Exploring the FullInformation Bifactor Model in Vertical Scaling

Exploring the Full-Information Bifactor Model in Vertical Scaling With Construct Shift Ying Li and Robert W. Lissitz

contents Questions Methods Results

Vertical scaling 1. Unidimensionality of tests at each grade level 2. Test construct invariance across grade ╳ bifactor model

Questions 1. Computational simplicity for estimation 2. Ease of interpretation 3. Vertical scaling problems across grades

Purpose • to propose and evaluate a bifactor model for IRT vertical scaling that can incorporate construct shift across grades while extracting a common scale, • to evaluate the robustness of the UIRT model in parameter recovery, and • to compare parameter estimates from the bifactor and UIRT models

Method • Bifactor Model • Gibbons and Hedeker (1992) generalized the work of Holzinger and Swineford (1937) to derive a bifactor model for dichotomously scored item response data. 0: general factor or ability i 0 : discrimination parameter for general factor s: group special factor or ability is : discrimination parameter for group-special factor di : overall multidimensional item difficulty

simulation • Common item design • Bifactor model for generating data • Concurrent calibration – Stable results

• Sample size • Number or percentage of common items – Test length: 60 • Variance of grade-specific factor: degree of construct shift • 100 replication

Data generation Item discrimination parameters were set deliberately and repeatedly at 1. 2, 1. 4, 1. 6, 1. 8, 2. 0, and 2. 2 for the general dimension, and fixed at 1. 7 for grade-specific

Identifications of Bifactor Model Estimation • For the general dimension, the variance of the general latent dimension was fixed to 1, and the discrimination parameters (loadings) were freely estimated in the study • For the grade-specific dimensions (s = 1, 2, . . . , k), the discrimination parameters (s = 1, 2, . . . , k) (loadings) were fixed to the true parameter value 1. 7, so that the variances of the grade-specific dimensions could be freely estimated • the common items answered by multiple groups were restricted so that they would have unique item parameters in the multiple-group concurrent calibration

Model Estimation • Multigroup concurrent calibration was implemented • The computer program IRTPRO (Cai, Thissen, & du Toit, 2011), using marginal maximumlikelihood estimation (MML) with an EM algorithm, was used to estimate the models

Evaluation criteria

Results RMSE = root mean square error; SE = standard error; SS = sample size; CI = common item; VR = variance of grade-specific factor.

Person parameter • person parameter estimates of the general dimension were better recovered than that of the grade-specific dimensions when the degree of construct shift was small or moderate • sample size the estimation accuracy

Group parameter overestimated

UIRT • discrimination: overestimated • Difficulty: well recovered • construct shift person & group mean

ANOVA Effects for the Simulated Factors • Three-way tests of between-subject effects (ANOVA) – bias, RMSE, and SE Sample size Degree of Construct shift Percentage of common item Bifactor model small~ moderate No~small Small Bias in d UIRT small~ moderate Large : & Group Mean ability Large bias in d & d & SE Group Mean ability grade-specific variance parameter Large: SE

comparison • UIRT: overestimate discrimation parameter • person and group mean parameter: Less accurate

Real data • 2006 fall Michigan mathematics assessments • Grade 3, 4, 5 • Randomly 4000 examinees • Bifactor vs UIRT

Variance estimation

R=0. 983

Discussion • sample size the estimation accuracy & stability • Variance of grade-specific dimension • Be caution about construct shift • Polytomounsly/mixed item format • Incorporate covariates • longitudinal studies stability

• Common item measure two group-specific ability？ • the item discriminate parameter fixed to the true value • Multidimensional IRT?