R: Positive Negative Binomial Distribution Family Function

posnegbinomial {VGAM} | R Documentation

## Positive Negative Binomial Distribution Family Function

### Description

Maximum likelihood estimation of the two parameters of a positive negative binomial distribution.

### Usage

    posnegbinomial(zero = "size", type.fitted = c("mean", "munb", "prob0"),
                   mds.min = 0.001, nsimEIM = 500, cutoff.prob = 0.999,
                   eps.trig = 1e-07, max.support = 4000, max.chunk.MB = 30,
                   lmunb = "loglink", lsize = "loglink", imethod = 1,
                   imunb = NULL, iprobs.y = NULL,
                   gprobs.y = ppoints(8), isize = NULL,
                   gsize.mux = exp(c(-30, -20, -15, -10, -6:3)))

### Arguments

`lmunb` | Link function applied to the `munb` parameter, which is the mean munb of an ordinary negative binomial distribution. See `Links` for more choices.
`lsize` | Parameter link function applied to the dispersion parameter, called `k`. See `Links` for more choices.
`isize` | Optional initial value for `k`, an index parameter. The value `1/k` is known as a dispersion parameter. If failure to converge occurs try different values (and/or use `imethod`). If necessary this vector is recycled to length equal to the number of responses. A value `NULL` means an initial value for each response is computed internally using a range of values.
`nsimEIM, zero, eps.trig` | See `CommonVGAMffArguments`.
`mds.min, iprobs.y, cutoff.prob` | Similar to `negbinomial`.
`imunb, max.support` | Similar to `negbinomial`.
`max.chunk.MB, gsize.mux` | Similar to `negbinomial`.
`imethod, gprobs.y` | See `negbinomial`.
`type.fitted` | See `CommonVGAMffArguments` for details.

### Details

The positive negative binomial distribution is an ordinary negative binomial distribution but with the probability of a zero response being zero. The other probabilities are scaled to sum to unity.

This family function is based on `negbinomial` and most details can be found there. To avoid confusion, the parameter `munb` here corresponds to the mean of an ordinary negative binomial distribution `negbinomial`. The mean of `posnegbinomial` is

munb / (1-p(0))

where p(0) = (k/(k + munb))^k is the probability an ordinary negative binomial distribution has a zero value.

The parameters `munb` and `k` are not independent in the positive negative binomial distribution, whereas they are in the ordinary negative binomial distribution.

This function handles multiple responses, so that a matrix can be used as the response. The number of columns is the number of species, say, and setting `zero = -2` means that all species have a `k` equalling a (different) intercept only.

### Value

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

### Warning

This family function is fragile; at least two cases will lead to numerical problems. Firstly, the positive-Poisson model corresponds to `k` equalling infinity. If the data is positive-Poisson or close to positive-Poisson, then the estimated `k` will diverge to `Inf` or some very large value. Secondly, if the data is clustered about the value 1 because the `munb` parameter is close to 0 then numerical problems will also occur. Users should set `trace = TRUE` to monitor convergence. In the situation when both cases hold, the result returned (which will be untrustworthy) will depend on the initial values.

The negative binomial distribution (NBD) is a strictly unimodal distribution. Any data set that does not exhibit a mode (in the middle) makes the estimation problem difficult. The positive NBD inherits this feature. Set `trace = TRUE` to monitor convergence.

See the example below of a data set where `posbinomial()` fails; the so-called solution is extremely poor. This is partly due to a lack of a unimodal shape because the number of counts decreases only. This long tail makes it very difficult to estimate the mean parameter with any certainty. The result too is that the `size` parameter is numerically fraught.

This VGAM family function inherits the same warnings as `negbinomial`. And if `k` is much less than 1 then the estimation may be slow.

### Note

If the estimated k is very large then fitting a `pospoisson` model is a good idea.

If both `munb` and k are large then it may be necessary to decrease `eps.trig` and increase `max.support` so that the EIMs are positive-definite, e.g., `eps.trig = 1e-8` and `max.support = Inf`.

### Author(s)

Thomas W. Yee

### References

Barry, S. C. and Welsh, A. H. (2002). Generalized additive modelling and zero inflated count data. Ecological Modelling, 157, 179–188.

Williamson, E. and Bretherton, M. H. (1964). Tables of the logarithmic series distribution. Annals of Mathematical Statistics, 35, 284–297.

### See Also

`gatnbinomial.mlm`, `Gaitnbinom.mlm`, `rposnegbin`, `pospoisson`, `negbinomial`, `zanegbinomial`, `rnbinom`, `CommonVGAMffArguments`, `corbet`, `logff`, `simulate.vlm`.

### Examples

    pdata <- data.frame(x2 = runif(nn <- 1000))
    pdata <- transform(pdata, y1 = rposnegbin(nn, munb = exp(0+2*x2), size = exp(1)),
                              y2 = rposnegbin(nn, munb = exp(1+2*x2), size = exp(3)))
    fit <- vglm(cbind(y1, y2) ~ x2, posnegbinomial, data = pdata, trace = TRUE)
    coef(fit, matrix = TRUE)
    dim(depvar(fit))  # Using dim(fit@y) is not recommended

    # Another artificial data example
    pdata2 <- data.frame(munb = exp(2), size = exp(3)); nn <- 1000
    pdata2 <- transform(pdata2, y3 = rposnegbin(nn, munb = munb, size = size))
    with(pdata2, table(y3))
    fit <- vglm(y3 ~ 1, posnegbinomial, data = pdata2, trace = TRUE)
    coef(fit, matrix = TRUE)
    with(pdata2, mean(y3))  # Sample mean
    head(with(pdata2, munb/(1-(size/(size+munb))^size)), 1)  # Population mean
    head(fitted(fit), 3)
    head(predict(fit), 3)

    # Example: Corbet (1943) butterfly Malaya data
    fit <- vglm(ofreq ~ 1, posnegbinomial, weights = species, data = corbet)
    coef(fit, matrix = TRUE)
    Coef(fit)
    (khat <- Coef(fit)["size"])
    pdf2 <- dposnegbin(x = with(corbet, ofreq), mu = fitted(fit), size = khat)
    print(with(corbet, cbind(ofreq, species, fitted = pdf2*sum(species))), dig = 1)
    ## Not run:
    with(corbet,
    matplot(ofreq, cbind(species, fitted = pdf2*sum(species)), las = 1,
            xlab = "Observed frequency (of individual butterflies)",
            type = "b", ylab = "Number of species", col = c("blue", "orange"),
            main = "blue 1s = observe; orange 2s = fitted"))

    ## End(Not run)

    ## Not run:
    # This data (courtesy of Maxim Gerashchenko) causes posbinomial() to fail
    pnbd.fail <- data.frame(
     y1 = c(1:16, 18:21, 23:28, 33:38, 42, 44, 49:51, 55, 56, 58,
     59, 61:63, 66, 73, 76, 94, 107, 112, 124, 190, 191, 244),
     ofreq = c(130, 80, 38, 23, 22, 11, 21, 14, 6, 7, 9, 9, 9, 4, 4, 5, 1,
               4, 6, 1, 3, 2, 4, 3, 4, 5, 3, 1, 2, 1, 1, 4, 1, 2, 2, 1, 3,
               1, 1, 2, 2, 2, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1))
    fit.fail <- vglm(y1 ~ 1, weights = ofreq, posnegbinomial,
                   trace = TRUE, data = pnbd.fail)

    ## End(Not run)

[Package VGAM version 1.1-3 Index]