∣T∗
≥ tj ): hazard function
S∗
(j) = P(T∗
> tj ): survival function
Observed outcomes
T: discrete survival time
h(j) = P(T = tj ∣T ≥ tj ): hazard function
S(j) = P(T > tj ): survival function
11 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Notations
True outcomes
T∗
: discrete survival time
h∗
(j) = P(T∗
= tj ∣T∗
≥ tj ): hazard function
S∗
(j) = P(T∗
> tj ): survival function
Observed outcomes
T: discrete survival time
h(j) = P(T = tj ∣T ≥ tj ): hazard function
S(j) = P(T > tj ): survival function
h(t) = lim
dt→0
Pr{t ≤ T < t + dt∣T ≥ t}
dt
S(t) = e− ∫
t
0 h(s)ds
11 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Proposed Method
Our method uses accuracy of the diagnostic tool to build a
bridge between the mismeasured outcomes and the true
outcomes.
12 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Why Accuracy?
Gold standard may not be routinely used due to cost or other
reasons.
● Gold standard test for Alzheimer diagnosis: autopsy
13 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Why Accuracy?
● Gold standard test for HCV: viral load ≤ 50 IU/ml
● Routine standard test for HCV: viral load ≤ 600 IU/ml
14 / 33
Time
ViralLoad(IU/ml)
0 t1 t2 t3 t4 t5 t6 t7 t8

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
How to Model Accuracy?
Exact Relationship: The true survival function can be
expressed as
S∗
(j) = (1 − ..τj ){1 − S(j)} + ..γj S(j).
15 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
expressed as
S∗
(j) = (1 − ..τj ){1 − S(j)} + ..γj S(j).
● τj = P(T∗
≤ tj ∣T ≤ tj ): ..
positive predicted value (PPV) at tj
.

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
expressed as
S∗
(j) = (1 − ..τj ){1 − S(j)} + ..γj S(j).
● τj = P(T∗
≤ tj ∣T ≤ tj ): ..
● γj = P(T∗
> tj ∣T > tj ): ..
negative predicted value (NPV) at tj
.

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
expressed as
S∗
(j) = (1 − ..τj ){1 − S(j)} + ..γj S(j).
● τj = P(T∗
≤ tj ∣T ≤ tj ): ..
● γj = P(T∗
> tj ∣T > tj ): ..
negative predicted value (NPV) at tj
.
15 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
The Model of the PPV
The model for the probability that true failure happen until tj ,
given observed failure occurrence at a certain previous time
point tk for tk ≤ tj , is
P(T∗
≤ tj ∣T = tk) = 1 − {1 − τ0}
(tj −t1)ω1+(tj −tk )ω2+1
,
for j = 1,2,... and known τ0 where ω1 ≥ 0 and ω2 ≥ 0.
● Key idea: increases as the true time moves further away from
the observed event time.
16 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
The Model of the PPV
The model for the probability that true failure happen until tj ,
given observed failure occurrence at a certain previous time
point tk for tk ≤ tj , is
P(T∗
≤ tj ∣T = tk) = 1 − {1 − τ0}
(tj −t1)ω1+(tj −tk )ω2+1
,
for j = 1,2,... and known τ0 where ω1 ≥ 0 and ω2 ≥ 0.
● Key idea: increases as the true time moves further away from
the observed event time.
τj = 1 −
∑
j
k=1 P(T = tk){1 − τ0}
(tj −t1)ω1+(tj −tk )w2+1
1 − S(j)
.
16 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Assumption 1
Assumption 1: P(T∗
≥ T) = 1
17 / 33
Time
ViralLoad(IU/ml)
0 t1 t2 t3 t4 t5 t6 t7 t8
50600
q
q
q

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Assumption 1
Assumption 1: P(T∗
≥ T) = 1
The “Truth” cannot happen before the “Observed”
17 / 33
Time
ViralLoad(IU/ml)
0 t1 t2 t3 t4 t5 t6 t7 t8
50600
q
q
q

