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The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
The Rasch Model and its Potential forEmpirical Economics Research
Dr. Carolin Strobl
Institut fur Statistik, Ludwig-Maximilians-Universitat Munchen
Empirical Economics and EconometricsResearch Seminar 2011
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
The Rasch Model and its extensionsfrom Item Response Theory (IRT)
Objective
Model specification
Parameter estimation
Model diagnostics
Extended models
Example: Consumer survey
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Georg Raschwas a Danish mathematicianwho eventually became profes-sor of statistics at the economicsdepartment of the University ofCopenhagen, but: “It would bewrong to say that Rasch’s pro-fessorship was a indisputable suc-cess. [...] Rasch developed thecourse in statistics. This changewas very welcome to a seg-ment of students and scientists,namely the sociologists. But alarger segment of people, namelythe economists, found that thecourse in statistics had becomenext to useless.”(http://www.rasch.org)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
You are in good company...
citations of Rasch’s book from 1960 (reprinted 1980)
(ISI web of knowledge)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
You are in good company...
(http://www.sueddeutsche.de)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
You are in good company...
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
You are in good company...
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
You are in good company...
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Objective
measurement of a latent trait
I intelligence
I math skills
I ...
I attitude
I consumer satisfaction
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Data
report for each subject and each item
I was the item answered correctly?or
I did the subject agree to the item?
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Data
itemsubject 1 2 3 4 5 6
1 0 1 0 1 0 12 0 1 1 0 1 13 0 1 1 1 0 04 1 0 0 1 0 0
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Model specification
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Model
whether a subject can answer an item correctly dependson both
I the ability of the subject θi and
I the difficulty of the item βj
P(uij = 1|θi , βj) =eθi−βj
1 + eθi−βj
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Item-characteristic-curves (ICCs)
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
θθi
P(u
ij=1|
θθ i,ββ
j)
ββj=6
P(uij = 1|θi , βj): probability that subject i “beats” item j
for θi = βj : P(uij = 1|θi , βj) = 0.5
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Item-characteristic-curves
0 2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
θθi
P(u
ij=1|
θθ i,ββ
j)
items: ← easy hard→subjects: ← stupid smart→
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Specific objectivity
0 2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
P(u
ij=1|
θθ i,ββ
j)
the ordering of the subjects does not depend onwhich item is used for the comparison
(but the discriminatory power is higher in the center)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Specific objectivity
0 2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
P(u
ij=1|
θθ i,ββ
j)
Pax
1− Pax:
Pbx
1− Pbx=
Pay
1− Pay:
Pby
1− Pby
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Local stochastic independence
for a given ability (i.e. for one subject or several subjectswith the same ability) the probability of answering oneitem does not depend on answering another item
and vice versa
⇒ allows us to compute joint probability as product ofindividual probabilities
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Local stochastic independence
for a given ability (i.e. for one subject or several subjectswith the same ability) the probability of answering oneitem does not depend on answering another item
and vice versa
⇒ allows us to compute joint probability as product ofindividual probabilities
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Local stochastic independence
could be violated if, e.g.,
I solving one item is crucial for solving another one
I subjects copy each others solutions
or
I the latent trait is not unidimensional(subsets of items measure different latent traits⇒ scores correlated)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Sufficient statistics
row and column sums are sufficient statisticsfor person and item parameters
itemsubject 1 2 3 4 5 6 ri
1 0 1 0 1 0 1 32 0 1 1 0 1 1 43 0 1 1 1 0 0 34 1 0 0 1 0 0 2
sj 1 3 2 3 1 2
sufficient statistics contain all information on parameters
⇒ allows us to condition on row sums in ML estimation
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Sufficient statistics
row and column sums are sufficient statisticsfor person and item parameters
itemsubject 1 2 3 4 5 6 ri
1 0 1 0 1 0 1 32 0 1 1 0 1 1 43 0 1 1 1 0 0 34 1 0 0 1 0 0 2
sj 1 3 2 3 1 2
sufficient statistics contain all information on parameters
⇒ allows us to condition on row sums in ML estimation
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Derivation of the model
models equivalent to the Rasch model can be derivedfrom
I continuous and strictly monotone ICCs
and
I local stochastic independence and
I sufficient statistics
or
I specific objectivity
see Fischer and Molenaar (1995, ch. 2) for a summary
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Derivation of the model
models equivalent to the Rasch model can be derived
P(uij = 1|θi , βj) =ea(θi−βj )+b
1 + ea(θi−βj )+b
and have the properties of interval scales of measurementwith the same unit a for person and item parameters
the common form
P(uij = 1|θi , βj) =eθi−βj
1 + eθi−βj
with a = 1 would have the properties of a differencescale, but a = 1 is not testable (Fischer and Molenaar,1995, ch. 2)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Parameter estimation
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
ML estimation for Rasch Models
two kinds of parameters: person and item parameters
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Joint ML estimation
maximize joint Likelihood Lu(θ,β) w.r.t. θ and βsimultaneously
I problem: not consistent!(# parameters increases with sample size)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Joint ML estimation
maximize joint Likelihood Lu(θ,β) w.r.t. θ and βsimultaneously
I problem: not consistent!(# parameters increases with sample size)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Sufficient statistics
itemsubject 1 2 3 4 5 6 ri
1 0 1 0 1 0 1 32 0 1 1 0 1 1 43 0 1 1 1 0 0 34 1 0 0 1 0 0 2
sj 1 3 2 3 1 2
P(ui |θi ,β) = P(ui |ri , θi ,β) · P(ri |θi ,β)= P(ui |ri ,β) · P(ri |θi ,β)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Conditional ML estimation
I step 1: estimate item parameters β conditional onsufficient statistics r for subject parameters θ
Lu(r,θ,β)suff. stat.
