Ruth Keogh - How can landmarking be useful

How can landmarking be useful in
case-control studies?
Ruth Keogh
Department of Medical Statistics
London School of Hygiene and Tropical Medicine
Sir David Cox’s 90th Birthday Symposium
Introduction
General setting
Studying how an individual’s exposures impact on their survival using
observational data
I
It is often not cost effective to obtain all measurements for all
individuals in a cohort
I
I
I
Some exposure measures are expensive to obtain
Some resources are finite and we don’t want to waste them
So we use a case-control-type sample from the cohort instead
Introduction
General setting
Studying how an individual’s exposures impact on their survival using
observational data
I
It is often not cost effective to obtain all measurements for all
individuals in a cohort
I
I
I
Some exposure measures are expensive to obtain
Some resources are finite and we don’t want to waste them
So we use a case-control-type sample from the cohort instead
Introduction
General setting
Studying how an individual’s exposures impact on their survival using
observational data
I
It is often not cost effective to obtain all measurements for all
individuals in a cohort
I
I
I
Some exposure measures are expensive to obtain
Some resources are finite and we don’t want to waste them
So we use a case-control-type sample from the cohort instead
A nested case-control study
A nested case-control study
A nested case-control study
Analysis
Cox proportional hazards model
h(t; x) = h0 (t) exp{β x}
Partial likelihood
PL = ∏
tj
exp{β xij }
∑R˜ j exp{β xk }
Analysis
Cox proportional hazards model
h(t; x) = h0 (t) exp{β x}
Partial likelihood
PL = ∏
tj
exp{β xij }
∑R˜ j exp{β xk }
Extensions
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There remain questions that can be addressed using full cohort
data that have not been extended to case-control studies.
I
Making dynamic predictions of survival
I
Landmarking is one solution to this
Dynamic prediction of survival
I
Biological measures, e.g. blood pressure, lung function
I
Transition to a new state, e.g. received a transplant
Dynamic prediction of survival
I
Biological measures, e.g. blood pressure, lung function
I
Transition to a new state, e.g. received a transplant
Dynamic prediction of survival
I
Biological measures, e.g. blood pressure, lung function
I
Transition to a new state, e.g. received a transplant
Dynamic prediction of survival
I
Biological measures, e.g. blood pressure, lung function
I
Transition to a new state, e.g. received a transplant
How can we make dynamic predictions of survival?
I
Cox proportional hazards model with time-updated covariates?
h(t; x(t)) = h0 (t)eβ
T x (t )
I
Jointly modelling the time-varying covariates x(t) and the
survival process
I
Modelling transitions between multiple states
I
Landmarking
How can we make dynamic predictions of survival?
I
Cox proportional hazards model with time-updated covariates?
h(t; x(t)) = h0 (t)eβ
T x (t )
I
Jointly modelling the time-varying covariates x(t) and the
survival process
I
Modelling transitions between multiple states
I
Landmarking
How can we make dynamic predictions of survival?
I
Cox proportional hazards model with time-updated covariates?
h(t; x(t)) = h0 (t)eβ
T x (t )
I
Jointly modelling the time-varying covariates x(t) and the
survival process
I
Modelling transitions between multiple states
I
Landmarking
How can we make dynamic predictions of survival?
I
Cox proportional hazards model with time-updated covariates?
h(t; x(t)) = h0 (t)eβ
T x (t )
I
Jointly modelling the time-varying covariates x(t) and the
survival process
I
Modelling transitions between multiple states
I
Landmarking
What is landmarking?
Andersen, Cain, Gelber. Analysis of survival by tumor response.
Journal of Clinical Oncology 1983.
What is landmarking?
Andersen, Cain, Gelber. Analysis of survival by tumor response.
Journal of Clinical Oncology 1983.
What is landmarking?
Andersen, Cain, Gelber. Analysis of survival by tumor response.
Journal of Clinical Oncology 1983.
Landmarking for dynamic prediction
Van Houwelingen. Dynamic prediction by landmarking in event
history analysis. Scandinavian Journal of Statistics 2007.
Van Houwelingen, Putter. Dynamic prediction in clinical survival
analysis. CRC Press 2012.
Landmarking for dynamic prediction
Landmarking for dynamic prediction
Landmarking for dynamic prediction
Landmarking for dynamic prediction
Landmarking for dynamic prediction
Estimation
A separate model for each landmark
hLM (t; x(LM)) = h0LM (t) exp{βLM x(LM)}
Using time horizons
A separate model for each landmark
hLM (t; x(LM)) = h0LM (t) exp{βLM x(LM)}
Using time horizons
A separate model for each landmark
hLM (t; x(LM)) = h0LM (t) exp{βLM x(LM)}
Using time horizons
A separate model for each landmark
hLM (t; x(LM)) = h0LM (t) exp{βLM x(LM)}
Estimation
A ‘supermodel’
hLM (t; x(LM)) = h0LM (t) exp{β (LM)x(LM)}
Estimation
An extended ‘supermodel’
hLM (t; x(LM)) = h0 (t)eθ (LM ) exp{β (LM)x(LM)}
Predicting survival
T : time of death/event
Dynamic prediction of death within 5 years
Pr (T < LM + 5|T ≥ LM, x(LM)) = 1 −
b
S(LM
+ 5|x(LM))
b
S(LM|x(LM))
b |x(LM)))
b |x(LM)) = − log(S(T
H(T
Predicting survival
T : time of death/event
Dynamic prediction of death within 5 years
Pr (T < LM + 5|T ≥ LM, x(LM)) = 1 −
b
S(LM
+ 5|x(LM))
b
S(LM|x(LM))
b |x(LM)))
b |x(LM)) = − log(S(T
H(T
A simple example - full cohort
I
Cohort of 10,000 individuals
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Some individuals change state: x(t)=0,1
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10% event rate: h0 (t) exp{β x(t)}
A simple example - full cohort
A simple example - full cohort
A simple example - full cohort
A simple example - full cohort
A simple example - full cohort
A simple example - full cohort
Dynamic prediction in a case-control setting
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Joint modelling of the time-varying exposure x(t) and survival?
