Local response dependence and the Rasch factor model - …publicifsv.sund.ku.dk/~kach/Rasch6.pdf ·...

Local response dependence and the Rasch factormodel

Karl Bang Christensen

Dept. of Biostatistics, Univ. of CopenhagenRasch6

Cape Town

Karl Bang Christensen Local response dependence and the Rasch factor model

Uni-dimensional latent variable model

X1

TREATMENTδT

%%

X2

AGEδA

,, Θ

KK

33

##

��

X3

X4

Latent variable Θ, X1, . . . ,X4 items. Monotone relationships.[Holland and Rosenbaum, 1986]


Uni-dimensional latent variable model - Rasch model

X1

TREATMENTδT

%%

X2

AGEδA

,, Θ

KK

33

##

��

X3

X4

Latent variable Θ, X1, . . . ,X4 items. Monotone relationships.Rasch: invariance, sufficiency


Rasch model for item response Xi

Item parameters β̄i = (βi0, βi1, βi2, . . .)

P(Xi = x |Θ = θ) = exp(xθ + βix)Ki (θ)−1 (1)

for x = 0, 1, . . . ,mi .βi0 = 0 for convenience and Ki (θ) =

∑mih=0 exp(hθ + βih)


Assumption of local independence

Response vector X̄ = (Xi )i∈I

P(X̄ = x̄ |Θ = θ)∗=∏i∈I

P(Xi = x |Θ = θ)

yielding

P(X̄ = x̄ |Θ = θ) = exp

(Rθ +

∑i∈I

βixi

)K (θ)−1 (2)

where K (θ) =∏

i∈I Ki (θ)R =

∑i∈I Xi sufficient.


Marginal probability

P(X̄ = x̄) =

∫P(X̄ = x̄ |Θ = θ)ϕ(θ)dθ

= exp

(∑i∈I

βixi

)∫exp(Rθ)K (θ)−1ϕσ(θ)dθ


Conditional probability

P(X̄ = x̄ |R = r) =exp

(∑i∈I βixi

)γR( ¯̄β)

(3)

¯̄β = (β̄i )i∈I , (3) independent of θ


Likelihood functions, inference

MML

lMML( ¯̄β, σ) =∑v

∑i∈I

βixi + log

∫exp(Rθ)K (θ)−1ϕσ(θ)dθ (4)

CML (complete cases)

lCML( ¯̄β) =∑v

∑i∈I

βixvi − log γRv ( ¯̄β) (5)

Pairwise conditional

lPW ( ¯̄β) =∑v

∑i 6=i ′

βixvi − log γR

(i,i′)v

( ¯̄β) (6)

Implementation:ConQuest [Wu et al., 2007], DIGRAM [Kreiner, 2003], RUMM[Andrich et al., 2010]. Standard software [Christensen, 2006].


Uni-dimensional latent variable model

X1

TREATMENTδT

%%

X2

AGEδA

,, Θ

KK

33

##

��

X3

X4

Latent variable Θ, X1, . . . ,X4 items. Monotone relationships.[Holland and Rosenbaum, 1986]


Uni-dimensional latent variable model, local dependence

X1

TREATMENTδT

%%

X2

AGEδA

,, Θ

KK

33

##

��

X3

X4

Latent variable Θ, X1, . . . ,X4 items. Monotone relationships.[Andrich, 1991]


Uni-dimensional latent variable model, local dependence

X1

TREATMENTδT

%%

X2

AGEδA

,, Θ

KK

33

##

��

X3

X4

Latent variable Θ, X1, . . . ,X4 items. Monotone relationships.Rasch: invariance, sufficiency. [Kreiner and Christensen, 2007]


Testing LD

1 Generalized Tjur test [Tjur, 1982]

2 Include interaction term in (9), i.e. loglinear Rasch model[Kelderman, 1984]

3 Item splitting [for dichotomous items]

4 Correlation structure in residuals


Testing LD1 Generalized Tjur test

Result for three dichotomous Rasch items [Tjur, 1982]:

X1 ⊥ X2|X1 + X3

generalization [Kreiner and Christensen, 2004]:

X1 ⊥ X2|R(1) and X1 ⊥ X2|R(2)

where R(1) =∑

i 6=1 Xi and R(2) =∑

i 6=2 Xi are rest scores.(Note: two p-values, need to control type I error rate).

