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Evidence synthesis of competing interventions
when there is inconsistency in how effectiveness
outcomes are measured across studies Nicola Cooper
Centre for Biostatistics and Genetic Epidemiology, Department of Health Sciences, University of Leicester,
UK.http://www.hs.le.ac.uk/group/bge/
Acknowledgements: Tony Ades, Guobing Lu, Alex Sutton & Nicky Welton
MULTIPLE EVENT OUTCOMES
• Often the main clinical outcome differs across trials
Inconsistent reporting (e.g. mean, median)
Change in outcomes used over time
• Possible to use the available data to inform the estimation of others
EXAMPLE: Anti-viral for influenza• Three antiviral treatments for influenza
• Amantadine
• Oseltamivir
• Zanamivir
• No direct comparisons of the different antiviral treatments
• All trials compare antiviral to standard care
• Different outcome measures
• Time to alleviation of fever
• Time to alleviation of ALL symptoms
• Different summary statistics reported
• Median time to event
• Mean time to event
Time to alleviation of:
Number of trials Fever Symptoms
Amantadine vs. standard care 6 Oseltamivir vs. standard care 8 Zanamivir vs. standard care 5
• Oseltamivir and Zanamivir trials report median time to event of interest
• Amantadine trials report mean time to event of interest
• No direct comparison trials of all antivirals & standard care. Important to preserve within-trial randomised treatment comparison of each trial whilst combining all available comparisons between treatments (i.e. maintain randomisation).
DATA AVAILABLE
• In clinical studies with time to event data as the principal outcome, median time to event usually reported.
• However, for economic evaluations the statistic of interest is the mean => Area under survival curve
(i.e. provides best estimate of expected time to an event)
• Often mean time to an event canNOT be determined from observed data alone due to right-censoring
(i.e. actual time to an event for some individuals unknown either due to loss of follow-up or event not incurred by end of study)
BACKGROUND
Time to symptoms alleviated (hours)
pro
po
rtio
n w
ith s
ymp
tom
s
0 100 200 300 400 500
0.0
0.2
0.4
0.6
0.8
1.0
control75mg bid
PROBLEM: Mean
undefined
Last observation
censored => mean
undefined
Trt 1
Trt 2
CALCULATING MEAN FROM MEDIAN TIME
• Simplest approach is to assume an Exponential distribution for time with influenza, thus assuming a constant hazard function over time,
• Probability of still having influenza at time t,
• At median time,
)texp()t(S
medmedmed t
)2ln(
t
)5.0ln()texp(5.0
• Mean time to event = 1/ & its corresponding variance = 1/2r, where r = number of events incurred during the study period
THREE STATE MARKOV MODEL
1 = transition rate (hazard) from influenza onset to alleviation of fever
2 = transition rate from alleviation of fever to alleviation of symptoms
1/1 = expected time from influenza onset to alleviation of fever
1/2 = expected time from alleviation of fever to alleviation of symptoms
(1/1 + 1/2) = expected time from influenza onset to alleviation of symptoms
Influenza Alleviation of fever
Alleviation of symptoms
h1 h2 Alleviation of ALL symptoms
Alleviation of ALL symptoms
1 2
Assumptions:
• Equal treatment effects in each period, jk
• Baseline hazard during second period same as in first period plus an additional random effect term, j
EVIDENCE SYNTHESIS MODEL
ln(i1) = j + jk = -ln(1) # i1, flu to fev
alleviated
ln(i2) = j + j + jk = -ln(2) # i2, fev to sym alleviated
i3 = i1 + i2 = 1/1 + 1/2 # i3, flu to sym alleviated
jk ~ Normal(dk , 2) # log hazard ratio
j ~ Normal(g , Vg) # random effect
where i = trial arms, j = trials, k = treatments.
