The Working Correlation Structure Selection Criteria

3.2 Assuming that Y i = (Yi1, Yi2, Yi3, . . . , Yini )T , i = 1, 2, 3 . . . , K, is indicative of the K which are response vectors and the Yit observation is identified as the t-th term of the i-th subject in t = 1, 2, 3, . . . , ni, then the design matrix of i-th subject can be expressed as: X i = (Xi1 , Xi2 , Xi3 ,. . . , X ini )T , while the Yit covariates for the p dimensional vector can be denoted as X it= (xit1 , xit2 , xit3 ,. . . , xitp )T. The ni observations of each of the i subjects are generally inter-correlated and are also correlated with the working correlation matrix denoted as R(α) as is identified by the GEE approach developed by Liang and Zeger (1986) which implies that the s dimensional vector α = (α1 , α2 , α3 , . . . , αs )T can be used to define the R(α). Considering the assumption that ni = n, then it is possible to delineate the four common structures that will be used in this paper, namely: (1) The Independence structure (IN), where, R(α) = In , the identity matrix is denoted as In when there is no inter-subject correlation; (2) The Exchangeable structure (EX) where α has an unknown parameter; (3) The One-order autoregressive structure (AR-1), where α has an unknown parameter; as well as (4) the Stationary structure (ST), where α has n ˆ’ 1 unknown parameters. The matrices of the respective structures can therefore be expressed as: IN =(–ˆ( 1 @ 0 @‹@ 0 )–ˆ(0 @1 @‹@0 )–ˆ(‹¯@‹¯@‹±@‹¯)–ˆ(0 @0 @‹@1 ))EX =(–ˆ( 1 @α@‹@α)–ˆ(α@1 @‹@α)–ˆ(‹¯@‹¯@‹±@‹¯)–ˆ(α@α@‹@1 )) AR-1 =(–ˆ( 1 @α@‹@α^(n-1) )–ˆ(α@ 1 @‹@‹¯)–ˆ(α^2@α@‹@α^2 )–ˆ(‹¯@‹¯@‹±@‹¯)–ˆ(α^(n-1)@α^(n-2)@‹@1 ))ST =(–ˆ( 1 @α_1@‹@α_(n-2)@α_(n-1) )–ˆ(α_1@ 1 @‹@α_(n-3)@α_(n-2) )–ˆ(α_2@α_1@‹±@‹¯@‹¯)–ˆ(‹¯@‹¯@‹@‹±@α_1 )–ˆ(α_(n-1)@α_(n-2)@‹@α_1@ 1)) The Yit conditional expectation according to the GEE method is E(Yit|Xit ) =µit (β) which can also be expressed as g ˆ’1 (X_it^T β) where g is the link function of the GLM model while β is the unknown regression parameter of the p dimensional vector which is of interest. Additionally, it can further be assumed that the Var (Yit ) = v(µit (β))•, which can also be expressed as ƒ_it^2•, while Vi =A_i^½R(α)A_i^½• is the Y i working covariance structure with the Ai diagonal matrix consisting ofƒ_it^2 , t = 1,2,3, . . . , ni to the diagonal while • is the parameter of overdispersion. 3.2.1 The Quasi-Likelihood under the Independence Model Criterion (QIC) One of the most popular criterions used in the likelihood-based selection of models is AIC. Despite its prominence, criterion such as AIC cannot be applied to the GEE method due to the fact that GEE is not likelihood-based. Pan (2001) however proposes a quasi-likelihood criterion commonly known as QIC, which can be used in the selection of the appropriate mean model as well as the working correlation structure. According to Hardin and Hilbe (2003), the quasi-likelihood function can be expressed as: Q(μ,•;y)= ˆ«_y^μ–’ã–(•(y-μ^*))/(v(μ^*)) dμ^* ã— Pan (2001) further argues that when it is assumed that both the subjects as well as the time points are independent, the quasi-likelihood of longitudinal data should be calculated as: Q(μ,•)= -2 ˆ‘_(i =1 )^K–’ˆ‘_(t=1)^(n_i)–’Q(μ,•;Y_it ) As a result, the quasi-likelihood function of the i cluster in observation t which is evaluated using the β regression parameters can be expressed as Q(β, •; Yit, xit) = Qit/•, with Qitbeing considered to defer to the distributions that are commonly used which can be better demonstrated through tabulation as is shown in Table 1 below. Exponential family distribution Link function Variance function Qit Binomial ln {µit/(1 “µit)} µit(1 “µit) yit ln{µit/(1 “µit)} + ln(1 “µit) Normal µit 1 “ ½(yit“ µit)2 Poisson ln µit µit log µit“µit Inverse Gaussian 1ã–/µã—_it^2 µ_it^3 “yit/(2µ_it^2) + log 1/µit Gamma 1/µit µ_it^2 “yit/µit“ log µit Table 1: The link functions, variance functions, as well as the quasi-likelihoods of common distributions If the working assumption is that both the clusters as well as the observations are independent, then the