= Lu(r,β)
maximize w.r.t. β
I step 2: estimate subject parameters θ with estimatesfor item parameters β plugged in
Lu(θ, β)
maximize w.r.t. θ
I problem: uncertainty from estimating β usually notaccounted for (Tsutakawa and Johnson, 1990)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Conditional ML estimation
I step 1: estimate item parameters β conditional onsufficient statistics r for subject parameters θ
Lu(r,θ,β)suff. stat.
= Lu(r,β)
maximize w.r.t. β
I step 2: estimate subject parameters θ with estimatesfor item parameters β plugged in
Lu(θ, β)
maximize w.r.t. θ
I problem: uncertainty from estimating β usually notaccounted for (Tsutakawa and Johnson, 1990)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Conditional ML estimation
I step 1: estimate item parameters β conditional onsufficient statistics r for subject parameters θ
Lu(r,θ,β)suff. stat.
= Lu(r,β)
maximize w.r.t. β
I step 2: estimate subject parameters θ with estimatesfor item parameters β plugged in
Lu(θ, β)
maximize w.r.t. θ
I problem: uncertainty from estimating β usually notaccounted for (Tsutakawa and Johnson, 1990)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Procedure
I estimate a few tens of item parameters from a largeperson sample = test calibration
I estimate the person parameter of one subject from afew tens of items
gives consistent estimates
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Procedure
I estimate a few tens of item parameters from a largeperson sample = test calibration
I estimate the person parameter of one subject from afew tens of items
gives consistent estimates
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Procedure
I estimate a few tens of item parameters from a largeperson sample = test calibration
I estimate the person parameter of one subject from afew tens of items
gives consistent estimates
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Procedure
I estimate a few tens of item parameters from a largeperson sample = test calibration
I estimate the person parameter of one subject from afew tens of items
gives consistent estimates
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Marginal ML estimationdifferent approach to “get rid of” the subject parametersθ for estimating the item parameters β:
I step 1: assume a distribution (usually the normal)F (θ)
I step 2: integrate the θ out
Lu(β) =
∫ΘLu(θ,β) ∂F (θ)
maximize w.r.t. β + constraints for identifiabilityI step 3: estimate subject parameters θ with estimates
for item parameters β plugged in
Lu(θ, β)
maximize w.r.t. θI problems
I uncertainty from estimating β not accounted forI distribution assumption may be wrong
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Marginal ML estimationdifferent approach to “get rid of” the subject parametersθ for estimating the item parameters β:
I step 1: assume a distribution (usually the normal)F (θ)
I step 2: integrate the θ out
Lu(β) =
∫ΘLu(θ,β) ∂F (θ)
maximize w.r.t. β + constraints for identifiability
I step 3: estimate subject parameters θ with estimatesfor item parameters β plugged in
Lu(θ, β)
maximize w.r.t. θI problems
I uncertainty from estimating β not accounted forI distribution assumption may be wrong
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Marginal ML estimationdifferent approach to “get rid of” the subject parametersθ for estimating the item parameters β:
I step 1: assume a distribution (usually the normal)F (θ)
I step 2: integrate the θ out
Lu(β) =
∫ΘLu(θ,β) ∂F (θ)
maximize w.r.t. β + constraints for identifiabilityI step 3: estimate subject parameters θ with estimates
for item parameters β plugged in
Lu(θ, β)
maximize w.r.t. θ
I problemsI uncertainty from estimating β not accounted forI distribution assumption may be wrong
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Marginal ML estimationdifferent approach to “get rid of” the subject parametersθ for estimating the item parameters β:
I step 1: assume a distribution (usually the normal)F (θ)
I step 2: integrate the θ out
Lu(β) =
∫ΘLu(θ,β) ∂F (θ)
maximize w.r.t. β + constraints for identifiabilityI step 3: estimate subject parameters θ with estimates
for item parameters β plugged in
Lu(θ, β)
maximize w.r.t. θI problems
I uncertainty from estimating β not accounted forI distribution assumption may be wrong
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Alternative estimation approaches
I based on Bayesian MCMC:assume marginal distribution+ prior on every parameter
I ...
