bayesian approaches for addressing missing data in cost ... · bayesian approaches for addressing...

Bayesian approaches for addressing missing data in Cost-EffectivenessAnalysis (alongside Randomised Controlled Trials)

Gianluca Baio

(Thanks to Andrea Gabrio and Alexina Mason)

University College LondonDepartment of Statistical Science

[email protected]

http://www.ucl.ac.uk/statistics/research/statistics-health-economics/http://www.statistica.it/gianluca

https://github.com/giabaio

Network of Applied Statisticians in Health Seminar SeriesUniversity College London

Monday 29 October 2018

Gianluca Baio (UCL) Bayesian models for missing data in CEA NASH Seminar, 29 Oct 2018 1 / 24

http://www.ucl.ac.uk/statistics/research/statistics-health-economics/

http://www.statistica.it/gianluca

https://github.com/giabaio

Dress code


Dress code

J. Nash


Dress code

J. Nash A rather applied statistician “in health” (?)...


Dress code Smart casual...

J. Nash A rather applied statistician “in health” (?)...


Outline

1. Health economic evaluation– What is it?– How does it work?

2. Statistical modelling– Standard approach– The importance of being a Bayesian

3. Bayesian modelling for missing data in HTA– Modelling & advantages– Bayesian nature of dealing with missing data

4. Motivating example– Data (and their weird features...) & Bayesian modelling– Results

5. Conclusions


Health technology assessment (HTA)

Objective: Combine costs & benefits of a given intervention into a rational scheme forallocating resources, increasingly often under a Bayesian framework

Statisticalmodel

Economicmodel

Decisionanalysis

Uncertaintyanalysis

• Estimates relevant populationparameters θ

• Varies with the type ofavailable data (& statisticalapproach!)

• Combines the parameters to obtaina population average measure forcosts and clinical benefits

• Varies with the type of availabledata & statistical model used

• Summarises the economic modelby computing suitable measures of“cost-effectiveness”

• Dictates the best course ofactions, given current evidence

• Standardised process

∆e = fe(θ)

∆c = fc(θ)

. . .

ICER = g(∆e,∆c)

EIB = h(∆e,∆c; k)

. . .

• Assesses the impact of uncertainty (eg inparameters or model structure) on theeconomic results

• Mandatory in many jurisdictions (includingNICE)

• Fundamentally Bayesian!


HTA alongside RCTs Individual level data

• The available data usually look something like this:

Demographics HRQL data Resource use data Clinical outcomeID Trt Sex Age . . . u0 u1 . . . uJ c0 c1 . . . cJ y0 y1 . . . yJ

1 1 M 23 . . . 0.32 0.66 . . . 0.44 103 241 . . . 80 y10 y11 . . . y1J2 1 M 21 . . . 0.12 0.16 . . . 0.38 1 204 1 808 . . . 877 y20 y21 . . . y2J3 2 F 19 . . . 0.49 0.55 . . . 0.88 16 12 . . . 22 y30 y31 . . . y3J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

– yij = Survival time, event indicator (eg CVD), number of events, continuousmeasurement (eg blood pressure), . . .

– uij = Utility-based score to value health (eg EQ-5D, SF-36, Hospital Anxiety &Depression Scale), . . .

– cij = Use of resources (drugs, hospital, GP appointments, . . . )

• Usually aggregate longitudinal measurements into a cross-sectional summary and foreach individual consider the pair (ei, ci)

• HTA preferably based on utility-based measures of effectiveness• Quality Adjusted Life Years (QALYs) are a measure of disease burden, combining

– Quantity of life (ie the amount of time spent in a given health state)– Quality of life (ie the utility value attached to that state)


(“Standard”) Statistical modelling Individual level data

1 Compute individual QALYs and total costs as

ei =J∑j=1

(uij + uij−1) δj2 and ci =J∑j=0

cij ,[

with: δj = Timej − Timej−1

Unit of time

]

Time (years)

Qu

alit

y o

f lif

e (

sca

le 0

-1)

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

δj

uij+uij−12

QALYi = “Area under the curve”

2 (Often implicitly) assume normality and linearity and model independentlyindividual QALYs and total costs by controlling for (centered) baseline values, egu∗0i = (u0i − u0) and c∗0i = (c0i − c0)

ei = αe0 + αe1u∗0i + αe2Trti + εei [+ . . .], εei ∼ Normal(0, σe)

ci = αc0 + αc1c∗0i + αc2Trti + εci [+ . . .], εci ∼ Normal(0, σc)

