Negative Binomial Canonical Link Function
Computes the negative binomial canonical link transformation, including its inverse and the first two derivatives.
nbcanlink(theta, size = NULL, wrt.param = NULL, bvalue = NULL,
inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)theta |
Numeric or character. Typically the mean of a negative binomial distribution (NBD). See below for further details. |
size, wrt.param |
|
bvalue |
Details at |
inverse, deriv, short, tag |
Details at |
The NBD canonical link is log(theta/(theta + k)) where theta is the NBD mean. The canonical link is used for theoretically relating the NBD to GLM class.
This link function was specifically written for
negbinomial and
negbinomial.size,
and should not be used elsewhere
(these VGAM family functions have code that
specifically handles nbcanlink().)
For deriv = 0, the above equation
when inverse = FALSE, and if inverse = TRUE then
kmatrix / expm1(-theta) where theta ie really eta.
For deriv = 1, then the function returns
d eta / d theta as a function of theta
if inverse = FALSE,
else if inverse = TRUE then it returns the reciprocal.
This function works with negbinomial but care
is needed because it is numerically fraught.
In particular, the first linear/additive predictor must have
negative values, and finding good initial values may be
difficult, leading to it crashing at the start.
Hence the NB-C model is sensitive to the initial values and may
converge to a local solution.
Pages 210 and 309 of Hilbe (2011) notes convergence difficulties (of
Newton-Raphson type algorithms), and some of that
this applies here.
Setting trace = TRUE is a good idea, as is
trying various values of imethod
in negbinomial.
While theoretically nice, this function is not recommended
in general since its value is always negative (linear predictors
ought to be unbounded in general). A loglink
link for argument lmu is recommended instead.
Numerical instability may occur when theta is close to 0 or 1.
Values of theta which are less than or equal to 0 can be
replaced by bvalue
before computing the link function value.
See Links.
Victor Miranda and Thomas W. Yee.
Miranda, V. S. and Yee, T. W. (2018). On mean function modelling for several one-parameter discrete distributions. Manuscript in preparation.
Yee, T. W. (2014). Reduced-rank vector generalized linear models with two linear predictors. Computational Statistics and Data Analysis, 71, 889–902.
Hilbe, J. M. (2011). Negative Binomial Regression, 2nd Edition. Cambridge: Cambridge University Press.
nbcanlink("mu", short = FALSE)
mymu <- 1:10 # Test some basic operations:
kmatrix <- cbind(runif(length(mymu)))
eta1 <- nbcanlink(mymu, size = kmatrix)
ans2 <- nbcanlink(eta1, size = kmatrix, inverse = TRUE)
max(abs(ans2 - mymu)) # Should be 0
## Not run: mymu <- seq(0.5, 10, length = 101)
kmatrix <- matrix(10, length(mymu), 1)
plot(nbcanlink(mymu, size = kmatrix) ~ mymu, las = 1,
type = "l", col = "blue", xlab = expression({mu}))
## End(Not run)
# Estimate the parameters from some simulated data
ndata <- data.frame(x2 = runif(nn <- 100))
ndata <- transform(ndata, eta1 = -1 - 1 * x2, # eta1 < 0
size1 = exp(1),
size2 = exp(2))
ndata <- transform(ndata,
mu1 = nbcanlink(eta1, size = size1, inverse = TRUE),
mu2 = nbcanlink(eta1, size = size2, inverse = TRUE))
ndata <- transform(ndata, y1 = rnbinom(nn, mu = mu1, size = size1),
y2 = rnbinom(nn, mu = mu2, size = size2))
summary(ndata)
nbcfit <- vglm(cbind(y1, y2) ~ x2,
negbinomial(lmu = "nbcanlink", imethod = 1), # Try this
# negbinomial(lmu = "nbcanlink", imethod = 2), # Try this
data = ndata, trace = TRUE)
coef(nbcfit, matrix = TRUE)
summary(nbcfit)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.