Heywood Cases in Unidimensional Factor Models and Item Response Models for Binary Data

Abstract

Heywood cases are known from linear factor analysis literature as variables with communalities larger than 1.00, and in present day factor models, the problem also shows in negative residual variances. For binary data, factor models for ordinal data can be applied with either delta parameterization or theta parametrization. The former is more common than the latter and can yield Heywood cases when limited information estimation is used. The same problem shows up as non convergence cases in theta parameterized factor models and as extremely large discriminations in item response theory (IRT) models. In this study, we explain why the same problem appears in different forms depending on the method of analysis. We first discuss this issue using equations and then illustrate our conclusions using a small simulation study, where all three methods, delta and theta parameterized ordinal factor models (with estimation based on polychoric correlations and thresholds) and an IRT model (with full information estimation), are used to analyze the same datasets. The results generalize across WLS, WLSMV, and ULS estimators for the factor models for ordinal data. Finally, we analyze real data with the same three approaches. The results of the simulation study and the analysis of real data confirm the theoretical conclusions.

Keywords

Heywood cases item response model factor model

Get full access to this article

View all access options for this article.

References

Asparouhov

Muthen

(2020). IRT in Mplus. https://www.statmodel.com/download/MplusIRT.pdf

Bock

R. D.

Gibbons

Muraki

(1988). Full-information item factor analysis. Applied Psychological Measurement, 12(3), 261–280. https://doi.org/10.1177/014662168801200305

Bolt

D. M.

(2005). Limited- and full-information estimation of item response theory models. In Maydeu-Olivares

McArdle

J. J.

(Eds.), Contemporary psychometrics (pp. 27–71): Lawrence Erlbaum.

Braeken

Tuerlinckx

De Boeck

(2007). Copulas for residual dependencies. Psychometrika, 72(3), 393–411. https://doi.org/10.1007/s11336-007-9005-4

Browne

M. W.

(1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37(1), 62–83. https://doi.org/10.1111/j.2044-8317.1984.tb00789.x

Chen

Bollen

K. A.

Paxton

Curran

P. J.

Kirby

J. B.

(2001). Improper solutions in structural equation models: Causes, consequences, and strategies. Sociological Methods & Research, 29(4), 468–508. https://doi.org/10.1177/0049124101029004003

Edwards

M. C.

Houts

C. R.

Cai

(2018). A diagnostic procedure to detect departures from local independence in item response theory models. Psychological Methods, 23(1), 138–149. https://doi.org/10.1037/met0000121

Forero

C. G.

Maydeu-Olivares

(2009). Estimation of IRT graded response models: Limited versus full information methods. Psychological Methods, 14(3), 275–299. https://doi.org/10.1037/a0015825

Harman

H. H.

Fukuda

(1966). Resolution of the Heywood case in the minres solution. Psychometrika, 31(4), 563–571. https://doi.org/10.1007/bf02289525

10.

Heywood

(1931). On finite sequences of real numbers. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 134(824), 486–501.

11.

Hoskens

De Boeck

(1997). A parametric model for local dependence among test items. Psychological Methods, 2(3), 261–275. https://doi.org/10.1037/1082-989x.2.3.261

12.

E. H.

(2000). Adjusting for information inflation due to local dependency in moderately large item clusters. Psychometrika, 65(1), 73–91. https://doi.org/10.1007/bf02294187

13.

E. H.

(2002). Locally dependent latent trait model and the Dutch identity revisited. Psychometrika, 67(3), 367–386. https://doi.org/10.1007/bf02294990

14.

Jöreskog

K. G.

Lawley

D. N.

(1968). New methods in maximum likelihood factor analysis. British Journal of Mathematical and Statistical Psychology, 21(1), 85–96. https://doi.org/10.1111/j.2044-8317.1968.tb00399.x

15.

Kamata

Bauer

D. J.

(2008). A note on the relation between factor analytic and item response theory models. Structural Equation Modeling: A Multidisciplinary Journal, 15(1), 136–153. https://doi.org/10.1080/10705510701758406

16.

Martin

J. K.

McDonald

R. P.

(1975). Bayesian estimation in unrestricted factor analysis: A treatment for Heywood cases. Psychometrika, 40(4), 505–517. https://doi.org/10.1007/bf02291552

17.

Muthen

L. K.

Muthen

B. O.

(1998-2017). Mplus user’s guide (8th ed.): Muthen & Muthen.

18.

Paek

Cui

Öztürk Gübeş

Yang

(2018). Estimation of an IRT model by Mplus for dichotomously scored responses under different estimation methods. Educational and Psychological Measurement, 78(4), 569–588. https://doi.org/10.1177/0013164417715738

19.

Rizopoulos

(2006). ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1–25. http://www.jstatsoft.org/v17/i05/

20.

Rosseel

(2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1–36. https://doi.org/10.18637/jss.v048.i02

21.

Savalei

(2006). Logistic approximation to the normal: The KL rationale. Psychometrika, 71, 763–767. https://doi.org/10.1007/s11336-004-1237-y

22.

Spitzer

R. L.

Kroenke

Williams

J. B.W.

(1999). Patient health questionnaire study group. Validity and utility of a self-report version of PRIME-MD: The PHQ primary care study. JAMA, 282(18), 1737–1744. https://doi.org/10.1001/jama.282.18.1737

23.

Takane

de Leeuw

(1987). On the relationship between item response theory and factor analysis of discretized variables. Psychometrika, 52(3), 393–408. https://doi.org/10.1007/bf02294363

24.

Thissen

Steinberg

Mooney

J. A.

(1989). Trace lines for testlets: A use of multiple-categorical-response models. Journal of Educational Measurement, 26(3), 247–260. https://doi.org/10.1111/j.1745-3984.1989.tb00331.x

25.

Tuerlinckx

De Boeck

(2001). The effect of ignoring item interactions on the estimated discrimination parameters in item response theory. Psychological Methods, 6(2), 181–195. https://doi.org/10.1037/1082-989x.6.2.181

26.

Venables

W. N.

Ripley

B. D.

(2002). Modern applied statistics with S (4th ed.): Springer. https://www.stats.ox.ac.uk/pub/MASS4/

27.

Yang-Wallentin

Joreskog

K. G.

Luo

(2010). Confirmatory factor analysis of ordinal variables with misspecified models. Structural Equation Modeling, 17(3), 392–423. https://doi.org/10.1080/10705511.2010.489003

28.

Yotebieng

K. A.

Fokong

Yotebieng

(2017). Depression, retention in care, and uptake of PMTCT service in kinshasa, the democratic republic of Congo: A prospective cohort. AIDS Care, 29(3), 285–289. https://doi.org/10.1080/09540121.2016.1255708

29.

Yotebieng

Thirumurthy

Mohalaracco

K. E.

Kawende

Chalachala

J. L.

Wenzi

L. K.

Behets

(2016). Conditional cash transfers and uptake of and retention in prevention of mother-to-child HIV transmission care: A randomized controlled trial. Lancet HIF, 3(2), e85–e93.