Comparing two frameworks for parametric multi-state modelling, - - PowerPoint PPT Presentation

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

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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/

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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

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Data structure

We observe all dates of

◮ hospital admission ◮ ICU admission ◮ discharge or death

plus “censoring”, either:

◮ still in hospital ◮ alive, but unknown if still in

hospital/discharged at date of data extraction Multi-state model

1. Hospital

• 2. ICU
• 3. Death

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)

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Outcomes of hospital stay for these patients

(at Aug 2, 2020)

Died Discharged Still in hospital care Unknown final

• utcome

Total 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

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Data summary: outcome proportions by age and gender

Statistical models will allow outcome probabilities to be estimated while adjusting for censoring

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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

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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. . .

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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 λrs(t) of

transition to state s.

◮ We use Semi-Markov model, where hazard depends on time t

since entry to state r.

◮ Interpret λrs(t) as hazards of latent times Trs of transition to

each state s, where minu Tru is the transition that actually happens.

◮ In parametric formulation, λrs(t) defined by a parametric

time-to-event / survival model (e.g. gamma, Weibull. . .)

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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

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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)

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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

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Quantities of interest from cause-specific hazard models

Given fitted cause-specific hazard functions λrs(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 λrs(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.

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Quantities of interest from cause-specific hazard models

Given fitted cause-specific hazard functions λrs(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 λrs(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.

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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 prs that the next state after r is state s. ◮ A parametric distribution for the time Srs 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 Srs to the transition that

actually happens

◮ instead of hazards of latent times Trs 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:

◮ prsfrs(θ|yi), where frs is the density of the parametric model

for the time to event s (if transition to this event is observed at yi)

◮ a mixture model s prs(1 − Frs(θ|yi)) if the next event after state

r is unobserved, thus transition time is right-censored at yi

◮ 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 prs and parameters of time to event distributions.

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flexsurv R package

◮ Survival models with a wide range of flexible parametric

distributions

◮ Standard models (Weibull, Gamma etc.) or very flexible

models based on splines (Royston and Parmar).

◮ Users can define their own distributions ◮ Multi-state models with cause-specific hazards or using

mixture models

◮ Any parameter of any distribution can be modelled on

covariates (defining e.g. non-proportional hazards models)

◮ On CRAN, or latest features at

https://github.com/chjackson/flexsurv-dev

◮ Similar facilities in Stata (see Crowther and Lambert paper and

references)

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Full multi-state model for COVID-19 hospital data

Multi-state model

1. Hospital

• 2. ICU
• 3. Death

4. Discharge

Multi-state model comprised of:

• 1. Model for event after hospital admission (ICU, discharge, death)

◮ one mixture model with three components ◮ or three cause-specific hazard models

• 2. Model for event after ICU admission (discharge or death)

◮ one mixture model with two components ◮ or two cause-specific hazard models Chris Jackson 18/ 29

Goodness of fit of selected parametric models

Models following hospital admission Probability that event of each type has happenened by time t Compared with nonparametric (Aalen-Johansen) estimates

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Goodness of fit: models after ICU

For the overall multi-state model, the competing risks / cause-specific hazards version has better fit than the mixture model (AIC 45454 vs 46732)

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Probability of next event after admission

Male, older: higher risk of death ICU admission rates lower in oldest ages by policy Mixture model estimates slightly higher risks of death

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Time to next event after hospital admission

Distribution between patients in the time to the next event after hospital admission, given that event happens (mean, median,

10-90% quantiles)

Some disagreement between selected parametric models, driven by behaviour for censored data where eventual event is unknown

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Probability of death following ICU

◮ Probabilities of death for a person just admitted to ICU ◮ Clear increase in risk with age, and genders similar ◮ Combine with probabilities of events following hospital

admission . . .

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Time to death or discharge after ICU admission

◮ Times to next event for a person just admitted to ICU ◮ Consistent between models ◮ Combine with times to events following hospital admission . . .

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Overall probability of death given hospital admission

◮ Estimates from full multi-state model ◮ Probabilities of death from COVID-19 for a person just

admitted to hospital.

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Time to final outcome, given that outcome

◮ Predictions of time from hospital admission to final outcome

under full multi-state model

◮ Upper tail / mean of time-to-death distribution unstable

under mixture model (which has worse overall fit)

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Comparison of modelling frameworks in light of application

◮ Mixture model produces parameters of interest directly ◮ Mixture model harder to fit (need to maximise likelihoods over

more parameters)

◮ Cause-specific hazards model with “cure” distributions fitted

best in this application Purpose of our analysis was to summarise outcomes for population subgroups (age groups and gender). Other applications might focus on estimating effects of many covariates.

◮ Lots of literature about contrasting approaches: regression

models on cause specific hazards vs. on cumulative incidence function (Fine and Gray, and parametric equivalents)

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Ongoing work

Our estimates from these data have been presented to UK government advisory groups In ongoing work we are

◮ using similar methods to estimate hospital outcomes from

• ther administrative datasets (UK and Italy)

◮ estimates vary according to cohort selection

◮ investigating the effects of other covariates ◮ combining with other sources of evidence (e.g. case-fatality

and case-hospitalisation risk, transmission) to inform epidemic

• utcomes for wider population

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Thanks for listening!

chris.jackson@mrc-bsu.cam.ac.uk www.mrc-bsu.cam.ac.uk

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