negbinomial

Negative Binomial Distribution Family Function

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

Usage

negbinomial(zero = "size", parallel = FALSE, deviance.arg = FALSE,
            type.fitted = c("mean", "quantiles"),
            percentiles = c(25, 50, 75),
            mds.min = 1e-3, nsimEIM = 500, cutoff.prob = 0.999,
            eps.trig = 1e-7, max.support = 4000, max.chunk.MB = 30,
            lmu = "loge", lsize = "loge",
            imethod = 1, imu = NULL, iprobs.y = NULL,
            gprobs.y = ppoints(6), isize = NULL,
            gsize.mux = exp(c(-30, -20, -15, -10, -6:3)))
polya(zero = "size", type.fitted = c("mean", "prob"),
      mds.min = 1e-3, nsimEIM = 500, cutoff.prob = 0.999,
      eps.trig = 1e-7, max.support = 4000, max.chunk.MB = 30,
      lprob = "logit", lsize = "loge", imethod = 1, iprob = NULL,
      iprobs.y = NULL, gprobs.y = ppoints(6), isize = NULL,
      gsize.mux = exp(c(-30, -20, -15, -10, -6:3)), imunb = NULL)
polyaR(zero = "size", type.fitted = c("mean", "prob"),
       mds.min = 1e-3, nsimEIM = 500,  cutoff.prob = 0.999,
       eps.trig = 1e-7, max.support = 4000, max.chunk.MB = 30,
       lsize = "loge", lprob = "logit", imethod = 1, iprob = NULL,
       iprobs.y = NULL, gprobs.y = ppoints(6), isize = NULL,
       gsize.mux = exp(c(-30, -20, -15, -10, -6:3)), imunb = NULL)

Details

The negative binomial distribution (NBD) can be motivated in several ways, e.g., as a Poisson distribution with a mean that is gamma distributed. There are several common parametrizations of the NBD. The one used by negbinomial() uses the mean \(\mu\) and an index parameter \(k\), both which are positive. Specifically, the density of a random variable \(Y\) is $$f(y;\mu,k) = {y + k - 1 \choose y} \, \left( \frac{\mu}{\mu+k} \right)^y\, \left( \frac{k}{k+\mu} \right)^k $$ where \(y=0,1,2,\ldots\), and \(\mu > 0\) and \(k > 0\). Note that the dispersion parameter is \(1/k\), so that as \(k\) approaches infinity the NBD approaches a Poisson distribution. The response has variance \(Var(Y)=\mu+\mu^2/k\). When fitted, the fitted.values slot of the object contains the estimated value of the \(\mu\) parameter, i.e., of the mean \(E(Y)\). It is common for some to use \(\alpha=1/k\) as the ancillary or heterogeneity parameter; so common alternatives for lsize are negloge and reciprocal.

For polya the density is $$f(y;p,k) = {y + k - 1 \choose y} \, \left( 1 - p \right)^y\, p^k $$ where \(y=0,1,2,\ldots\), and \(k > 0\) and \(0 < p < 1\).

Family function polyaR() is the same as polya() except the order of the two parameters are switched. The reason is that polyaR() tries to match with rnbinom closely in terms of the argument order, etc. Should the probability parameter be of primary interest, probably, users will prefer using polya() rather than polyaR(). Possibly polyaR() will be decommissioned one day.

The NBD can be coerced into the classical GLM framework with one of the parameters being of interest and the other treated as a nuisance/scale parameter (this is implemented in the MASS library). The VGAM family function negbinomial() treats both parameters on the same footing, and estimates them both by full maximum likelihood estimation.

The parameters \(\mu\) and \(k\) are independent (diagonal EIM), and the confidence region for \(k\) is extremely skewed so that its standard error is often of no practical use. The parameter \(1/k\) has been used as a measure of aggregation. For the NB-C the EIM is not diagonal.

These VGAM family functions handle multiple responses, so that a response matrix can be inputted. 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.

Note

These 3 functions implement 2 common parameterizations of the negative binomial (NB). Some people called the NB with integer \(k\) the Pascal distribution, whereas if \(k\) is real then this is the Polya distribution. I don't. The one matching the details of rnbinom in terms of \(p\) and \(k\) is polya().