3 Estimate population average cost and effectiveness differentials– Under this model specification, these are ∆e = αe2 and ∆c = αc2

4 Quantify impact of uncertainty in model parameters on the decision making process– In a fully frequentist analysis, this is done using resampling methods (eg bootstrap)


(“Standard”) Statistical modelling Individual level data

1 Compute individual QALYs and total costs as

ei =J∑j=1

(uij + uij−1) δj2 and ci =J∑j=0

cij ,[

with: δj = Timej − Timej−1

Unit of time

]

2 (Often implicitly) assume normality and linearity and model independentlyindividual QALYs and total costs by controlling for (centered) baseline values, egu∗0i = (u0i − u0) and c∗0i = (c0i − c0)

ei = αe0 + αe1u∗0i + αe2Trti + εei [+ . . .], εei ∼ Normal(0, σe)

ci = αc0 + αc1c∗0i + αc2Trti + εci [+ . . .], εci ∼ Normal(0, σc)

3 Estimate population average cost and effectiveness differentials– Under this model specification, these are ∆e = αe2 and ∆c = αc2

4 Quantify impact of uncertainty in model parameters on the decision making process– In a fully frequentist analysis, this is done using resampling methods (eg bootstrap)


What’s wrong with this?... (and why should you be Bayesian?)

1 Potential correlation between costs & clinical benefits– Strong positive correlation — effective treatments are innovative and result from

intensive and lengthy research ⇒ are associated with higher unit costs– Negative correlation — more effective treatments may reduce total care pathway costs

e.g. by reducing hospitalisations, side effects, etc.– Because of the way in which standard models are set up, bootstrapping generally only

approximates the underlying level of correlation — Bayesian methods usually do abetter job!

2 Joint/marginal normality not realistic– Costs usually skewed and benefits may be bounded in [0; 1]– Can use transformation (e.g. logs) — but care is needed when back transforming to

the natural scale– Should use more suitable models (e.g. Beta, Gamma or log-Normal) — generally

easier under a Bayesian framework

3 ... and of course Partially Observed data– Can have item and/or unit non-response– Missingness may occur in either or both benefits/costs– The missingness mechanisms may also be correlated

– Focus in decision-making, not inference — Bayesian approach particularly suited forthis!


Bayesian approach to HTA

• In general can represent the joint distribution as p(e, c) = p(e)p(c | e) = p(c)p(e | c)




ci

φicτc

µc[. . .]

ei

φie

τe

µe [. . .]

β1

Conditional model for c

Marginal model for e

ei ∼ p(e | φei, τe)ge(φei) = α0 [+ . . .]µe = g−1

e (α0)

φei = locationτe = ancillary

φci = locationτc = ancillary

φei = marginal meanτe = marginal variance

φci = conditional meanτc = conditional variance

φei = marginal meanτe = marginal scale

φci = conditional meanτc = shapeτc/φci = rate

ci ∼ p(c | e, φci, τc)gc(φci) = β0 + β1(ei − µe) [+ . . .]µc = g−1

c (β0)

• For example:

• Combining “modules” and fully characterising uncertainty about deterministicfunctions of random quantities is relatively straightforward using MCMC• Prior information can help stabilise inference (especially with sparse data!), eg

– Cancer patients are unlikely to survive as long as the general population– ORs are unlikely to be greater than ±5




ci

φicτc

µc[. . .]

ei

φie

τe

µe [. . .]β1

Conditional model for cMarginal model for e


e (α0)








c (β0)

• For example:






ci

φicτc

µc[. . .]

ei

φie

τe

µe [. . .]β1



e (α0)








c (β0)

• For example:ei ∼ Normal(φei, τe), φei = α0 [+ . . . ], µe = α0

ci | ei ∼ Normal(φci, τc), φci = β0 + β1(ei − µe) [+ . . .], µc = β0





• In general can represent the joint distribution as p(e, c) = p(c | e) = p(c)p(e | c)

ci

φicτc

µc[. . .]

ei

φie

τe

µe [. . .]β1



e (α0)








c (β0)

• For example:ei ∼ Beta(φeiτe, (1− φei)τe), logit(φei) = α0 [+ . . . ], µe = exp(α0)

1+exp(α0)ci | ei ∼ Gamma(τc, τc/φci), log(φci) = β0 + β1(ei − µe) [+ . . .], µc = exp(β0)






ci

φicτc

µc[. . .]

ei

φie

τe

µe [. . .]β1



e (α0)








c (β0)



• Combining “modules” and fully characterising uncertainty about deterministicfunctions of random quantities is relatively straightforward using MCMC

• Prior information can help stabilise inference (especially with sparse data!), eg





ci

φicτc

µc[. . .]

ei

φie

τe

µe [. . .]β1



e (α0)








c (β0)






The problems with missing data...

