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Abstract

A latent multivariate regression model is developed that employs a generalized asymmetric Laplace (GAL) prior distribution for regression coefficients. The model is designed for high-dimensional applications where an approximate sparsity condition is satisfied, such that many regression coefficients are near zero after accounting for all the model predictors. The model is applicable to large-scale assessments such as the National Assessment of Educational Progress (NAEP), which includes hundreds of student, teacher, and school predictors of latent achievement. Monte Carlo evidence suggests that employing the GAL prior provides more precise estimation of coefficients that equal zero in comparison to a multivariate normal (MVN) prior, which translates to more accurate model selection. Furthermore, the GAL yielded less biased estimates of regression coefficients in smaller samples. The developed model is applied to mathematics achievement data from the 2011 NAEP for 175,200 eighth graders. The GAL and MVN NAEP estimates were similar, but the GAL was more parsimonious by selecting 12 fewer (i.e., 83 of the 148) variable groups. There were noticeable differences between estimates computed with a GAL prior and plausible value regressions with the AM software (beta version 0.06.00). Implications of the results are discussed for test developers and applied researchers.

Keywords

multivariate regression Bayesian Lasso National Assessment of Educational Progress multivariate generalized asymmetric Laplace distribution probit model

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References

Albert

(1992). Bayesian estimation of normal ogive item response curves using Gibbs sampling. Journal of Educational and Behavioral Statistics, 17, 251–269.

Albert

Chib

(1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669–679.

American Institutes for Research. (2002–2012). AM statistical software. Retrieved from http://am.air.org/default.asp

Béguin

A. A.

Glas

C. A.

(2001). MCMC estimation and some model-fit analysis of multidimensional IRT models. Psychometrika, 66, 541–561.

Cowles

M. K.

(1996). Accelerating Monte Carlo Markov chain convergence for cumulative-link generalized linear models. Statistics and Computing, 6, 101–111.

Eddelbuettel

(2013). Seamless R and C++ integration with Rcpp. New York: Springer-Verlag.

Fox

J. P.

(2010). Bayesian item response modeling. New York, NY: Springer.

Fox

J. P.

Glas

C. A.

(2001). Bayesian estimation of a multilevel IRT model using Gibbs sampling. Psychometrika, 66, 271–288.

Frey

Hartig

Rupp

A. A.

(2009). An NCME instructional module on booklet designs in large-scale assessments of student achievement: Theory and practice. Educational Measurement: Issues and Practice, 28, 39–53.

10.

Gonzalez

Rutkowski

(2010). Practical approaches for choosing multiple-matrix sample designs. IEA-ETS Research Institute Monograph, 3, 125–156.

11.

Hans

(2009). Bayesian Lasso regression. Biometrika, 96, 835–845.

12.

Hastie

Tibshirani

Wainwright

(2015). Statistical learning with sparsity: The Lasso and generalizations. Boca Raton, FL: CRC Press.

13.

Hoff

P. D.

(2009). A first course in Bayesian statistical methods. Dordrecht: Springer.

14.

Johnson

E. G.

(1992). The design of the National Assessment of Educational Progress. Journal of Educational Measurement, 29, 95–110.

15.

Johnson

M. S.

(2002). A Bayesian hierarchical model for multidimensional performance assessments. Paper presented at the annual meeting of the National Council on Measurement in Education, New Orleans, LA.

16.

Johnson

M. S.

Jenkins

(2004). A Bayesian hierarchical model for large-scale educational surveys: An application to the national assessment of educational progress. Princeton, NJ: Educational Testing Service.

17.

Johnson

M. S.

Sinharay

(2015). Does the NAEP model adequately predict the achievement gap? Paper presented at the annual meeting of the National Council on Measurement in Education, Chicago, IL.

18.

Kozubowski

T. J.

Podgórski

Rychlik

(2013). Multivariate generalized Laplace distribution and related random fields. Journal of Multivariate Analysis, 113, 59–72.

19.

Kyung

Gill

Ghosh

Casella

(2010). Penalized regression, standard errors, and Bayesian Lassos. Bayesian Analysis, 5, 369–411.

20.

Lawrence

Bingham

Liu

Nair

V. N.

(2008). Bayesian inference for multivariate ordinal data using parameter expansion. Technometrics, 50, 182–191.

21.

Maydeu-Olivares

Drasgow

Mead

A. D.

(1994). Distinguishing among parametric item response models for polychotomous ordered data. Applied Psychological Measurement, 18, 245–256.

22.

Mislevy

Johnson

Muraki

(1992). Scaling procedures in NAEP. Journal of Educational Statistics, 17, 131–154.

23.

Muraki

(1992). A generalized partial credit model: Application of an EM algorithm. Applied psychological measurement, 16, 159–176.

24.

Park

Casella

(2008). The Bayesian Lasso. Journal of the American Statistical Association, 103, 681–686.

25.

Patz

R. J.

Junker

B. W.

(1999). Applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses. Journal of Educational and Behavioral Statistics, 24, 342–366.

26.

Rubin

(1987). Multiple imputation for nonresponse in surveys. New York, NY: Wiley.

27.

Sinharay

von Davier

(2005). Extension of the NAEP BGROUP program to higher dimensions (Tech. Rep. No. RR-05-27). Princeton, NJ: Educational Testing Service.

28.

Thomas

(2000). Assessing model sensitivity of the imputation methods used in the National Assessment of Educational Progress. Journal of Educational and Behavioral Statistics, 25, 351–371.

29.

Thomas

Gan

(1997). Generating multiple imputations for matrix sampling data analyzed with item response models. Journal of Educational and Behavioral Statistics, 22, 425–445.

30.

Tibshirani

(1996). Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society. Series B (Methodological), 58, 267–288.

31.

von Davier

Sinharay

(2004). Application of the stochastic EM method to latent regression models (Tech. Rep. No. RR-04-34). Princeton, NJ: Educational Testing Service.

32.

von Davier

Sinharay

(2007). An importance sampling EM algorithm for latent regression models. Journal of Educational and Behavioral Statistics, 32, 233–251.

33.

von Davier

Sinharay

(2010). Stochastic approximation methods for latent regression item response models. Journal of Educational and Behavioral Statistics, 35, 174–193.

34.

Yuan

Lin

(2006). Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68, 49–67.

35.

Zwick

(1987). Assessing the dimensionality of NAEP reading data. Journal of Educational Measurement, 24, 293–308.

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