1Implementation: Standard software or the item screening in DIGRAMKarl Bang Christensen Local response dependence and the Rasch factor model

Testing LD2 Loglinear Rasch model

Rasch model (9):

P(X̄ = x̄ |Θ = θ) = exp

(Rθ +

∑i∈I

βixi

)K (θ)−1

include interaction term [Kelderman, 1984]:

P(X̄ = x̄ |Θ = θ) = exp

(Rθ +

∑i∈I

βixi + δ(x1, x2)

)K̃ (θ)−1 (7)

compare (9) and (7) using likelihood ratio test.

2Implementation: DIGRAMKarl Bang Christensen Local response dependence and the Rasch factor model

Testing LD3 Item splitting

Splitting dependent item X2 into X ∗i2(0) and X ∗i2(1).[Andrich and Kreiner, 2010]

Illustration (dichotomous items)

ID X1 X21 1 12 0 13 1 04 1 15 0 0

.

.

.

.

.

.

.

.

.

−→

ID X1 X∗2 (0) X∗

2 (1)1 1 . 12 0 1 .3 1 . 04 1 . 15 0 0 .

.

.

.

.

.

.

.

.

.

.

.

.

Fit model for items X ∗2 (0),X ∗2 (1),X3, . . . compare itemparameters (∼ DIF test).

Need asymptotic (co)variance for formal test.

3Implementation: RUMM, DIGRAMKarl Bang Christensen Local response dependence and the Rasch factor model

Item splitting

X1

TREATMENTδT

%%

X2

AGEδA

,, Θ

KK

33

##

��

X3

X4


Item splitting

X1

TREATMENTδT

%%

X2

AGEδA

,, Θ

33

##

��

X3

X4


Testing LD4 Residuals

Local dependence for (X1,X2). [Yen, 1984, Q3]

Observe (X1v ,X2v )v=1,...,N

Estimate (θv )v=1,...,N

Compute standardized residuals Riv = Xiv−E(Xi |θ̂v )

V (Xi |θ̂v )and

ρOBS = corr(R1,R2)

Test LD: Is observed value of ρOBS due to random variation

4Implementation: RUMMKarl Bang Christensen Local response dependence and the Rasch factor model

Testing LD using residuals




Compute residuals Riv = Xiv − E (Xi |θ̂v ) and

ρOBS = corr(R1,R2)


But we don’t know the null distribution of ρ ...


Testing LD using residuals




Compute residuals Riv = Xiv − E (Xi |θ̂v ) and

ρOBS = corr(R1,R2)


But we don’t know the null distribution of ρ ...

No formal test, rule-of-thumb ρOBS > ρ0,ρ0 = 0.2, 0.3, 0.6, . . .


The problem with residuals

we don’t know the distribution of θ̂

θ̂ is a biased estimate of θ

Riv can only take mi + 1 different values

under the Rasch model corr(R1,R2) < 0

0 1

0 .

1 .

[Kreiner and Christensen, 2011a]


Data example (9 items from the SF36, acute leukemia)

How much of the time during the past 4 weeks

Did you feel full of pep?

Have you been a very nervous person?

Have you felt so down in the dumps that nothing could cheer you up?

Have you felt calm and peaceful?

Did you have a lot of energy?

Have you felt downhearted and blue?

Did you feel worn out?

Have you been a happy person?

Did you feel tired?

Response options: All of the Time, Most of the Time, A Good Bitof the Time, Some of the Time, A Little of the Time, None of theTime


Generalized Tjur test

X9 X8

X7 X6

Θ

OO GG

77 44

//

��

**

X5

X4

X1 X2 X3

Latent variable Θ, X1, . . . ,X9 items. Results from item screening[Kreiner and Christensen, 2011b]


Log linear Rasch model

X9 X8

X7 X6

Θ

OO GG

77 44

//

��

**

X5

X4

X1 X2 X3

Latent variable Θ, X1, . . . ,X9 items.Same as Tjur + five additional (neg. LD also found, not shown)