(Fischer and Molenaar, 1995, ch. 3)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Restrictions
P(uij = 1|θi , βj) =ea(θi−βj )+b
1 + ea(θi−βj )+b
for a unique solution
I fix a = 1 and
I fix b by means of∑
j βj = 0 or set one βj = 0
+ for conditional ML: zero and perfect scores mustbe excluded from the data matrix
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Information of an item
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
θθi
P(u
ij=1|
θθ i,ββ
j)
ββj=6
in the Rasch model the information(discriminatory power) of an item j is its gradient
Ij(θi ) = ∂∂θi
P(uij = 1|θi , βj)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Information of an item
for all items the information adds up
I(θi ) =∑j
Ij(θi )
the information is the inverse of the variance, so that theconfidence interval for the ML estimator θi isθi ± z1−α
2
1√I(θi )
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Model diagnostics
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Graphical model testidea: item parameter estimates should not depend on theperson-sample
I split person sample, e.g., at the median of the rawscores ri
I plot βgroup 1 against βgroup 2
⇒ accept model if confidence ellipses cover bisector
●
●●
●
●
●
●
●
●
●
●
●
−3 −2 −1 0 1 2 3
−3
−2
−1
01
23
Geschlecht = Mann
Ges
chle
cht =
Fra
u 1
234
5
6
7
89
10
11
12
●
●●
●
●
●
●
●
●
●
●
●
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Andersen’s likelihood ratio (LR) testidea: item parameter estimates should not depend on theperson-sample
I split person sample into K subsamples based on,e.g., the raw scores
I compare ML-estimates βk from k = 1, . . . ,Ksubsamples and β from entire sample
LR =Lu(r, β)∏K
k=1 Luk(rk, βk)=
∏Kk=1 Luk(rk, β)∏Kk=1 Luk(rk, βk)
I T = −2 log LRas.∼ χ2 ((K − 1) · (M − 1)− (M − 1))
for M items
I H0 : model holds (LR = 1, T = 0)H1 : model violated (LR < 1, T >> 0)⇒ accept model if p-value is large
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Wald test
idea: item parameter estimates should not depend on theperson-sample
I split person sample into K (usually K = 2)subsamples based on, e.g., the raw scores
I compare ML-estimates β1 and β2
W = (β1 − β2)′(Σ1 + Σ2)−1(β1 − β2)
I Was.∼ χ2
I H0 : model holds (W = 0)H1 : model violated (W >> 0)⇒ accept model if p-value is large
note: LR and Wald tests, as well as Lagrange-Multiplier(LM) tests, are asymptotically equivalent
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Item specific Wald test
idea: item parameter estimates should not depend on theperson-sample
I split person sample into K (usually K = 2)subsamples based on, e.g., the raw scores
I compare ML-estimates βj ,1 and βj ,2
Wj =(βj ,1 − βj ,2)2
σ2j ,1 + σ2
j ,2
I sign(βj ,1 − βj ,2)√
Wjas.∼ N(0, 1)
I H0 : model holds for item j (Wj = 0)H1 : model violated for item j (|Wj | >> 0)(note: two-sided test)⇒ exclude item j if p-value is small (< 0.05)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Sample splitting
for graphical, LR and Wald tests: sample can be splitbased on
I the raw scores
I any other criterion, including covariates such asgender, age etc.
