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Estimation of Discrete Survival Function Throughthe Modeling of Diagnostic Accuracy for

Mismeasured Outcome Data

Abidemi K. Adeniji, PhD

The 10th ICSA International ConferenceShanghai 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

. . . . . . . . . . . .Results and Discussion

Mismeasured Outcome Data

Misclassified outcomes can occur from false diagnostic test results.

● Alzheimer’s Disease:Imaging vs. AutopsyDubois B, [2014]; Raman MR, [2014]; Johnson KA, [2012]; Fearing MA, [2007].

● Hepatitis:Quantitative assays vs. Qualitative assaysKhuroo MS, [2014]; Franzeck FC, [2013]; Kamili S, [2012]; Conjeevaram HS, [2006].

● Oncology:Investigator reads vs. Central readsFloquet A, [2015]; Dodd LE, [2011]; Amit O, [2010]; FDA, [2007].

● Ebola:Antigen rapid test vs. RT-PCRSchieffelin J, [2016]; Iwen PC, [2014]; WHO Ebola Response Team, [2014].

● Helicobacter pylori:Urea breathe test vs. BiopsyKalali B, [2015]; Queiroz DMM, [2013]; Choi J, [2011]; Gatta L, [2006]; Gisbert JP, [2006].

4 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Mismeasured Outcome Data

Misclassified outcomes can occur from false diagnostic test results.

● Alzheimer’s Disease:Imaging vs. AutopsyDubois B, [2014]; Raman MR, [2014]; Johnson KA, [2012]; Fearing MA, [2007].

● Hepatitis:Quantitative assays vs. Qualitative assaysKhuroo MS, [2014]; Franzeck FC, [2013]; Kamili S, [2012]; Conjeevaram HS, [2006].

● Oncology:Investigator reads vs. Central readsFloquet A, [2015]; Dodd LE, [2011]; Amit O, [2010]; FDA, [2007].

● Ebola:Antigen rapid test vs. RT-PCRSchieffelin J, [2016]; Iwen PC, [2014]; WHO Ebola Response Team, [2014].

● Helicobacter pylori:Urea breathe test vs. BiopsyKalali B, [2015]; Queiroz DMM, [2013]; Choi J, [2011]; Gatta L, [2006]; Gisbert JP, [2006].

4 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Mismeasured Outcome Data

Misclassified outcomes can occur from false diagnostic test results.

● Alzheimer’s Disease:Imaging vs. AutopsyDubois B, [2014]; Raman MR, [2014]; Johnson KA, [2012]; Fearing MA, [2007].

● Hepatitis:Quantitative assays vs. Qualitative assaysKhuroo MS, [2014]; Franzeck FC, [2013]; Kamili S, [2012]; Conjeevaram HS, [2006].

● Oncology:Investigator reads vs. Central readsFloquet A, [2015]; Dodd LE, [2011]; Amit O, [2010]; FDA, [2007].

● Ebola:Antigen rapid test vs. RT-PCRSchieffelin J, [2016]; Iwen PC, [2014]; WHO Ebola Response Team, [2014].

● Helicobacter pylori:Urea breathe test vs. BiopsyKalali B, [2015]; Queiroz DMM, [2013]; Choi J, [2011]; Gatta L, [2006]; Gisbert JP, [2006].

4 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Mismeasured Outcome Data

Misclassified outcomes can occur from false diagnostic test results.

● Alzheimer’s Disease:Imaging vs. AutopsyDubois B, [2014]; Raman MR, [2014]; Johnson KA, [2012]; Fearing MA, [2007].

● Hepatitis:Quantitative assays vs. Qualitative assaysKhuroo MS, [2014]; Franzeck FC, [2013]; Kamili S, [2012]; Conjeevaram HS, [2006].

● Oncology:Investigator reads vs. Central readsFloquet A, [2015]; Dodd LE, [2011]; Amit O, [2010]; FDA, [2007].