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
True Survival Function under the model of PPV
Under P(T∗
≥ T) = 1 and the model of τj , the true survival
function is
S∗
(j) = S(j) +
j
∑
k=1
P(T = tk){1 − τ0}
(tj −t1)ω1+(tj −tk )ω2+1
18 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Two Data Scenarios
Scenario 1: obtaining three parameters, ω1, ω2 and τ0, from
medical experts.
Scenario 2: estimating ω1, ω2 and τ0 directly from the
on-going clinical study.
We ﬁrst need to obtain the “pilot data” (complete data) only
on a small and randomly selected number of participants.
This data is used to estimate ω1, ω2 and τ0.
The remaining (unselected) participants in the clinical study
would only have the error-prone outcomes, and this set of
observations is called the “analysis data”.
Under this setting, the pilot data and the analysis data are
independent.
19 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Framework of the approximation process
20 / 33
………
………
………
………
………
Investigator
30% ………
Independent

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
The Estimation Procedure under Scenario 2
Using the pilot dataset, obtain (ˆω1, ˆω2, ˆτ0) as follows
(ˆω1, ˆω2, ˆτ0) = argmin
ω1,ω2,τ0
{
K
∑
k=1
w(k)(S∗
P(k) − ˆS∗
(k))
2
},
where S∗
P(k) is the estimated true survival rates, ˆS∗
(k) is the
estimated approximated survival function, and the weight
w(k) is {ˆS∗
(k)}ρ1
{1 − ˆS∗
(k)}ρ2
for 0 ≤ ρ1,ρ2 ≤ 1 and
k = 1,2,...,K.
Using (ˆω1, ˆω2, ˆτ0), we then obtain the approximated “true”
survival function for the analysis data.
21 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Gamma Process
Gamma Process
△ Continuous r.v T ∼ Gamma(a,b), where mean(T) = ab.
△ Suppose α(t) is an increasing and right continuous function
on [0,∞) with α(0) = 0.
△ Let W = {Wt, t ≥ 0} be a Gamma process (GP), denoted by
W ∼ GP(α(t), b) with the following properties:
● W0 = 0
● W has independent increments in disjoint intervals
● for t > s, Wt − Ws ∼ Gamma(α(t) − α(s), b), where b > 0 is a
constant
● W ∼ GP(α(t), b) where b > 0 is a constant
W ∗
j = Wj − E[Wj ] and assume that we only observe Wj at
discrete integer times, i.e., j = 1,2,3,....
22 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Survival function with a lower detection level under
Gamma Process
Under the assumption that the course of viral load follows
Gamma process, closed-form expression of the survival
function with a lower detection limit as c can be derived
analytically.
Let W = {Wj , j ≥ 0} be a GP(j,1), where Wj = X1 + ⋯ + Xj
and the Xj are i.i.d. from Gamma(1,1) for j = 1,....
Under the discrete Gamma process, the survival function at
time tj with low detection limit as c is
23 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Survival function with a lower detection level under
Gamma Process
Under the assumption that the course of viral load follows
Gamma process, closed-form expression of the survival
function with a lower detection limit as c can be derived
analytically.
Let W = {Wj , j ≥ 0} be a GP(j,1), where Wj = X1 + ⋯ + Xj
and the Xj are i.i.d. from Gamma(1,1) for j = 1,....
Under the discrete Gamma process, the survival function at
time tj with low detection limit as c is
Sc (j) = P(X1 ≥ 1 + c, X1 + X2 ≥ 2 + c, . . . , X1 + ⋯ + Xj ≥ n + c)
=
j(j−1)
(j − 1)!
exp {−(c + j)}.
23 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Survival Functions Under Gamma Process with c∗
= −0.8
and c = −0.4
True survival function:
Sc∗ (j) =
j(j−1)
(j − 1)!
exp{−(c∗
+ j)}.
Observed survival function:
Sc(j) =
j(j−1)
(j − 1)!
exp{−(c + j)}.
Approximated survival function: S∗
c (j).
24 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Survival functions with c∗
= −0.8, c = −0.4 for ρ1 = ρ2 = 0.5
If c∗
≤ c then P(T∗
≥ T) = 1.
25 / 33
Time
SurvivalRate
Time
SurvivalRate
Time
SurvivalRate
t1 t2 t3 t4 t5 t6 t7 t8
0.00.20.40.60.81.0
G−S
E−P
Approximated

