negbinomial: Negative Binomial Distribution Family Function

Description

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

Usage

negbinomial(lmu = "loge", lsize = "loge",
            imu = NULL, isize = NULL, probs.y = 0.75,
            nsimEIM = 100, cutoff = 0.995,
            Maxiter = 5000, deviance.arg = FALSE, imethod = 1,
            parallel = FALSE, shrinkage.init = 0.95, zero = -2)
polya(lprob = "logit", lsize = "loge",
      iprob = NULL, isize = NULL, probs.y = 0.75, nsimEIM = 100,
      imethod = 1, shrinkage.init = 0.95, zero = -2)

Arguments

lmu, lsize, lprob

Link functions applied to the $\mu$, $k$ and $p$ parameters. See Links for more choices. Note that the $\mu$, $k$ and $p$ parameters are the mu, size and prob

imu, isize, iprob

Optional initial values for the mean and $k$ and $p$. For $k$, if failure to converge occurs then try different values (and/or use imethod). For a $S$-column response, isize can be of length $S$. A value NULL

probs.y

Passed into the probs argument of quantile when imethod = 3 to obtain an initial value for the mean.

nsimEIM

This argument is used for computing the diagonal element of the expected information matrix (EIM) corresponding to $k$. See CommonVGAMffArguments for more information and the

cutoff

Used in the finite series approximation. A numeric which is close to 1 but never exactly 1. Used to specify how many terms of the infinite series for computing the second diagonal element of the EIM are actually used. The sum of the probabilites

Maxiter

Used in the finite series approximation. Integer. The maximum number of terms allowed when computing the second diagonal element of the EIM. In theory, the value involves an infinite series. If this argument is too small then the value may be inac

deviance.arg

Logical.

If TRUE, the deviance is computed after convergence. It only works in the NB-2 model. It is also necessary to set criterion = "coefficients" or half.step = FALSE since one cannot use th

imethod

An integer with value 1 or 2 or 3 which specifies the initialization method for the $\mu$ parameter. If failure to converge occurs try another value and/or else specify a value for shrinkage.init

parallel

See CommonVGAMffArguments for more information. Setting parallel = TRUE is useful in order to get something similar to quasipoisso

shrinkage.init

How much shrinkage is used when initializing $\mu$. The value must be between 0 and 1 inclusive, and a value of 0 means the individual response values are used, and a value of 1 means the median or mean is used. This argument is used in conjunctio

zero

Integer valued vector, usually assigned $-2$ or $2$ if used at all. Specifies which of the two linear/additive predictors are modelled as an intercept only. By default, the $k$ parameter (after lsize is applied) is modelled as a single

Value

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

Warning

The Poisson model corresponds to $k$ equalling infinity. If the data is Poisson or close to Poisson, numerical problems will occur. Possibly choosing a log-log link may help in such cases, otherwise use poissonff or quasipoissonff.

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.

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

Details

The negative binomial distribution 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 negative binomial distribution. 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 negative binomial distribution 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 $0 < p < 1$ and $k > 0$.

The negative binomial distribution 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. Simulated Fisher scoring is employed as the default (see the nsimEIM argument).

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.

These VGAM family functions handle multivariate 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.

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.

Examples

Run this code

# Example 1: apple tree data
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")
fit <- vglm(y ~ 1, negbinomial(deviance = TRUE), data = appletree,
            weights = w, half.step = FALSE)  # Alternative method
summary(fit)
coef(fit, matrix = TRUE)
Coef(fit)
deviance(fit)  # NB2 only; needs 'crit = "coef"' & 'deviance = TRUE' above

# Example 2: simulated data with multivariate response
ndata <- data.frame(x2 = runif(nn <- 500))
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)

# Example 3: large counts so definitely use the nsimEIM argument
ndata <- transform(ndata, y3 = rnbinom(nn, mu = exp(12+x2), size = exp(1)))
with(ndata, range(y3))  # Large counts
fit2 <- vglm(y3 ~ x2, negbinomial(nsimEIM = 100), data = ndata, trace = TRUE)
coef(fit2, matrix = TRUE)

# Example 4: a NB-1 to estimate a negative binomial with Var(Y) = phi0 * mu
nn <- 1000  # 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))
plot(y3 ~ x2, data = mydata, pch = "+", col = 'blue',
     main = paste("Var(Y) = ", phi0, " * mu", sep = ""), las = 1)
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)", "log(size)"] -
           cnb1["(Intercept)", "log(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

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