# Function to fit the limiting form of a Faddy distribution for under-dispersed counts. Outputs probabilities for the limiting form of a Faddy distribution modeled by means and scale-factors i.e. with the design matrices for mean and scale-factor input together with data and offsets. ```r Model.FaddyJMV.limiting(parameter, link, covariates.matrix.mean, covariates.matrix.scalef, offset.mean, offset.scalef, vnmax) ``` ## Arguments - `parameter`: A vector of the parameters of the model which is set to initial estimates on function call. - `link`: Takes one of one values i.e., 'log'. The default is 'log'. - `covariates.matrix.mean`: A matrix of covariates for the mean where rows are the number of values in listcounts and columns the covariates. This matrix is extracted from the formulae in function CountsEPPM. However, in the accompanying example it is shown how it can be constructed independently of function CountsEPPM. - `covariates.matrix.scalef`: A matrix of covariates for the scale factor where rows are the number of values in listcounts and columns the covariates. The default is a vector of ones. This matrix is extracted from the formulae in function CountsEPPM. However, in the accompanying example it is shown how it can be constructed independently of function CountsEPPM. - `offset.mean`: An offset vector for the mean. The default is a vector of ones. - `offset.scalef`: An offset vector for the scale-factor. The default is a vector of ones. - `vnmax`: A vector of the maximum counts for each vector in list.counts i.e. the list of grouped counts. ## Returns The list output with elements - **model**: The model being fitted - **estimate**: Estimates of parameters - **probabilities**: Vector of probabilities ## References Faddy M, Smith D. (2011). Analysis of count data with covariate dependence in both mean and variance. **Journal of Applied Statistics**, 38 , 2683-2694. tools:::Rd_expr_doi("10.1002/bimj.201100214") . ## Author(s) David M. Smith ## Examples ```r all.counts=c(rep(0,5),352,479,530,291,101,17) nmax1 <- length(all.counts) nmax <- nmax1 - 1 cnum <- 0:nmax ncount <- sum(all.counts) all.mean <- sum(cnum*all.counts)/ncount all.scalef <- ((sum(cnum*cnum*all.counts) - ncount*all.mean*all.mean) / (ncount - 1)) / all.mean alldata <- data.frame(all.mean, all.scalef) mf <- model.frame(formula = all.mean~1, data = alldata) covariates.matrix.mean <- model.matrix(attr(mf, "terms"), data = mf) mf <- model.frame(formula = all.scalef~1, data = alldata) covariates.matrix.scalef <- model.matrix(attr(mf, "terms"), data = mf) list.counts <- list(all.counts) parameter <- c(1.8388023, 0.6009881) names(parameter) <- c("beta0 log(mean)", "beta0 log(scale-factor)") link <- "log" attr(link, which = "mean") <- make.link(link) offset.mean <- matrix(c(rep(0, nrow(covariates.matrix.mean))), ncol=1) offset.scalef <- matrix(c(rep(0, nrow(covariates.matrix.mean))), ncol=1) output <- Model.FaddyJMV.limiting(parameter, link, covariates.matrix.mean, covariates.matrix.scalef, offset.mean, offset.scalef, vnmax = c(10)) print(output) ```

Model.FaddyJMV.limiting function