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Estimating IRT models with - gllamm -. Herbert Matschinger University of Leipzig – Department für Psychiatry e-mail: math@medizin.uni-leipzig.de. “Health-Related Quality Of Life” (HRQOL) . 5 3 - categorical items to assess health related quality of lifi. dichotomized {1} {2,3} Mobility

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Estimating IRT models with - gllamm -

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Estimating IRT models with - gllamm -

Herbert Matschinger

University of Leipzig – Department für Psychiatry

e-mail: math@medizin.uni-leipzig.de


“Health-Related Quality Of Life” (HRQOL)

  • 5 3 - categorical items to assess health related quality of lifi.

  • dichotomized {1} {2,3}

    • Mobility

    • Self-care

    • Usual (Daily) activities

    • Pain/discomfort

    • Anxiety/depression


Data

  • European Study of the Epidemiology of Mental Disorders (ESEMeD)

  • 2001 - 2003

  • 6 European countries :N = 21425

    • Belgium2419

    • France2894

    • Germany.3555

    • Italy4712

    • The Netherlands2372

    • Spain5473


Research Questions

  • Do these 5 items measure one single dimension ?

  • Is the sum of endorsements a sufficient statistic?

  • Do the item parameters differ between countries ?

  • Do the discrimination parameters differ between countries ?

  • How much dimensions should be assumed


Random Intercept Modell

2 sets of predictors :1) X – fixed effects

2) Z – random effects

De Boeck,P. and Wilson,M. (2004). Exploratory Item Response Models: A General Linear and Nonlinear Approach. New York, Berlin: Springer.


1 and 2- Parameter IRT

ηpi = log(πpi/(1-πpi)

Logit[Pr(ypi = 1; θp)] = - βi + θp

random intercept θp θp0Zi0

fixe Effekte -βi 

Logit[Pr(ypi = 1; θp)] = -βi + λiθp

Logit[Pr(ypi = 1; θp)] = X´pi βi + θp X´pi λi


Preparation of the data

  • egen pattern = group(eurod1-eurod5 countryn)

  • drop if pattern ==.

  • contract eurod1-eurod5 countryn pattern,f(wt2)

  • reshape long eurod,i(pattern)j(item)

  • for num 1/5:gen itemX=0 \ replace itemX=1 if item ==X


Structure of the data (long format)


Frequencies


1 -Parameter model for all 6 countries

gllamm eurod item1-item5, nocons link(logit) fam(bin) i(pattern) w(wt) adapt dot

Intercept (-β) SE

item1 ;-3.528638.043579

item2 ;-5.576689.061522

item3 ;-3.851309.045826

item4 ;-1.809436.033516

item5 ;-4.449809.050426

var(θ) = 6.4802599 .16565955

log likelihood = -31545.688


gllapred raschu,u -[posterior means]

(means and standard deviations will be stored in raschum1 raschus1)

Non-adaptive log-likelihood: -31522.338

-3.155e+04 -3.155e+04 -3.155e+04 -3.155e+04 -3.155e+04

Log-likelihood:-31545.694

gllapred raschmu,mu - [response probabilities]

(mu will be stored in raschmu)

Non-adaptive log-likelihood: -31522.338

-3.155e+04 -3.155e+04 -3.155e+04 -3.155e+04 -3.155e+04

Log-likelihood:-31545.694


gr7 raschmu raschum1,s([item]) t2("1 - Parameter No Differential Item Functioning - no country effect") ylab(0(0.1)1) yline(.5) psize(120) xline(0)


2 – Parameter (Birnbaum) model

eq discrim: item1-item5

gllamm eurod item1-item5, nocons link(logit) fam(bin) i(pattern) w(wt) eqs(discrim) adapt dot

e(b)[1,10]

eurod: eurod: eurod: eurod: eurod: pat1_1l: pat1_1l:

item1 item2 item3 item4 item5 item2 item3

y1 -4.8818004 -8.507911 -7.8598477 -1.6312255 -2.9198215 1.1467799 1.562892

pat1_1l: pat1_1l: pat1_1:

item4 item5 item1

y1 .55341432 .27273694 3.8073739

constraint def 1 [pat1_1]item1=1

gllamm eurod item1-item5, nocons link(logit) fam(bin) i(pattern) w(wt) constr(1) frload(1) eqs(discrim) adapt dot