● Ebola:Antigen rapid test vs. RT-PCRSchieffelin J, [2016]; Iwen PC, [2014]; WHO Ebola Response Team, [2014].

● Helicobacter pylori:Urea breathe test vs. BiopsyKalali B, [2015]; Queiroz DMM, [2013]; Choi J, [2011]; Gatta L, [2006]; Gisbert JP, [2006].

4 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Mismeasured Outcome Data

Misclassified outcomes can occur from false diagnostic test results.

● Alzheimer’s Disease:Imaging vs. AutopsyDubois B, [2014]; Raman MR, [2014]; Johnson KA, [2012]; Fearing MA, [2007].

● Hepatitis:Quantitative assays vs. Qualitative assaysKhuroo MS, [2014]; Franzeck FC, [2013]; Kamili S, [2012]; Conjeevaram HS, [2006].

● Oncology:Investigator reads vs. Central readsFloquet A, [2015]; Dodd LE, [2011]; Amit O, [2010]; FDA, [2007].

● Ebola:Antigen rapid test vs. RT-PCRSchieffelin J, [2016]; Iwen PC, [2014]; WHO Ebola Response Team, [2014].

● Helicobacter pylori:Urea breathe test vs. BiopsyKalali B, [2015]; Queiroz DMM, [2013]; Choi J, [2011]; Gatta L, [2006]; Gisbert JP, [2006].

4 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Mismeasured Outcome Data

Misclassified outcomes can occur from false diagnostic test results.

● Alzheimer’s Disease:Imaging vs. AutopsyDubois B, [2014]; Raman MR, [2014]; Johnson KA, [2012]; Fearing MA, [2007].

● Hepatitis:Quantitative assays vs. Qualitative assaysKhuroo MS, [2014]; Franzeck FC, [2013]; Kamili S, [2012]; Conjeevaram HS, [2006].

● Oncology:Investigator reads vs. Central readsFloquet A, [2015]; Dodd LE, [2011]; Amit O, [2010]; FDA, [2007].

● Ebola:Antigen rapid test vs. RT-PCRSchieffelin J, [2016]; Iwen PC, [2014]; WHO Ebola Response Team, [2014].

● Helicobacter pylori:Urea breathe test vs. BiopsyKalali B, [2015]; Queiroz DMM, [2013]; Choi J, [2011]; Gatta L, [2006]; Gisbert JP, [2006].

4 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Mismeasured Outcome Data

In time to event analysis with a binary outcome, eventmisclassification 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 McKeownand Jewell [2010].

5 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Mismeasured Outcome Data

In time to event analysis with a binary outcome, eventmisclassification 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 McKeownand Jewell [2010].

5 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Mismeasured Outcome Data

In time to event analysis with a binary outcome, eventmisclassification 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 McKeownand Jewell [2010].

5 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

Two Types of Discrete Time

...1 Derived Discrete

...2 Intrinsically Discrete

6 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

Discrete time survival data are common in social science,behavior science, economics, and biomedical science.

Derived Discrete:

◇ Example Data 1: Obeysekara, 2013

● Host selection of Tiphia● Observing grubs once a week for three weeks● Endpoint: Time to death of a grub

. .Time.0 .t1 .t2 .t3

.

.

.Event

● By grouping, recorded into a discrete time

7 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

Discrete time survival data are common in social science,behavior science, economics, and biomedical science.

Derived Discrete:

◇ Example Data 1: Obeysekara, 2013

● Host selection of Tiphia● Observing grubs once a week for three weeks● Endpoint: Time to death of a grub

. .Time.0 .t1 .t2 .t3

.

.

.Event

● By grouping, recorded into a discrete time

7 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

Discrete time survival data are common in social science,behavior science, economics, and biomedical science.

Derived Discrete:

◇ Example Data 1: Obeysekara, 2013

● Host selection of Tiphia● Observing grubs once a week for three weeks● Endpoint: Time to death of a grub

. .Time.0

.t1 .t2 .t3.