• We plan to observe nplanned data points, but end up with a (much) lower number ofobservations nobserved

– What is the proportion of missing data? Does it matter?...

• We typically don’t know why the unobserved points are missing and what theirvalue might have been

– Missingness can be differential in treatment/exposure groups

• ... Basically, not very very much we can do about it!– Any modelling based on at least some untestable assumptions– Cannot check model fit to unobserved data– Have to accept inherent uncertainty in our analysis!


Missing data mechanism: Missing Completely At Random (MCAR)

yi

µi xi

β

σ mi

πi

δ

Partially observed dataUnobservable parametersDeterministic function of random quantitiesFully observed, unmodelled dataFully observed, modelled data

Model of analysis Model of missingness

For example:yi ∼ Normal(µi, σ)µi = xiβ

For example:mi ∼ Bernoulli(πi)logit(πi) = δ

• yi = Outcome subject to missingness• mi = 1 if yi missing or 0 if yi observed (“missingness indicator”)• θ =

(θMoA,θMoM) = model parameters

– θMoA = (β, σ)– θMoM = δ


Missing data mechanism: Missing At Random (MAR)

yi

µi xi

β

σ mi

πi

δ


Model of analysisModel of missingness


For example:mi ∼ Bernoulli(πi)logit(πi) = xiδ





Missing data mechanism: Missing Not At Random (MNAR)

yi

µi xi

β

σ mi

πi

δ


Model of analysisModel of missingness


For example:mi ∼ Bernoulli(πi)logit(πi) = xiδ + f(yi)





Missing data analysis methods

• Complete Case Analysis– Elimination of partially observed cases– Simple but reduce efficiency and possibly bias parameter estimates

• Single Imputation– Imputation of missing data with a single value (mean, median, LVCF)– Does not account for the uncertainty in the imputation process

• Multiple Imputation (MI)– Missing data imputed T times to obtain T different imputed datasets– Each dataset is analysed and T sets of estimates are derived– Parameter estimates are combined into a single quantity– The uncertainty due to imputation is incorporated but the validity relies on the correct

specification of the imputation model

• “Full Bayesian”– Basically extends MI to model formally the missing mechanism


(Bayesian) Modelling for missing data

• Effectively, need to model a bivariate outcome (y,m), depending on the modelparameters

p(y,m | θ) = p(y | m,θMoA) p (m | θMoM) (Pattern mixture model)

= p(m | y,θMoM) p (y | θMoA) (Selection model)

– Common assumption: the two blocks of model parameters are independent — at leasta priori, eg assume p(θMoM,θMoA) = p(θMoM)p(θMoA)

– NB: They may not be so in the posterior distribution, eg in generalp(θMoM,θMoA | y,m) 6= p(θMoM | y,m)p(θMoA | y,m)

• Pattern mixture models– Needs to model the full possible missingness “patterns” m using a marginal distribution– Models for data more natural

• Selection models– Models directly the marginal distribution of the observable data– Needs to figure out how the missingness model may be affected by it


Missing data in HTA

• Missing data are complicated in any context– But are fairly established in medical/bio-statistical research

• In HTA it’s even more complicated...– Bivariate outcome, usually correlated– Normality not reasonable (skewness)– Other features of the data (“spikes”)– Main objective: decision-making, not inference!


Missing data in HTA Gabrio et al. (2017). PharmacoEconomics Open, 1(2), 79-97

Missing cost (2003-2009) Missing cost (2009-2015)

Missing effectiveness (2003-2009) Missing effectiveness (2009-2015)


Missing data in HTA Selection models

MCAR (e, c)

ci

φicψc

µcxci

ei

φie ψe

µe xei

mei

πei

γe

mci

πci

γc

Model of analysis for (c, e) Model of missingness for eModel of missingness for c