2 - Parameter Model

Intercept SE

item1 ; -4.885114 .1492511

item2 ; -8.501162 .3370249

item3 ; -7.840406 .3939964

item4 ; -1.630332 .0350471

tem5 ; -2.919229 .0394554

var(1): 1 (0)

loadings for random effect 1

item1: 3.8137476 (.13478418)

item2: 4.3643911 (.20426133)

item3: 5.9397444 (.32311677)

item4: 2.1078878 (.05010409)

item5: 1.0385574 (.03638325)

LL = -30547.222


gr7 intmu intum1 ,s([item]) t2("2 - Parameter No Differential Item Functioning - no country effect") ylab(0(0.1)1) yline(.5) psize(120)


1 –Parameter model / effect on θ

  • Generate two sets of indicator variables for two different reference categories

  • Estimate the model twice for different reference categories

  • Compare the two results with respect to the differences of the fixed parameters (item difficulties)


1 –Parameter model / effect on θ

  • char countryn[omit] 1 or

  • char countryn[omit] 2

  • xi3:eq f1: i.country

  • gllamm eurod item1 item2 item3 item4 item5,link(logit) fam(bin) adapt dot i(pattern) w(wt) nocons geqs(f1)

Caveat:The output does not tell you what contrast you have employed


1 –Parameter model effect on θ (reference group is Belgium (1))

Intercept (-β) SEeffect SE

Reference = Belgium

item1 | -3.510938 .0803299 France .54711696 (.0949265)

item2 | -5.55874 .0912916 Germany -.11129763 (.0931433)

item3 | -3.833834 .0815657 Italy -.19944865 (.08868224)

item4 | -1.788838 .0753648 Netherlands .14494306 (.10088877)

item5 | -4.432465 .0842277 Spain -.21903409 (.08672157)

var(1): 6.4690974 (.16557056)

LL = -31485.737 26


1 –Parameter model effect on θ (reference group is France (2))

Intercept (-β) SEeffect SE

Reference = France

item1 | -2.96341 .0696799 Belgium -.54803974 (.09490968)

item2 | -5.01124 .0813064Germany -.65890334 (.08620927)

item3 | -3.28631 .0709361Italy -.74705435 (.08142842)

item4 | -1.24127 .0651058Netherlands -.40267533 (.09432633)

item5 | -3.88495 .0737165 Spain -.76663319 (.07938433)

var(1): 6.4690974 (.16557056)

LL = -31485.737


Differences between „fixed“ parameters (β)

France

-2.96341

-5.011237

-3.28631

-1.241268

-3.884948

BelgiumDifference

-3.510938

-5.55874

-3.833834 0.548 for each item

-1.788838

-4.432465


Systematics in differences

  • The „fixed“ parameter depend on the contrast employed for the predictor.

  • The „fixed“ parameter are the item difficulties for the reference category of the predictor.

  • The difference in difficulties between the two estimates are the differences between the two reference categories (countries)

  • These differences are the same for all items


Modeling country differences via „fixed“ effects / effects on β

  • The „fixed“ effects depend on the reference category

  • Choose category 1 (Belgium) for reference

  • Define all possible interaction effects between the items and the 5 dummies (France to Spain)

  • Constrain all the 5 interaction effects to be equal for each item


Interactions and constraints

char countryn[omit]

xi3:i.countryn*item1 i.countryn*item2 i.countryn*item3 i.countryn*item4 i.countryn*item5

for A in num 2/5 \

B in num 1/4:constraint def A _Ico2Xit = _IBco2Xit

for A in num 6/9 \

B in num 1/4:constraint def A _Ico3Xit = _IBco3Xit

for A in num 10/13 \

B in num 1/4:constraint def A _Ico4Xit = _IBco4Xit

for A in num 14/17 \

B in num 1/4:constraint def A _Ico5Xit = _IBco5Xit

for A in num 18/21 \

B in num 1/4:constraint def A _Ico6Xit = _IBco6Xit


gllamm syntax

gllamm eurod item1- item5 _Ico2Xit- _I4co6Xit, link(logit) fam(bin)i(pattern) w(wt) nocons adapt dot constr(2/21)

gllamm model with constraints:

( 1) [eurod]_Ico2Xit - [eurod]_I1co2Xit = 0

( 2) [eurod]_Ico2Xit - [eurod]_I2co2Xit = 0

( 3) [eurod]_Ico2Xit - [eurod]_I3co2Xit = 0

( 4) [eurod]_Ico2Xit - [eurod]_I4co2Xit = 0

( 5) [eurod]_Ico3Xit - [eurod]_I1co3Xit = 0

( 6) [eurod]_Ico3Xit - [eurod]_I2co3Xit = 0

( 7) [eurod]_Ico3Xit - [eurod]_I3co3Xit = 0

( 8) [eurod]_Ico3Xit - [eurod]_I4co3Xit = 0

( 9) [eurod]_Ico4Xit - [eurod]_I1co4Xit = 0

(10) [eurod]_Ico4Xit - [eurod]_I2co4Xit = 0

(11) [eurod]_Ico4Xit - [eurod]_I3co4Xit = 0

(12) [eurod]_Ico4Xit - [eurod]_I4co4Xit = 0

(13) [eurod]_Ico5Xit - [eurod]_I1co5Xit = 0

(14) [eurod]_Ico5Xit - [eurod]_I2co5Xit = 0

(15) [eurod]_Ico5Xit - [eurod]_I3co5Xit = 0

(16) [eurod]_Ico5Xit - [eurod]_I4co5Xit = 0

(17) [eurod]_Ico6Xit - [eurod]_I1co6Xit = 0

(18) [eurod]_Ico6Xit - [eurod]_I2co6Xit = 0

(19) [eurod]_Ico6Xit - [eurod]_I3co6Xit = 0

(20) [eurod]_Ico6Xit - [eurod]_I4co6Xit = 0


Results

item1 | -3.510822 .0803441

item2 | -5.558637 .0913007

item3 | -3.833719 .081579

item4 | -1.788679 .0753829

item5 | -4.432352 .0842393

France Ico2Xit | .5468687 .0949383

Germany Ico3Xit | -.1115037 .0931599

Italy Ico4Xit | -.1996552 .0887003

Netherlands Ico5Xit | .1447117 .1009002

Spain Ico6Xit | -.2192302 .0867413

...........................................

...........................................

I4co2Xit | .5468687 .0949383

I4co3Xit | -.1115037 .0931599

I4co4Xit | -.1996552 .0887003

I4co5Xit | .1447117 .1009002

I4co6Xit | -.2192302 .0867413

Belgium

Item 1

Item 5


var(1): 6.4695502 (.16557072)LL= -31485.71012285193Compare with results from slide 19

These two models are equal ! Both models assume equal item functioning with respect to country differences


Modeling country differences without constraints on „fixed“ effects

  • Results for a model without these 20 constraints are completely different.

    • LL = -3115 compared to –31484 df(15)

  • Deviations from Belgium are different for each item

  • Item-difficulties are heterogeneous with respect to countries


Results

item1 | -3.510822 .0803441

item2 | -5.558637 .0913007

item3 | -3.833719 .081579

item4 | -1.788679 .0753829

item5 | -4.432352 .0842393

France Ico2Xit | .2672621 .1360669

Germany Ico3Xit | .4485834 .1301335

Italy Ico4Xit | -.3667433 .1284983

Netherlands Ico5Xit | -.1668095 .1474595

Spain Ico6Xit | .1085338 .122575

France I1co2Xit | .1939196 .1898053

GermanyI1co3Xit | -.6497205 .1954673

ItalyI1co4Xit | -.3702901 .1787872

NetherlandsI1co5Xit | -.2827915 .2075728

SpainI1co6Xit | -.1267169 .169957

Belgium

Item 1

Item 2


Results cont.

