Zero-Altered Poisson Distribution

Fits a zero-altered Poisson distribution based on a conditional model involving a Bernoulli distribution and a positive-Poisson distribution.

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 $p_0$, 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 $p_0$, else $Y$ has a positive-Poisson($\lambda)$ distribution with probability $1-p_0$. Thus $0 < p_0 < 1$, 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 $(logit(p_0), \log(\lambda))^T$.

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.

Value

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 $$\mu = (1-p_0) \lambda / [1 - \exp(-\lambda)].$$ If type.fitted = "pobs0" then $p_0$ 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

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 $p_0$ 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).

Examples

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))
#> y
#>   0   1   2   3   4   5   6   7   8   9  10 
#> 372 257 178  95  51  31   9   4   1   1   1 
fit <- vglm(y ~ x2, zapoisson, data = zdata, trace = TRUE)
#> Iteration 1: loglikelihood = -1481.0057
#> Iteration 2: loglikelihood = -1472.0233
#> Iteration 3: loglikelihood = -1471.8381
#> Iteration 4: loglikelihood = -1471.838
#> Iteration 5: loglikelihood = -1471.838
fit <- vglm(y ~ x2, zapoisson, data = zdata, trace = TRUE, crit = "coef")
#> Iteration 1: coefficients = 
#> -1.21382490, -0.20257475,  1.17945726,  1.63484348
#> Iteration 2: coefficients = 
#> -1.05772117, -0.47415044,  1.04619515,  1.92254122
#> Iteration 3: coefficients = 
#> -1.06226401, -0.52675882,  1.05178684,  1.98427337
#> Iteration 4: coefficients = 
#> -1.06226713, -0.52811861,  1.05179094,  1.98601528
#> Iteration 5: coefficients = 
#> -1.0622671, -0.5281274,  1.0517909,  1.9860295
#> Iteration 6: coefficients = 
#> -1.06226713, -0.52812747,  1.05179094,  1.98602963
#> Iteration 7: coefficients = 
#> -1.06226713, -0.52812748,  1.05179094,  1.98602964
head(fitted(fit))
#>        [,1]
#> 1 1.4697027
#> 2 0.9858453
#> 3 1.1257438
#> 4 1.6722408
#> 5 1.6443012
#> 6 1.6259878
head(predict(fit))
#>   logitlink(pobs0) loglink(lambda)
#> 1       -0.3584078       0.8009251
#> 2       -1.0532867      -0.5111702
#> 3       -0.7116293       0.1339596
#> 4       -0.2342826       1.0353028
#> 5       -0.2495983       1.0063832
#> 6       -0.2599054       0.9869208
head(predict(fit, untransform = TRUE))
#>       pobs0    lambda
#> 1 0.4113450 2.2276007
#> 2 0.2585945 0.5997933
#> 3 0.3292389 1.1433466
#> 4 0.4416958 2.8159588
#> 5 0.4379224 2.7356886
#> 6 0.4353870 2.6829605
coef(fit, matrix = TRUE)
#>             logitlink(pobs0) loglink(lambda)
#> (Intercept)        -1.062267      -0.5281275
#> x2                  1.051791       1.9860296
summary(fit)
#> 
#> Call:
#> vglm(formula = y ~ x2, family = zapoisson, data = zdata, trace = TRUE, 
#>     crit = "coef")
#> 
#> Coefficients: 
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept):1 -1.06227    0.13945  -7.618 2.58e-14 ***
#> (Intercept):2 -0.52813    0.09261  -5.703 1.18e-08 ***
#> x2:1           1.05179    0.23486   4.478 7.52e-06 ***
#> x2:2           1.98603    0.13092  15.169  < 2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Names of linear predictors: logitlink(pobs0), loglink(lambda)
#> 
#> Log-likelihood: -1471.838 on 1996 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 7 
#> 
#> No Hauck-Donner effect found in any of the estimates
#> 

# 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")
#> Iteration 1: coefficients = 1.25650524, 0.12608894
#> Iteration 2: coefficients =  1.14002466, -0.14124406
#> Iteration 3: coefficients =  1.14356045, -0.16530315
#> Iteration 4: coefficients =  1.14356368, -0.16572625
#> Iteration 5: coefficients =  1.14356368, -0.16572636
#> Iteration 6: coefficients =  1.14356368, -0.16572636
coef(fit3, matrix = TRUE)
#>             logitlink(pobs0) loglink(lambda)
#> (Intercept)         1.143564      -0.1657264
Coef(fit3)  # Estimate lambda (they get 0.6997 with SE 0.1520)
#>     pobs0    lambda 
#> 0.7583333 0.8472781 
head(fitted(fit3), 1)
#>        [,1]
#> 1 0.3583333
with(Abdata, mean(yy))  # Compare this with fitted(fit3)
#> [1] 0.3583333