A profile likelihood approach to longitudinal data Speaker: Ziqi Chen Central South University Workshop on Flexible Models for Longitudinal and Survival Data with Applications in Biostatistics July, 27th, 2015 A profile likelihood approach to longitudinal data
Introduction Longitudinal data arise frequently in biomedical and health studies in which repeated measurements from the same subject are correlated. The consistency and efficiency of estimators of the regression parameters are important for longitudinal data analysis. A profile likelihood approach to longitudinal data
Introduction Liang and Zeger (1986) developed the generalized estimating equation (GEE) for correlated data. GEE estimators are efficient when the working correlation structure is correctly specified, which is not an easy task however. Misspecification of the correlation structure may lead to a great loss of efficiency (Wang and Carey 2003), although consistency remains holds. Qu, Lindsay, and Li (2000) proposed the quadratic inference function (QIF) to improve efficiency of the GEE by combining strength of several correlation structures. A profile likelihood approach to longitudinal data
Introduction Ye and Pan (2006) proposed the simultaneous GEE equations to estimate both the mean regression coefficients and covariance structure parameters. Leung, Wang and Zhu (2009) proposed a hybrid method that combines multiple GEEs based on different working correlation models, using the empirical likelihood method (Qin and Lawless, 1994). In order to attain efficient estimators (i.e., their asymptotic variances achieve the efficiency bounds), the aforementioned papers all need correct specification of the correlation structure for longitudinal data. A profile likelihood approach to longitudinal data
Introduction If we know the full likelihood function of the longitudinal data, we could achieve the maximum likelihood estimators of the regression parameters, which is efficient. Motivated by the modified Cholesky decomposition, Pourahmadi (1999,2000) proposed to estimate the mean regression coefficients efficiently by simultaneously estimating the covariance matrices based on the the multivariate normal distribution. However, it is difficult to specify the full likelihood function when responses are non-normal because of the correlated nature of longitudinal data. A profile likelihood approach to longitudinal data
Introduction By regressing the error on its predecessors (Pourahmadi, 1999), we treat the prediction error density as an unknown nonparametric function and propose to estimate it by kernel smoothing. With the estimated prediction error density, we achieve the estimators of the regression parameters by maximizing the so-called profile likelihood. Our proposed method performs well without specification of the likelihood as well as the correlation structure. A profile likelihood approach to longitudinal data
The profile likelihood methods Suppose the response variable for the i − th subject is measured m times, y i = ( y i 1 , · · · , y im ) T , where y i ’s are independently distributed, i = 1 , · · · , n , where n is the sample size. The covariate corresponding to y ij is x ij , which is a p − dimensional vector. Denote X i = ( x i 1 , · · · , x im ) T , which is m × p − dimensional matrix for the i − th subject. Let µ ij = E ( y ij | X i ) = x T ij β , where β is a p − dimensional parameter vector with true value being β ∗ . ij β ∗ and ǫ i = ( ǫ i 1 , · · · , ǫ im ) T . We assume Denote ǫ ij = y ij − x T that the error vector ǫ i is independent of the covariate matrix X i , and ǫ i ( i = 1 , · · · , n ) are independently and identically distributed. A profile likelihood approach to longitudinal data
The profile likelihood methods The independence estimating procedure We first naively assume that the responses of the i − th subject are independent of each other, for i = 1 , · · · , n . The maximum likelihood estimator of β could be got if the density functions of ǫ 1 j ( j = 1 , · · · , m ) are known. However, these density functions are typically unknown in practice. Let ǫ ij ( β ) = y ij − x T ij β . We propose to estimate the density function of ǫ 1 j ( β ) for given β by kernel smoothing as n f ǫ 1 j ( β ) ( u ) = 1 K ( ǫ ij ( β ) − u ˆ � ) , for j = 1 , · · · , m , nh h i =1 where K is a scalar kernel and h is an appropriate bandwidth. A profile likelihood approach to longitudinal data
