Multi-logit Link Function

Description

Computes the multilogit transformation, including its inverse and the first two derivatives.

Usage

multilogit(theta, refLevel = "last", M = NULL, whitespace = FALSE,
           bvalue = NULL, inverse = FALSE, deriv = 0,
           short = TRUE, tag = FALSE)

Arguments

Details

The multilogit() link function is a generalization of the logit link to M levels/classes. It forms the basis of the multinomial logit model. It is sometimes called the multi-logit link or the multinomial logit link. When its inverse function is computed it returns values which are positive and add to unity.

Value

For multilogit with deriv = 0, the multilogit of theta, i.e., log(theta[, j]/theta[, M+1]) when inverse = FALSE, and if inverse = TRUE then exp(theta[, j])/(1+rowSums(exp(theta))).

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.

Here, all logarithms are natural logarithms, i.e., to base e.

Note

Numerical instability may occur when theta is close to 1 or 0 (for multilogit). One way of overcoming this is to use, e.g., bvalue. Currently care.exp() is used to avoid NAs being returned if the probability is too close to 1.

Author(s)

Thomas W. Yee

References

McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London: Chapman & Hall.

See Also

Links, multinomial, logit, normal.vcm, CommonVGAMffArguments.

Examples

pneumo <- transform(pneumo, let = log(exposure.time))
fit <- vglm(cbind(normal, mild, severe) ~ let,
            multinomial, trace = TRUE, data = pneumo)  # For illustration only!
fitted(fit)
predict(fit)

multilogit(fitted(fit))
multilogit(fitted(fit)) - predict(fit)  # Should be all 0s

multilogit(predict(fit), inverse = TRUE)  # rowSums() add to unity
multilogit(predict(fit), inverse = TRUE, refLevel = 1)  # For illustration only
multilogit(predict(fit), inverse = TRUE) - fitted(fit)  # Should be all 0s

multilogit(fitted(fit), deriv = 1)
multilogit(fitted(fit), deriv = 2)