Positive Poisson Distribution Family Function

Fits a positive Poisson distribution.

pospoisson(link = "loglink", type.fitted = c("mean", "lambda",
           "prob0"), expected = TRUE, ilambda = NULL, imethod = 1,
           zero = NULL, gt.1 = FALSE)

Arguments

link

Link function for the usual mean (lambda) parameter of an ordinary Poisson distribution. See Links for more choices.

expected

Logical. Fisher scoring is used if expected = TRUE, else Newton-Raphson.

ilambda, imethod, zero

See CommonVGAMffArguments for information.

type.fitted

See CommonVGAMffArguments for details.

gt.1

Logical. Enforce lambda > 1? The default is to enforce lambda > 0.

Details

The positive Poisson distribution is the ordinary Poisson distribution but with the probability of zero being zero. Thus the other probabilities are scaled up (i.e., divided by \(1-P[Y=0]\)). The mean, \(\lambda / (1 - \exp(-\lambda))\), can be obtained by the extractor function fitted applied to the object.

A related distribution is the zero-inflated Poisson, in which the probability \(P[Y=0]\) involves another parameter \(\phi\). See zipoisson.

Warning

Under- or over-flow may occur if the data is ill-conditioned.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, rrvglm and vgam.

References

Coleman, J. S. and James, J. (1961). The equilibrium size distribution of freely-forming groups. Sociometry, 24, 36–45.

Author

Thomas W. Yee

Note

This family function can handle multiple responses.

Yet to be done: a quasi.pospoisson which estimates a dispersion parameter.

See also

Gaitdpois, gaitdpoisson, posnegbinomial, poissonff, zapoisson, zipoisson, simulate.vlm, otpospoisson, Pospois.

Examples

# Data from Coleman and James (1961)
cjdata <- data.frame(y = 1:6, freq = c(1486, 694, 195, 37, 10, 1))
fit <- vglm(y ~ 1, pospoisson, data = cjdata, weights = freq)
Coef(fit)
#>    lambda 
#> 0.8924961 
summary(fit)
#> 
#> Call:
#> vglm(formula = y ~ 1, family = pospoisson, data = cjdata, weights = freq)
#> 
#> Coefficients: 
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept) -0.11373    0.02678  -4.248 2.16e-05 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Name of linear predictor: loglink(lambda) 
#> 
#> Log-likelihood: -2304.659 on 5 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 4 
#> 
#> No Hauck-Donner effect found in any of the estimates
#> 
fitted(fit)
#>       [,1]
#> 1 1.511762
#> 2 1.511762
#> 3 1.511762
#> 4 1.511762
#> 5 1.511762
#> 6 1.511762

pdata <- data.frame(x2 = runif(nn <- 1000))  # Artificial data
pdata <- transform(pdata, lambda = exp(1 - 2 * x2))
pdata <- transform(pdata, y1 = rgaitdpois(nn, lambda, truncate = 0))
with(pdata, table(y1))
#> y1
#>   1   2   3   4   5   6 
#> 537 286 108  43  19   7 
fit <- vglm(y1 ~ x2, pospoisson, data = pdata, trace = TRUE, crit = "coef")
#> Iteration 1: coefficients =  0.87505175, -1.16582910
#> Iteration 2: coefficients =  0.98859772, -1.80736456
#> Iteration 3: coefficients =  1.0136741, -1.9781525
#> Iteration 4: coefficients =  1.0150203, -1.9881407
#> Iteration 5: coefficients =  1.0150246, -1.9881733
#> Iteration 6: coefficients =  1.0150246, -1.9881733
coef(fit, matrix = TRUE)
#>             loglink(lambda)
#> (Intercept)        1.015025
#> x2                -1.988173