×

An alternative method for evaluating stationarity in transition models. (English) Zbl 07192105

Summary: Transition models are an important framework that can be used to model longitudinal categorical data. A relevant issue in applying these models is the condition of stationarity, or homogeneity of transition probabilities over time. We propose two tests to assess stationarity in transition models: Wald and likelihood-ratio tests, which do not make use of transition probabilities, using only the estimated parameters of the models in contrast to the classical test available in the literature. In this paper, we present two motivating studies, with ordinal longitudinal data, to which proportional odds transition models are fitted and the two proposed tests are applied as well as the classical test. Additionally, their performances are assessed through simulation studies. The results show that the proposed tests have good performance, being better for control of type-I error and they present equivalent power functions asymptotically. Also, the correlations between the Wald, likelihood-ratio and the classical test statistics are positive and large, an indicator of general concordance. Additionally, both of the proposed tests are more flexible and can be applied in studies with qualitative and quantitative covariates.

MSC:

62F03 Parametric hypothesis testing
62J12 Generalized linear models (logistic models)

Software:

ordinal; markovchain; VGAM; R
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Molenbergs G, Verbeke G. Models for discrete longitudinal data. New York: Springer-Verlag; 2005. [Google Scholar]
[2] Ware JH, Lipsitz S, Speizer FE.Issues in the analysis of repeated categorical outcomes. Stat Med. 1988;7:95-107. doi: 10.1002/sim.4780070113[Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[3] De Rooij M. Transitional modelling of experimental longitudinal data with missing values. Advances in Data Analysis and Classification. Available from link.springer.com/content/pdf/10.1007. [Google Scholar] · Zbl 1414.62013
[4] Lindsey JK. Statistical analysis of stochastic processes in time. New York: Cambridge University Press; 2004. [Crossref], [Google Scholar]
[5] Anderson TW, Goodman LA.Statistical Inference about Markov Chains. Ann Math Stat. 1957;28:89-110. doi: 10.1214/aoms/1177707039[Crossref], [Google Scholar] · Zbl 0087.14905
[6] Cox DR. The analysis of binary data. Methuen: London; 1970. [Google Scholar] · Zbl 0199.53301
[7] Korn EL, Whittemore AS.Methods for analysing panel studies of acute health effects of air pollution. Biometrics. 1979;35:715-802. doi: 10.2307/2530111[Crossref], [Web of Science ®], [Google Scholar]
[8] Azzalini A.Maximum likelihood estimation of order m for stationary stochastic processes. Biometrika. 1983;70:381-388. doi: 10.1093/biomet/70.2.381[Crossref], [Web of Science ®], [Google Scholar]
[9] Zeger SL, Liang KY, Self SG.The analysis of binary longitudinal data with time-independent covariates. Biometrika. 1985;72:31-38. [Web of Science ®], [Google Scholar] · Zbl 0557.62076
[10] Bonney GE.Logistic Regression for dependent binary observations. Biometrics. 1987;43:951-973. doi: 10.2307/2531548[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 0707.62153
[11] Fitzmaurice GM, Laird NM.A likelihood-based method for analysing longitudinal binary responses. Biometrika. 1993;80:141-151. doi: 10.1093/biomet/80.1.141[Crossref], [Web of Science ®], [Google Scholar] · Zbl 0775.62296
[12] Diggle PJ, Heagerty PJ, Liang KY, et al. New York: Oxford University Press; 2002. [Google Scholar]
[13] Lee K, Daniels MJ.A class of Markov Models for longitudinal data. Biometrics. 2007;63:1060-1067. doi: 10.1111/j.1541-0420.2007.00800.x[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1274.62482
[14] Heagerty PJ.Marginalized transition models and likelihood inference for categorical data. Biometrics. 2002;58:342-351. doi: 10.1111/j.0006-341X.2002.00342.x[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1209.62158
[15] De Rooij M.Transitional ideal point models for longitudinal multinomial outcomes. Statist Model. 2011;11(2):15-135. doi: 10.1177/1471082X1001100202[Crossref], [Web of Science ®], [Google Scholar]
[16] Agresti A. Analysis of ordinal categorical data. 2nd ed. New York: Wiley; 2010. [Crossref], [Google Scholar] · Zbl 1263.62007
[17] Koch GC, Carr GJ, Amara IA, et al. Categorical data analysis. In: Berry, D.A. Statistical methodology in the pharmaceutical sciences. New York: Marcel Dekker; 1990. Chapter 13, p. 389-473. [Google Scholar]
[18] Castro AC. Comportamento e desempenho sexual de suínos reprodutores em ambientes enriquecidos [unpublished doctoral dissertation]. Piracicaba (Brazil): University of São Paulo, 2016. [Google Scholar]
[19] Zeger SL, Liang KY.An overview of methods for the analysis of longitudinal data. Stat Med. 1992;11:1825-1839. doi: 10.1002/sim.4780111406[Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[20] Nelder JA, Wedderburn RWM.Generalized linear models. J R Statist Soc Ser A. 1972;135:370-384. doi: 10.2307/2344614[Crossref], [Web of Science ®], [Google Scholar]
[21] Agresti A. Categorical data analysis. 3rd ed. Wiley: New York; 2012. [Google Scholar] · Zbl 0716.62001
[22] McCullagh P.Regression methods for ordinal data. J R Statist Soc. Ser B. 1980;42:109-142. [Google Scholar] · Zbl 0483.62056
[23] Wald A.Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans Amer Math Soc. 1943;54:426-482. doi: 10.1090/S0002-9947-1943-0012401-3[Crossref], [Google Scholar] · Zbl 0063.08120
[24] Rayner JCW, Best DJ. Smooth tests of goodness fit. New York: Oxford University Press; 1989. [Google Scholar] · Zbl 0731.62064
[25] Rayner JCW.The asymptotically optimal tests. J R Statist Soc (Statist). 1997;46:337-346. doi: 10.1111/1467-9884.00087[Crossref], [Google Scholar]
[26] Buse A.The likelihood-ratio, Wald and Lagrange multiplier tests: an expository note. Amer Stat. 1982;36:153-157. [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
[27] Liao TF.Comparing social groups: Wald statistics for testing equality among multiple logit models. Int J Comparative Sociol. 2004;45(12):3-16. doi: 10.1177/0020715204048308[Crossref], [Google Scholar]
[28] R Development Core Team. A language and environment for statistical computing, 3.2. Vienna, Austria, 2015. Available from http://www.R-project.org. [Google Scholar]
[29] Yee TW.The VGAM package for categorical data analysis. J Statist Softw. 2010;32:1-34. doi: 10.18637/jss.v032.i10[Crossref], [Web of Science ®], [Google Scholar]
[30] Christensen RHB. Analysis of ordinal data with cumulative link models estimation with the R-package ordinal. 2011. Available from: http://www.R-project.org. [Google Scholar]
[31] Spedicato GA.markovchain: discrete time Markov chains made easy, 2015. Available from: http://www.R-project.org. [Google Scholar]
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.