zibinomial {VGAM} | R Documentation

## Zero-Inflated Binomial Distribution Family Function

### Description

Fits a zero-inflated binomial distribution by maximum likelihood estimation.

### Usage

    zibinomial(lpstr0 = "logit", lprob = "logit",
               type.fitted = c("mean", "prob", "pobs0", "pstr0", "onempstr0"),
               ipstr0 = NULL, zero = NULL, multiple.responses = FALSE,
               imethod = 1)
    zibinomialff(lprob = "logit", lonempstr0 = "logit",
                 type.fitted = c("mean", "prob", "pobs0", "pstr0", "onempstr0"),
                 ionempstr0 = NULL, zero = "onempstr0",
                 multiple.responses = FALSE, imethod = 1)

### Arguments

`lpstr0, lprob` | Link functions for the parameter phi and the usual binomial probability prob parameter. See `Links` for more choices. For the zero-deflated model see below.

`type.fitted` | See `CommonVGAMffArguments` and `fittedvlm`.
`ipstr0` | Optional initial values for phi, whose values must lie between 0 and 1. The default is to compute an initial value internally. If a vector then recyling is used.
`lonempstr0, ionempstr0` | Corresponding arguments for the other parameterization. See details below.

`multiple.responses` | Logical. Currently it must be `FALSE` to mean the function does not handle multiple responses. This is to remain compatible with the same argument in `binomialff`.
`zero, imethod` | See `CommonVGAMffArguments` for information. Argument `zero` changed its default value for version 0.9-2.

### Details

These functions are based on

P(Y=0) = phi + (1- phi) * (1-prob)^N,

for y=0, and

P(Y=y) = (1-phi) * choose(N,Ny) * prob^(N*y) * (1-prob)^(N*(1-y)).

for y=1/N,2/N,…,1. That is, the response is a sample proportion out of N trials, and the argument `size` in `rzibinom` is N here. The parameter phi is the probability of a structural zero, and it satisfies 0 < phi < 1. The mean of Y is E(Y) = (1-phi) * prob and these are returned as the fitted values by default. By default, the two linear/additive predictors for `zibinomial()` are (logit(phi), logit(prob))^T.

The VGAM family function `zibinomialff()` has a few changes compared to `zibinomial()`. These are: (i) the order of the linear/additive predictors is switched so the binomial probability comes first; (ii) argument `onempstr0` is now 1 minus the probability of a structural zero, i.e., the probability of the parent (binomial) component, i.e., `onempstr0` is `1-pstr0`; (iii) argument `zero` has a new default so that the `onempstr0` is intercept-only by default. Now `zibinomialff()` is generally recommended over `zibinomial()`. Both functions implement Fisher scoring.

### Value

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

### Warning

Numerical problems can occur. Half-stepping is not uncommon. If failure to converge occurs, make use of the argument `ipstr0` or `ionempstr0`, or `imethod`.

### Note

The response variable must 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 needs large values of N and prob>0, i.e., the larger N and prob are, the better. If N = 1 then the model is unidentifiable since the number of parameters is excessive.

Setting `stepsize = 0.5`, say, may aid convergence.

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

The zero-deflated binomial distribution might be fitted by setting `lpstr0 = identitylink`, albeit, not entirely reliably. See `zipoisson` for information that can be applied here. Else try the zero-altered binomial distribution (see `zabinomial`).

### Author(s)

T. W. Yee

### References

Welsh, A. H., Lindenmayer, D. B. and Donnelly, C. F. (2013) Fitting and interpreting occupancy models. PLOS One, 8, 1–21.

### See Also

`rzibinom`, `binomialff`, `posbinomial`, `rbinom`.

### Examples

    size <- 10  # Number of trials; N in the notation above
    nn <- 200
    zdata <- data.frame(pstr0 = logit( 0, inverse = TRUE),  # 0.50
                        mubin = logit(-1, inverse = TRUE),  # Mean of usual binomial
                        sv    = rep(size, length = nn))
    zdata <- transform(zdata,
                       y = rzibinom(nn, size = sv, prob = mubin, pstr0 = pstr0))
    with(zdata, table(y))
    fit <- vglm(cbind(y, sv - y) ~ 1, zibinomialff, data = zdata, trace = TRUE)
    fit <- vglm(cbind(y, sv - y) ~ 1, zibinomialff, data = zdata, trace = TRUE,
                stepsize = 0.5)

    coef(fit, matrix = TRUE)
    Coef(fit)  # Useful for intercept-only models
    head(fitted(fit, type = "pobs0"))  # Estimate of P(Y = 0)
    head(fitted(fit))
    with(zdata, mean(y))  # Compare this with fitted(fit)
    summary(fit)

[Package VGAM version 1.0-4 Index]