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A note on exploratory item factor analysis by singular value decomposition. (English) Zbl 1458.62282

Summary: We revisit a singular value decomposition (SVD) algorithm given in [Y. Chen et al., Psychometrika 84, No. 1, 124–146 (2019; Zbl 1431.62524)] for exploratory item factor analysis (IFA). This algorithm estimates a multidimensional IFA model by SVD and was used to obtain a starting point for joint maximum likelihood estimation in [loc. cit.]. Thanks to the analytic and computational properties of SVD, this algorithm guarantees a unique solution and has computational advantage over other exploratory IFA methods. Its computational advantage becomes significant when the numbers of respondents, items, and factors are all large. This algorithm can be viewed as a generalization of principal component analysis to binary data. In this note, we provide the statistical underpinning of the algorithm. In particular, we show its statistical consistency under the same double asymptotic setting as in [loc. cit.]. We also demonstrate how this algorithm provides a scree plot for investigating the number of factors and provide its asymptotic theory. Further extensions of the algorithm are discussed. Finally, simulation studies suggest that the algorithm has good finite sample performance.

MSC:

62P15 Applications of statistics to psychology
62H25 Factor analysis and principal components; correspondence analysis

Citations:

Zbl 1431.62524

Software:

mirtjml
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Full Text: DOI arXiv

References:

[1] Bartholomew, DJ; Moustaki, I.; Galbraith, J.; Steele, F., Analysis of multivariate social science data (2008), Boca Raton, FL: CRC Press, Boca Raton, FL · Zbl 1162.62096
[2] Bock, RD; Aitkin, M., Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm, Psychometrika, 46, 443-459 (1981)
[3] Bock, RD; Gibbons, R.; Muraki, E., Full-information item factor analysis, Applied Psychological Measurement, 12, 261-280 (1988)
[4] Browne, MW, An overview of analytic rotation in exploratory factor analysis, Multivariate Behavioral Research, 36, 111-150 (2001)
[5] Cai, L., High-dimensional exploratory item factor analysis by a Metropolis-Hastings Robbins-Monro algorithm, Psychometrika, 75, 33-57 (2010) · Zbl 1272.62113
[6] Cai, L., Metropolis-Hastings Robbins-Monro algorithm for confirmatory item factor analysis, Journal of Educational and Behavioral Statistics, 35, 3, 307-335 (2010)
[7] Chatterjee, S., Matrix estimation by universal singular value thresholding, The Annals of Statistics, 43, 177-214 (2015) · Zbl 1308.62038
[8] Chen, Y.; Fan, J.; Ma, C.; Yan, Y., Inference and uncertainty quantification for noisy matrix completion, Proceedings of the National Academy of Sciences, 116, 22931-22937 (2019) · Zbl 1431.90117
[9] Chen, Y.; Li, X.; Zhang, S., Joint maximum likelihood estimation for high-dimensional exploratory item factor analysis, Psychometrika, 84, 124-146 (2019) · Zbl 1431.62524
[10] Chen, Y., Li, X., & Zhang, S. (2019c). Structured latent factor analysis for large-scale data: Identifiability, estimability, and their implications. Journal of the American Statistical Association. 10.1080/01621459.2019.1635485.
[11] Chiu, C-Y; Köhn, H-F; Zheng, Y.; Henson, R., Joint maximum likelihood estimation for diagnostic classification models, Psychometrika, 81, 1069-1092 (2016) · Zbl 1367.62313
[12] Friedman, J.; Hastie, T.; Tibshirani, R., The elements of statistical learning (2001), New York, NY: Springer, New York, NY
[13] Haberman, SJ, Maximum likelihood estimates in exponential response models, The Annals of Statistics, 5, 815-841 (1977) · Zbl 0368.62019
[14] Haberman, S. J. (2004). Joint and conditional maximum likelihood estimation for the Rasch model for binary responses. ETS Research Report Series RR-04-20.
[15] Jöreskog, KG, On the estimation of polychoric correlations and their asymptotic covariance matrix, Psychometrika, 59, 381-389 (1994) · Zbl 0830.62059
[16] Kaiser, HF, The varimax criterion for analytic rotation in factor analysis, Psychometrika, 23, 187-200 (1958) · Zbl 0095.33603
[17] Katsikatsou, M.; Moustaki, I.; Yang-Wallentin, F.; Jöreskog, KG, Pairwise likelihood estimation for factor analysis models with ordinal data, Computational Statistics and Data Analysis, 56, 4243-4258 (2012) · Zbl 1255.62172
[18] Lee, S-Y; Poon, W-Y; Bentler, P., Full maximum likelihood analysis of structural equation models with polytomous variables, Statistics and Probability Letters, 9, 91-97 (1990) · Zbl 0687.62043
[19] Lee, S-Y; Poon, W-Y; Bentler, PM, Structural equation models with continuous and polytomous variables, Psychometrika, 57, 89-105 (1992) · Zbl 0760.62099
[20] Muraki, E.; Carlson, JE, Full-information factor analysis for polytomous item responses, Applied Psychological Measurement, 19, 73-90 (1995)
[21] Muthén, B., A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators, Psychometrika, 49, 115-132 (1984)
[22] O’Rourke, S.; Vu, V.; Wang, K., Random perturbation of low rank matrices: Improving classical bounds, Linear Algebra and its Applications, 540, 26-59 (2018) · Zbl 1380.65076
[23] Reckase, M., Multidimensional item response theory (2009), New York, NY: Springer, New York, NY · Zbl 1291.62023
[24] Stewart, G.; Sun, J., Matrix perturbation theory (1990), Cambridge, MA: Academic Press, Cambridge, MA
[25] Stock, JH; Watson, MW, Forecasting using principal components from a large number of predictors, Journal of the American Statistical Association, 97, 1167-1179 (2002) · Zbl 1041.62081
[26] Wall, ME; Rechtsteiner, A.; Rocha, LM; Berrar, DP; Dubitzky, W.; Granzow, M., Singular value decomposition and principal component analysis, A practical approach to microarray data analysis, 91-109 (2003), New York, NY: Springer, New York, NY
[27] Xia, D., & Yuan, M. (2019). Statistical inferences of linear forms for noisy matrix completion. arXiv preprint arXiv:1909.00116.
[28] Zhang, S., Chen, Y., & Li, X. (2018). mirtjml: Joint maximum likelihood estimation for high-dimensional item factor analysis. R package version, 1.2.
[29] Zhang, S.; Chen, Y.; Liu, Y., An improved stochastic EM algorithm for large-scale full-information item factor analysis, British Journal of Mathematical and Statistical Psychology, 73, 44-71 (2020) · Zbl 1440.62089
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