The Annals of Statistics

A likelihood ratio test for $MTP_2$ within binary variables

Francesco Bartolucci and Antonio Forcina

Full-text: Open access

Abstract

Multivariate Totally Positive $(MTP_2)$ binary distributions have been studied in many fields, such as statistical mechanics, computer storage and latent variable models. We show that $MTP_2$ is equivalent to the requirement that the parameters of a saturated log-linear model belong to a convex cone, and we provide a Fisher-scoring algorithm for maximum likelihood estimation.We also show that the asymptotic distribution of the log-likelihood ratio is a mixture of chi-squares (a distribution known as chi-bar-squared in the literature on order restricted inference); for this we derive tight bounds which turn out to have very simple forms. A potential application of this method is for Item Response Theory (IRT) models, which are used in educational assessment to analyse the responses of a group of subjects to a collection of questions (items): an important issue within IRT is whether the joint distribution of the manifest variables is compatible with a single latent variable representation satisfying local independence and monotonicity which, in turn, imply that the joint distribution of item responses is $MTP_2$.

Article information

Source
Ann. Statist., Volume 28, Number 4 (2000), 1206-1218.

Dates
First available in Project Euclid: 12 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1015956713

Digital Object Identifier
doi:10.1214/aos/1015956713

Mathematical Reviews number (MathSciNet)
MR1811325

Zentralblatt MATH identifier
1105.62351

Subjects
Primary: 62H20: Measures of association (correlation, canonical correlation, etc.)
Secondary: 62G10: Hypothesis testing 62H15: Hypothesis testing 62H17: Contingency tables

Keywords
Chi-bar-squared distribution conditional association item response models order-restricted inference stochastic ordering

Citation

Bartolucci, Francesco; Forcina, Antonio. A likelihood ratio test for $MTP_2$ within binary variables. Ann. Statist. 28 (2000), no. 4, 1206--1218. doi:10.1214/aos/1015956713. https://projecteuclid.org/euclid.aos/1015956713


Export citation

References

  • Dardanoni, V. and Forcina, A. (1998). A unified approach to likelihood inference on stochastic orderings in a nonparametric context. J. Amer. Statist. Assoc. 93 1112-1123.
  • Darroch, J. N., Fienberg, S. E., Glonek, G. F. V. and Junker, B. W. (1993). A threesample multiple-recapture approach to census population estimation with heterogeneous catchability. J. Amer. Statist. Assoc. 88 1137-1148.
  • Dykstra, R. L. (1983). An Algorithmfor Restricted least Squares. J. Amer. Statist. Assoc. 78 837-842.
  • Fortuin, C. M., Kasteleyn, P. W. and Ginibgre, J. (1971). Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 89-103.
  • Hambleton, R. K. and Swaminathan, H. (1985). Item Response Theory: Principles and Applications. Kluwer, Boston.
  • Hoijtink, H. and Molenaar, I. W. (1997). A multidimensional item response model: constrained latent class analysis using the Gibbs sampler and posterior predictive checkes. Psychometrika 62 171-189.
  • Holland, P. W. and Rosenbaum, P. R. (1986). Conditional association and unidimensionality in monotone latent variable models. Ann. Statist. 14 1523-1543.
  • Junker, B. W. and Ellis, J. L. (1997). A characterization of monotone unidimensional latent variable models. Ann. Statist. 25 1327-1343.
  • Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal. 10 467-498.
  • Lawson, L. L. and Hanson, R. J. (1995). Solving Least Squares Problems. SIAM, Philadelphia.
  • Lindsay, B., Clogg, C. and Grego, J. (1991). Semiparametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis, J. Amer. Statist. Assoc. 86 86-107.
  • Perlman, M. D. (1969). One-sied testing problems in multivariate analysis. Ann. Math. Statist. 40 549-567.
  • Rosenbaum, P. R. (1984). Testing the conditional independence and monotonicity assumption of itemresponse theory. Psychometrika 49 425-435.
  • Shapiro, A. (1985). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika 72 13-144.
  • Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis. Internat. Statist. Rev. 56 49-62.
  • van den Berg, J. and Gandolfi, A. (1992). LRU is better than FIFO under the independent reference model. J. Appl. Probab. 29 239-243.
  • van den Berg, J. and Gandolfi, A. (1995). A triangle inequality for covariances of binary FKG randomvariables. Ann. Appl. Probab. 5 322-326. Wolak, F. A. (1989a). Local and global testing of linear and nonlinear inequality constraints in nonlinear econometric models. Econom. Theory 5 1-35. Wolak, F. A. (1989b). Testing inequality constraints in linear econometric models. J. Economtrics 41 205-235.