• mei ∼ Bernoulli(πei); logit(πei) = γe0

• mci ∼ Bernoulli(πci); logit(πci) = γc0



MAR (e, c)

ci

φicψc

µcxci

ei

φie ψe

µe xei

mei

πei

γe

mci

πci

γc



• mei ∼ Bernoulli(πei); logit(πei) = γe0 +∑K

k=1 γekxeik

• mci ∼ Bernoulli(πci); logit(πci) = γc0 +∑H

h=1 γchxcih



MNAR (e, c)

ci

φicψc

µcxci

ei

φie ψe

µe xei

mei

πei

γe

mci

πci

γc




k=1 γekxeik + γeK+1ei


h=1 γchxcih + γcH+1ci



MNAR e; MAR c

ci

φicψc

µcxci

ei

φie ψe

µe xei

mei

πei

γe

mci

πci

γc




k=1 γekxeik + γeK+1ei


h=1 γchxcih



MAR e; MNAR c

ci

φicψc

µcxci

ei

φie ψe

µe xei

mei

πei

γe

mci

πci

γc




k=1 γekxeik


h=1 γchxcih + γcH+1ci


Motivating example: MenSS trial

• The MenSS pilot RCT evaluates the cost-effectiveness of a new digital interventionto reduce the incidence of STI in young men with respect to the SOC

– QALYs calculated from utilities (EQ-5D 3L)– Total costs calculated from different components (no baseline)

QALYs

Fre

quen

cy

0.5 0.6 0.7 0.8 0.9 1.0

02

46

810 n1 = 27

mean = 0.904median = 0.931

costs (£)

Fre

quen

cy

0 200 400 600 800 1000

02

46

810 n1 = 27

mean = 208 median = 123

QALYs

Fre

quen

cy

0.5 0.6 0.7 0.8 0.9 1.0

02

46

810 n2 = 19

mean = 0.902 median = 0.943

costs (£)

Fre

quen

cy

0 200 400 600 800 1000

02

46

810 n2 = 19



Motivating example: MenSS trial Partially observed data



Time Type of outcome observed (%) observed (%)Control (n1=75) Intervention (n2=84)

Baseline utilities 72 (96%) 72 (86%)3 months utilities and costs 34 (45%) 23 (27%)6 months utilities and costs 35 (47%) 23 (27%)

12 months utilities and costs 43 (57%) 36 (43%)Complete cases utilities and costs 27 (44%) 19 (23%)

QALYs

Fre

quen

cy

0.5 0.6 0.7 0.8 0.9 1.0

02

46

810 n1 = 27

mean = 0.904median = 0.931

costs (£)

Fre

quen

cy

0 200 400 600 800 1000

02

46

810 n1 = 27


QALYs

Fre

quen

cy

0.5 0.6 0.7 0.8 0.9 1.0

02

46

810 n2 = 19

mean = 0.902 median = 0.943

costs (£)

Fre

quen

cy

0 200 400 600 800 1000

02

46

810 n2 = 19



Motivating example: MenSS trial Skewness & “structural values”



QALYs

Fre

quen

cy

0.5 0.6 0.7 0.8 0.9 1.0

02

46

810 n1 = 27

mean = 0.904median = 0.931

costs (£)

Fre

quen

cy

0 200 400 600 800 1000

02

46

810 n1 = 27


QALYs

Fre

quen

cy

0.5 0.6 0.7 0.8 0.9 1.0

02

46

810 n2 = 19

mean = 0.902 median = 0.943

costs (£)

Fre

quen

cy

0 200 400 600 800 1000

02

46

810 n2 = 19



Modelling1 Bivariate Normal

– Simpler and closer to “standard” frequentist model– Account for correlation between QALYs and costs

2 Beta-Gamma– Account for– Model the relevant ranges: QALYs ∈ (0, 1) and costs ∈ (0,∞)– But: needs to rescale observed data e∗it = (eit − ε) to avoid spikes at 1

3 Hurdle model– Model eit as a mixture to account for correlation between outcomes, model the

relevant ranges and account for structural values– May expand to account for partially observed baseline utility u0it

cit

φictψct

µct βt

eit

φiet

ψet

µet u∗0it αtMarginal model for e

eit ∼ Normal(φeit, ψet)φeit = µet + αt(u0it − u0t)φeit = µet + αtu

∗0it

Conditional model for c | ecit | eit ∼ Normal(φcit, ψct)φcit = µct + βt(eit − µet)




2 Beta-Gamma– Account for correlation between outcomes– Model the relevant ranges: QALYs ∈ (0, 1) and costs ∈ (0,∞)– But: needs to rescale observed data e∗it = (eit − ε) to avoid spikes at 1



cit

φictψct

µct βt

e∗it

φiet

ψet

µet u∗0it αt

Marginal model for e∗

e∗it ∼ Beta (φeitψet, (1− φeit)ψet)

logit(φeit) = µet + αt(u0it − u0t)φeit = µet + αtu

∗0it

Conditional model for c | e∗

cit | e∗it ∼ Gamma(ψctφcit, ψct)

log(φcit) = µct + βt(e∗it − µet)