Prior distributions specified for dk , g , 2, Vg
Exp(dk) is the ratio of hazards of recovery at any time for an individual on treatment k relative to an individual on the standard treatment
If exp(dk) < 1 then treatment k is superior
If exp(dk) > 1 then standard treatment is superior
Model fitted in WinBUGS and evaluated using MCMC simulation
HAZARD RATIO / RELATIVE HAZARD
caterpillar plot: e.d
0.5 0.6 0.7 0.8 0.9 1.0
CATERPILLAR PLOT OF HAZARD RATIOS
Hazard Ratio
Improvement in rate of recovery
Oseltamivir
Amantadine
Zanamivir
RANKING TREATMENTSBest 0% Best 33%
Best 67% Best 1%
standard care
rank
1 2 3 4
0.0
0.5
1.0
Oseltamivir
rank
1 2 3 4
0.0
0.5
1.0
Amantadine
rank
1 2 3 4
0.0
0.5
1.0
Zanamivir
rank
1 2 3 4
0.0
0.5
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Days
Pro
po
rtio
n o
f in
div
idu
als
PROPORTION OF INDIVIDUALS IN EACH STATE
Standard careOseltamivirAmantadineZanamivir
InfluenzaAll symptoms alleviated
Fever alleviated
Ose
lta m
ivir
tria
ls 2
1 da
ys (
8 -22
%)
Zan
amiv
ir tr
ials
28
days
(7-
25%
)
WEIBULL MODEL• Relaxes assumption of constant hazard ( = shape)
Time (t)
Ha
zard
= 1
0 < < 1
= 2
Exponential
WEIBULL MODEL (cont.)
• Probability of still having influenza at time t,
S(t)=e -(t/) >0 (shape) >0 (scale)
• At median time,
0.5=e -(tmed/) tmed= (ln(2))1/
• If r out of n individuals still had symptoms at X days (i.e. end of trial), the proportion of censored individuals can be expressed as:
S(t(X)) =r/n=e -(t/)
Time to symptoms alleviated (hours)
pro
po
rtio
n w
ith s
ymp
tom
s
0 100 200 300 400 500
0.0
0.2
0.4
0.6
0.8
1.0
control75mg bid
AVAILABLE TRIAL DATA
Trt 1
Trt 20.5
tmed X
r/n
WEIBULL MODEL (cont.)
• Important to calculate E(S(t(X))|tmed) to take
account of the correlation between median time
(tmed) to alleviation of illness and proportion of
participants (r/n) still ill at X days as they are from the same trial dataset.
• Mean time to event (i.e. statistic of interest)
(1+1/)
EVIDENCE SYNTHESIS MODEL
Assumptions:
• Equal treatment effects in each period, jk
• Baseline hazard during second period same as in first period plus an additional random effect term, j
ln(i1) = j + jk = ln(i1) + ln( (1+1/1)) # i1, flu to fev alleviated
ln(i2) = j + j + jk # i2, fev to sym alleviated
i3 = i1 + i2 = i3 (1+1/3) # i3, flu to sym alleviated
jk ~ Normal(dk, 2) # log hazard ratio
j ~ Normal(g, Vg) # random effect
where i = trial arms, j = trials, k = treatments.
Prior distributions specified for dk , g , 2, Vg
EVIDENCE SYNTHESIS MODEL (cont.)
• and are the shape and scale parameters of a Weibull distribution respectively
• Model assumes the shape parameters are the same for time to alleviation of fever, 1, regardless of antiviral treatment & similarly for time to alleviation of symptoms, 3
• Due to lack of data, to estimate 1 set equal to 1 (i.e. exponential distribution). Could set constraint to ensure proportion still with fever at X days (i.e. end of trial) proportion still with symptoms
caterpillar plot: e.d
0.4 0.6 0.8 1.0
CATERPILLAR PLOT OF HAZARD RATIOS
Improvement in rate of recovery
Zanamivir
Oseltamivir
Amantadine
RANKING TREATMENTS
Best 0%
Best 1%
Best 20%
Best 79%
standard care
rank
1 2 3 4
0.0
0.5
1.0
zanamivir
rank
1 2 3 4
0.0
0.5
1.0
oseltamivir
rank
1 2 3 4
0.0
0.5
1.0
amantadine
rank
1 2 3 4
0.0
0.5
1.0
CONCLUSIONS• Although amantadine is ranked “best” it does have serious
side effects (e.g. gastrointestinal symptoms, central nervous system) which are not taken into account in this analysis
• This type of model could inform the effectiveness parameters of a cost effectiveness decision model
• Allows multiple outcomes & indirect comparisons to be modelled within a single framework
• Appropriate model for nested outcomes (e.g. progression-free survival & overall survival)
• If non-nested outcomes, then a multivariate meta-analysis model, as developed for surrogate outcomes, more appropriate