## The order-restricted association model: two estimation algorithms and issues in testing.(English)Zbl 1306.62414

Summary: This paper presents a row-column (RC) association model in which the estimated row and column scores are forced to be in agreement with an a priori specified ordering. Two efficient algorithms for finding the order-restricted maximum likelihood (ML) estimates are proposed and their reliability under different degrees of association is investigated by a simulation study. We propose testing order-restricted RC models using a parametric bootstrap procedure, which turns out to yield reliable $$p$$ values, except for situations in which the association between the two variables is very weak. The use of order-restricted RC models is illustrated by means of an empirical example.

### MSC:

 62P15 Applications of statistics to psychology 62F40 Bootstrap, jackknife and other resampling methods

### Software:

AS 253; bootstrap; CML
Full Text:

### References:

 [1] Agresti, A. (2002).Categorical data analysis. New York: Wiley. · Zbl 1018.62002 [2] Agresti, A., & Chuang, C. (1986). Bayesian and maximum likelihood approaches to order restricted inference for models from ordinal categorical data. In R. Dykstra & T. Robertson (Eds.),Advances in ordered statistical inference (pp. 6–27). Berlin: Springer-Verlag. · Zbl 0621.62058 [3] Agresti, A., Chuang, C., & Kezouh, A. (1987). Order-restricted score parameters in association models for contingency tables.Journal of the American Statistical Association, 82, 619–623. · Zbl 0625.62041 [4] Bartolucci, F., & Forcina, A. (2002). Extended RC association models allowing for order restrictions and marginal modeling.Journal of the American Statistical Association, 97, 1192–1199. · Zbl 1041.62049 [5] Becker, M.P. (1990). Algorithm AS253: Maximum likelihood estimation of the RC(M) association model.Applied Statistics, 39, 152–167. · Zbl 0715.62111 [6] Clogg, C.C. (1982). Some models for the analysis of association in multi-way cross-classifications having ordered categories.Journal of the American Statistical Association, 77, 803–815. [7] Clogg, C.C., & Shihadeh, E.S. (1994).Statistical models for ordinal data. Thousand Oaks, CA: Sage Publications. [8] Croon, M.A. (1990). Latent class analysis with ordered latent classes.British Journal of Mathematical and Statistical Psychology, 43, 171–192. [9] Dardanoni, V. & Forcina A. (1998). A unified approach to likelihood inference on stochastic orderings in a nonparametric context.Journal of the American Statistical Association, 93, 1112–1123. · Zbl 1063.62547 [10] Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977). Maximum likelihood estimation from incomplete data via the EM algorithm (with discussion).Journal of the Royal Statistical Society, Series B.,39, 1–38. · Zbl 0364.62022 [11] Efron, B., & Tibshirani, R.J. (1993).An introduction to the Bootstrap. London: Chapman and Hall. · Zbl 0835.62038 [12] Geyer, C.J. (1995). Likelihood ratio tests and inequality constraints. Technical Report No. 610, University of Minnesota. [13] Gill, P.E., & Murray, W. (1974).Numerical methods for constrained optimization. London: Academic Press. · Zbl 0297.90082 [14] Goodman, L.A. (1979). Simple models for the analysis of association in cross-classifications having ordered categories.Journal of the American Statistical Association, 74, 537–552. [15] Hoijtink, H., & Molenaar, I.W. (1997). A multidimensional item response model: Constrained latent class analysis using the Gibbs sampler and posterior predictive checks.Psychometrika, 62, 171–190. · Zbl 1003.62548 [16] Langeheine, R., Pannekoek, J., & Van de Pol, F. (1996). Bootstrapping goodness-of-fit measures in categorical data analysis.Sociological Methods and Research, 24, 492–516. [17] Ritov, Y., & Gilula, Z. (1991). The order-restricted RC model for ordered contingency tables: Estimation and testing for fit.Annals of Statistics, 19, 2090–2101. · Zbl 0745.62058 [18] Ritov, Y., & Gilula, Z. (1993). Analysis of contingency tables by correspondence models subject to order constraints.Journal of the American Statistical Association, 88, 1380–1387. · Zbl 0792.62049 [19] Robertson, T., Wright, F.T., & Dykstra, R.L. (1988).Order restricted statistical inference. Chichester: Wiley. · Zbl 0645.62028 [20] Schoenberg, R. (1997). CML: Constrained maximum likelihood estimation.The Sociological Methodologist, Spring 1997, 1–8. [21] Shapiro, A. (1985). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints.Biometrika, 72, 133–144. · Zbl 0596.62019 [22] Vermunt, J.K. (1999). A general nonparametric approach to the analysis of ordinal categorical data.Sociological Methodology, 29, 197–221. [23] Vermunt, J.K. (2001). The use restricted latent class models for defining and testing nonparametric and parametric item response theory models.Applied Psychological Measurement, 25, 283–294. [24] Wang, Y. (1996). A likelihood ratio test against stochastic ordering in several populations.Journal of the American Statistical Association, 91, 1676–1683. · Zbl 0881.62019
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