VGAM (version 0.8-1)

polf: Poisson-Ordinal Link Function

Description

Computes the Poisson-ordinal transformation, including its inverse and the first two derivatives.

Usage

polf(theta, earg = stop("'earg' must be given"), inverse = FALSE,
     deriv = 0, short = TRUE, tag = FALSE)

Arguments

theta

Numeric or character. See below for further details.

earg

Extra argument for passing in additional information. This must be list with component cutpoint. The cutpoints should be non-negative integers. If polf() is used as the link function in

inverse

Logical. If TRUE the inverse function is computed.

deriv

Order of the derivative. Integer with value 0, 1 or 2.

short

Used for labelling the blurb slot of a vglmff-class object.

tag

Used for labelling the linear/additive predictor in the initialize slot of a vglmff-class object. Contains a little more information if TRUE.

Value

• See Yee (2007) for details.

Warning

Prediction may not work on vglm or vgam etc. objects if this link function is used.

Details

The Poisson-ordinal link function (POLF) can be applied to a parameter lying in the unit interval. Its purpose is to link cumulative probabilities associated with an ordinal response coming from an underlying Poisson distribution. If the cutpoint is zero then a complementary log-log link is used.

The arguments short and tag are used only if theta is character.

See Links for general information about VGAM link functions.

References

Yee, T. W. (2007) Ordinal ordination with normalizing link functions for count data, (in preparation).

See Also

Links, ordpoisson, poissonff, nbolf, golf, cumulative.

Examples

earg = list(cutpoint=2)
polf("p", earg=earg, short=FALSE)
polf("p", earg=earg, tag=TRUE)

p = seq(0.01, 0.99, by=0.01)
y = polf(p, earg=earg)
y. = polf(p, earg=earg, deriv=1)
max(abs(polf(y, earg=earg, inv=TRUE) - p)) # Should be 0

par(mfrow=c(2,1), las=1)
plot(p, y, type="l", col="blue", main="polf()")
abline(h=0, v=0.5, col="red", lty="dashed")

plot(p, y., type="l", col="blue",
     main="(Reciprocal of) first POLF derivative")


# Rutherford and Geiger data
ruge = data.frame(yy = rep(0:14,
          times=c(57,203,383,525,532,408,273,139,45,27,10,4,0,1,1)))

with(ruge, length(yy))  # 2608 1/8-minute intervals
cutpoint = 5
ruge = transform(ruge, yy01 = ifelse(yy <= cutpoint, 0, 1))
earg = list(cutpoint=cutpoint)
fit = vglm(yy01 ~ 1, binomialff(link="polf", earg=earg), ruge)
coef(fit, matrix=TRUE)
exp(coef(fit))


# Another example
pdat = data.frame(x2 = sort(runif(nn <- 1000)))
pdat = transform(pdat, x3 = runif(nn))
pdat = transform(pdat, mymu = exp( 3 + 1 * x2 - 2 * x3))
pdat = transform(pdat, y1 = rpois(nn, lambda=mymu))
cutpoints = c(-Inf, 10, 20, Inf)
pdat = transform(pdat, cuty = Cut(y1, breaks=cutpoints))
with(pdat, plot(x2, x3, col=cuty, pch=as.character(cuty)))
with(pdat, table(cuty) / sum(table(cuty)))
fit = vglm(cuty ~ x2 + x3, fam = cumulative(link="polf",
           reverse=TRUE, parallel=TRUE, intercept.apply=TRUE,
           mv=TRUE, earg=list(cutpoint=cutpoints[2:3])),
           pdat, trace=TRUE)
head(fit@y)
head(fitted(fit))
head(predict(fit))
coef(fit)
coef(fit, matrix=TRUE)
constraints(fit)
fit@misc$earg