QIC can be indicated as: QIC(R)= -2 ˆ‘_(i =1 )^K–’ˆ‘_(t=1)^n–’ã–Q(β,•;Y_it,x_it ) + 2tr {„¦V_T (R)} ã— Whereby tr denotes the total diagonal elements within the matrix while „¦ =ˆ‘_(i= 1)^K–’ã–D_i^T A_i^(- 1) ã— D_i. Pan (2001) also posits that QIC can be expressed as follows: QIC(R)= -2Q(β Ì‚,• Ì‚ )+ 2tr („¦ Ì‚_I V Ì‚_r) The first term in the above equation is the quasi-likelihood, and can further be expressed as a function of β Ì‚ through the substitution of μ Ì‚ ˆ. Therefore, it is possible to obtain „¦ Ì‚_Iby substituting β, • and α with their respective estimates. On the other hand, V Ì‚_r is the robust variance estimate which according to Pan (2001) can be used in the selection of the correlation structure with a minimum QIC(R) value in accordance with the working correlation structure. 3.2.2 The Rotnitzky-Jewell’s Criterion (RJC) According to Rotnitzky and Jewell (1990), the test statistics can be used to support the premise that the regression coefficients vector is equivalent to a specified β. With reference to the pertinent theorem on the test statistics, Rotnitzky and Jewell (1990) define Ψ0, Ψ1, and Ψ as: Ψ_0= 1/K ˆ‘_(i = 1)^K–’D_i^T V_i^(- 1) S_i S_i^T V_i^(- 1) D_i Ψ_1= 1/K ˆ‘_(i = 1)^K–’D_i^T V_i^(- 1) D_i Ψ= Ψ_0^(- 1) Ψ_1 In this case, Ψ is equivalent to an identity matrix if the working correlation structure has been correctly specified. In line with the assessment of Hin et al (2007), the RJC for the selection of the working correlation structure can therefore be expressed as: RJC(R) = [{1-tr(Ψ)/p}^2+ {1-tr(Ψ^2)/p}^2 ]^(1/2) 3.2.3. Correlation Information Criterion (CIC) The proposal by Hin and Wang (2009) to modify the QIC in order to improve its performance yields the correlation information criterion which can be expressed as: CIC(R) = tr {„¦V_r (R)} CIC is only contingent on the second term of QIC which is the crux of the QIC penalty in accordance with the QIC Equation, where the first term represents the sum of the quasi-likelihoods of all the observations, with the assumption that the presented subjects as well as the time points are independent. As a result, the CIC does not factor in the first term in the comparison of different working correlation structures due to the fact that the term is primarily not dependent on the specified R. Taking into consideration that the main intention of Hin and Wang (2009) in constructing the CIC is to improve QIC performance by minimizing CIC(R) in the selection of the correlation structure, the working correlation structure of CIC can therefore be expressed as:CIC(R) = tr („¦ Ì‚_I V Ì‚_r ). It is possible to obtain „¦ Ì‚_I by substituting β, † and α with their respective estimates in the QIC Equation which is similar to calculating the QIC, with V Ì‚_r as the estimate of robust variance that corresponds to the variance matrix equation that is obtained using the GEE method. 3.2.4. The Gosho Criterion (DEW) In line with the proposal by Hin and Wang (2009), Gosho et al (2011) also suggest the selection of the correlation structure with the minimal DEW(R), in accordance with the working correlation structure which Gosho et al (2011) expresses as: DEW(R)= tr [{(1/K ˆ‘_(i = 1)^K–’ã–S_i Sã—_i^T ) (1/K ˆ‘_(i = 1)^K–’V_i )^(- 1)- I}^2 ] The identity matrix is denoted as I in the DEW equation while DEW(R) is the criterion that is used to directly measure the difference between the estimator of the covariance matrix and the covariance matrix of the specified working criterion. 