# Estimated mean and variance of the average change in CAL for each subject The estimated Monte Carlo mean and variance of the average change in clinical attachment level (CAL) for each subject ```r MC_var_yibar_mis(mu, Sigma, sigma1, lambda, nu, sigma0, Num, a0, b0, cutoff) ``` ## Arguments - `mu`: Mean matrix, where row represents each treatment path, and column represents each cluster unit - `Sigma`: Within-mouth teeth covariance matrix - `sigma1`: Standard deviation of the residual for the continuous outcome $Y_{it}$ - `lambda`: The skewness parameter of the residual for the continuous outcome $Y_{it}$ - `nu`: The degree freedom, or kurtosis parameter of the residual for the continuous outcome $Y_{it}$ - `sigma0`: Standard deviation of the residual for the binary outcome $M_{it}$ - `Num`: Number of samples to estimate mean or variance of $\bar{Y}_i$ - `a0`: Intercept parameter in the probit model for the binary outcome $M_{it}$ - `b0`: Slope parameter corresponding to the spatial random effect in the probit model for the binary outcome $M_{it}$ - `cutoff`: Cut-off value in the binary outcome regression ## Returns The simulated dataset of CAL change "$Y_{it}$", missingness "$M_{it}$" and function inside the indicator of "$M_{it} I_{it}$" for each tooth of each patient, with the corresponding estimated mean "$mY_i$", variance "$VarY_i$" and missing proportion "PM" for each patient ## Details **MC_var_yibar_mis** computes the Monte-Carlo estimates of expectation and variance of the sample mean among the teeth within each mouth, i.e $\bar{Y}_i = \sum Y_{it}(1 - M_{it})/\sum(1 - M_{it})$, where $Y_{it}$ is the change in CAL (measured in mm) for patient $i$ and tooth $t$, and $M_{it}$ is the misingness indicator, i.e., $M_{it} = 1$ implies tooth $t$ in subject $i$ is mising. The joint regression models for $Y_{it}$ and $M_{it}$ are available in Reich & Bandyopadhyay (2010, **Annals of Applied Statistics**). ## Examples ```r m <- 28 Num <- 1000 cutoff <- 0 sigma1 <- 0.95 sigma0 <- 1 lambda <- 0 nu <- Inf b0 <- 0.5 a0 <- -1.0 rho <- 0.975 tau <- 0.85 del1 <- 0.5 del2 <- 2 Sigma <- CAR_cov_teeth(m, rho, tau) Sigma_comp <- array(Sigma, c(m, m, 4)) Sigma_sim <- array(Sigma, c(m, m, 10)) mu_comp <- array(0, c(2, m, 2)) mu_comp[, , 1] <- rbind(rep(0, m), rep(del1, m)) mu_comp[, , 2] <- rbind(rep(0, m), rep(del2, m)) VarYitd1R = MC_var_yibar_mis(mu = mu_comp[1, , 1], Sigma = Sigma, sigma1 = sigma1, lambda = lambda, nu = nu, sigma0 = sigma0, Num = Num, a0 = a0, b0 = b0, cutoff = cutoff) PM <- VarYitd1R$PM VarYid1R <- VarYitd1R$VarYi mYid1R <- VarYitd1R$mYi VarYitd1NR <- MC_var_yibar_mis(mu = mu_comp[2, , 1], Sigma = Sigma, sigma1 = sigma1, lambda = lambda, nu = nu, sigma0 = sigma0, Num = Num, a0 = a0, b0 = b0, cutoff = cutoff) PM <- VarYitd1NR$PM VarYid1NR <- VarYitd1NR$VarYi mYid1NR <- VarYitd1NR$mYi VarYitd3R <- MC_var_yibar_mis(mu = mu_comp[1, , 2], Sigma = Sigma, sigma1 = sigma1, lambda = lambda, nu = nu, sigma0 = sigma0, Num = Num, a0 = a0, b0 = b0, cutoff = cutoff) PM <- VarYitd3R$PM VarYid3R <- VarYitd3R$VarYi mYid3R <- VarYitd3R$mYi VarYitd3NR <- MC_var_yibar_mis(mu = mu_comp[2,,2], Sigma = Sigma, sigma1 = sigma1, lambda = lambda, nu = nu, sigma0 = sigma0, Num = Num, a0 = a0, b0 = b0, cutoff = cutoff) PM <- VarYitd3NR$PM VarYid3NR <- VarYitd3NR$VarYi mYid3NR <- VarYitd3NR$mYi ``` ## 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 CAR_cov_teeth, SampleSize_SMARTp ## Author(s) Jing Xu, Dipankar Bandyopadhyay, Douglas Azevedo, Bibhas Chakraborty

MC_var_yibar_mis function