oiposbinomial {VGAMdata}

One-Inflated Positive Binomial Distribution Family Function

Description

Fits a one-inflated positive binomial distribution by maximum likelihood estimation.

Usage

oiposbinomial(lpstr1 = "logitlink", lprob = "logitlink",
    type.fitted = c("mean", "prob", "pobs1", "pstr1", "onempstr1"),
    iprob = NULL, gpstr1 = ppoints(9), gprob  = ppoints(9),
    multiple.responses = FALSE, zero = NULL)

Arguments

lpstr1, lprob

Link functions for the parameter \phi and the positive binomial probability \mu parameter. See Links for more choices. See CommonVGAMffArguments also. For the one-deflated model see below.

type.fitted

See CommonVGAMffArguments and fittedvlm.

iprob, gpstr1, gprob

For initial values; see CommonVGAMffArguments.

multiple.responses

Logical. See binomialff and posbinomial.

zero

See CommonVGAMffArguments for information.

Details

These functions are based on

P(Y=y) = \phi + (1-\phi) N \mu (1-\mu)^N / (1-(1-\mu)^N),

for y=1/N, and

P(Y=y) = (1-\phi) {N \choose Ny} \mu^{Ny} (1-\mu)^{N(1-y)} / (1-(1-\mu)^N).

for y=2/N,\ldots,1. That is, the response is a sample proportion out of N trials, and the argument size in roiposbinom is N here. Ideally N > 2 is needed. The parameter \phi is the probability of a structural one, and it satisfies 0 < \phi < 1 (usually). The mean of Y is E(Y)=\phi + (1-\phi) \mu / (1-(1-\mu)^N) and these are returned as the default fitted values. By default, the two linear/additive predictors for oiposbinomial() are (logit(\phi), logit(\mu))^T.

Value

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

Note

The response variable should have one of the formats described by binomialff, e.g., a factor or two column matrix or a vector of sample proportions with the weights argument specifying the values of N.

To work well, one ideally needs large values of N and \mu much greater than 0, i.e., the larger N and \mu are, the better. If N = 1 then the model is unidentifiable since the number of parameters is excessive.

Estimated probabilities of a structural one and an observed one are returned, as in zipoisson.

The one-deflated positive binomial distribution might be fitted by setting lpstr1 = "identitylink", albeit, not entirely reliably. See zipoisson for information that can be applied here.

Author(s)

T. W. Yee

See Also

roiposbinom, posbinomial, binomialff, rbinom.

Examples

size <- 10  # Number of trials; N in the notation above
nn <- 200
odata <- data.frame(pstr1  = logitlink( 0, inv = TRUE),  # 0.50
              mubin1 = logitlink(-1, inv = TRUE),  # Binomial mean
              svec   = rep(size, length = nn),
              x2     = runif(nn))
odata <- transform(odata,
                   mubin2 = logitlink(-1 + x2, inv = TRUE))
odata <- transform(odata,
              y1 = roiposbinom(nn, svec, pr = mubin1, pstr1 = pstr1),
              y2 = roiposbinom(nn, svec, pr = mubin2, pstr1 = pstr1))
with(odata, table(y1))
fit1 <- vglm(y1 / svec ~  1, oiposbinomial, data = odata,
             weights = svec, trace = TRUE, crit = "coef")
fit2 <- vglm(y2 / svec ~ x2, oiposbinomial, data = odata,
             weights = svec, trace = TRUE)

coef(fit1, matrix = TRUE)
Coef(fit1)  # Useful for intercept-only models
head(fitted(fit1, type = "pobs1"))  # Estimate of P(Y = 1)
head(fitted(fit1))
with(odata, mean(y1))  # Compare this with fitted(fit1)
summary(fit1)