Joint Modelling of Event Counts and Survival Times:
Example Using Data from the MESS Trial
J.K. Rogers, J.L. Hutton & K. Hemming
To treat or not to treat
• Antiepileptic drugs (AEDs) come with side effects.
• In the case of early epilepsy are AEDs absolutely necessary?
• Does seizure type have an effect on risk of future
seizure, particularly risk of future tonic-clonic seizure
(regarded as a major seizure)?
Time to first seizure
The survival time, t, for an individual, is a realisation of the non-negative random variable T with underlying probability density function f (t). The survivor
function and hazard function are given by:
S(t) = P(T ≥ t) = 1 − F (t).
(
)
P(t ≤ T ≤ t + δt | T ≥ t)
h(t) = lim
.
δt→0
δt
1.0
0.8
0.6
All Seizures
Partial
Partial with 2 Degree T−C
Generalised
Other
Immediate
Deferred
0.0
0
2
4
6
8
Time in years
Partial Seizures are localised and involve only part of
the brain
Generalised seizures involve all of the brain and include tonic-clonic seizures
• Analysis of data which is in the form of times from
a well defined start point, up to a particular event of
interest
• The survivor function - probability that an individual’s survival time is greater than or equal to some
value t
• The hazard function - instantaneous event rate
Post-randomisation survival times - following randomisation to an antiepileptic treatment, time to first
seizure is measured for each individual.
0.2
• Seizure recurrence rate after a single untreated
seizure is around 50%
• Around 20-30% of epilepsy sufferers never achieve
long-term remission (3)
1.0
Time to first tonic−clonic seizure
0.8
0.6
0.4
Survival
0.2
All Seizures
Partial
Partial with 2 Degree T−C
Generalised
Other
0
2
4
Immediate
Deferred
Xi | νi ∼ Poisson(λi uiνi ),
Yi | νi ∼ Exponential(λiψi νi ),
νi ∼ Gamma(α, α).
6
• Treatment policy not influential on both time to first
seizure and time to first tonic-clonic seizure for those
experiencing only partial seizures
• For other seizure types immediate treatment is
favoured
• Seizure type also appears to be influential. For time
to first seizure of any type partial performs worst but
for time to first tonic-clonic seizure generalised perform
worst
The log-likelihood for the observed data D on all the n individuals is given by
n x
i −1
X
X
ℓ(α, β1, β2 | D) =
ln(α + k) + δi ln(xi + α) + xi ln(ui)
k=0
+α ln(α) + (xi + δi) ln(λi) + δi ln(ψi)
− ln(xi!) − (xi + α + δi ) ln(λiui + λiψi yi + α) .
Department of Statistics, University of Warwick, Coventry, CV4 7AL, J.K.Rogers@warwick.ac.uk
We fit simple models to the pre-randomisation seizure
rates and time to first seizure separately, and two versions
of the joint model, the first just considering the overall
affect of treatment on the seizure rate and the second
considering the effects of other influential variables.
We can predict what percentage of seizures experienced
by an individual pre-randomisation will be experienced
post-randomisation for the different seizure types and
treatment policies:
Seizure
type
Partial
2◦ Gen
Generalised
Other
Percentage (95% C.I.)
Immediate
Deferred
5.9 (2.9,8.8) 0.9 (0.6,1.4)
2.5 (1.7,3.5) 3.8 (2.7,5.4)
1.9 (1.4,2.6) 2.7 (2.0,3.6)
1.2 (0.3,4.5) 11.0 (4.0,30.4)
So for a person with 2◦ generalised seizures, on immediate treatment we would expect seizures to occur about
2.5% as often as before randomisation.
The estimated regression coefficients for the joint model are:
λi β1,0
β1,age
β1,2◦ gen
β1,gen
β1,other
α 0.800 (0.033)
-2.793 (0.151) ψi β2,0
-0.006 (0.002)
β2,trt
β2,age
-0.541 (0.161)
-0.351 (0.151)
β2,trt×age
β2,2◦ gen
-1.036 (0.348)
β2,gen
β2,other
β2,trt×2◦ gen
β2,trt×gen
β2,trt×other
-2.992 (0.281)
-1.716 (0.351)
-0.003 (0.004)
0.008 (0.006)
-0.710 (0.292)
-0.977 (0.280)
-1.456 (0.709)
2.152 (0.383)
2.067 (0.366)
3.954 (0.873)
• Cure rate models
Therefore the joint model is specified by the following equations:
(λi uiνi )xi exp(−λiuiνi )
fX|ν (xi | νi ; λi, ui) =
,
xi !
fY |ν (yi | νi ; λi, ψi) = λi ψi νi exp(−λiψi νi yi ),
αα νiα−1 exp(−ανi )
,
gν (νi; α) =
Γ(α)
with λi = exp(β1′ z1i) and ψi = exp(β2′ z2i). The covariate z1i contains baseline
explanatory variables and z2i contains a treatment indicator, with other explanatory variables and interaction terms; δi is the censoring indicator. The parameters
β1 and β2 are the vectors of regression coefficients.
i=1
Application to the MESS Trial
Further Work
8
Time in years
• High proportion cured
Most standard survival analysis may treat the prerandomisation event count information as a covariate,
possibly measuring this quantity as a covariate with error. We consider a technique that jointly models the
pre-randomisation seizure counts and post-randomisation
survival times.
We allow the pre-randomisation seizure count to depend
on the influential baseline covariates, random effects encapsulate additional heterogeneity within the population.
We present the simple case of including time to first seizure post-randomisation
only (2), but have developed a model that allows the joint modelling of time to
first and second seizure post-randomisation, with the pre-randomisation seizure
count.
We assume a Poisson process for seizures with rate λi νi , where λi is related to
the individuals baseline covariates, and νi follows a Gamma(α, α) distribution,
where α measures the degree of heterogeneity. Consequently, the event count
over a period ui for individual i, Xi, follows a Poisson distribution with rate
λi νi. The time to first seizure following randomisation, Yi, is then Exponential,
with rate λiψi νi , updated to allow for a treatment effect.
0.0
Survival Analysis
Pre-randomisation event count - the number of seizures
an individual has experienced in a given period of time
prior to entry to the trial, and
0.4
Early Epilepsy and MESS
• For those individuals who have had only a single
seizure, or have infrequent and mild seizures the question of whether to withhold treatment until absolutely
necessary becomes an interesting one
• MESS (1) randomised 1443 patients to either immediate or deferred treatment
• Outcomes of interest - time to first seizure, time to
first tonic-clonic seizure
Data arrives in two parts:
Survival
Further details of the mathematical concepts will be presented in small print.
Joint Modelling of Event Counts
and Survival Times
Exploratory Analysis
• Inclusion of further survival times
• Modifications to ψ
References
[1] MRC Multicentre Trial for Early Epilepsy and Single Seizures.
[2] Cowling, B. J., Hutton, J. L., and Shaw, J. E. H.
Joint modelling of event counts and survival times. JRRSC
Appl. Statist. 55, 1 (2006), 31–39.
[3] Marson, A., Jacoby, A., Johnson, A., Kim, L.,
Gamble, C., and Chadwick, D. Immediate versus
deferred antiepileptic drug treatment for early epilepsy and
single seizures: a randomised control trial. The Lancet 365
(2005), 2007–2013.