Correlation matrix of observed residuals

1.00 −0.32 0.06 −0.12 0.16 −0.21 −0.16 0.14 −0.131.00 0.05 0.01 −0.31 0.39 0.04 −0.12 −0.13

1.00 0.19 0.10 0.02 −0.02 0.01 0.021.00 −0.11 −0.08 −0.22 0.12 −0.31

1.00 −0.25 −0.03 −0.05 0.121.00 −0.00 −0.09 −0.19

1.00 −0.39 0.161.00 −0.25

1.00


LD summary

Generalized Tjur tests and residuals disagreex x

x

x

x

x xx

x

x

x

All models incorrect. Agreement about the most highlysignificant pair

type I error known for Tjur tests.

need to know the (multivariate) distribution of the correlationmatrix (or the distribution of residuals) under the Raschmodel.


Rasch measurement

X1

TREATMENTδT

%%

X2

AGEδA

,, Θ

KK

33

##

��

X3

X4

Latent variable Θ, X1, . . . ,X4 items.


Rasch measurement, multidimensionality

X1

TREATMENT

δ1,T

,,

δ2,T

��

Θ1

FF

++OO

��

X2

AGE

δ1,A22

δ2,A

%%Θ2

++

''

X3

X4

Latent variable Θ, X1, . . . ,X4 items.


Two-dimensional model

Set of items split into d disjoint sets. For d = 2: θ̄ =

[θ1

θ2

]I = I1 ∪ I2, (8)

with items in I1 and I2 measuring latent variables θ1 and θ2

respectively.

P(X = x |Θ̄ = θ̄) =exp

(R1θ1 + R2θ2 +

∑i∈I ηixi

)K (θ̄)

=exp

(∑d Rdθd +

∑i∈I ηixi

)K (θ̄)

where Rd =∑

i∈Id xi and K (θ̄) =∏

d

∏i∈Id Ki (θd)


Marginal probability

P(X̄ = x̄) = exp

(∑i∈I

βixi

)∫exp

(∑d

Rdθd

)K (θ)−1ϕΣ(θ̄)d θ̄

can write d dim. marginal likelihood function. Unidim. likelihoodfunction (4) nested within, but singular.can’t compare nested models using LRT[Christensen et al., 2002, Harrell-Williams and Wolfe, 2014]


Conditional probability

P(X̄ = x̄ |R̄ = r̄) =exp

(∑i∈I βixi

)∏d γ

(d)rd ((β̄i )i∈Id )

d dim. conditional likelihood function is the sum∑d

lCML(β̄(d)).

Unidim. likelihood function (4) nested within ?


Rasch measurement, multidimensionality

Patterns in residuals

t-test

observed vs. expected sub score correlationn

Rasch factor model

Summary of tests and diagnostics by Horton et al[Horton et al., 2013]



1 -0.61 0.12 -0.212 0.76 0.14 0.053 -0.05 0.14 0.804 0.02 0.59 0.525 -0.56 -0.27 0.246 0.67 0.17 -0.157 0.27 -0.64 0.128 -0.32 0.63 -0.269 -0.11 -0.69 0.11



2 0.76 0.14 0.056 0.67 0.17 -0.157 0.27 -0.64 0.128 -0.32 0.63 -0.264 0.02 0.59 0.521 -0.61 0.12 -0.215 -0.56 -0.27 0.249 -0.11 -0.69 0.113 -0.05 0.14 0.80


Tests currently avaliable

Use patterns in residuals or Tjur tests to provide hypothetises

I1 ∪ I2

beware of circular logic

t-test, observed vs. expected sub score correlation and othertests are significant for many hypothetical I = I1 ∪ I2

Lack of overview, many tests, risk of type I error..

(still) need to know the (multivariate) distribution of thematrix of residuals under the Rasch model.