.

.Event

● By grouping, recorded into a discrete time

7 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

Discrete time survival data are common in social science,behavior science, economics, and biomedical science.

Derived Discrete:

◇ Example Data 1: Obeysekara, 2013

● Host selection of Tiphia● Observing grubs once a week for three weeks● Endpoint: Time to death of a grub

. .Time.0 .t1

.t2 .t3.

.

.Event

● By grouping, recorded into a discrete time

7 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

Discrete time survival data are common in social science,behavior science, economics, and biomedical science.

Derived Discrete:

◇ Example Data 1: Obeysekara, 2013

● Host selection of Tiphia● Observing grubs once a week for three weeks● Endpoint: Time to death of a grub

. .Time.0 .t1 .t2

.t3.

.

.Event

● By grouping, recorded into a discrete time

7 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

Discrete time survival data are common in social science,behavior science, economics, and biomedical science.

Derived Discrete:

◇ Example Data 1: Obeysekara, 2013

● Host selection of Tiphia● Observing grubs once a week for three weeks● Endpoint: Time to death of a grub

. .Time.0 .t1 .t2 .t3

.

.

.Event

● By grouping, recorded into a discrete time

7 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

Discrete time survival data are common in social science,behavior science, economics, and biomedical science.

Derived Discrete:

◇ Example Data 1: Obeysekara, 2013

● Host selection of Tiphia● Observing grubs once a week for three weeks● Endpoint: Time to death of a grub

. .Time.0 .t1 .t2 .t3

.

.

.Event

● By grouping, recorded into a discrete time

7 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

Discrete time survival data are common in social science,behavior science, economics, and biomedical science.

Derived Discrete:

◇ Example Data 1: Obeysekara, 2013

● Host selection of Tiphia● Observing grubs once a week for three weeks● Endpoint: Time to death of a grub

. .Time.0 .t1 .t2 .t3

.

.

.Event

● By grouping, recorded into a discrete time

7 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

Discrete time survival data are common in social science,behavior science, economics, and biomedical science.

Derived Discrete:

◇ Example Data 1: Obeysekara, 2013

● Host selection of Tiphia● Observing grubs once a week for three weeks● Endpoint: Time to death of a grub

. .Time.0 .t1 .t2 .t3

.

.

.Event

● By grouping, recorded into a discrete time

7 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

Discrete time survival data are common in social science,behavior science, economics, and biomedical science.

Derived Discrete:

◇ Example Data 1: Obeysekara, 2013

● Host selection of Tiphia● Observing grubs once a week for three weeks● Endpoint: Time to death of a grub

. .Time.0 .t1 .t2 .t3

.

.

.Event

● By grouping, recorded into a discrete time

7 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

◇ Example Data 2: SEER (Surveillance, Epidemiology, and EndResults) breast cancer data in Joeng et al., 2015.

● 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

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

◇ Example Data 2: SEER (Surveillance, Epidemiology, and EndResults) breast cancer data in Joeng et al., 2015.

● 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

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

◇ Example Data 2: SEER (Surveillance, Epidemiology, and EndResults) breast cancer data in Joeng et al., 2015.

● 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

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

◇ Example Data 2: SEER (Surveillance, Epidemiology, and EndResults) breast cancer data in Joeng et al., 2015.

● 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

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

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 careerdevelopment of biomedical scientists

● Endpoint: Time to get a degree

● The event can not happen between the units of survival time

9 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

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 careerdevelopment of biomedical scientists

● Endpoint: Time to get a degree● The event can not happen between the units of survival time

9 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

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 careerdevelopment of biomedical scientists

● Endpoint: Time to get a degree● The event can not happen between the units of survival time

9 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

◇ 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 ofsurvival time.

10 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Discrete Time Survival Data

◇ 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 ofsurvival time.

