two-way mapping of eq-5d-3l and eq-5d-5l: a copula-based method with application to the evaluation...

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Copula-mixture modelCost-e�ectiveness

Copula-based modelling of self-reported health scalesEQ-5D and the evaluation of drug therapies for rheumatic disease

Mónica Hernández-AlavaScHaRR, University of She�eld

Steve PudneyISER, University of Essex

York 14 Apr 2016

Hernandez-Pudney Copula models

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Economic evaluation of competing therapies

EQ-5D links clinical outcomes to (HR)QoL and is used in manyevaluations for NICE

Two components:5-dimensional description Y of health stateutility scale υ(Y ) to evaluate QoL

Quality of

life

Mortality

Quality-

adjusted life

years (QALY)

Marginal cost per QALY, ICER, etc

Treatment

cost

EQ-5D

health state

EQ-5D

utility scale

Clinical

trial


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Not everyone sees the bene�ts...

�Economics, second only to management, may

just be the biggest fraud ever perpetrated on the

world�

Richard Horton, Editor, The LancetTwitter Dec 31, 2012

https://twitter.com/richardhorton1/status/285694937792647168)


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Objectives

Old (3L) version of EQ-5D is being replaced by more sensitive(5L) version. How do we use old 3L and new 5L evidence?

Our objectives:1 Test hypothesis that 3L and 5L variants of the EQ-5D

instrument are mutually consistent descriptors of health states2 Develop more satisfactory method of mapping between 3L and

5L (or any other QoL scales). �Satisfactory� means it should:use a �exible statistical approachtreat the two scales symmetricallyseparate health description from utility scoring (so samemapping can be used with alternative scoring systems)deal appropriately with the �holes� in the EQ-5D-3L scale

3 Make recommendations to NICE and develop Stata software

4 Examine mapping from 3L to 5L in an existingcost-e�ectiveness study


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Three-level version of EQ-5D

MobilityI have no problems in walking aboutI have some problems in walking aboutI am con�ned to bed

Self-careI have no problems with self-careI have some problems washing or dressing myselfI am unable to wash or dress myself

Usual activities (e.g. work, study, housework, family or leisure activities)I have no problems with performing my usual activitiesI have some problems with performing my usual activitiesI am unable to perform my usual activities

Pain/discomfortI have no pain or discomfortI have moderate pain or discomfortI have extreme pain or discomfort

Anxiety/depressionI am not anxious or depressedI am moderately anxious or depressedI am extremely anxious or depressed


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3L and 5L versions of EQ-5D

Examples of 3L and 5L: mobility and pain/discomfort

3-level 5-levelMobilityI have no problems in walking about I have no problems in walking about

I have slight problems in walking aboutI have some problems in walking about I have moderate problems in walking about

I have severe problems in walking aboutI am con�ned to bed I am unable to walk aboutPain/discomfortI have no pain or discomfort I have no pain or discomfort

I have slight pain or discomfortI have moderate pain or discomfort I have moderate pain or discomfort

I have severe pain or discomfortI have extreme pain or discomfort I have extreme pain or discomfort


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The National Data Bank for Rheumatic Diseases

NDBRD is a register of patients with rheumatoid disease:

referred by US and Canadian rheumatologists

data supplied by patients + records from hospitals and physicians

data collection in Jan and Jul each year from 1998

records include very detailed clinical assessments (HAQ, etc)

also EQ-5D, with switch from 3L to 5L in Jan 2011 (n = 5,192cases)

both versions included in 27-page questionnaire for Jan 2011 sweep:

EQ-5D-5L on page 11EQ-5D-3L on page 22


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NDBRD response distributions (Jan 2011 sweep)

Mobility, Self-care, Usual activities Pain, Anxiety/depression

Self care, Anxiety/depression � �no problems� category is dominant

Pain/discomfort � dominant middle categories

Mobility, Usual activities � intermediate

⇒ Shouldn't presume same model for each domain?


