CAR_cov_teeth {SMARTp}

The within-mouth covariance matrix with conditional autoregressive structure

Description

The covariance matrix of individual teeth measures for each subject follows a Conditional Autoregressive model (CAR) density

Usage

CAR_cov_teeth(m, rho, tau)

Arguments

m

Maximum number of units in each cluster, i.e., 28 teeth in each mouth (the 4 third-molars are usually ignored)

rho

Association parameter of the CAR model

tau

Variation parameter of the CAR model

Details

CAR_cov_teeth gives the covariance matrix among the teeth within each mouth based on the CAR structure (Besag et al., 1991), given the maximum number of teeth for each subject (m), the variance (\tau), and the association (\rho) parameters.

The CAR covariance matrix can be expressed as \Sigma_{28\times 28} = \tau^2 (W - \rho D)^{-1}, where \tau^2 > 0, and \rho \in [0, 1] are the parameters that control the magnitude of variation and the degree of spatial association, respectively. For matrix D, the element D_{tt'} is 1 if locations t and t' are adjacent and 0 otherwise. The matrix W is diagonal with diagonal elements W_{tt} = \sum_{t'} D_{tt'}. Note, the argument \tau in CAR_cov_teeth is the variance, and not the standard deviation.

Value

The covariance matrix among the teeth in each mouth (assuming full dentition, i.e., 28 teeth) based on a CAR model.

Author(s)

Jing Xu, Dipankar Bandyopadhyay, Douglas Azevedo, Bibhas Chakraborty

References

Besag, J., York, J. & Mollie, A. (1991), "Bayesian image restoration, with two applications in spatial statistics (With Discussion)", Annals of the Institute of Statistical Mathematics 43, 159.

Reich, B. & Bandyopadhyay, D. (2010), "A latent factor model for spatial data with informative missingness", The Annals of Applied Statistics 4, 439–459.

See Also

MC_var_yibar_mis, SampleSize_SMARTp

Examples

m <- 28
rho <- 0.975
tau <- 0.85
Sigma <- CAR_cov_teeth(m = m, rho = rho, tau = tau)