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Two-way mapping of EQ-5D-3L and EQ-5D-5L: A copula-based method with application to the evaluation...
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Transcript of Two-way mapping of EQ-5D-3L and EQ-5D-5L: A copula-based method with application to the evaluation...
PreambleData
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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Copula-mixture modelCost-e�ectiveness
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
Hernandez-Pudney Copula models
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Copula-mixture modelCost-e�ectiveness
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)
Hernandez-Pudney Copula models
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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?
Hernandez-Pudney Copula models
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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
Hernandez-Pudney Copula models
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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
Domain-speci�c model Joint modelCoe�cient Std. error Coe�cient Std. error
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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parameter estimates continued
Domain-speci�c model Joint modelCoe�cient Std. error Coe�cient Std. error
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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parameter estimates continued
Domain-speci�c model Joint modelCoe�cient Std. error Coe�cient Std. error
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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parameter estimates continued
Domain-speci�c model Joint modelCoe�cient Std. error Coe�cient Std. error
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
Hernandez-Pudney Copula models