Comparing two frameworks for parametric multi-state modelling, applied to hospital admissions with COVID-19 Christopher Jackson MRC Biostatistics Unit, University of Cambridge With Brian Tom, Peter Kirwan, Shaun Seaman, Kevin Kunzmann, Anne Presanis, Daniela De Angelis (MRC Biostatistics Unit) Sema Mandal (Joint Modelling Cell and Epidemiology Cell at Public Health England) Canadian Statistical Sciences Institute October 2020 Chris Jackson 1/ 29
Overview ◮ CHESS: COVID-19 hospital admissions data in England ◮ Infer disease severity and hospital resource usage ◮ Methods for multi-state modelling based on ◮ cause-specific hazards ◮ mixture modelling ◮ Application of methods to CHESS data ◮ Differences and advantages of different modelling frameworks Chris Jackson 2/ 29
MRC Biostatistics Unit work on the course of the COVID-19 epidemic “Nowcasting” and forecasting ◮ Transmission models informed by deaths and other data ◮ Estimate, incidence, deaths, “R” number, regional differences ◮ Number of deaths corrected for reporting delay Outcomes for people with COVID-19 ◮ after onset of infection / symptoms ◮ after admission to hospital. this work ◮ probabilities of / times to ICU admission, death, discharge ◮ using multi-state modelling of hospital data ◮ insights into disease severity, hospital resource usage Details of this and related work at https://www.mrc-bsu.cam.ac.uk/tackling-covid-19/ Chris Jackson 3/ 29
COVID-19 Hospitalisation in England Surveillance System (CHESS) ◮ Daily individual patient-level data ◮ on every intensive care unit (ICU) admission with COVID-19 ◮ from all National Health Service “trusts” (areas). ◮ Subset of trusts (“sentinel”) report ◮ individual data on all hospitalised cases of COVID-19 ◮ presented in this talk . . . ◮ Restrict here to those with community-acquired infections ◮ swab confirming positive test taken 2 days after hospital admission or sooner ◮ excludes those infected while in hospital for other conditions ◮ Linked to official data on registered deaths Chris Jackson 4/ 29
Data structure We observe all dates of plus “censoring”, either: ◮ hospital admission ◮ still in hospital ◮ ICU admission ◮ alive, but unknown if still in hospital/discharged ◮ discharge or death at date of data extraction Multi-state model 3. Death 1. 2. ICU Hospital 4. Discharge (Note: patients return to hospital ward after discharge from ICU, but limited data on ICU exit dates, so “ICU” state includes post-ICU ward stay) Chris Jackson 5/ 29
Outcomes of hospital stay for these patients (at Aug 2, 2020) Died Discharged Still in Unknown Total hospital final care outcome ICU 417 476 143 32 1068 (20%) No ICU 1216 2769 336 81 4402 (80%) Total 1633 3245 479 113 5470 (30%) (59%) (9%) (2%) ◮ “Unknown” final outcomes: unknown whether discharged or still in hospital, but know they are alive. ◮ Tiny number of events with unknown date excluded Chris Jackson 6/ 29
Data summary: outcome proportions by age and gender Statistical models will allow outcome probabilities to be estimated while adjusting for censoring Chris Jackson 7/ 29
Data summary: times to events by age group Next event following hospital admission ◮ Moderate differences between ages ◮ Large times of right-censoring from some people still in hospital Chris Jackson 8/ 29
Multi-state modelling Compare two different frameworks for building a multi-state model 1. Cause-specific hazards for competing risks 2. Mixture models Both in fully parametric formulations ◮ stabilise estimation at later times when data sparse ◮ provide inputs for Bayesian epidemic models on outcomes for wider population . . . Chris Jackson 9/ 29
Multi-state modelling using cause-specific hazards of competing risks as in, e.g. Andersen and Keiding (Stat Meth Med Res 2002), Putter et al. (Stat. Med, 2007) ◮ Individual in state r subject to cause-specific hazards λ rs ( t ) of transition to state s . ◮ We use Semi-Markov model, where hazard depends on time t since entry to state r . ◮ Interpret λ rs ( t ) as hazards of latent times T rs of transition to each state s , where min u T ru is the transition that actually happens. ◮ In parametric formulation, λ rs ( t ) defined by a parametric time-to-event / survival model (e.g. gamma, Weibull . . . ) Chris Jackson 10/ 29