10 / 33

Outline

.. .1 Introduction

.. .2 Methods

.. .3 Results and Discussion

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Notations

True outcomes

T ∗: discrete survival time

h∗(j) = P(T ∗ = tj ∣T ∗ ≥ tj): hazard function

S∗(j) = P(T ∗ > tj): survival functionObserved outcomes

T : discrete survival time

h(j) = P(T = tj ∣T ≥ tj): hazard function

S(j) = P(T > tj): survival function

h(t) = limdt→0

Pr{t ≤ T < t + dt ∣T ≥ t}dt

S(t) = e− ∫t0 h(s)ds

11 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Notations

True outcomes

T ∗: discrete survival time

h∗(j) = P(T ∗ = tj ∣T ∗ ≥ tj): hazard function

S∗(j) = P(T ∗ > tj): survival functionObserved outcomes

T : discrete survival time

h(j) = P(T = tj ∣T ≥ tj): hazard function

S(j) = P(T > tj): survival function

h(t) = limdt→0

Pr{t ≤ T < t + dt ∣T ≥ t}dt

S(t) = e− ∫t0 h(s)ds

11 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Notations

True outcomes

T ∗: discrete survival time

h∗(j) = P(T ∗ = tj ∣T ∗ ≥ tj): hazard function

S∗(j) = P(T ∗ > tj): survival functionObserved outcomes

T : discrete survival time

h(j) = P(T = tj ∣T ≥ tj): hazard function

S(j) = P(T > tj): survival function

h(t) = limdt→0

Pr{t ≤ T < t + dt ∣T ≥ t}dt

S(t) = e− ∫t0 h(s)ds

11 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Proposed Method

Our method uses accuracy of the diagnostic tool to build abridge between the mismeasured outcomes and the trueoutcomes.

12 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Why Accuracy?

Gold standard may not be routinely used due to cost or otherreasons.

● Gold standard test for Alzheimer diagnosis: autopsy

13 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Why Accuracy?

Gold standard may not be routinely used due to cost or otherreasons.

● Gold standard test for Alzheimer diagnosis: autopsy

13 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Why Accuracy?

● Gold standard test for HCV: viral load ≤ 50 IU/ml

● Routine standard test for HCV: viral load ≤ 600 IU/ml

14 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Why Accuracy?

● Gold standard test for HCV: viral load ≤ 50 IU/ml

● Routine standard test for HCV: viral load ≤ 600 IU/ml

14 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

How to Model Accuracy?

Exact Relationship: The true survival function can beexpressed as

S∗(j) = (1 − ..τj ){1 − S(j)} + ..γj S(j).

● τj = P(T ∗ ≤ tj ∣T ≤ tj): ..positive predicted value (PPV) at tj

● γj = P(T ∗ > tj ∣T > tj): ..

negative predicted value (NPV) at tj

.

15 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

How to Model Accuracy?

Exact Relationship: The true survival function can beexpressed as

S∗(j) = (1 − ..τj ){1 − S(j)} + ..γj S(j).

● τj = P(T ∗ ≤ tj ∣T ≤ tj): ..positive predicted value (PPV) at tj

● γj = P(T ∗ > tj ∣T > tj): ..

negative predicted value (NPV) at tj

.

15 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

How to Model Accuracy?

Exact Relationship: The true survival function can beexpressed as

S∗(j) = (1 − ..τj ){1 − S(j)} + ..γj S(j).

● τj = P(T ∗ ≤ tj ∣T ≤ tj): ..positive predicted value (PPV) at tj

● γj = P(T ∗ > tj ∣T > tj): ..

negative predicted value (NPV) at tj

.

15 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

How to Model Accuracy?

Exact Relationship: The true survival function can beexpressed as

S∗(j) = (1 − ..τj ){1 − S(j)} + ..γj S(j).

