Resultados de la política de IS

Let yi(s) be the CAL response for subject i = 1, . . . , N at spatial location s = 1, . . . , S.

For each subject, there are S = 168 potential measurement locations. Denote yi =

[yi(1), . . . , yi(S)]T, the response vector for subject i. Typical for any PD data, either

all 6 measurements from a tooth are observed, or all observations are missing (given that probing doesn’t happen for missing tooth). We first develop the SNI-CAR model

assuming all observations are present, and then extend it to include missingness. Under a standard linear mixed model (LMM) setup, the observed CAL for subject i can be written as:

yi = µi+ εi, (3.4)

µ_i = X>_i β + θi,

where µ_i = [µi(1), . . . , µi(S)]> is the vector of true CAL values for subject i for all

the available observations, εi ∼ N(0, σ2IS) is the vector of random errors εi(s), X>i

is p-vector of subject-level (say, age) and site-level (say, site in gap) covariates, β are the corresponding regression parameters of dimensions p × 1 and θi is the vector of

random effects. Now, to accommodate possible spatial referencing, the latent vector θi = [θi(1), . . . , θi(S)]>can follow a S-dimensional multivariate normal distribution with

mean zero (E[θi] = 0) and a CAR (Besag, 1974) covariance matrix. The CAR covariance

of θi, denoted by Σi, is given by τ2Q(ρi)−1, where Q(ρi) = M − ρiD, where D is S × S

the adjacency matrix of the underlying graph whose elements Dss0 equals 1 if locations

s and s0 are adjacent, and 0 otherwise; M is a S × S diagonal matrix with diagonal elements Mss=

P

s0D_ss0representing the number of neighbors for site s; ρ_i ∈ [0, 1] is the

smoothing parameter controlling the degree of spatial association and τ2 > 0 controls the magnitude of spatial variation. In the adjacent matrix, we model the adjacent sites on the same tooth and sites that share a gap between teeth as ‘neighbors’. For issues with identifiability, henceforth, we assume ρi= ρ for all i, i.e., all subjects have the same

spatial variation. This assumption is not unrealistic from a clinical standpoint, given that the set of subjects from the GAAD data are all Type-2 diabetic with extremely homogeneous socio-economic features (Johnson-Spruill et al., 2009). Now, due to the

presence of possible asymmetry (skewness and thick-tails), we assume θi follows a SNI-

CAR density of dimension S, which we write as θi ∼ SNI-CARS(µ, Σ, Λ, H(·; ν)), where

Σ is the CAR covariance, Λ is a diagonal matrix associated with the skewness parameter, and H(·; ν) denotes one of the distributions presented in Subsection 3.3.1. Centering θi to have zero mean, we assume the location parameter µ = −

q

2

πκ1λ. Thus, we

have θi ∼ SNI-CARS(−

r 2

πκ1(ν)λ1S, Σ, λI, H(·; ν)), where the skewness parameter λ is chosen to be a scalar to avoid over-parametrization and identifiability problems. This representation partitions the skewness component and the spatial component, and hence provides a flexible way to incorporate multivariate asymmetric spatial random effects into our modeling.

However, in reality, substantial proportion of missing data is observed from PD stud- ies, and the GAAD dataset is no exception (Reich and Bandyopadhyay, 2010). Hence, the complete response vector yi for subject i is incomplete, and can be decomposed

into yo_i and ym_i , the observed and missing components respectively, according to the missingness process ∆, which is assumed non-ignorable, i.e., missingness is induced due to unobserved responses. For instance, in the GAAD study, subjects with higher level of PD tend to have teeth that had fallen out due to previous incidence of PD. Further- more, the missingness is monotone, i.e., a missing tooth is never going to come back, and is different from the non-monotone assumption typical in longitudinal studies. In this situation, it has been shown that ignoring the missingness process and analyzing ‘only available’ data can lead to biased parameter estimates (Follmann and Wu, 1995; Reich and Bandyopadhyay, 2010). Hence, joint modeling of the observed CAL data and the missingness process is indicated, and we achieve this via the popular shared parameter models (SPM), where a set of (spatial) random effects induces the interdependence of the two processes (Follmann and Wu, 1995; Tsonaka et al., 2009).

we cannot have an observed site and a missing site from the same tooth. Let δi(t) = 1 if

tooth t is missing for subject i, and 0 otherwise, and ∆idenotes the corresponding vector

for subject i. Under this SPM framework, the joint density of yi and ∆ (suppressing i)

can be factored as:

f (yo, ym, ∆|Ω) = Z

f1(yo, ym|θ, Ω)f2(∆|θ, Ω)g(θ|Ω)dθ

where f, f1and f2are the respective probability density functions, θ is the vector of SNI-

CAR random effects, and Ω is the parameter vector. From this factorization, it follows that given θ, the processes y and ∆ are independent. Now, the missing tooth locations are not random, but are related to the periodontal health of that region inside the mouth. Hence, for subject i, we allow the missing tooth indicator δi(t) ∼ Bernoulli(pit),

such that

logit(pit) = a0+ b0Z>t θi (3.5)

where Z>_tθi is the mean of θi at the six observations on tooth t, t = 1, . . . , 28, with

Zt(s) equal 1/6 if site s is on tooth t, and 0 otherwise, and a0 and b0 relate the latent

process to the missing tooth indicator (Reich and Bandyopadhyay, 2010). Note that since θi(s) is included in both the model for presence of and value of the responses,

both presence and value of the data contribute to the posterior of θi(s), and thus the

posterior of Ω, the full parameter vector under consideration. Also note that b0 = 0

corresponds to independence between the latent true CAL and the location of missing teeth, in which case the location of missing teeth does not contribute to estimating Ω. Note that in our current formulation, we assume the missingness process is dependent only on the (latent) spatial random effects, and not on any covariates. This was assumed for simplicity of interpretation, and also to avoid identifiability issues; however this can certainly be relaxed in our estimation framework. Assuming θi ∼ G (the distribution

function), the joint density of the observed data vector (yi, ∆i) for the ith subject is

obtained from the following marginalization:

f (yi, ∆i|G, Ω) =

Z

f1(yi|θi, Ω)f2(∆i|θi, Ω)dG(θi)