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Brownian Motion(BM) Process
Let W = {Wt,t ≥ 0} be a standard BM process satisfying the
following properties:
△ W0 = 0.
△ W (tk ) − W (ts) ∼ N(0,
tk − ts
60000
), for 0 ≤ ts < tk ≤ tK .
△ Wt1 ,Wt2 − Wt1 ,...,Wtk
− Wts are independent.
△ Wk ∼ N(0,k).
26 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Simulation Study under BM Process
Generating Bi = (Bij )′
as Bij ∼ N(0,
1
60000
) where i = 1,...,n
and j = 1,...,J.
Obtaining Witk
=
tk
∑
j=1
Bij , for i = 1,2,3,...,n and
j = 1,2,3,...,60000.
Setting Wi0 = 0, tK = 60000, we have
Witk
− Wi0 ∼ N(0,
tk
60000
).
27 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
as Bij ∼ N(0,
1
60000
) where i = 1,...,n
and j = 1,...,J.
Obtaining Witk
=
tk
∑
j=1
Bij , for i = 1,2,3,...,n and
j = 1,2,3,...,60000.
Witk
− Wi0 ∼ N(0,
tk
60000
).
Taking K time points of Witk
, deﬁned as tk = 1000k for
k = 1,2,...,K.
27 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
as Bij ∼ N(0,
1
60000
) where i = 1,...,n
and j = 1,...,J.
Obtaining Witk
=
tk
∑
j=1
Bij , for i = 1,2,3,...,n and
j = 1,2,3,...,60000.
Witk
− Wi0 ∼ N(0,
tk
60000
).
Taking K time points of Witk
, deﬁned as tk = 1000k for
k = 1,2,...,K.
Obtaining true and observed survival times as
Ti = min{k ∶ Witk
≤ −0.025} and T∗
i = min{k ∶ Witk
≤ −0.04},
where n = 400, K = 8, and J = 60000.
27 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Survival Functions with c∗
= −0.04 and c = −0.025 for
ρ1 = ρ2 = 0.5 using n0 = 40 (a) and n0 = 80 (b)
Time
SurvivalRate
Time
SurvivalRate
Time
SurvivalRate
t1 t2 t3 t4 t5 t6 t7 t8
0.00.20.40.60.81.0
G−S
E−P
Approximated
Time
SurvivalRate
Time
SurvivalRate
Time
SurvivalRate
t1 t2 t3 t4 t5 t6 t7 t8
0.00.20.40.60.81.0
G−S
E−P
Approximated
(a) (b)
28 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Survival Functions with c∗
= −0.04 and c = −0.025 for
ρ1 = 1, and ρ2 = 0 using n0 = 40 (a) and n0 = 80 (b)
Time
SurvivalRate
Time
SurvivalRate
Time
SurvivalRate
t1 t2 t3 t4 t5 t6 t7 t8
0.00.20.40.60.81.0
G−S
E−P
Approximated
Time
SurvivalRate
Time
SurvivalRate
Time
SurvivalRate
t1 t2 t3 t4 t5 t6 t7 t8
0.00.20.40.60.81.0
G−S
E−P
Approximated
(a) (b)
29 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Data
372 subjects from the VIRAHEP-C (Viral Resistance to
Antiviral Therapy of Chronic Hepatitis C) Study
True event at time tj :
E∗
j = I{ viral load ≤ 50 IU/ml at tj }
Observed event at time tj :
Ej = I{ viral load ≤ 600 IU/ml at tj }
30 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Survival Functions for n0 = 37 with ρ1 = ρ2 = 0.5 (a) and
ρ1 = 1 and ρ2 = 0 (b)
Time
SurvivalRate
Time
SurvivalRate
Time
SurvivalRate
t1 t2 t3 t4 t5 t6 t7 t8
0.00.20.40.60.81.0
G−S
E−P
Approximated
Time
SurvivalRate
Time
SurvivalRate
Time
SurvivalRate
t1 t2 t3 t4 t5 t6 t7 t8
0.00.20.40.60.81.0
G−S
E−P
Approximated
(a) (b)
31 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Concluding Remarks
Our discrete-time survival estimator:
.
..1 allows for the cumulative probability of correctly classifying a G-S event
to increase with time given the prior occurrence of an E-P event,
.
..2 allows for the inclusion of estimated model parameters ˆD1 = (ˆω1, ˆω2, ˆτ0)
through a validation subsample (“pilot dataset”).
32 / 33

Introduction
. . . . . . . . . .
Methods
. . . . . . . . . . . .
Thank You!
We offer a more flexible strategy to the conduct of clinical trials,
there is a possibility to reduce cost and improve efficiency.
33 / 33

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