How Should We Assess the Fit of Rasch-Type Models?

Approximating the Power of Goodness-of-fit Statistics in Categorical Data Analysis

Alberto Maydeu-OlivaresRosa Montano

OutlineIntroductionRasch-Type Models for Binary DataRationale of Goodness-of-Fit Statistics

◦Full Picture◦M2, R1 and R2

Estimating the PowerEmpirical Comparison of R1, R2 and M2Numerical ExamplesDiscussion and Conclusion

IntroductionTwo properties of Rasch-Type models

◦ Sufficient statistics◦ Specific objectivity

Estimation methods◦ Specific for Rasch-Type models (CML)◦ General procedures (MML via EM)

Goodness-of-fit testing procedures◦ Specific to Rasch-Type models◦ General to IRT or multivariate discrete data

models

IntroductionCompare the performance of certain

goodness-of-fit statistics to test Rasch-Type models in MML via EM◦ Binary data◦ 1PL (random effects)

R1 and R2 for 1PLM2 for multivariate discrete data

Rasch model and 1PL

Fixed effects◦ The distribution of ability is not specified

Random effects◦ Specify a standard normal distribution for

ability◦ The less restrictive definition of specific

objectivity still hold

Rationale

(000)

(100)

(010)

(001)

(110)

(101)

(011)

(111)

1 0 1 0 0 0 0 0 0

2 0 0 0 0 0 1 0 0

3 0 0 0 0 1 0 0 0

Marginal Total for each cell > 5

1. High-dimensional contingency table

C = 2^n cells which n is the number of items.For example, 20 items testC = 2^20 = 1048576 cellsTo fulfill the rule of thumb >5, at least 1048576*5 sample size is needed.

(000)

(100)

(010)

(001)

(110)

(101)

(011)

(111)

1 0 1 0 0 0 0 0 0

2 0 0 0 0 0 1 0 0

…

Marginal Total 10 17 21 32 15 8 12 19 134

Observed proportion 0.07

ProbabilityUnder Model

0.11

2.

When order r = 2, Mr -> M2M2 used the univariate and bivariate

informationThe degree of freedom is It is statistics of choice for testing IRT models

3. Limited information approach (M2)

Pooling cells of the contingency table

Degree of freedom is n(n-2)Specific to the monotone increasing and

parallel item response functions assumptions

3. Limited information approach (R1 and R2)

Degree of freedom is (n(n-2)+2)/2Specific to the unidimensionality assumption

Estimating the Asymptotic Power RateUnder the sequence of local

alternatives◦The noncentrality parameter of a chi-

square distribution can be calculated given the df for M2, R1 and R2

The Kullback-Leibler discrepancy function can be used◦The minimizer of DKL is the same as

the maximizer of the maximum likelihood function between a “true” model and a null model

Study 1: Accuracy of p-values under correct model

df = Mean; df = ½ Var Another Study by Montano (2009), M2 is better than

R1 and the discrepancies between the empirical and asymptotic rate were not large.

Group the sum scores ->

The degree of freedom is also adjustAn iterative procedureWhen appropriate score ranges are used, the

empirical rejection rate of R1 should be closely match the theoretical rejection rates.

This should be also done in R2

Study 2: Asymptotic Power to reject a 2PL

Study 3: Empirical Power to reject a 2PL

Study 4: Asymptotic Power to reject a 3PL

Study 5: Asymptotic Power to reject a multidimensional model

Empirical Example 1: LSAT 7 Data

The agreement in ordering between value/df ratio and power

Empirical Example 2: Chilean Mathematical Proficiency Data

Discussion and ConclusionsGenerally, M2 is more powerful

than R1, R2.That is, the R1 and R2 which

developed specific to Rasch-type models is not superior than the general M2