● τj = P(T ∗ ≤ tj ∣T ≤ tj): ..positive predicted value (PPV) at tj

● γj = P(T ∗ > tj ∣T > tj): ..

negative predicted value (NPV) at tj

.

15 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

How to Model Accuracy?

Exact Relationship: The true survival function can beexpressed as

S∗(j) = (1 − ..τj ){1 − S(j)} + ..γj S(j).

● τj = P(T ∗ ≤ tj ∣T ≤ tj): ..positive predicted value (PPV) at tj

● γj = P(T ∗ > tj ∣T > tj): ..

negative predicted value (NPV) at tj

.

15 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

How to Model Accuracy?

Exact Relationship: The true survival function can beexpressed as

S∗(j) = (1 − ..τj ){1 − S(j)} + ..γj S(j).

● τj = P(T ∗ ≤ tj ∣T ≤ tj): ..positive predicted value (PPV) at tj

● γj = P(T ∗ > tj ∣T > tj): ..

negative predicted value (NPV) at tj

.

15 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

The Model of the PPV

The model for the probability that true failure happen until tj ,given observed failure occurrence at a certain previous timepoint 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

. . . . . . . . . . . .Results and Discussion

The Model of the PPV

The model for the probability that true failure happen until tj ,given observed failure occurrence at a certain previous timepoint 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

. . . . . . . . . . . .Results and Discussion

The Model of the PPV

The model for the probability that true failure happen until tj ,given observed failure occurrence at a certain previous timepoint 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

. . . . . . . . . . . .Results and Discussion

Assumption 1

Assumption 1: P(T ∗ ≥ T ) = 1

The “Truth” cannot happen before the “Observed”

17 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Assumption 1

Assumption 1: P(T ∗ ≥ T ) = 1

The “Truth” cannot happen before the “Observed”

17 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Assumption 1

Assumption 1: P(T ∗ ≥ T ) = 1

The “Truth” cannot happen before the “Observed”

17 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

True Survival Function under the model of PPV

Under P(T ∗ ≥ T ) = 1 and the model of τj , the true survivalfunction is

S∗(j) = S(j) +j

∑k=1

P(T = tk){1 − τ0}(tj−t1)ω1+(tj−tk)ω2+1

18 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Two Data Scenarios

Scenario 1: obtaining three parameters, ω1, ω2 and τ0, frommedical experts.

Scenario 2: estimating ω1, ω2 and τ0 directly from theon-going clinical study.

We first need to obtain the “pilot data” (complete data) onlyon 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 studywould only have the error-prone outcomes, and this set ofobservations is called the “analysis data”.Under this setting, the pilot data and the analysis data areindependent.

19 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Two Data Scenarios

Scenario 1: obtaining three parameters, ω1, ω2 and τ0, frommedical experts.

Scenario 2: estimating ω1, ω2 and τ0 directly from theon-going clinical study.

We first need to obtain the “pilot data” (complete data) onlyon 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 studywould only have the error-prone outcomes, and this set ofobservations is called the “analysis data”.Under this setting, the pilot data and the analysis data areindependent.

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Framework of the approximation process

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

The Estimation Procedure under Scenario 2

Using the pilot dataset, obtain (ω1, ω2, τ0) as follows

(ω1, ω2, τ0) = argminω1,ω2,τ0

{K

∑k=1

w(k)(S∗P(k) − S∗(k))2},

where S∗P(k) is the estimated true survival rates, S∗(k) is theestimated approximated survival function, and the weightw(k) is {S∗(k)}ρ1{1 − S∗(k)}ρ2 for 0 ≤ ρ1, ρ2 ≤ 1 andk = 1,2, . . . ,K .

Using (ω1, ω2, τ0), we then obtain the approximated “true”survival function for the analysis data.