FranceI2co2Xit | -.2079065 .141196

GermanyI2co3Xit | -.4445011 .1373677

ItalyI2co4Xit | -.51551 .1304282

NetherlandsI2co5Xit | .1836286 .1448181

SpainI2co6Xit | -.1787188 .1247335

FranceI3co2Xit | .6728065 .1156087

GermanyI3co3Xit | -.0709686 .1138256

ItalyI3co4Xit | -.1572684 .1080919

NetherlandsI3co5Xit | .5405529 .1222693

SpainI3co6Xit | -.5646274 .1067354

FranceI4co2Xit | 1.640549 .1511489

GermanyI4co3Xit | -.7005679 .1700497

ItalyI4co4Xit | .4469239 .1478239

NetherlandsI4co5Xit | -.9392069 .196039

SpainI4co6Xit | .1914548 .1461186

Item 3

Item 4

Item 5


2 –Parameter Modell effect on θ

  • Now we choose the e – contrast of xi3

  • The model was estimated twice

    • Reference category 6 (Spain)

      • char countryn[omit] 6

    • Reference category 1 (Belgium)

      • char countryn[omit] 1

  • „Fixed“ effects keep virtually the same

  • „Loadings“ keep almost the same

  • The effect for the reference category is the negativ sum of all the other effects


2 –Parameter Modell effect on θ

xi3:eq f1:e.country

eq discrim: item1-item5

constraint def 2 [pat1_1]item1=1

gllamm eurod item1 item2 item3 item4 item5,link(logit) fam(bin) adapt dot i(pattern) w(wt) nocons geqs(f1) eqs(discrim) constr(2) frload(1)

gllapred birncontu,u

(means and standard deviations will be stored in birncontum1 birncontus1)

gllapred birncontmu,mu

(mu will be stored in birncontmu)


2 –Parameter Modell effect on θ

Intercept SE Loading SE

item 1: -4.801975 .1460378 item 1: 3.8015702 .13333549

item 2: -8.457937 .3378112 item 2: 4.3896022 .20675589

item 3: -7.658351 .3787267 item 3: 5.8764124 .31438171

item 4: -1.599355 .0351327 item 4: 2.1281491 .05062713

item 5: -2.903158 .0392471 item 5: 1.0433228 .03642365

Deviation SE

(1) Belgium: .00116297 .0249041 LL=-30506

(2) France: .13703951 .0233686

(3) Germany: -.02622923 .0212044

(4) Italy:-.10642868 .0196428

(5) Netherlands: .06823874 .0252931

(6) Spain: -.07378098 .0181395


gr7 birncontmu birncontum1 ,s([countryn]) ylab(0(0.1)1) yline(0.5) t1("2 - Parameter No Differential Item Functioning - effect on theta") xline(0)


LL für 1- and 2-parameter models


DIF for difficulties β(i-contrast)

xi3:i.countryn*item1 i.countryn*item2 i.countryn*item3 i.countryn*item4 i.countryn*item5

i.countryn _Icountryn_1-6 (naturally coded; _Icountryn_1 omitted)

unrecognized command: _Icountryn_

r(199);

gllamm eurod item1-item5 _Ico2Xit- _I4co6Xit, nocons link(logit) fam(bin) i(pattern) w(wt) constr(2) frload(1) eqs(discrim) adapt dot

.

Caveat:You will never know later on what contrast you have employed


Belgium

------------------------------

| Coef. Std. Err.

------+-----------------------

item1 | -5.060236 .2046231

item2 | -8.197401 .3721319

item3 | -7.625611 .4503168

item4 | -1.636369 .08049

item5 | -3.142707 .0942269


Item 1

----------------------------------

| Coef. Std. Err.

------+---------------------------

_Ico2Xit | .3273206 .1845059 France

_Ico3Xit | .6340536 .1725151 Germany

_Ico4Xit | -.4935873 .1723957 Italy

_Ico5Xit | -.1209027 .1988732 Netherlands

_Ico6Xit | .1167805 .1615509 Spain


Item 2

----------------------------------

| Coef. Std. Err.

------+---------------------------

_I1co2Xit | .2694966 .2662902 France

_I1co3Xit | -.8450162 .2730304 Germany

_I1co4Xit | -.6450119 .2538839 Italy

_I1co5Xit | -.2683981 .2922587 Netherlands

_I1co6Xit | -.2463331 .2377217 Spain


Item 3

----------------------------------

| Coef. Std. Err.