3.2.5. Gaussian Pseudo-Likelihood Criteria (GPC) The Gaussian pseudo-likelihood criterion (GPC) constructed by Carey and Wang (2011) is predicated on the extended quasi-likelihood function that was developed by Hall and Severini (1998). The extended quasi-likelihood function offers the estimating function for both β and α, with Hall and Severini (1998) further establishing the consistency as well as the asymptotic multivariate normality properties of the related parameters that are estimated using the extend quasi-likelihood (QL) functions. The extended QL function has similar features to the multivariate Gaussian likelihood function. Vi can therefore be denoted as: ã– Wã—_i (α) = A_i^(1/2) R(α)A_i^(1/2) •, while, E(Y_i”‚X_i )=µ_iwhereby, Wi (α) is the estimated working covariance structure of the estimating equations. The extended quasi-likelihood function can therefore be expressed as: LG= -1/2 ˆ‘_i–’{(Y_i-µ_i )^T ã– Wã—_i^(- 1) (Y_i-µ_i )+log¡(|W_i | ) } The Gaussian pseudo-likelihood criterion proposed by Carey and Wang (2011) is therefore GPC = ˆ’2LG, and if the estimations of β and α are extracted from the GEE, both µi(β Ì‚ ) as well asR_i (α Ì‚) can be fitted. Model selection can therefore be completed by selecting the candidate model that minimizes the GPC. Carey and Wang (2011), use GPC to make a distinction between the EX and AR-1 working correlation matrices in which both structures have one unknown R(α)parameter. As a result, GPC is not highly efficient in model selection between different structures with varying numbers of free parameters. Hall and Severini (1998) consider GPC to be an extended QL function which also comprises of likelihood oriented properties, with notably high performances in the selection of working correlation matrices predicated on both Gaussian as well as Lognormal responses. Taking into account the fact that GEE lacks a likelihood function, the AIC can be modified by substituting the likelihood factor with a quasi-likelihood factor in a working independence model, in deference to works by Pan (2001). In addition, Pan (2001) modified the discrepancy through the adjustment of the QIC penalty term, implying that modified criteria can be improved by substituting the likelihood factor in AIC with the extended quasi-likelihood factor. As a result, the AIC-based modified criterion can be expressed as: AGPC = -2LG + 2 dim(θ) = GPC + 2 dim(θ) The Bayes-based criterion BIC developed by Schwarz (1978) is similar to the AIC with the only difference occurring in the penalty term. By maintaining a similar penalty term to that of BIC, the BIC can therefore be modified to form the BGPC criterion which is expressed as: BGPC = -2LG + log¡(K) dim(θ) = GPC + log¡(K) dim(θ) θ is the estimated free parameters vector which can also be expressed as: (β^T,α^T )^T. Schwarz (1978) notes that there is a significant disparity between AIC and BIC in evaluating a large number of observations which implies that AGPC and BGPC should also present a significant difference in the simulation studies. Unlike the QIC proposed Pan (2001), the parametric likelihood term in AIC and BIC is directly substituted with the extended quasi-likelihood function while omitting additional estimates and derivations. It is also important to note that unlike the QIC, the AGPC and BGPC criteria do not assume the independence working correlation matrix. 3.2.6 Empirical Likelihood AIC (EAIC) and BIC (EBIC) The approach used by Chen and Lazar (2012) which substitutes the empirical likelihood in AIC and BIC with parametric likelihood to form two additional criteria for selecting the working correlation matrix demonstrates the two constructed criteria to be more efficient when compared to QIC and CIC. As a result, it is essential to further investigate and evaluate the performance of the EL-based criteria against the other criteria discussed in this chapter. In their evaluation pertaining to of EAIC and EBIC, Chen and Lazar (2012) primarily focus on deriving the empirical likelihood ratio (ELR) of a full model under the assumption that the stationary (ST) working correlation matrix RF (α) has the free parameters p+nˆ’1 inserted into θ^T=(β^T,α_1,α_2,α_3,¦ ,α_(n“ 1)). Under the assumed model, obtaining the model ELR requires the initial definition of the (g^F (ˆ™) )estimating function as: g^F ((Y_i ,X_i ), β, α_1,α_2,α_3,¦ ,α_(n“ 1); R_F (α) ) = [– ((ˆ‚μ_i/ˆ‚β^T )^T A_i^(-1/2) R_F^(- 1) (α_1,α_2,α_3,¦ ,α_(n“ 1))A_i^(-1/2) ã– (Yã—_i“μ_i)@– (ˆ‘_(t = 1)^(n- 1)–’ã–e_it (β) e_(i,t + 1) (β)-α_1 • Ì‚(β)(n-1-p/K) ã—@– (‹@ˆ‘_(t = 1)^1–’ã–e_it (β) e_(i,t + n-1) (β)-α_(n-1) • Ì‚(β)(1-p/K) ã—)))]_((p + n- 1)X 1) With the Pearson