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QoL measures

De�ne:5-dimensional vector of ordinal responses for EQ-5D-3L system: Y3

Utility valuation for 3L health state: υ3(Y3)

5L health state and utility score: Y5, υ5(Y5)

EQ-5D-3L has 35 = 243 states; EQ-5D-5L has 55 = 3125 states

Dolan (1997) for 3L & Devlin & van Hout (2015) for 5L estimated utilityvalues using time tradeo� and discrete choice experiments for core states+ regression extrapolation

Scales normalised to 0 = death, 1 = perfect health (worst states haveutilities -0.594 for 3L, -0.205 for 5L)

5L score more correlated with expected in�uences than 3L

Distribution of QoL score υ(Y ) smoother for 5L than 3L

Heavier extreme left-hand tail for 3L - including states worse than death


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EQ-5D distributions

01

23

45

-.5 0 .5 1

EQ-5D-3L EQ-5D-5L


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Multi-equation ordinal response model

Reported outcomes for EQ-5D for domains d = 1 . . .53L: Y3id ∈ {1,2,3}5L: Y5id ∈ {1,2,3,4,5}System of 5 pairs of latent regressions

Y ∗3id = Xidβ3d +U3id

Y ∗5id = Xidβ5d +U5id

⎫⎪⎪⎬⎪⎪⎭

d = 1...5

Observed responses:

Ykid = q i� Γkqd ≤Y∗kid < Γk(q+1)d ; q = 1...k ; k = 3,5

X contains: clinical outcomes (quadratic in HAQ generaldisability & pain scale); demographics (age, gender)


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Copula speci�cation for residual distribution

Factor residual structure:

Ukid = ψkdVi +εkid

Normal mixture speci�cation for V :

G(V ) = πΦ((V −µ1)/σ1)+ [1−π]Φ((V −µ2)/σ2)

Copula speci�cation of Fd(ε3d ,ε5d)Fd(ε3d ,ε5d) = cd (G3d(ε3d),G5d(ε5d);θd)

θd is scalar dependency parameter

Normal mixture speci�cation for marginal df of G3d(.),G5d(.)

⇒ New Stata command bicop for estimating bivariate system


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Dependence patterns

By Avraham - Own work, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=38108215


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Five copula forms

Gaussian (symmetric, uniform dependence):

c(ε3,ε5) = Φ(Φ−1(ε3),Φ−1(ε5);θ) −1 ≤ θ ≤ 1

Frank (symmetric dependence, weaker in tails than Gaussian):

c(ε3,ε5) = −1

θln

⎛⎜⎝1+

(e−θε3 −1)(e−θε5 −1)e−θ −1

⎞⎟⎠

θ ≠ 0

Clayton (+ve dependence, strong in left tail):

c(ε3,ε5) = [max{ε−θ3 +ε

−θ5 −1,0}]

−1/θ

0 < θ ≤∞

Gumbel (+ve dependence, strong in right tail):

c(ε3,ε5) = exp(−[(− lnε3)θ +(− lnε5)θ ]1/θ

) θ ≥ 1

Joe (+ve dependence, stronger in right tail than Gumbel):

c(ε3,ε5) = 1−[(1−ε3)θ +(1−ε5)θ −(1−ε3)θ (1−ε5)θ ]1/θ

θ ≥ 1


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Domain-speci�c bivariate models: choice of

copula

Gaussian Frank Clayton Gumbel JoeMobility domainLog-likelihood -6656.54 -6665.73 -6727.46 -6669.82 -6736.73χ2(7) for H0 ∶ β3 = β5 29.02∗∗∗ 29.49∗∗∗ 23.82∗∗∗ 33.64∗∗∗ 37.14∗∗∗

Self-care domainLog-likelihood -4221.35 -4212.35 -4248.89 � �χ2(7) for H0 ∶ β3 = β5 8.31 5.98 5.35

Usual activities domainLog-likelihood -6772.96 -6796.04 -6866.11 -6785.64 -6829.65χ2(7) for H0 ∶ β3 = β5 10.87 10.22 10.89 11.23 11.53