2 Beta-Gamma– Account for correlation between outcomes– Model the relevant ranges: QALYs ∈ (0, 1) and costs ∈ (0,∞)– But: needs to rescale observed data e∗it = (eit − ε) to avoid spikes at 1



cit

φictψct

µct βt

e<1it

φiet ψet

µ<1et

u∗0it αt

e1it

e∗itπit

ditXit ηt

µet

Model for the structural onesdit := I(eit = 1) ∼ Bernoulli(πit)

logit(πit) = Xitηt

Mixture model for ee1it := 1

e<1it∼ Beta (φeitψet, (1− φeit)ψet)

logit(φeit) = µ<1et

+ αt(u0it − u0t)logit(φeit) = µ<1

et+ αtu

∗0it

e∗it = πite

1it + (1− πit)e<1

it

µet = (1− πt)µ<1et

+ πt

Conditional model for c | e∗

cit | e∗it ∼ Gamma(ψctφcit, ψct)

log(φcit) = µct + βt(e∗it − µet)


Bayesian multiple imputation (under MAR)

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Bivariate Normal

Individuals (n1 = 75) Individuals (n2 = 84)

—•— Imputed, observed baseline—•— Imputed, missing baseline× Observed



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MNAR Basic/preliminary analysis

• We observe n01 = 13 and n02 = 22 individuals with u0it = 1 and ujit = NA, forj > 1• For those individuals, we cannot compute directly the structural one indicator dit

and so need to make assumptions/model this– Sensitivity analysis to alternative MNAR departures from MAR

MNAR1. Set dit = 1 for all individuals with unit observed baseline utilityMNAR2. Set dit = 0 for all individuals with unit observed baseline utilityMNAR3. Set dit = 1 for the n01 = 13 individuals with u0i1 = 1 and dit = 0 for the

n02 = 22 individuals with u0i2 = 1MNAR4. Set dit = 0 for the n01 = 13 individuals with u0i1 = 1 and dit = 1 for the

n02 = 22 individuals with u0i2 = 1





MNAR1. Set dit = 1 for all individuals with unit observed baseline utility

MNAR2. Set dit = 0 for all individuals with unit observed baseline utilityMNAR3. Set dit = 1 for the n01 = 13 individuals with u0i1 = 1 and dit = 0 for the







MNAR1. Set dit = 1 for all individuals with unit observed baseline utilityMNAR2. Set dit = 0 for all individuals with unit observed baseline utility

MNAR3. Set dit = 1 for the n01 = 13 individuals with u0i1 = 1 and dit = 0 for then02 = 22 individuals with u0i2 = 1



Results — MNAR

Probability of structural ones0.0 0.1 0.2 0.3 0.4 0.5 0.6

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—•— Control (t = 1)—•— Intervention (t = 2)


Cost-effectiveness analysis


missingHE: a R package to deal with missing data in HTA

Objective: Run a set of complex models to account for different level of complexity &missingness

selection hurdle

summary plot

pic diagnostics

⇒ BCEA

Gabrio et al. (2018). https://arxiv.org/abs/1801.09541https://github.com/giabaio/missingHE


Conclusions

• A full Bayesian approach to handling missing data extends standard“imputation methods”

– Can consider MAR and MNAR with relatively little expansion to the basic model

• Particularly helpful in cost-effectiveness analysis, to account for– Asymmetrical distributions for the main outcomes– Correlation between costs & benefits– Structural values (eg spikes at 1 for utilities or spikes at 0 for costs)

• Need specialised software + coding skills– R package missingHE under development to implement a set of general models– Preliminary work available at https://github.com/giabaio/missingHE

prior <- list("mu.prior.e"=mu.prior.e,"delta.prior.e"=delta.prior.e)model <- run_model(data=data ,model.eff=e˜1,model.cost=c˜1,

dist_e ="norm",dist_c ="norm",type=" MNAR_eff",stand=FALSE ,program ="JAGS",forward=FALSE ,prob=c(0.05 ,0.95) ,n.chains=2,n.iter =20000 ,n.burnin=floor (20000/2) ,inits=NULL ,n.thin=1, save_model=FALSE ,prior=prior)

– Eventually, will be able to combine with existing packages (eg BCEA:http://www.statistica.it/gianluca/BCEA; https://github.com/giabaio/BCEA) toperform the whole economic analysis


https://github.com/giabaio/missingHE

http://www.statistica.it/gianluca/BCEA

https://github.com/giabaio/BCEA

Thank you!


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