Higher Order Markov Structure-Based Logistic Model and Likelihood

Journal of Modern Applied Statistical Methods Volume 10 | Issue 2 Article 12 11-1-2011 Higher Order Markov Structure-Based Logistic Model and Likelihood Inference for Ordinal Data Soma Chowdhury Biswas University of Chittagong, Chittagong, Bangladesh, [email protected] M. Ataharul Islam University of Dhaka, Bangladesh, [email protected] Jamal Nazrul Islam University of Chittagong, Chittagong, Bangladesh, [email protected] Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Biswas, Soma Chowdhury; Islam, M. Ataharul; and Islam, Jamal Nazrul (2011) "Higher Order Markov Structure-Based Logistic Model and Likelihood Inference for Ordinal Data," Journal of Modern Applied Statistical Methods: Vol. 10 : Iss. 2 , Article 12. DOI: 10.22237/jmasm/1320120660 Available at: http://digitalcommons.wayne.edu/jmasm/vol10/iss2/12 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Journal of Modern Applied Statistical Methods Copyright © 2011 JMASM, Inc. November 2011, Vol. 10, No. 2, 528-538 1538 – 9472/11/$95.00 Higher Order Markov Structure-Based Logistic Model and Likelihood Inference for Ordinal Data Soma Chowdhury Biswas M. Ataharul Islam Jamal Nazrul Islam University of Chittagong, University of Dhaka, University of Chittagong, Chittagong, Bangladesh Dhaka, Bangladesh Chittagong, Bangladesh Azzalini (1994) proposed a first order Markov chain for binary data. Azzalini’s model is extended for ordinal data and introduces a second order model. Further, the test statistics are developed and the power of the test is determined. An application using real data is also presented. Key words: Markov chain, serial correlation, longitudinal data, ordinal data, covariate dependence, repeated measures. Introduction studies. Islam and Chowdhury (2007) reviewed The Markov chain model is one of the most the first order model of Muenz and Rubinstem important and effective model classes for the (1985) and developed a general procedure based assessment of probability for time dependent on the Chapman-Kolmogorov equation for processes. A number of models have been transition, reverse transition and repeated proposed for analyzing repeated categorical and transition. Lee and Daniels (2007) extended ordinal data. Muenz and Rubinstein (1985) Heagerty’s (2002) MTM to accommodate employed a logistic regression model to analyze longitudinal ordinal data. Ching, Fung and Ng the transitional probabilities from one state to (2004) generalized the Raftery and Tavare another. Azzalini (1994) introduced a Markov (1994) model by allowing Q = {qij} to vary with chain model that incorporated serial dependence different lags; they also developed an efficient and facilitated expression of covariate effects on method to estimate the model parameters. marginal features. Raftery & Tavare (1994) Ching, Ng and Fung (2007) extended their 2004 suggested a Markov chain model of order higher results (Ching, Fung & Ng, 2004) and proposed than one that involves only one parameter for a higher-order multivariate Markov model for each extra lag variable: Heagerty and Zeger multiple categorical data sequences. (2000) and Heagerty (2002) extended that work Azzalini’s (1994) Markov structure to a qth-order marginalized transition model based regression model for ordinal data is (MTM). These models are based on binary data extended here, and a second order model is and do not address the more general issue of proposed. Likelihood based inferences are ordinal data that arises in many biomedical possible because the model is fully specified so that resulting estimators are consistent and fully efficient. The proposed methods are applied to real data collected at successive time points from Soma Chowdhury Biswas is a Professor in the diabetic patients registered at Bangladesh Department of Statistics. Email him at: Institute of Research and Rehabilitation in [email protected]. M. Ataharul Islam is Diabetes, Endocrine and Metabolic disorders a Professor in the Department of Statistics, (BIRDEM) in Bangladesh. Biostatistics and Informatics. Email him at: [email protected]. Jamal Nazrul Islam is First Order Covariate Dependent Markov{ Model} Professor Emeritus, RCMPS. Email him at: Consider a stationary process Yij for [email protected]. individual i (i = 1, 2, …, n) at follow-up j (j = 1, 2, …, n) representing past and present responses 528 BISWAS, ISLAM & ISLAM where at time t the response p(Y= k / X) j gX()= In == k = Yk(k0,1,2)j . If the transition models for p(Y 0 / X) =β +β + +β which the conditional distribution of Yij given k0 k1X 1 ... kp X p (1) the prior observations Y...Yij−− 1 ij r is considered the model of order r, then the first order Markov Azzalini (1994) searched for a model can be expressed as: θ = () parameterization such that E Yt , which is Pr() Y Y= Pr() Y Y , ...,Y . free from a parameter and that regulates serial ij ij−−− 1 ij i, j r i, j 1 dependence. The odds ratio Ψ is a quantity that measures the dependence between successive For a three state Markov chain the observations. A technical reason in favor of this corresponding transition probability matrix is choice was provided by Fitzmaurice and Laird given by (1993) who stated that, when the association between observations is modeled using an odds p00 p01 p02 ratio, the estimates of the mean are relatively insensitive to changes of the association P = p p p 10 11 12 parameter. Moreover, the range of feasible Ψ θ p20 p21 p22 values for is independent of the value. The above three stated Markov models and the transition probabilities are: for non-stationary cases are parameterized as θ = θ + −θ == = t P1 t−1 P0 (1 t−1 ) (2) PP0it,0itit1 PYY0()= and == = θ = P θ − + P ()1−θ − (3) PP1it,1itit1 PYY1()− t 2 t 1 0 t 1 where, P0 , P1 and P2 will vary with t and (t = 2, == = 3, 4, 5, …, T). For t = 1, PP2it,2itit1 PYY2.()− ()===θ ( ) EY111Pr Y 1 .and odds ratios: Consider a vector of covariates for the ith ′ = person Xij [1,X i1 , ...,X ip ] and the P ()1 − P ψ = 1 1 (4) corresponding vector of parameters 1 ()− P0 1 P0 β′ = [ββ , ,..., β ] . The transition kk0k1kp probabilities can be expressed in terms of P ()1− P conditional probabilities (Hosmer & Lemeshow, ψ = 2 2 (5) 2 ()− 1989) as follows: P0 1 P0 ===− For a given value of β the sequence of pPYjYj1,Xij() i,j i,j−− 1 i,j 1 θ can be determined by (1) and solving (1) and gXj = e (3); (2) and (4) and after algebraic manipulation, m1− , gXj results in, e k0= where 529 HIGHER ORDER MARKOV BASED LOGISTIC MODEL FOR ORDINAL DATA θψ= Let the parameterization of mean and odds ratio ti,for 1 for second order can be extended as δ−11 +()() ψ − θ −θ itt1− ()()ψ− −θ− 2111it θ=pp θ +(1 −θ ) P;1,= ψ ≠ it it,1 it−− 2 it ,0 it 2 ; −δ+()( ψ − θ +θ − θθ ) 11itt1tt1−− 2 Ρ+Ρ+Ρ+Ρ+Ρ + j =θ001 101 201 011 111 ()()ψ− θ −θ t−2 211it1t1−− +Ρ +Ρ +Ρ +Ρ 211 021 121 221 Ρ+Ρ+Ρ+Ρ+Ρ (6) +−θ000 100 200 010 110 ()1 +Ρ +Ρ +Ρ +Ρ t−2 210 020 120 220 where (7) 2 θ= θ + −θ ()θ−θ Ψ itpp it,2 it−− 2 it ,0(1 it 2 ) δ=+Ψ−2 ()tt1i− 11i Ρ+Ρ+Ρ+Ρ+Ρ 2 002 102 202 012 112 −θ−θ()()−− +2 θ+θ =θ tt1 tt1 +Ρ +Ρ +Ρ +Ρ t−2 and 212 022 122 222 Ρ+Ρ+Ρ+Ρ+Ρ +−θ000 100 200 010 110 () 1 t−2 δ−1( + ψ− 1)( θ −θ − ) +Ρ +Ρ +Ρ +Ρ tt1 210 020 120 220 2(ψ− 1)(1 −θ ) = t1− (8) Py t1− 1(1)(2)−δ+ ψ− θ +θ − θθ + tt1tt1−− yt1− − 2(ψ− 1) θ (1 −θ ) pit,1 (1 Pit,1 ) t1−− t1 ψ = (9) 1 − pit,0 (1 Pit,0 ) for t > 1 and for t = 1, p =θ. yit 1 p | (1− P ) ψ = it,2 it,2 (10) Second Order Covariate Dependent Markov 2 p |(1− P ) Model it,0 it,0 A second order Markov model assumes and that the current response variable is dependent on the history not only through the immediate δ−11 +()() Ψ − θ −θ Ρ=P = iit−2 previous response but also on the previous two j y t2− 211()()Ψ− −θ responses, that is, it−2 ()()(δ−11 + Ψ − θ +θ − 2 θθ ) + iittt−−22 = j Pr()YY−= , Y Pr() YY −− , Y , ..., Y − . 211()()Ψ− θ −θ it it21 ij it it 12 it it n it−−22 t Ψ≠ for i 1, The transition probabilities for the three state second order Markov Chain can be written as: where Pp==Pr() yy | = 0, y = 0 0,0it it it−− 1 t 2 2 =+Ψ−()()()() θ−θ22 Ψ−θ−θ + θ+θ δ 11ittitttt{ −−−222 2 } ==() = = Pp1,1itPr yy it | it−− 1 0, y t 2 1 Ψ = λ Ψ = λ and log i 1 and log 2 2 . These relationships generate a process ==() = = having the desired properties. Upon taking Pr(yt Pp2,2itPr yy it | it−− 1 0, y t 2 2 . = 1) = θ and generating y , …, y via a non- 1 2 t homogeneous Markov chain with transition probabilities Pj, a sequence is obtained such that 530 BISWAS, ISLAM & ISLAM E(yt) = θt for t = 1, 2, ..., t and the odds ratios for observations on the same individual must be (yt-2, yt) are equal to ψ . taken into account, it is plausible that adjacent data are more strongly correlated than data that Estimation are separated by time and that different The conditional likelihood function for a individuals behave independently on the log sample of n independent observations is: likelihood, this is given by: n 4 y0i y1i y2i ()β λ = β λ L()β = ∏[][]P()0 x []P()1 x []P()2 x l , lt ( , ) = i=1 T 1 (11) where score vectors are: The log-likelihood function can be written as ∂l T n ∂l = it ∂β ∂β yPx01 log() 0 t==11i n and lyPx()β= + log() 1 ti 1 T n = ∂l ∂l i 1 = it + yPxlog() 2 ∂λ ∂λ 2i t==11i Ρ=()y 0 y log and the variance of the estimate is approximated 0i ()−Ρ() = by 10y n Ρ=() −1 y 1 T =+y log ∂∂ll 1i −=() tt i=1 11Py n ∂β ∂β βλˆˆ = V ( , ) Ρ=()y 2 = ∂∂ + i 1 lltt y2i log 12−=Py() ∂λ ∂λ β =βˆˆ,λ=λ (12) and The quantity inside the square brackets approximates the Fisher Information for large n.

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