Pain/discomfort domainLog-likelihood -6148.63 -6148.07 -6190.84 -6147.80 -6199.63χ2(7) for H0 ∶ β3 = β5 29.75∗∗∗ 30.26∗∗∗ 32.71∗∗∗ 29.09∗∗∗ 26.82∗∗∗

Anxiety/depression domainLog-likelihood -6243.59 -6238.86 -6300.55 -6244.72 -6302.70χ2(7) for H0 ∶ β3 = β5 12.05∗ 8.56 5.10 10.66 11.86

⇒ Consistency hypothesis rejected for mobility and pain domains


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Domain-speci�c bivariate models: non-normal

marginals

Gaussian marginals for Mobility, Self-care, Anxiety/depression domains

For Usual activities and Pain/discomfort domains:

Gaussian marginals Non-Gaussian marginalsPreferred H0 ∶ β3 = β5

Domain AIC BIC mixture AIC BIC χ2(7)

Usual activities1 13587.9 13725.5 equal 13550.5 13707.8 8.39(Gaussian copula)

Pain/discomfort2 12337.6 12475.3 unequal 12252.9 12429.9 40.91∗∗∗

(Gumbel copula)

Statistical signi�cance: * = 10%, ** = 5%, *** = 1%


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Non-normality

Non-normality insigni�cant for mobility, self-care anddepression/anxiety domainsVery slight leptokurtic non-normality for usual activitiesdomainFor pain/discomfort domain, di�erent leptokurtic 3L and 5Lresidual distributions

0.2

.4.6

-3 -2 -1 0 1 2 3

Mixture N(0,1)

0.1

.2.3

.4

-3 -2 -1 0 1 2 3

Mixture N(0,1)

(a) 3-level (b) 5-level


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Full 10-equation model

Two di�erent modelsnormal for V and marginal df of G(εkd)normal for V and equal mixture for marginal df of G(εkd)

Adding V improves AIC, BIC - mixture model bestCoe�cients of V

highly signi�cantnot signi�cantly di�erent within dimensions - exceptionpain/discomfortdi�erent across dimensionsproportion of the overall unobserved variance ranges from 0.09to 0.52 (lowest: self-care and anxiety/depression; largest: usualactivities)

Conclusions about equality of β s within dimensions unchanged� Mobility and Pain/discomfort di�er signi�cantly


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Estimates of full model

Type of mixture in ε

None Equal UnequalLog-likelihood -29197.46 -29136.23 -29132.50Number of parameters 115 118 124AIC 58624.91 58508.46 58513.00BIC 59378.73 59281.93 59325.80Mobility domain

Equality of β χ2(7) 26.59∗∗∗ 26.53∗∗∗ 25.69∗∗∗

Self-care domain

Equality of β χ2(7) 4.14 3.50 3.99

Usual activities domain

Equality of β χ2(7) 8.81 7.93 9.39

Pain/discomfort domain

Equality of β χ2(7) 31.64∗∗∗ 30.19∗∗∗ 36.58∗∗∗

Anxiety/depression domain

Equality of β χ2(7) 9.27 8.70 9.36

Statistical signi�cance: * = 10%, ** = 5%, *** = 1%.


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Mapping

Large stock of existing NICE recommendations based on 3L version

�Mapping� or �Cross-walking� ⇒ prediction of utility score under onesystem (e.g. υ5(Y5)) when a di�erent descriptive system is observed(e.g. Y3)

Common procedure:regress υ5(Y5) on υ3(Y3) in a validation dataset where both are observedfor the same subjectsuse regression estimates to make linear prediction for υ5(Y5) in theprimary dataset where only υ3(Y3) is observed

Drawbacks:linear regression often �ts poorlydistribution of υ3(Y3) is very non-normal with large ranges of zeroprobability ⇒ gives poor approximation to distribution of υ5(Y5)

Instead: predict υ5(Y5) using the conditional probability Pr(Y5∣Y3,X)and knowledge of the valuation scale υ5(.):

