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 I 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 I Some individuals change state: x(t)=0,1 I 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 I Joint modelling of the time-varying exposure x(t) and survival? I 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 I 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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