Poisson Regression

Family function for a generalized linear model fitted to Poisson responses.

poissonff(link = "loglink", imu = NULL, imethod = 1,
          parallel = FALSE, zero = NULL, bred = FALSE,
          earg.link = FALSE, type.fitted = c("mean", "quantiles"),
          percentiles = c(25, 50, 75))

Arguments

Link function applied to the mean or means. See Links for more choices and information.

parallel

A logical or formula. Used only if the response is a matrix.

imu, imethod

See CommonVGAMffArguments for more information.

zero

Can be an integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2,...,\(M\)}, where \(M\) is the number of columns of the matrix response. See CommonVGAMffArguments for more information.

Details at CommonVGAMffArguments. Setting bred = TRUE should work for multiple responses and all VGAM link functions; it has been tested for loglink, identity but further testing is required.

type.fitted, percentiles

Details at CommonVGAMffArguments.

Details

\(M\) defined above is the number of linear/additive predictors. With overdispersed data try negbinomial.

Value

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

References

McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.

Author

Thomas W. Yee

Note

This function will handle a matrix response automatically.

Regardless of whether the dispersion parameter is to be estimated or not, its value can be seen from the output from the summary() of the object.

Warning

With multiple responses, assigning a known dispersion parameter for each response is not handled well yet. Currently, only a single known dispersion parameter is handled well.

Examples

poissonff()
#> Family:  poissonff 
#> Informal classes: poissonff, VGAMglm, VGAMcategorical 
#> 
#> Poisson distribution
#> 
#> Link:     loglink(lambda)
#> Variance: lambda

set.seed(123)
pdata <- data.frame(x2 = rnorm(nn <- 100))
pdata <- transform(pdata, y1 = rpois(nn, exp(1 + x2)),
                          y2 = rpois(nn, exp(1 + x2)))
(fit1 <- vglm(cbind(y1, y2) ~ x2, poissonff, data = pdata))
#> 
#> Call:
#> vglm(formula = cbind(y1, y2) ~ x2, family = poissonff, data = pdata)
#> 
#> 
#> Coefficients:
#> (Intercept):1 (Intercept):2          x2:1          x2:2 
#>     0.9175762     1.0204439     1.0471179     1.0084276 
#> 
#> Degrees of Freedom: 200 Total; 196 Residual
#> Residual deviance: 235.6729 
#> Log-likelihood: -387.3006 
(fit2 <- vglm(y1 ~ x2, poissonff(bred = TRUE), data = pdata))
#> 
#> Call:
#> vglm(formula = y1 ~ x2, family = poissonff(bred = TRUE), data = pdata)
#> 
#> 
#> Coefficients:
#> (Intercept)          x2 
#>   0.9203612   1.0466136 
#> 
#> Degrees of Freedom: 100 Total; 98 Residual
#> Residual deviance: 116.6895 
#> Log-likelihood: -191.1954 
coef(fit1, matrix = TRUE)
#>             loglink(E[y1]) loglink(E[y2])
#> (Intercept)      0.9175762       1.020444
#> x2               1.0471179       1.008428
coef(fit2, matrix = TRUE)
#>             loglink(lambda)
#> (Intercept)       0.9203612
#> x2                1.0466136

nn <- 200
cdata <- data.frame(x2 = rnorm(nn), x3 = rnorm(nn), x4 = rnorm(nn))
cdata <- transform(cdata, lv1 = 0 + x3 - 2*x4)
cdata <- transform(cdata, lambda1 = exp(3 - 0.5 *  (lv1-0)^2),
                          lambda2 = exp(2 - 0.5 *  (lv1-1)^2),
                          lambda3 = exp(2 - 0.5 * ((lv1+4)/2)^2))
cdata <- transform(cdata, y1 = rpois(nn, lambda1),
                          y2 = rpois(nn, lambda2),
                          y3 = rpois(nn, lambda3))
if (FALSE)  lvplot(p1, y = TRUE, lcol = 2:4, pch = 2:4, pcol = 2:4, rug = FALSE)  # \dontrun{}