Pr (υ5(Y5) ≤ Υ∣Y3,X) = ∑Y5∈U(Υ)

Pr(Y5∣Y3,X)

(where U(Υ) = {Y ∶ υ5(Y ) ≤Υ})


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Mapped distributions

NDBRD data: map 3L → 5L prior to 2011 switch, 5L → 3L post-switch

3L → 5L mapping gives 2010 predictive distribution similar to 2012empirical distribution

5L → 3L less successful: 2012 predictive distribution misses the heavy lefttail of the 2010 empirical distribution

⇒ NICE needs to adopt EQ-5D-5L fully, not mapped scores from 5L to 3L

0.2

.4.6

.81

-.5 0 .5 1utility

EQ-5D-3L EQ-5D-5L

0.2

.4.6

.81

-.5 0 .5 1utility

EQ-5D-3L EQ-5D-5L

(a) Jan 2010 - 3L observed (b) Jan 2012 - 5L observed


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Example - the CARDERA cost-e�ectiveness study

Combination of Anti-Rheumatic Drugs in Early Rheumatoid Artyhritis(CARDERA) study (Choy et al 2008):

2-year, factorial design, double-blind, randomised, placebo-controlled trial

n = 467 (241 complete) cases with duration ≤ 24 months

4 treatment groups combining anti-rheumatic drugs methotrexate (MTX)

and ciclosporin (CS) and steroid prednisolone (PNS):Monotherapy (MTX)combination MTX and CSMTX + decreasing dose PNStriple therapy: MTX + CS + PNS

EQ-5D-3L and treatment costs observed 6-monthly for 2 years

CARDERA is UK-based, so we're applying US-based NDBRD results in adi�erent country


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Robustness of the CARDERA study

Monotherapy Combination therapiesMTX MTX+CS MTX+PNS MTX+CS+PNS

Total costs ¿7,503 ¿6,829 ¿6,323 ¿6,203EQ-5D-3L from trial data

Total QALYs 1.238 1.093 1.152 1.320ICER (for col therapy vs. row therapy)MTX only - ¿4,648 ¿13,714 -¿15,929MTX+CS ¿4,648 - -¿8,597 -¿2,765MTX+PNS ¿13,714 -¿8,597 - -¿714

EQ-5D-5L mapped from 3L trial data (independent domains model)Total QALYs 1.452 1.368 1.397 1.523MTX only - ¿8,021 ¿21,476 -¿18,254MTX+CS ¿8,021 - -¿17,440 -¿4,037MTX+PNS ¿21,476 -¿17,440 - -¿952

EQ-5D-5L mapped from 3L trial data (joint model)Total QALYs 1.440 1.343 1.375 1.504MTX only - ¿6,930 ¿18,100 -¿20,141MTX+CS ¿6,930 - -¿15,819 -¿3,873MTX+PNS ¿18,100 -¿15,819 - -¿926


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CARDERA study � conclusions

Policy criterion based on Incremental Cost-E�ectiveness Ratio:

ICER = di� in costs

di� in QALY

Commissioning rule used by NICE: recommend treatments withICER < ¿20000 (approx)

Wailoo et al (2014) found:

(i) triple therapy dominates all others (lower cost, higher QALY)(ii) MTX+PNS dominates MTX+CS(iii) Monotherapy more e�ective and costly than MTX+PNS with ICER of

¿13,721 ⇒ monotherapy cost-e�ective

Using EQ-5D-5L mapped from 3L data, result (iii) changes: ICER formonotherapy vs MTX+PNS rises to ¿18,100/¿21,455 ⇒ just under/nolonger cost-e�ective


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APPENDIX


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Existing descriptive work on 3L and 5L

Measurement properties of 5L vs 3L (Janssen et al 2013 andseveral other studies after)

reduces ceiling e�ect: better for measuring population healthbetter discriminative ability

General conclusionsHealth problems are more common due to lower ceiling e�ectbut less severe (Craig et al 2014)

increased prevalence of pain/discomfort andanxiety/depressionmost responses in these domains "move-up"