Multi-state modelling using cause-specific hazards (2) Here we define cause-specific distributions for the time from ◮ hospital – ICU (observed death/discharge considered as censoring) ◮ hospital – death (observed ICU/discharge considered as censoring) ◮ hospital – discharge (observed ICU/death considered as censoring) ◮ ICU – death (observed discharge considered as censoring) ◮ ICU – discharge (observed death considered as censoring) Full likelihood defined by the product of the five transition-specific likelihoods ◮ Five likelihoods can be maximised separately, as their parameters are distinct ◮ Can use standard survival modelling software to maximise each likelihood Chris Jackson 11/ 29
Parametric distributions Select among flexible parametric distributions defining the transition-specific hazards, using AIC ◮ generalised Gamma (Prentice 1975) ◮ 3-parameter distribution encompassing log-normal, Weibull and gamma ◮ fitted best for hospital-discharge, ICU-discharge ◮ cure models formed by mixing a standard parametric model with cure fraction: people who will never have the event. ◮ e.g. mixture of generalised gamma and point mass on t = ∞ ◮ fitted best for hospital-ICU, hospital-death, ICU-death ◮ represents “long tail” of survivors with long hospital stays ◮ don’t use these for transitions to discharge. assume everyone gets discharged eventually Some model parameters depend additively on age group and gender (selection determined by AIC) Chris Jackson 12/ 29
Checks of fit of cause-specific hazard models Against Kaplan-Meier estimates of time from hospital admission to latent event Note the survival curve converges to a value > 0 under the “cure” models Chris Jackson 13/ 29
Quantities of interest from cause-specific hazard models Given fitted cause-specific hazard functions λ rs ( t ) for each r → s transition, we want to estimate: 1. Probability that the next event after state r is state s ◮ Calculated as the solution to a differential equation (Kolmogorov forward equation) generalisation of principle that survival S ( t ) = exp ( − H ( t )) in a standard survival model 2. Expected time to next event (and between-person variability in this time) ◮ Not the mean of the distribution defining λ rs ( t )! Mean time to event s given that it happens before the competing events. ◮ Calculated by simulating a large population from the fitted model and summarising the sample. ◮ Simulate latent times for each competing event from the cause-specific parametric distribution, and take the minimum to be the one that happens. Chris Jackson 14/ 29
Quantities of interest from cause-specific hazard models Given fitted cause-specific hazard functions λ rs ( t ) for each r → s transition, we want to estimate: 1. Probability that the next event after state r is state s ◮ Calculated as the solution to a differential equation (Kolmogorov forward equation) generalisation of principle that survival S ( t ) = exp ( − H ( t )) in a standard survival model 2. Expected time to next event (and between-person variability in this time) ◮ Not the mean of the distribution defining λ rs ( t )! Mean time to event s given that it happens before the competing events. ◮ Calculated by simulating a large population from the fitted model and summarising the sample. ◮ Simulate latent times for each competing event from the cause-specific parametric distribution, and take the minimum to be the one that happens. Chris Jackson 14/ 29
Multi-state modelling using a mixture model see e.g. Larson and Dinse (JRSSC 1985), after Cox (1959), see also Lau, Cole and Gange (Amer. J. Epi, 2009) ◮ Constructed by ◮ Probabilities p rs that the next state after r is state s . ◮ A parametric distribution for the time S rs to the transition, given that the transition happens, for each possible transition r → s ◮ exactly the quantities we want to know! ◮ Note difference from cause-specific hazards model ◮ specify a distribution for the time S rs to the transition that actually happens ◮ instead of hazards of latent times T rs to competing events Chris Jackson 15/ 29
Multi-state modelling using a mixture model (2) Likelihood: Contribution to the likelihood of parameters θ , for an individual i in state r , defined by either: ◮ p rs f rs ( θ | y i ), where f rs is the density of the parametric model for the time to event s (if transition to this event is observed at y i ) ◮ a mixture model � s p rs (1 − F rs ( θ | y i )) if the next event after state r is unobserved, thus transition time is right-censored at y i ◮ Does not factorise nicely into terms for different competing risks, so is slower to maximise. Generalized gamma or log-normal models selected by minimum AIC Age group and gender effects estimated on both probability of event p rs and parameters of time to event distributions. Chris Jackson 16/ 29
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