rasch modeling v3 - biblio iab mda 04-2011

7/31/2019 Rasch Modeling V3 - Biblio IAB MDA 04-2011

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References

Tesio L. 2003, Measuring behaviours andperceptions: Rasch analysis as a tool for

rehabilitation research, J Rehabil Med. Illustrates nicely, but some confusing wordings

Bond T. & Fox. C. 2007, Applying the Rasch

model. Good introductory book for understanding Rasch possibilities, but

short on theoretical bases

Wright B. & Stone M. 1999, Measurementessentials. http://www.rasch.org/memos.htm#measess

http://www.rasch.org/memos.htm

http://www.rasch.org/memos.htm



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Outline

Introductory examples

Rasch basics Illustrating Rasch analysis: maps, fit, DIF

Extensions : IRT, latent regression, SEM

Software & conclusion



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Ex. 1: students answers to math

quizzes

Person 2/5+5/6 3,2² 105-66 sin(pi/3) 6+10 94 mod 7 log 1000 73 x 24

Person

score

1 1 1 1 0 1 0 0 1 5

2 0 0 0 0 1 0 0 0 1

3 1 1 1 0 1 1 0 1 6

4 1 1 1 1 1 1 1 0 7

5 0 0 1 0 1 0 0 0 2

6 1 0 1 0 1 0 0 1 4

Item

score 4 3 5 1 6 2 1 3

Correct answer : 1, incorrect : 0



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Ex. 1 aim: derive a measure of

each student’s math ability Measures of math abilities 1, 2, … should

be « objective » = invariant with respect toany subset of test items

Educational testing: context of initial Raschapplications

Limitations of Classical Test Theory : personraw scores depend on set of items (low forhard items, high for easy items).



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Ex.2 : fictitious mobility quiz –

expected answers

S u

b j e c t a b i l i t y

Item difficulty


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Ex.2: fictitious mobility quiz –

unexpected answers



Ex. 3: disabled children’s

participation in daily activities

Items: eating meals, getting in/out of bed,

dressing, … Responses: 3 levels = with a lot of difficulty,

with some difficulty, without difficulty

Measure of participation?



Ex. 4: rehabilitation patients’

progress

Barthel Index:

With help Independent

Feeding 5 10

Moving fromwheelchair to bed 5-10 15

Personal toilet 0 5

…



Other applications

Sociology: political opinions, crime

propensity Psychiatry: Geriatric Depression Scale, Mini-

mental Status

Quality of Life

…



Outline


Rasch basics Illustrating Rasch analysis: maps, fit, DIF





Rasch basics – terminology

uni response of person n to item i

Dichotomous response: uni{0;1} Failure/success. Endorses/does not endorse.

Person « ability » h n = latent trait

Item « difficulty » t i

Pni= Pr(uni=1 | n , t i)



Rasch preliminary assumptions

Unidimensionality of latent construct: h nR

Local independence : responses to items are mutually independent

once ability h n is known.



… lead to a logistic probability

distribution

)exp(1

)exp(] / 1[Pr

in

in

nnini uP i

t h

t h h

Ability h n t i

t i t i+



Logistic regression

interpretation

niinni

ni

ni PP

P h t t h .1itlog1

logslopeintercept

Ability h n t i

t i t i +



Mathematical consequences

Distances between person abilities h n-h m are

preserved with any item subset

Distances between item difficulties also

Each person’s score hn(=number of

successes) is a sufficient statistic to estimatehis/her ability h n.

Calibrate table relating scores to abilities.



Estimation of model parameters

Person abilities and item difficulties may be

estimated conjointly by maximum likelihood

But ML estimators are asymptotically biased

Item difficulties estimated by conditional ML

= Lc(

i

|t 1,…,t

N

) where t n

=sufficient statistics

of the abilities n

Iterative procedure



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t 3

Rasch models for polytomous

responses Each item has K ordered categories of

responses1:Strongly Disagree; 2:Disagree; 3:Agree; 4:Strongly Agree

Step k = difficulty of endorsing a category

higher than k (offset relative to the average

difficulty t i of item i).