For polya() the code may fail when \(p\) is close to 0 or 1. It is not yet compatible with cqo or cao.

Suppose the response is called ymat. For negbinomial() the diagonal element of the expected information matrix (EIM) for parameter \(k\) involves an infinite series; consequently SFS (see nsimEIM) is used as the backup algorithm only. SFS should be better if max(ymat) is large, e.g., max(ymat) > 1000, or if there are any outliers in ymat. The default algorithm involves a finite series approximation to the support 0:Inf; the arguments max.memory, min.size and cutoff.prob are pertinent.

Regardless of the algorithm used, convergence problems may occur, especially when the response has large outliers or is large in magnitude. If convergence failure occurs, try using arguments (in recommended decreasing order) max.support, nsimEIM, cutoff.prob, iprobs.y, imethod, isize, zero, max.chunk.MB.

The function negbinomial can be used by the fast algorithm in cqo, however, setting eq.tolerances = TRUE and I.tolerances = FALSE is recommended.

In the first example below (Bliss and Fisher, 1953), from each of 6 McIntosh apple trees in an orchard that had been sprayed, 25 leaves were randomly selected. On each of the leaves, the number of adult female European red mites were counted.

There are two special uses of negbinomial for handling count data. Firstly, when used by rrvglm this results in a continuum of models in between and inclusive of quasi-Poisson and negative binomial regression. This is known as a reduced-rank negative binomial model (RR-NB). It fits a negative binomial log-linear regression with variance function \(Var(Y)=\mu+\delta_1 \mu^{\delta_2}\) where \(\delta_1\) and \(\delta_2\) are parameters to be estimated by MLE. Confidence intervals are available for \(\delta_2\), therefore it can be decided upon whether the data are quasi-Poisson or negative binomial, if any.

Secondly, the use of negbinomial with parallel = TRUE inside vglm can result in a model similar to quasipoissonff. This is named the NB-1 model. The dispersion parameter is estimated by MLE whereas glm uses the method of moments. In particular, it fits a negative binomial log-linear regression with variance function \(Var(Y) = \phi_0 \mu\) where \(\phi_0\) is a parameter to be estimated by MLE. Confidence intervals are available for \(\phi_0\).

Warning

Poisson regression corresponds to \(k\) equalling infinity. If the data is Poisson or close to Poisson, numerical problems may occur. Some corrective measures are taken, e.g., \(k\) is effectively capped (relative to the mean) during estimation to some large value and a warning is issued. And setting stepsize = 0.5 for half stepping is probably a good idea too when the data is extreme.

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

These functions are fragile; the maximum likelihood estimate of the index parameter is fraught (see Lawless, 1987). In general, the quasipoissonff is more robust. Other alternatives to negbinomial are to fit a NB-1 or RR-NB (aka NB-P) model; see Yee (2014). Also available are the NB-C, NB-H and NB-G. Assigning values to the isize argument may lead to a local solution, and smaller values are preferred over large values when using this argument.

If one wants to force SFS to be used on all observations, then set max.support = 0 or max.chunk.MB = 0. If one wants to force the exact method to be used for all observations, then set max.support = Inf. If the computer has much memory, then trying max.chunk.MB = Inf and max.support = Inf may provide a small speed increase. If SFS is used at all, then the @weights slot of the fitted object will be a matrix; otherwise that slot will be a 0 x 0 matrix.

Yet to do: write a family function which uses the methods of moments estimator for \(k\).

References

Lawless, J. F. (1987) Negative binomial and mixed Poisson regression. The Canadian Journal of Statistics 15, 209--225.

Hilbe, J. M. (2011) Negative Binomial Regression, 2nd Edition. Cambridge: Cambridge University Press.

Bliss, C. and Fisher, R. A. (1953) Fitting the negative binomial distribution to biological data. Biometrics 9, 174--200.

Yee, T. W. (2014) Reduced-rank vector generalized linear models with two linear predictors. Computational Statistics and Data Analysis, 71, 889--902.