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Multistate modelling?
Borgan. Estimation of covariate-dependent Markov transition
probabilities from nested case-control data. Statistical Methods in
Medical Research 2002.
Landmarking in a case-control setting
1. Fit a series of Cox models starting at a series of new landmarks
2. Find estimated survival probabilities
T : time of death/event
Pr (T < LM + 5|T ≥ LM, x(LM)) = 1 −
b
S(LM
+ 5|x(LM))
b
S(LM|x(LM))
Baseline cumulative hazard
b 0 (t) =
H
∑ ∑˜
tj ≤t
Rj
1
exp{β xk } × No. at risk at tj /size of sampled risk set
Landmarking in a case-control setting
1. Fit a series of Cox models starting at a series of new landmarks
2. Find estimated survival probabilities
T : time of death/event
Pr (T < LM + 5|T ≥ LM, x(LM)) = 1 −
b
S(LM
+ 5|x(LM))
b
S(LM|x(LM))
Baseline cumulative hazard
b 0 (t) =
H
∑ ∑˜
tj ≤t
Rj
1
exp{β xk } × No. at risk at tj /size of sampled risk set
Landmarking in a case-control setting
1. Fit a series of Cox models starting at a series of new landmarks
2. Find estimated survival probabilities
T : time of death/event
Pr (T < LM + 5|T ≥ LM, x(LM)) = 1 −
b
S(LM
+ 5|x(LM))
b
S(LM|x(LM))
Baseline cumulative hazard
b 0 (t) =
H
∑ ∑˜
tj ≤t
Rj
1
exp{β xk } × No. at risk at tj /size of sampled risk set
Landmarking in a case-control setting
1. Fit a series of Cox models starting at a series of new landmarks
2. Find estimated survival probabilities
T : time of death/event
Pr (T < LM + 5|T ≥ LM, x(LM)) = 1 −
b
S(LM
+ 5|x(LM))
b
S(LM|x(LM))
Baseline cumulative hazard
b 0 (t) =
H
∑ ∑˜
tj ≤t
Rj
1
exp{β xk } × No. at risk at tj /size of sampled risk set
A simple example - using nested case-control data
I
Nested case-control sample with 1 control per case
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Using about 20% of the data
A simple example - using nested case-control data
A simple example - using nested case-control data
A simple example - using nested case-control data
A simple example - using nested case-control data
A simple example - using nested case-control data
Comments
I
There don’t appear to be any existing methods for making
dynamic predictions of survival in case-control studies
I
Landmarking is a simple way of doing this
I
I
I
I
when we have time-varying exposures
or time-varying exposure effects
...or both
It is fairly easy to apply in standard software
I
dynpred package in R
Comments
I
There don’t appear to be any existing methods for making
dynamic predictions of survival in case-control studies
I
Landmarking is a simple way of doing this
I
I
I
I
when we have time-varying exposures
or time-varying exposure effects
...or both
It is fairly easy to apply in standard software
I
dynpred package in R
Comments
I
There don’t appear to be any existing methods for making
dynamic predictions of survival in case-control studies
I
Landmarking is a simple way of doing this
I
I
I
I
when we have time-varying exposures
or time-varying exposure effects
...or both
It is fairly easy to apply in standard software
I
dynpred package in R
Comments
Refinements
I
Where to place the landmarks - are there different considerations
for nested case-control studies?
I
How to best handle missing data/error in exposure
measurements
I
... including visit times/frequencies determined by a patient
Further use of landmarking in case-control studies
I
To tackle questions about causality in a non-randomized setting
Gran, Rysland, Wolbers, Didelez, Sterne, Ledergerber, Furrer,
von Wyl, Aalen. A sequential Cox approach for estimating the causal
effect of treatment in the presence of time dependent confounding.
Statistics in Medicine 2010.
Comments
Refinements
I
Where to place the landmarks - are there different considerations
for nested case-control studies?
I
How to best handle missing data/error in exposure
measurements
I
... including visit times/frequencies determined by a patient
Further use of landmarking in case-control studies
I
To tackle questions about causality in a non-randomized setting
Gran, Rysland, Wolbers, Didelez, Sterne, Ledergerber, Furrer,
von Wyl, Aalen. A sequential Cox approach for estimating the causal
effect of treatment in the presence of time dependent confounding.
Statistics in Medicine 2010.
Happy birthday David

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