I usually the median is arbitrarily used for splitting
alternative approaches:
I “mixed” (mixture distribution) Rasch model
I Rasch trees
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Rasch trees
Strobl, Kopf, and Zeileis (2010a,b)
Geschlechtp < 0.001
1
Mann Frau
Alterp = 0.001
2
≤ 38 > 38
Node 3 (n = 25)
●
●
●
●
●
●
●
●
●●
●
●
1 2 3 4 5 6 7 8 9 10 11 12
−2.88
3.31Node 4 (n = 33)
● ●
●●
●
●
●
● ●●
●
●
1 2 3 4 5 6 7 8 9 10 11 12
−2.88
3.31Node 5 (n = 42)
●
● ●●
●
●
●
●●
●
●
●
1 2 3 4 5 6 7 8 9 10 11 12
−2.88
3.31
⇒ Achim’s Antrittsvorlesung
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Extended models
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
The Birnbaum (two parameter) model
0 2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
P(u
ij=1|
θ i,β
j,δj)
P(uij = 1|θi , βj , δj) =eδj (θi−βj )
1 + eδj (θi−βj )
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
The Birnbaum (two parameter) model
0 2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
P(u
ij=1|
θθ i,ββ
j,δδj)
comparison of items not specifically objective
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
The (Birnbaum) three-parameter model
0 2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
P(u
ij=1|
θθ i,ββ
j,δδj,γγ
)
P(uij = 1|θi , βj , δj , γj) = γj + (1− γj) ·
(eδj (θi−βj )
1 + eδj (θi−βj )
)
for multiple choice tests set γi = γ = 1number of options
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Partial credit model
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
P(u
ij=k|
θ i,β
jk)
P(uij = c|θi ,βj) =e c·θi−βjc∑mj
l=0 el ·θi−βjl
with c = 0, 1, . . . ,mj and βj0 = 0
(Masters, 1982)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Partial credit model
−5 0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
P(u
ij=k|
θ i,β
jk)
thresholds τj1, . . . , τjmj
(intersections of ICCs for categories 0 and 1, 1 and 2 etc.)
βj0 = 0, βjk =∑k
l=1 τjl
location τ(intersection of ICCs for categories 0 and mj)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Data
report for each subject and each item
I to what degree (for example 0 – 5 credits) was theitem answered correctly?or
I how strongly (on a scale from 0 – 5) did the subjectagree to the item?
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Model diagnostics
I if βjk and τjk are not ordered:item j violates model assumptions
I will be eliminated, e.g., by Wald test
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Other “ordinal” models
I Andrich’s rating scale modelspecial case of the partial credit model with a fixednumber of categories mj = m for each item
I Samejima’s graded response modelcumulative probabilities for passing the successivecategory thresholds
(see also Masters, 1982, for a comparison)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
The Rasch model as a generalized linear mixedmodel (GLMM)
I item parameters are considered as fixed effects
I person parameters are considered as random effects⇒ assume distribution (usually normal)θ ∼ N (0,Σ), i.e. the random effects θi aredeviations from the average
(Rijmen et al., 2003)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Example: Consumer survey
from Salzberger & Sinkovics (International MarketingReview, 2006)
I consumer study on technophobia (in ATM usage)
I five category Likert items
⇒ Partial Credit modelcategory ordering?
I samples from England (N = 278), Mexico (N = 200)and Austria (N = 449)
⇒ Differential Item Functioning?
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
Example: Consumer survey
from Salzberger & Sinkovics (International MarketingReview, 2006)
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
References and further reading I
Fischer, G. (Ed.) (1974). Einfuhrung in die Theoriepsychologischer Tests: Grundlagen und Anwendungen. Bern:Verlag Hans Huber.
Fischer, G. and I. Molenaar (Eds.) (1995). Rasch Models:Foundations, Recent Developments and Applications. NewYork: Springer.
Irtel, H. (1996). Entscheidungs- und testtheoretischeGrundlagen der Psychologischen Diagnostik. Frankfurt amMain: Verlag Peter Lang.
Masters, G. (1982). A Rasch model for partial credit scoring.Psychometrika 47(2), 149–174.
Rijmen, F., F. Tuerlinckx, P. De Boeck, and P. Kuppens(2003). A nonlinear mixed model framework for ItemResponse Theory. Psychological Methods 8(2), 185–205.
The Rasch Model
Carolin Strobl
Objective
Model specification
ICCs
Specific objectivity
Local stochastic independence
Sufficient statistics
Derivation of the model
Parameter estimation
ML estimation
Joint ML
Conditional ML
Marginal ML
Information of an item
Model diagnostics
Graphical test
LR test
Wald tests
Extended models
Birnbaum models
Models for ordinal data
The Rasch model as a GLMM
Example: Consumersurvey
References
References and further reading II
Strobl, C., J. Kopf, and A. Zeileis (2010a). A new method fordetecting differential item functioning in the Rasch model.Technical Report 92, Department of Statistics,Ludwig-Maximilians-Universitat Munchen, Germany.
Strobl, C., J. Kopf, and A. Zeileis (2010b). Wissen Frauenweniger oder nur das Falsche? – Ein statistisches Modell furunterschiedliche Aufgaben-Schwierigkeiten inTeilstichproben. In S. Trepte and M. Verbeet (Eds.),Allgemeinbildung in Deutschland – Erkenntnisse aus demSPIEGEL Studentenpisa-Test, Wiesbaden, pp. 255–272. VSVerlag.
Tsutakawa, R. and J. Johnson (1990). The effect ofuncertainty of item parameter estimation on abilityestimates. Psychometrika 55(2), 371–390.
Various authors (2007). Special volume: Psychometrics in R.Journal of Statistical Software 20.
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