LimitationsMainly comparison of frequenciesSometimes di�erent 3L and 5L samples with no adjustments(Feng et al 2015)


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Spearman correlations of EQ-5D scores

Variable EQ-5D-3L EQ-5D-5LEQ-5D-3L 1.000 0.849EQ-5D-5L 0.849 1.000Female -0.051 -0.070Age 0.035 0.056HAQ score(0-3) -0.735 -0.766Pain scale (0-10) -0.707 -0.711Overall RADAI score -0.737 -0.753Global severity (0-10) -0.698 -0.726Disease duration (months) -0.053 -0.067Polysymptomatic distress scale 0.462 0.487Fatigue scale (0-10) -0.633 -0.670Sleep disturbance scale (0-10) -0.506 -0.540Arthritis activity (general) -0.611 -0.630Physical component score (SF-6D) 0.727 0.767Mental component score (SF-6D) 0.475 0.523


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Domain comparisons0

.2.4

.6.8

1y

-2 -1 0 1 2 3 4 5 6 7

Mobility0

.2.4

.6.8

1y

-2 -1 0 1 2 3 4 5 6 7

Self-care

0.2

.4.6

.81

y

-2 -1 0 1 2 3 4 5 6 7 8

Usual activities

0.2

.4.6

.81

y

-2 -1 0 1 2 3 4 5 6 7

Pain/discomfort

0.2

.4.6

.81

y

-3 -2 -1 0 1 2 3 4

Anxiety/depression


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Full parameter estimates

Domain-speci�c model Joint modelCoe�cient Std. error Coe�cient Std. error

Mobility domain - 3 levelsmale 0.4601 0.0543 0.5125 0.0637age/10 -0.0117 0.0169 -0.0067 0.0197pain/10 2.4178 0.3205 2.8928 0.3826HAQ 1.2370 0.1092 1.3765 0.1347

HAQ2 -0.9591 0.3880 0.0987 0.0627

pain2 0.0593 0.0522 -1.2067 0.4554HAQ * pain -0.3067 0.1603 -0.3134 0.1907ψ 0.6494 0.0416

Mobility domain - 5 levelsmale 0.3390 0.0430 0.3839 0.0504age/10 0.0506 0.0137 0.0612 0.0159pain/10 1.9446 0.2525 2.4359 0.2964HAQ 1.2235 0.0841 1.4009 0.1010

HAQ2 -0.4122 0.3099 0.0610 0.0470

pain2 0.0458 0.0397 -0.6556 0.3606HAQ * pain -0.3969 0.1283 -0.4656 0.1527ψ 0.6279 0.0317Dependency θ 0.7074 0.0139 0.5956 0.0203


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parameter estimates continued


Self-care domain - 3 levelsmale 0.6103 0.0662 0.6438 0.0688age/10 -0.1067 0.0204 -0.1096 0.0210pain/10 1.0591 0.4462 1.4948 0.4722HAQ 1.8555 0.1966 1.9641 0.2226

HAQ2 -0.6821 0.4457 -0.0444 0.0790

pain2 -0.0314 0.0729 -1.0048 0.4603HAQ * pain 0.0428 0.2036 0.0040 0.2144ψ 0.3163 0.0347

Self-care domain - 5 levelsmale 0.6366 0.0536 0.6779 0.0569age/10 -0.0949 0.0167 -0.1006 0.0175pain/10 1.2139 0.3390 1.7335 0.3669HAQ 1.5870 0.1270 1.7245 0.1432

HAQ2 -0.7787 0.3644 0.0097 0.0561

pain2 0.0182 0.0519 -1.1726 0.3852HAQ * pain 0.0764 0.1583 0.0276 0.1686ψ 0.3806 0.0289Dependency θ 6.0530 0.3145 5.5022 0.3051


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Usual activities domain - 3 levelsmale 0.2409 0.0539 0.3278 0.0781age/10 -0.0582 0.0168 -0.0751 0.0240pain/10 2.6254 0.3175 4.1937 0.4879HAQ 1.7515 0.1164 2.6488 0.1936

