# Small Area Estimation using Hierarchical Bayesian under Negative Binomial Distribution This function is implemented to variable of interest $(y)$ that assumed to be a Negative Binomial Distribution. The data is a number of the Bernoulli process. The negative binomial is used to overcome an over dispersion from the discrete model. ```r NegativeBinomial( formula, iter.update = 3, iter.mcmc = 10000, coef, var.coef, thin = 2, burn.in = 2000, tau.u = 1, data ) ``` ## Arguments - `formula`: Formula that describe the fitted model - `iter.update`: Number of updates with default `3` - `iter.mcmc`: Number of total iterations per chain with default `10000` - `coef`: a vector contains prior initial value of Coefficient of Regression Model for fixed effect with default vector of `0` with the length of the number of regression coefficients - `var.coef`: a vector contains prior initial value of variance of Coefficient of Regression Model with default vector of `1` with the length of the number of regression coefficients - `thin`: Thinning rate, must be a positive integer with default `2` - `burn.in`: Number of iterations to discard at the beginning with default `2000` - `tau.u`: Prior initial value of inverse of Variance of area random effect with default `1` - `data`: The data frame ## Returns This function returns a list of the following objects: - **Est**: A vector with the values of Small Area mean Estimates using Hierarchical bayesian method - **refVar**: Estimated random effect variances - **coefficient**: A dataframe with the estimated model coefficient - **plot**: Trace, Dencity, Autocorrelation Function Plot of MCMC samples ## Examples ```r ## Data Generation set.seed(123) library(MASS) m <- 30 x <- runif(m, 0, 1) b0 <- b1 <- 0.5 u <- rnorm(m, 0, 1) Mu <- exp(b0 + b1 * x + u) theta <- 1 y <- MASS::rnegbin(m, Mu, theta) vardir <- Mu + Mu^2 / theta dataNegativeBinomial <- as.data.frame(cbind(y, x, vardir)) dataNegativeBinomialNs <- dataNegativeBinomial dataNegativeBinomialNs$y[c(3, 14, 22, 29, 30)] <- NA dataNegativeBinomialNs$vardir[c(3, 14, 22, 29, 30)] <- NA ## Compute Fitted Model ## y ~ x ## For data without any nonsampled area formula <- y ~ x v <- c(1, 1) c <- c(0, 0) dat <- dataNegativeBinomial ## Using parameter coef and var.coef saeHBNegbin <- NegativeBinomial(formula, coef = c, var.coef = v, iter.update = 10, data = dat) saeHBNegbin$Est # Small Area mean Estimates saeHBNegbin$refVar # Random effect variance saeHBNegbin$coefficient # coefficient # Load Library 'coda' to execute the plot # autocorr.plot(saeHBNegbin$plot[[3]]) is used to generate ACF Plot # plot(saeHBNegbin$plot[[3]]) is used to generate Density and trace plot ## Do not using parameter coef and var.coef saeHBNegbin <- NegativeBinomial(formula, data = dat) ## For data with nonsampled area use dataNegativeBinomialNs ```

NegativeBinomial function