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

The Estimation Procedure under Scenario 2

Using the pilot dataset, obtain (ω1, ω2, τ0) as follows

(ω1, ω2, τ0) = argminω1,ω2,τ0

{K

∑k=1

w(k)(S∗P(k) − S∗(k))2},

where S∗P(k) is the estimated true survival rates, S∗(k) is theestimated approximated survival function, and the weightw(k) is {S∗(k)}ρ1{1 − S∗(k)}ρ2 for 0 ≤ ρ1, ρ2 ≤ 1 andk = 1,2, . . . ,K .

Using (ω1, ω2, τ0), we then obtain the approximated “true”survival function for the analysis data.

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Outline

.. .1 Introduction

.. .2 Methods

.. .3 Results and Discussion

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

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, . . . .

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

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, . . . .

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Survival function with a lower detection level underGamma Process

Under the assumption that the course of viral load followsGamma process, closed-form expression of the survivalfunction with a lower detection limit as c can be derivedanalytically.

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 attime 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)}.

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Survival function with a lower detection level underGamma Process

Under the assumption that the course of viral load followsGamma process, closed-form expression of the survivalfunction with a lower detection limit as c can be derivedanalytically.

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 attime 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)}.

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Survival function with a lower detection level underGamma Process

Under the assumption that the course of viral load followsGamma process, closed-form expression of the survivalfunction with a lower detection limit as c can be derivedanalytically.

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 attime 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)}.

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Survival Functions Under Gamma Process with c∗ = −0.8and 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).

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Survival functions with c∗ = −0.8, c = −0.4 for ρ1 = ρ2 = 0.5

If c∗ ≤ c then P(T ∗ ≥ T ) = 1.

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Brownian Motion(BM) Process

Let W = {Wt , t ≥ 0} be a standard BM process satisfying thefollowing properties:

△ W0 = 0.△ W (tk) −W (ts) ∼ N(0,

tk − ts60000

), for 0 ≤ ts < tk ≤ tK .△ Wt1 ,Wt2 −Wt1 , . . . ,Wtk −Wts are independent.△ Wk ∼ N(0, k).

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Brownian Motion(BM) Process

Let W = {Wt , t ≥ 0} be a standard BM process satisfying thefollowing properties:

△ W0 = 0.△ W (tk) −W (ts) ∼ N(0,

tk − ts60000

), for 0 ≤ ts < tk ≤ tK .△ Wt1 ,Wt2 −Wt1 , . . . ,Wtk −Wts are independent.△ Wk ∼ N(0, k).

26 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Simulation Study under BM Process

Generating Bi = (Bij)′ as Bij ∼ N(0,1

60000) where i = 1, . . . ,n

and j = 1, . . . , J.

Obtaining Wi tk =tk

∑j=1

Bij , for i = 1,2,3, . . . ,n and

j = 1,2,3, . . . ,60000.Setting Wi0 = 0, tK = 60000, we have

Wi tk −Wi0 ∼ N(0,tk

60000).

Taking K time points of Witk , defined as tk = 1000k fork = 1,2, . . . ,K .

Obtaining true and observed survival times asTi = 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

. . . . . . . . . . . .Results and Discussion

Simulation Study under BM Process

Generating Bi = (Bij)′ as Bij ∼ N(0,1

60000) where i = 1, . . . ,n

and j = 1, . . . , J.

Obtaining Wi tk =tk

∑j=1

Bij , for i = 1,2,3, . . . ,n and

j = 1,2,3, . . . ,60000.Setting Wi0 = 0, tK = 60000, we have

Wi tk −Wi0 ∼ N(0,tk

60000).

Taking K time points of Witk , defined as tk = 1000k fork = 1,2, . . . ,K .

Obtaining true and observed survival times asTi = min{k ∶Witk ≤ −0.025} and T ∗i = min{k ∶Witk ≤ −0.04},where n = 400, K = 8, and J = 60000.

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Simulation Study under BM Process

Generating Bi = (Bij)′ as Bij ∼ N(0,1

60000) where i = 1, . . . ,n

and j = 1, . . . , J.