residuals expressed as: e_it (β)= (Y_it-μ_it (β) )/ˆš(v(μ_it (β) ) ) • Ì‚(β)= ˆ‘_(i = 1)^K–’ã–ˆ‘_(t = 1)^n–’e_it^2 /(Kn“p)ã— This makes it possible to subsequently express the ELR function by substituting the new estimating equation with the established GEE term resulting in the expression: R^F (β,α^T )=sup¡{ˆ_(i = 1)^K–’ã–ã–K‰ã—_i ˆ¶ ‰_i‰¥0,ˆ‘_(i = 1)^K–’ã–‰_i=1,ã—ã— ˆ‘_(i = 1)^K–’ã–ã–‰_i gã—^F ((Y_i ,X_i ),β,α^T; R_F (α) )=0ã—} The estimating equations for the respective candidate structures therefore have a similar working correlation matrix with a stationary structure which together with the (g^F (ˆ™) )estimating function facilitates the signing of ELRs with both different as well as analogous values pertinent to the different working correlation matrices. This is predicated on the fact that maximum empirical likelihood estimators are similar to GEE estimators exclusive of the modification which would with result in ELR values that are equivalent to 1. Chen and Lazar (2012) computed the θ Ì‚_GGEE estimates that correspond to each of four candidate working correlation structures as: (β Ì‚_IN,O^T )^Tfor an IN structure, (β Ì‚_EX^T,α Ì‚_EX^T )^T for an EX structure, (β Ì‚_(AR(1))^T,α Ì‚_(AR(1))^T )^Tfor an AR-1 structure, as well as (β Ì‚_ST^T,α Ì‚_ST^T )^T for an ST structure. The ELR values are subsequently obtained by inserting the GEE estimates into the ELR function expressionR^F (β,α^T ). Chen and Lazar (2012) further developed the criteria for selecting the working correlation matrix with resulting higher ELR values, namely: EAIC and EBIC, where: ã–EAIC =-2 log Rã—^F (θ Ì‚_G )+ 2 dim(θ) ã–EBIC =-2 log Rã—^F (θ Ì‚_G )+log¡(K)dim (θ) Here, dim (θ) is the element of the (β^T,α^T )^Tfree parameters that are being estimated, and the minimum values of EAIC and EBIC are indicative of the likely model. 3.3 Evaluation of the Identified Criteria All the model selection criteria that have been selected for further analysis in this chapter are likelihood-based, with the modified GPC methods as well as the EL-based criteria being considered as AIC and BIC extensions. While GPC directly utilizes the extended quasi-likelihood function as the selection criterion which effectively enhances its performance particularly in deference to Gaussian and Lognormal data, the modified AGPC and BGPC criteria factor in the number of free parameters and applies this number as the penalty term. The modification is logical, predominantly in improving the accuracy of the selection process which requires the estimation of a variety of parameters in the working correlation matrix. The selection of the identified criteria is also contingent on the fact that recent literature works have demonstrated enhanced accuracy and performance of the criteria. The performance of the identified criteria can be explored further through an extensive evaluation of the criteria by broadening the scope of the working correlation matrices as well as the distributions of the responses in both continuous and discrete terms. For instance, considering the fact that GPC has a propensity to select the more complicated ST structures exclusive of the free parameters penalty, it is expected that the efficiency of GPC will decline in selecting candidates that consist of ST structure since this structure has a higher number of free parameters when compared other candidates. The empirical likelihood term used in the EL-based approach is different from the extended QL term although both likelihood-based terms have effective properties. Generally the criteria have relatively simple computational complexity, with each having its unique set of incentives and merits depending on the circumstances. The performance of the identified criteria is investigate using simulation studies that are will discuss and conduct in Chapter 4. For a custom paper on the above topic, place your order now! What We Offer: ¢ On-time delivery guarantee ¢ PhD-level writers ¢ Automatic plagiarism check ¢ 100% money-back guarantee ¢ 100% Privacy and Confidentiality ¢ High Quality custom-written papers