t iDifficulty of endorsing

category within itemt i + 2 t i + 3 t i + 4

t 2 t 4

SD D A SA



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Logistic probability extended to

polytomous case

))(exp(1))(exp(] / [Pr ;k in

k innninik k uP k i

t h t h h t



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Outline


Rasch basics

Illustrating Rasch analysis: maps, fit, DIF





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Mappingpersons anditems

Patients with too high ability

to be accurately measured

by this set of items (too easy)



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Computer Adaptive Testing

Each examinee answers initially the same set

of calibrated standard difficulty items

The software Rasch-analyses the responses

on-the-fly, estimating the examinee’s ability

Next test items are adapted to the estimatedability of each examinee: not too easy nor too

hard.



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Fit statistics principles



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Differential Item Functioning

(DIF) principle Do the items relative difficulties differ

depending on the considered subpopulation?

I.e. is item invariance law not verified when

tested on 2 groups of subjects A and B ?

Test whether t i;B - t i;A lies within 95% CI

CI based on std errors of t i;A and t i;B estimations



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Graphical assessment of DIF

t i;A= t i;B

t i;A

t i;B

95% confidence interval



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DIF between admission and

discharge patients

For these

items

t i;A> t i;D



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Search for causes of DIF

The 3 items estimated as more difficult atadmission than at discharge :

Transfer bed/WC, bed/chair, wheelchair/walking :moving out of bed

At admission, patients in immediate post-acuteorthopedic / stroke conditions : many are initially

ordered to stay in bed (stabilization of fracture,optimization of anticoagulation, etc.)

« Artefactually » higher difficulty for these items

Should be evaluated once acute condition is over



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Outline


Rasch basics






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Item Response Theory

A.k.a « latent trait » models

Also postulates : Unidimensional latent trait h n indirectly

observed through binary/ordinal responses uni.

Independence of responses conditional on h n

Other shapes for probability of responses

Pni= Pr(uni=1 | h n)



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2-parameters IRT

Each item i has

difficulty t i and

discrimination orslope ai.

Pr(uni=1 | h n) =

F(ai (h n - t i))where F : logistic or

probit



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3 parameters IRT

Guessing

parameter ci

in

multiple choice:



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Crossing ICC in 2 and 3-

parameters IRT

Hierarchy of item

difficulties changesdepending on

ability range



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Pros and cons of 2 and 3-

parameters IRT Pros:

Model represents better data in some situations

Probit link : relationship with factor analysis

Cons:

Loose item invariance property

No item-person mapping



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Choose Rasch or IRT ?

In principle, goals differ:

IRT : descriptive, fit model to data

Rasch : prescriptive, check if data fit the

measurement model

In practice, frontier is fuzzy

Rasch when high stake measure, extensive

piloting ok (diagnose misfit and refine items)

IRT when need to adjust for item heterogeneity.



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Latent regression

Natural extension: use latent construct asoutcome in regression.

h n : random effect

Estimation : marginal maximum likelihood

edisturbanccovariate

;

covariate

1;10 ... nC nC nn x x h



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Measurement part of a

Structural Equations Model

.

Mealtimes

Participation

u.1

eat

u.2

prepare

food

u.3

choose

food

x.1

impairment

level

x.2

parentalstress

.

1

2

1

1

1

1

2

3

s 2



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Outline


Rasch basics






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Software

Rasch analysis: WINSTEPS (bundled with Bond & Fox book)

ConQuest, RUMM… SEM:

Mplus : + fast, simple syntax, quick author response indiscussion groups; - black box, terse documentation

GLLAMM in Stata: + great documentation, - slow

AMOS : no support for multilevel models? LISREL…

Freeware : R packages (a bit hard to get into) : eRm, ltm, sem

MINISTEPS = WINSTEPS demo (75 subjects, 25 items)



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Conclusion

Rasch modeling: do item responses allowinvariant measurement of a latent trait?

Fit statistics to diagnose unexpected behaviours

Differential Item Functioning to detect non-invariance

Latent trait on an interval-valued scale (additive) Natural integration into generalized linear

mixed models and SEM.



Thanks for your attention !

rasch modeling v3 - biblio iab mda 04-2011

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