------+---------------------------

_I2co2Xit | -.4286408 .2792733 France

_I2co3Xit | -.780114 .2661109 Germany

_I2co4Xit | -1.038312 .2641851 Italy

_I2co5Xit | .5058603 .2947703 Netherlands

_I2co6Xit | -.3856712 .2397361 Spain


Item 4

----------------------------------

| Coef. Std. Err.

------+---------------------------

_I3co2Xit | .607621 .1018758 France

_I3co3Xit | -.053771 .0992247 Germany

_I3co4Xit | -.1220413 .0946785 Italy

_I3co5Xit | .4768741 .1079205 Netherlands

_I3co6Xit | -.5042381 .0935174 Spain


Item 5

----------------------------------

| Coef. Std. Err.

------+---------------------------

_I4co2Xit | 1.061981 .1076839 France

_I4co3Xit | -.4796974 .1252521 Germany

_I4co4Xit | .3253115 .1061347 Italy

_I4co5Xit | -.6806979 .1476796 Netherlands

_I4co6Xit | .1797599 .104995 Spain


loadings for random effect 1

***level 2 (id)

var(1): 1 (0)

item1: 3.8973286 (.14143303)

item2: 4.3660867 (.20470644)

item3: 6.1098725 (.35212898)

item4: 2.1373606 (.05083788)

item5: 1.0655632 (.03739259)


DIF for difficulties β(i-contrast) LL=-30161.21


GermanyFranceSpainBelgium NetherlandsItaly


FranceBelgium Spain NetherlandsItalyGermany


NetherlandsBelgium Spain FranceGermanyItaly


FranceNetherlandsBelgium GermanyItaly Spain


FranceItalySpain Belgium Germany Netherlands


DIF for difficulty and discrimination (i-contrast)

eq discrimc: item1-item5 _Ico2Xit- _I4co6Xit

gllamm eurod item1- item5 _Ico2Xit- _I4co6Xit, link(logit) fam(bin) i(id) eqs(discrimc) constr(2) frload(1) w(wt) nocons adapt dot


Discrimination for Belgium

------------------------------

| Coef. Std. Err.

------+-----------------------

item1: 2.9217353 (.3955187)

item2: 3.5756061 (.41036239)

item3: 6.6334026 (.86959252)

item4: 2.0837394 (.14233753)

item5: .95331206 (.112838)


Item 1

_Ico2Xit: -.97672244 (.46282706) France

_Ico3Xit: -.14834368(.53854614)Germany

_Ico4Xit: 1.1158482 (.60224724) Italy

_Ico5Xit: .32108006 (.66483885)Netherlands

_Ico6Xit: .00363324 (.47251395)Spain


Item 2

_I1co2Xit: .05795306 (.57767685) France

_I1co3Xit: .79687968 (.7296182)Germany

_I1co4Xit: 1.7103824 (.73387108)Italy

_I1co5Xit: 1.0374762 (.86778915)Netherlands

_I1co6Xit: 1.178677 (.5917116)Spain


Item 3

_I2co2Xit: -1.6862693 (1.0787816)France

_I2co3Xit: -1.2215711 (1.1674393)Germany

_I2co4Xit: .12178561 (1.1594229)Italy

_I2co5Xit: -2.5445023 (1.0072729)Netherlands

_I2co6Xit: .7605634 (1.2962747) Spain


Item 4

_I3co2Xit: .20873075 (.21891064)France

_I3co3Xit: .05125628 (.18909945)Germany

_I3co4Xit: .07868985 (.18044014) Italy

_I3co5Xit: -.12776638 (.20528291) Netherlands

_I3co6Xit: .15262695 (.17397662)Spain


Item 5

_I4co2Xit: -.3214976 (.13368081)France

_I4co3Xit: .13574822 (.15949164)Germany

_I4co4Xit: .18829768 (.13714607)Italy

_I4co5Xit: -.14969769 (.18109017)Netherlands

_I4co6Xit: .50673288 (.13977853)Spain


Summary

  • The 5 Items of the HRQOL do not portray one single dimension

  • Anxiety/Depression measures a different dimension

  • By means of the precommands xi3 and constr many IRT models can be specified quite easily

  • gllamm is a perfect tool for specifying IRT models


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