Obtaining Wi tk =tk

∑j=1

Bij , for i = 1,2,3, . . . ,n and

j = 1,2,3, . . . ,60000.Setting Wi0 = 0, tK = 60000, we have

Wi tk −Wi0 ∼ N(0,tk

60000).

Taking K time points of Witk , defined as tk = 1000k fork = 1,2, . . . ,K .

Obtaining true and observed survival times asTi = 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

. . . . . . . . . . . .Results and Discussion

Simulation Study under BM Process

Generating Bi = (Bij)′ as Bij ∼ N(0,1

60000) where i = 1, . . . ,n

and j = 1, . . . , J.

Obtaining Wi tk =tk

∑j=1

Bij , for i = 1,2,3, . . . ,n and

j = 1,2,3, . . . ,60000.Setting Wi0 = 0, tK = 60000, we have

Wi tk −Wi0 ∼ N(0,tk

60000).

Taking K time points of Witk , defined as tk = 1000k fork = 1,2, . . . ,K .

Obtaining true and observed survival times asTi = 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

. . . . . . . . . . . .Results and Discussion

Survival Functions with c∗ = −0.04 and c = −0.025 forρ1 = ρ2 = 0.5 using n0 = 40 (a) and n0 = 80 (b)

Time

Sur

viva

l Rat

e

Time

Sur

viva

l Rat

e

Time

Sur

viva

l Rat

e

t1 t2 t3 t4 t5 t6 t7 t8

0.0

0.2

0.4

0.6

0.8

1.0

G−SE−PApproximated

Time

Sur

viva

l Rat

e

Time

Sur

viva

l Rat

e

Time

Sur

viva

l Rat

e

t1 t2 t3 t4 t5 t6 t7 t8

0.0

0.2

0.4

0.6

0.8

1.0

G−SE−PApproximated

(a) (b)

28 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Survival Functions with c∗ = −0.04 and c = −0.025 forρ1 = 1, and ρ2 = 0 using n0 = 40 (a) and n0 = 80 (b)

Time

Sur

viva

l Rat

e

Time

Sur

viva

l Rat

e

Time

Sur

viva

l Rat

e

t1 t2 t3 t4 t5 t6 t7 t8

0.0

0.2

0.4

0.6

0.8

1.0

G−SE−PApproximated

Time

Sur

viva

l Rat

e

Time

Sur

viva

l Rat

e

Time

Sur

viva

l Rat

e

t1 t2 t3 t4 t5 t6 t7 t8

0.0

0.2

0.4

0.6

0.8

1.0

G−SE−PApproximated

(a) (b)

29 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Data

372 subjects from the VIRAHEP-C (Viral Resistance toAntiviral 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}

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Survival Functions for n0 = 37 with ρ1 = ρ2 = 0.5 (a) andρ1 = 1 and ρ2 = 0 (b)

Time

Sur

viva

l Rat

e

Time

Sur

viva

l Rat

e

Time

Sur

viva

l Rat

e

t1 t2 t3 t4 t5 t6 t7 t8

0.0

0.2

0.4

0.6

0.8

1.0

G−SE−PApproximated

Time

Sur

viva

l Rat

e

Time

Sur

viva

l Rat

e

Time

Sur

viva

l Rat

e

t1 t2 t3 t4 t5 t6 t7 t8

0.0

0.2

0.4

0.6

0.8

1.0

G−SE−PApproximated

(a) (b)

31 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

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 D1 = (ω1, ω2, τ0)through a validation subsample (“pilot dataset”).

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Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

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 D1 = (ω1, ω2, τ0)through a validation subsample (“pilot dataset”).

32 / 33

Introduction. . . . . . . . . .Methods

. . . . . . . . . . . .Results and Discussion

Thank You!

We offer a more flexible strategy to the conduct of clinical trials,there is a possibility to reduce cost and improve efficiency.

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