# Ordinal Poisson Family Function Fits a Poisson regression where the response is ordinal (the Poisson counts are grouped between known cutpoints). ```r ordpoisson(cutpoints, countdata = FALSE, NOS = NULL, Levels = NULL, init.mu = NULL, parallel = FALSE, zero = NULL, link = "loglink") ``` ## Arguments - `cutpoints`: Numeric. The cutpoints, $K_l$. These must be non-negative integers. `Inf` values may be included. See below for further details. - `countdata`: Logical. Is the response (LHS of formula) in count-data format? If not then the response is a matrix or vector with values `1`, `2`, , `L`, say, where `L` is the number of levels. Such input can be generated with `cut` with argument `labels = FALSE`. If `countdata = TRUE` then the response is expected to be in the same format as `depvar(fit)` where `fit` is a fitted model with `ordpoisson` as the `VGAM` family function. That is, the response is matrix of counts with `L` columns (if `NOS = 1`). - `NOS`: Integer. The number of species, or more generally, the number of response random variates. This argument must be specified when `countdata = TRUE`. Usually `NOS = 1`. - `Levels`: Integer vector, recycled to length `NOS` if necessary. The number of levels for each response random variate. This argument should agree with `cutpoints`. This argument must be specified when `countdata = TRUE`. - `init.mu`: Numeric. Initial values for the means of the Poisson regressions. Recycled to length `NOS` if necessary. Use this argument if the default initial values fail (the default is to compute an initial value internally). - `parallel, zero, link`: See `poissonff`. See `CommonVGAMffArguments` for information. ## Details This `VGAM` family function uses maximum likelihood estimation (Fisher scoring) to fit a Poisson regression to each column of a matrix response. The data, however, is ordinal, and is obtained from known integer cutpoints. Here, $l=1,\ldots,L$ where $L$ ($L >= 2$) is the number of levels. In more detail, let $Y^*=l$ if c("$K_{l-1} < Y\n$", "$ <= K_{l}$") where the $K_l$ are the cutpoints. We have $K_0=-Inf$ and $K_L=Inf$. The response for this family function corresponds to $Y^*$ but we are really interested in the Poisson regression of $Y$. If `NOS=1` then the argument `cutpoints` is a vector $(K_1,K_2,\ldots,K_L)$ where the last value (`Inf`) is optional. If `NOS>1` then the vector should have `NOS-1` `Inf` values separating the cutpoints. For example, if there are `NOS=3` responses, then something like `ordpoisson(cut = c(0, 5, 10, Inf, 20, 30, Inf, 0, 10, 40, Inf))` is valid. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm` and `vgam`. ## References Yee, T. W. (2020). **Ordinal ordination with normalizing link functions for count data**, (in preparation). ## Author(s) Thomas W. Yee ## Note Sometimes there are no observations between two cutpoints. If so, the arguments `Levels` and `NOS` need to be specified too. See below for an example. ## Warning The input requires care as little to no checking is done. If `fit` is the fitted object, have a look at `fit@extra` and `depvar(fit)` to check. ## See Also `poissonff`, `polf`, `ordered`. ## Examples ```r set.seed(123) # Example 1 x2 <- runif(n <- 1000); x3 <- runif(n) mymu <- exp(3 - 1 * x2 + 2 * x3) y1 <- rpois(n, lambda = mymu) cutpts <- c(-Inf, 20, 30, Inf) fcutpts <- cutpts[is.finite(cutpts)] # finite cutpoints ystar <- cut(y1, breaks = cutpts, labels = FALSE) ## Not run: plot(x2, x3, col = ystar, pch = as.character(ystar)) ## End(Not run) table(ystar) / sum(table(ystar)) fit <- vglm(ystar ~ x2 + x3, fam = ordpoisson(cutpoi = fcutpts)) head(depvar(fit)) # This can be input if countdata = TRUE head(fitted(fit)) head(predict(fit)) coef(fit, matrix = TRUE) fit@extra # Example 2: multivariate and there are no obsns between some cutpoints cutpts2 <- c(-Inf, 0, 9, 10, 20, 70, 200, 201, Inf) fcutpts2 <- cutpts2[is.finite(cutpts2)] # finite cutpoints y2 <- rpois(n, lambda = mymu) # Same model as y1 ystar2 <- cut(y2, breaks = cutpts2, labels = FALSE) table(ystar2) / sum(table(ystar2)) fit <- vglm(cbind(ystar,ystar2) ~ x2 + x3, fam = ordpoisson(cutpoi = c(fcutpts,Inf,fcutpts2,Inf), Levels = c(length(fcutpts)+1,length(fcutpts2)+1), parallel = TRUE), trace = TRUE) coef(fit, matrix = TRUE) fit@extra constraints(fit) summary(depvar(fit)) # Some columns have all zeros ```

ordpoisson function