HAQ2 -1.3382 0.3756 -0.3058 0.0709

pain2 -0.1891 0.0503 -2.1676 0.5438HAQ * pain 0.0196 0.1594 -0.1170 0.2237ψ 1.0333 0.0819

Usual activities domain - 5 levelsmale 0.1923 0.0440 0.2462 0.0625age/10 -0.0751 0.0139 -0.0961 0.0195pain/10 2.4151 0.2616 3.7146 0.3862HAQ 1.6059 0.0925 2.2971 0.1437

HAQ2 -1.3418 0.3149 -0.1997 0.0581

pain2 -0.1386 0.0416 -2.0802 0.4497HAQ * pain 0.0367 0.1325 -0.0395 0.1881ψ 0.9943 0.0616Dependency θ 0.5560 0.0172 0.1019 0.0541

Common mixtureπ 0.0621 0.0461µ1 0.2841 0.4314µ2 -0.0188 0.0217

σ2

13.0482 0.8537

σ2

20.8587 0.0665


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Pain/discomfort domain - 3 levelsmale 0.1737 0.0472 0.2130 0.0562age/10 0.0332 0.0156 0.0274 0.0181pain/10 6.3976 0.4445 7.1520 0.4037HAQ 0.6059 0.0908 0.7806 0.1046

HAQ2 -2.3849 0.4493 -0.1176 0.0551

pain2 -0.1296 0.0488 -3.0418 0.4349HAQ * pain 0.4015 0.1796 0.1717 0.1849ψ 0.3705 0.0325π 0.5871 0.0787µ1 -0.0936 0.0528µ2 0.1331 0.0771

σ2

10.2850 0.0824

σ2

21.9866 0.2359

Pain/discomfort domain - 5 levelsmale 0.1085 0.0424 0.1278 0.0484age/10 -0.0504 0.0137 -0.0605 0.0155pain/10 6.0189 0.2887 6.9250 0.3362HAQ 0.6694 0.0819 0.7903 0.0936

HAQ2 -2.6218 0.3451 -0.1119 0.0460

pain2 -0.1042 0.0402 -3.0565 0.3848HAQ * pain 0.3632 0.1391 0.3352 0.1563ψ 0.5364 0.0301π 0.1075 0.0745µ1 0.1204 0.1985µ2 -0.0145 0.0195

σ2

12.6886 0.7068

σ2

20.7948 0.0830

Dependency θ 1.7094 0.0474 1.5660 0.0452


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Anxiety/depression domain - 3 levelsmale 0.0387 0.0491 0.0469 0.0495age/10 -0.1350 0.0148 -0.1355 0.0152pain/10 1.2087 0.2829 1.3453 0.2894HAQ 0.4322 0.0904 0.4549 0.0923

HAQ2 -0.2623 0.3495 -0.0663 0.0440

pain2 -0.0580 0.0436 -0.4026 0.3550HAQ * pain 0.1788 0.1471 0.1903 0.1478ψ 0.3257 0.0259

Anxiety/depression domain - 5 levelsmale -0.0137 0.0453 -0.0071 0.0462age/10 -0.1456 0.0137 -0.1482 0.0142pain/10 1.2094 0.2554 1.3614 0.2640HAQ 0.3731 0.0826 0.4139 0.0855

HAQ2 -0.4111 0.3179 -0.0526 0.0410

pain2 -0.0387 0.0401 -0.5557 0.3251HAQ * pain 0.2730 0.1354 0.2818 0.1377ψ 0.3554 0.0240Dependency θ 14.4849 0.5894 13.9413 0.5912

Common mixture - Joint modelπ 0.0250 0.0127µ1 -0.5004 0.2528µ2 0.0128 0.0072

σ2

15.6660 1.6944

σ2

20.8739 0.0286


two-way mapping of eq-5d-3l and eq-5d-5l: a copula-based method with application to the evaluation...

Health & Medicine