The profile likelihood methods The independence estimating procedure We propose to obtain the estimator of β by maximizing the profile likelihood equation (Chen, et al., 2014): m n β I ˆ � � log ˆ MPL = arg max f ǫ 1 j ( β ) ( ǫ ij ( β )) . (1) β j =1 i =1 A profile likelihood approach to longitudinal data
The profile likelihood methods The independence estimating procedure Since x ij is independent of ǫ ij , we know m � � f ǫ 1 j ( β ) ( u ) log f ǫ 1 j ( β ) ( u ) du j =1 m � � < f ǫ 1 j ( β ∗ ) ( u ) log f ǫ 1 j ( β ∗ ) ( u ) du , (2) j =1 for any β � = β ∗ . A profile likelihood approach to longitudinal data
The profile likelihood methods The independence estimating procedure We have that by simple calculations m n m 1 � � � log ˆ � f ǫ 1 j ( β ) ( ǫ ij ( β )) → P f ǫ 1 j ( β ) ( u ) log f ǫ 1 j ( β ) ( u ) du , (3) n j =1 i =1 j =1 uniformly in β , by Theorem 2.1 of Newey and McFadden (1994), we obtain ˆ β I MPL → P β ∗ . A profile likelihood approach to longitudinal data
The profile likelihood methods The independence estimating procedure Even though the consistency of ˆ β I MPL holds, it may not be efficient estimator, since ˆ β I MPL is got based on the independence assumption, that is, the within-subject correlation has not been taken into account. This is consistent with the literature in GEE that one could not get the fully efficient estimator when independence working correlation matrix is used (Liang and Zeger, 1986; Diggle, 2002). The asymptotical normality theory in the next subsection and the simulation studies further verify this point. A profile likelihood approach to longitudinal data
The profile likelihood methods The efficient estimating approach Let Cov( ǫ i ) = Σ. By the modified Cholesky decomposition (Pourahmadi, 1999), there exist a lower triangular matrix P with ones as diagonal entries and − φ jl as the ( j , l )th element and a diagonal matrix D = diag( σ 2 i 1 , · · · , σ 2 im ) such that P Σ P T = D . Based on this decomposition, one regresses ǫ ij on its predecessors ǫ i 1 , · · · , ǫ i ( j − 1) with corresponding regression coefficients being φ j 1 , · · · , φ j ( j − 1) and denotes η ij to be the corresponding prediction error, that is, j − 1 � η ij = ǫ ij − φ jl ǫ il , for j = 2 , · · · , m ; i = 1 , · · · , n . l =1 Let η i 1 = y i 1 − µ i 1 , for i = 1 , · · · , n . A profile likelihood approach to longitudinal data
The profile likelihood methods The efficient estimating approach We have that η i 1 , · · · , η im are uncorrelated random variables since their covariance matrix D is a diagonal matrix. The specification of φ jl ’s determines the correlation structure of the error ǫ i . For example, the error has the independence correlation structure when φ jl ’s are all zero and owns the AR-1 correlation structure if φ jl is zero for j − l ≥ 2. A profile likelihood approach to longitudinal data
The profile likelihood methods The efficient estimating approach ij β − � j − 1 il β ) φ jl , let � 0 ′ ′ Define η ij ( β, φ j ) = y ij − x l =1 ( y il − x l =1 equal zero throughout this paper and let φ j = ( φ j 1 , · · · , φ j ( j − 1) ). When the distribution of ǫ i is not available, instead of using the maximum likelihood method, we propose η 1 j follows completely nonparametric distribution for j = 1 , · · · , m and use the kernel nonparametric technique to estimate the density function of η 1 j . A profile likelihood approach to longitudinal data
The profile likelihood methods The efficient estimating approach We assume φ j ( j = 2 , · · · , m ) are known and propose to estimate the density function of η 1 j ( β, φ j ) for given β by kernel smoothing as n f η 1 j ( β,φ j ) ( u ) = 1 K ( η ij ( β, φ j ) − u ˆ � ) , for j = 1 , · · · , m , (4) nh h i =1 where K is a scalar kernel and h is an appropriate bandwidth. A profile likelihood approach to longitudinal data
The profile likelihood methods The efficient estimating approach With the estimated prediction error densities, we propose to get the estimate of β through the maximum profile likelihood (MPL; Chen et al., 2014), that is, ˆ β MPL is got through maximizing the following profile likelihood equation: m n log ˆ � � f η 1 j ( β,φ j ) ( η ij ( β, φ j )) . (5) j =1 i =1 A profile likelihood approach to longitudinal data
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