See Also

quasipoissonff, poissonff, zinegbinomial, negbinomial.size (e.g., NB-G), nbcanlink (NB-C), posnegbinomial, inv.binomial, rnbinom, nbolf, rrvglm, cao, cqo, CommonVGAMffArguments, simulate.vlm, ppoints, qnbinom.

Aliases

• negbinomial
• polya
• polyaR

Examples

# NOT RUN {
# Example 1: apple tree data (Bliss and Fisher, 1953)
appletree <- data.frame(y = 0:7, w = c(70, 38, 17, 10, 9, 3, 2, 1))
fit <- vglm(y ~ 1, negbinomial(deviance = TRUE), data = appletree,
            weights = w, crit = "coef")  # Obtain the deviance
fit <- vglm(y ~ 1, negbinomial(deviance = TRUE), data = appletree,
            weights = w, half.step = FALSE)  # Alternative method
summary(fit)
coef(fit, matrix = TRUE)
Coef(fit)  # For intercept-only models
deviance(fit)  # NB2 only; needs 'crit = "coef"' & 'deviance = TRUE' above

# Example 2: simulated data with multiple responses
# }
# NOT RUN {
ndata <- data.frame(x2 = runif(nn <- 200))
ndata <- transform(ndata, y1 = rnbinom(nn, mu = exp(3+x2), size = exp(1)),
                          y2 = rnbinom(nn, mu = exp(2-x2), size = exp(0)))
fit1 <- vglm(cbind(y1, y2) ~ x2, negbinomial, data = ndata, trace = TRUE)
coef(fit1, matrix = TRUE)
# }
# NOT RUN {
# Example 3: large counts implies SFS is used
# }
# NOT RUN {
ndata <- transform(ndata, y3 = rnbinom(nn, mu = exp(10+x2), size = exp(1)))
with(ndata, range(y3))  # Large counts
fit2 <- vglm(y3 ~ x2, negbinomial, data = ndata, trace = TRUE)
coef(fit2, matrix = TRUE)
head(fit2@weights)  # Non-empty; SFS was used
# }
# NOT RUN {
# Example 4: a NB-1 to estimate a negative binomial with Var(Y) = phi0 * mu
nn <- 200  # Number of observations
phi0 <- 10  # Specify this; should be greater than unity
delta0 <- 1 / (phi0 - 1)
mydata <- data.frame(x2 = runif(nn), x3 = runif(nn))
mydata <- transform(mydata, mu = exp(2 + 3 * x2 + 0 * x3))
mydata <- transform(mydata, y3 = rnbinom(nn, mu = mu, size = delta0 * mu))
# }
# NOT RUN {
plot(y3 ~ x2, data = mydata, pch = "+", col = "blue",
     main = paste("Var(Y) = ", phi0, " * mu", sep = ""), las = 1) 
# }
# NOT RUN {
nb1 <- vglm(y3 ~ x2 + x3, negbinomial(parallel = TRUE, zero = NULL),
            data = mydata, trace = TRUE)
# Extracting out some quantities:
cnb1 <- coef(nb1, matrix = TRUE)
mydiff <- (cnb1["(Intercept)", "loge(size)"] -
           cnb1["(Intercept)", "loge(mu)"])
delta0.hat <- exp(mydiff)
(phi.hat <- 1 + 1 / delta0.hat)  # MLE of phi
summary(nb1)
# Obtain a 95 percent confidence interval for phi0:
myvec <- rbind(-1, 1, 0, 0)
(se.mydiff <- sqrt(t(myvec) %*%  vcov(nb1) %*%  myvec))
ci.mydiff <- mydiff + c(-1.96, 1.96) * se.mydiff
ci.delta0 <- ci.exp.mydiff <- exp(ci.mydiff)
(ci.phi0 <- 1 + 1 / rev(ci.delta0))  # The 95 percent conf. interval for phi0

Confint.nb1(nb1)  # Quick way to get it

summary(glm(y3 ~ x2 + x3, quasipoisson, mydata))$disper  # cf. moment estimator
# }