Rasch factor model

UseP(X̄ = x̄ |θ) = P(X̄ = x̄ |R̄ = r̄)P(R̄ = r̄ |θ)︸︷︷︸

ν

to write extended likelihood

lEML( ¯̄β, ν) =∑d

lCML(β̄(d)) +∑v

log νv

with the probabilities of the marginal sub score vector distributionas unrestricted parameters ν.Compare nested models using LRT [Christensen et al., 2002]


Factor models

(1, 5, 8) + (2, 6) + (3, 4) + (7, 9)

(1, 2, 5, 6, 8) + (3, 4) + (7, 9) (1, 3, 4, 5, 8) + (2, 6) + (7, 9) (1, 5, 8) + (2, 6) + (3, 4, 7, 9)

(1, 2, 5, 6, 8) + (3, 4, 7, 9) (1, 3, 4, 5, 8) + (2, 6, 7, 9)

(1, 2, . . . , 9)

Compare nested models using LRT (or use BIC [Schwarz, 1978])


Factor models

l = . . . ,NPAR = 205

l = . . . ,NPAR = 184 l = . . . ,NPAR = 194 l = . . . ,NPAR = 190

l = . . . ,NPAR = 141 l = . . . ,NPAR = 159

l = . . . ,NPAR = (45− 1) + (45− 1) = 88

Compare nested models using LRT (or use BIC [Schwarz, 1978])


Andrich, D. (1991).Book review: Langeheine and Rost ’Latent Trait and LatentClass Models’.Psychometrika, 56(1):155–168.

Andrich, D. and Kreiner, S. (2010).Quantifying response dependence between two dichotomousitems using the rasch model.Applied Psychological Measurement, 34(3):181–192.

Andrich, D., Sheridan, B., and Luo, G. (2010).RUMM2030 [Computer software and manual].RUMM Laboratory, Perth, Australia.

Christensen, K. B. (2006).Fitting polytomous Rasch models in SAS.Journal of applied measurement, 7(4):407–17.


Christensen, K. B., Bjorner, J. B., S., K., and Petersen, J. H.(2002).Testing unidimensionality in polytomous Rasch models.Psychometrika, 67:563–574.

Harrell-Williams, L. and Wolfe, E. (2014).Performance of the likelihood ratio difference (g2 diff) test fordetecting unidimensionality in applications of themultidimensional rasch model.Journal of applied measurement, 15(3):267–275.

Holland, P. W. and Rosenbaum, P. R. (1986).Conditional association and unidimensionality in monotonelatent variable models.The Annals of Statistics, 14(4):1523–1543.

Horton, M., Marais, I., and Christensen, K. B. (2013).Dimensionality, pages 137–158.


John Wiley & Sons, Inc.

Kelderman, H. (1984).Loglinear Rasch model tests.Psychometrika, 49:223–245.

Kreiner, S. (2003).Introduction to digram.Research Report 10, Department of Statistics, University ofCopenhagen.

Kreiner, S. and Christensen, K. B. (2004).Analysis of local dependence and multidimensionality ingraphical loglinear rasch models.Communications in Statistics - Theory and Methods,33:1239–1276.

Kreiner, S. and Christensen, K. B. (2007).Validity and objectivity in health-related scales: Analysis bygraphical loglinear rasch models.


In Multivariate and Mixture Distribution Rasch Models,Statistics for Social and Behavioral Sciences, pages 329–346.Springer New York.

Kreiner, S. and Christensen, K. B. (2011a).Exact evaluation of bias in Rasch model residuals.In Baswell, editor, Advances in Mathematics Research vol.12,pages 19–40. Nova publishers.

Kreiner, S. and Christensen, K. B. (2011b).Item screening in graphical loglinear rasch models.Psychometrika, 76(2):228–256.

Schwarz, G. E. (1978).Estimating the dimension of a model.Annals of Statistics, 6:461–464.

Tjur, T. (1982).A connection between Rasch’s item analysis model and amultiplicative poisson model.


Scandinavian Journal of Statistics, 9:23–30.

Wu, M. L., Adams, R. J., Wilson, M. R., and Haldane, S. A.(2007).ACER ConQuest Version 2: Generalised item responsemodelling software.Australian Council for Educational Research, Camberwell.

Yen, W. M. (1984).Effects of local item dependence on the fit and equatingperformance of the three-parameter logistic model.Applied Psychological Measurement, 8(2):125–145.


Local response dependence and the Rasch factor model - …publicifsv.sund.ku.dk/~kach/Rasch6.pdf ·...

Documents

Transcript of Local response dependence and the Rasch factor model - …publicifsv.sund.ku.dk/~kach/Rasch6.pdf ·...