# Zero-Altered Poisson Distribution Fits a zero-altered Poisson distribution based on a conditional model involving a Bernoulli distribution and a positive-Poisson distribution. ```r zapoisson(lpobs0 = "logitlink", llambda = "loglink", type.fitted = c("mean", "lambda", "pobs0", "onempobs0"), imethod = 1, ipobs0 = NULL, ilambda = NULL, ishrinkage = 0.95, probs.y = 0.35, zero = NULL) zapoissonff(llambda = "loglink", lonempobs0 = "logitlink", type.fitted = c("mean", "lambda", "pobs0", "onempobs0"), imethod = 1, ilambda = NULL, ionempobs0 = NULL, ishrinkage = 0.95, probs.y = 0.35, zero = "onempobs0") ``` ## Arguments - `lpobs0`: Link function for the parameter $pobs0$, called `pobs0` here. See `Links` for more choices. - `llambda`: Link function for the usual $lambda$ parameter. See `Links` for more choices. - `type.fitted`: See `CommonVGAMffArguments` and `fittedvlm` for information. - `lonempobs0`: Corresponding argument for the other parameterization. See details below. - `imethod, ipobs0, ionempobs0, ilambda, ishrinkage`: See `CommonVGAMffArguments` for information. - `probs.y, zero`: See `CommonVGAMffArguments` for information. ## Details The response $Y$ is zero with probability $pobs0$, else $Y$ has a positive-Poisson($lambda)$ distribution with probability $1-pobs0$. Thus $0 < pobs0 < 1$, which is modelled as a function of the covariates. The zero-altered Poisson distribution differs from the zero-inflated Poisson distribution in that the former has zeros coming from one source, whereas the latter has zeros coming from the Poisson distribution too. Some people call the zero-altered Poisson a **hurdle** model. For one response/species, by default, the two linear/additive predictors for `zapoisson()` are c("$(logit(pobs0),\n$", "$ log(lambda))^T$"). The `VGAM` family function `zapoissonff()` has a few changes compared to `zapoisson()`. These are: (i) the order of the linear/additive predictors is switched so the Poisson mean comes first; (ii) argument `onempobs0` is now 1 minus the probability of an observed 0, i.e., the probability of the positive Poisson distribution, i.e., `onempobs0` is `1-pobs0`; (iii) argument `zero` has a new default so that the `onempobs0` is intercept-only by default. Now `zapoissonff()` is generally recommended over `zapoisson()`. Both functions implement Fisher scoring and can handle multiple responses. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`. The `fitted.values` slot of the fitted object, which should be extracted by the generic function `fitted`, returns the mean $mu$ (default) which is given by $$ \mu = (1-p_0) \lambda / [1 - \exp(-\lambda)].%mu = (1-pobs0) * lambda / [1 - exp(-lambda)]. $$ If `type.fitted = "pobs0"` then $pobs0$ is returned. ## References Welsh, A. H., Cunningham, R. B., Donnelly, C. F. and Lindenmayer, D. B. (1996). Modelling the abundances of rare species: statistical models for counts with extra zeros. **Ecological Modelling**, 88 , 297--308. Angers, J-F. and Biswas, A. (2003). A Bayesian analysis of zero-inflated generalized Poisson model. **Computational Statistics & Data Analysis**, 42 , 37--46. Yee, T. W. (2014). Reduced-rank vector generalized linear models with two linear predictors. **Computational Statistics and Data Analysis**, 71 , 889--902. ## Author(s) T. W. Yee ## Note There are subtle differences between this family function and `zipoisson` and `yip88`. In particular, `zipoisson` is a **mixture** model whereas `zapoisson()` and `yip88` are **conditional** models. Note this family function allows $pobs0$ to be modelled as functions of the covariates. This family function effectively combines `pospoisson` and `binomialff` into one family function. This family function can handle multiple responses, e.g., more than one species. It is recommended that `Gaitdpois` be used, e.g., `rgaitdpois(nn, lambda, pobs.mlm = pobs0, a.mlm = 0)` instead of `rzapois(nn, lambda, pobs0 = pobs0)`. ## See Also `Gaitdpois`, `rzapois`, `zipoisson`, `gaitdpoisson`, `pospoisson`, `posnegbinomial`, `spikeplot`, `binomialff`, `CommonVGAMffArguments`, `simulate.vlm`. ## Examples ```r zdata <- data.frame(x2 = runif(nn <- 1000)) zdata <- transform(zdata, pobs0 = logitlink( -1 + 1*x2, inverse = TRUE), lambda = loglink(-0.5 + 2*x2, inverse = TRUE)) zdata <- transform(zdata, y = rgaitdpois(nn, lambda, pobs.mlm = pobs0, a.mlm = 0)) with(zdata, table(y)) fit <- vglm(y ~ x2, zapoisson, data = zdata, trace = TRUE) fit <- vglm(y ~ x2, zapoisson, data = zdata, trace = TRUE, crit = "coef") head(fitted(fit)) head(predict(fit)) head(predict(fit, untransform = TRUE)) coef(fit, matrix = TRUE) summary(fit) # Another example ------------------------------ # Data from Angers and Biswas (2003) abdata <- data.frame(y = 0:7, w = c(182, 41, 12, 2, 2, 0, 0, 1)) abdata <- subset(abdata, w > 0) Abdata <- data.frame(yy = with(abdata, rep(y, w))) fit3 <- vglm(yy ~ 1, zapoisson, data = Abdata, trace = TRUE, crit = "coef") coef(fit3, matrix = TRUE) Coef(fit3) # Estimate lambda (they get 0.6997 with SE 0.1520) head(fitted(fit3), 1) with(Abdata, mean(yy)) # Compare this with fitted(fit3) ```

zapoisson function