cumulative: Ordinal Regression with Cumulative Probabilities

Description

Fits a cumulative link regression model to a (preferably ordered) factor response.

Usage

cumulative(link = "logit", parallel = FALSE, reverse = FALSE,
           mv = FALSE, whitespace = FALSE)

Arguments

link

Link function applied to the $J$ cumulative probabilities. See Links for more choices, e.g., for the cumulative probit/

parallel

A logical or formula specifying which terms have equal/unequal coefficients. See below for more information about the parallelism assumption. The default results in what some people call the generalized ordered logit model to be fitted.

reverse

Logical. By default, the cumulative probabilities used are $P(Y\leq 1)$, $P(Y\leq 2)$, ..., $P(Y\leq J)$. If reverse is TRUE then $P(Y\geq 2)$, $P(Y\geq 3)$, ..., $P(Y\geq J+1)$ are used.

This should be set to

Logical. Multivariate response? If TRUE then the input should be a matrix with values $1,2,\dots,L$, where $L=J+1$ is the number of levels. Each column of the matrix is a response, i.e., multivariate response. A suitable matrix can

whitespace

See CommonVGAMffArguments for information.

Value

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

Warning

No check is made to verify that the response is ordinal if the response is a matrix; see ordered.

Details

In this help file the response $Y$ is assumed to be a factor with ordered values $1,2,\dots,J+1$. Hence $M$ is the number of linear/additive predictors $\eta_j$; for cumulative() one has $M=J$.

This VGAM family function fits the class of cumulative link models to (hopefully) an ordinal response. By default, the non-parallel cumulative logit model is fitted, i.e., $$\eta_j = logit(P[Y \leq j])$$ where $j=1,2,\dots,M$ and the $\eta_j$ are not constrained to be parallel. This is also known as the non-proportional odds model. If the logit link is replaced by a complementary log-log link (cloglog) then this is known as the proportional-hazards model.

In almost all the literature, the constraint matrices associated with this family of models are known. For example, setting parallel = TRUE will make all constraint matrices (except for the intercept) equal to a vector of $M$ 1's. If the constraint matrices are equal, unknown and to be estimated, then this can be achieved by fitting the model as a reduced-rank vector generalized linear model (RR-VGLM; see rrvglm). Currently, reduced-rank vector generalized additive models (RR-VGAMs) have not been implemented here.

References

Agresti, A. (2002) Categorical Data Analysis, 2nd ed. New York: Wiley.

Agresti, A. (2010) Analysis of Ordinal Categorical Data, 2nd ed. New York: Wiley.

Dobson, A. J. and Barnett, A. (2008) An Introduction to Generalized Linear Models, 3rd ed. Boca Raton: Chapman & Hall/CRC Press.

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

Simonoff, J. S. (2003) Analyzing Categorical Data, New York: Springer-Verlag.

Yee, T. W. (2010) The VGAM package for categorical data analysis. Journal of Statistical Software, 32, 1--34. http://www.jstatsoft.org/v32/i10/.

Yee, T. W. and Wild, C. J. (1996) Vector generalized additive models. Journal of the Royal Statistical Society, Series B, Methodological, 58, 481--493.

Further information and examples on categorical data analysis by the VGAM package can be found at http://www.stat.auckland.ac.nz/~yee/VGAM/doc/categorical.pdf.

Examples

Run this code

# Fit the proportional odds model, p.179, in McCullagh and Nelder (1989)
pneumo <- transform(pneumo, let = log(exposure.time))
(fit <- vglm(cbind(normal, mild, severe) ~ let,
             cumulative(parallel = TRUE, reverse = TRUE), data = pneumo))
depvar(fit)  # Sample proportions (good technique)
fit@y        # Sample proportions (bad technique)
weights(fit, type = "prior")  # Number of observations
coef(fit, matrix = TRUE)
constraints(fit)  # Constraint matrices
apply(fitted(fit), 1, which.max)  # Classification
apply(predict(fit, newdata = pneumo, type = "response"),
      1, which.max)  # Classification

# Check that the model is linear in let ----------------------
fit2 <- vgam(cbind(normal, mild, severe) ~ s(let, df = 2),
             cumulative(reverse = TRUE), data = pneumo)
plot(fit2, se = TRUE, overlay = TRUE, lcol = 1:2, scol = 1:2)

# Check the proportional odds assumption with a LRT ----------
(fit3 <- vglm(cbind(normal, mild, severe) ~ let,
              cumulative(parallel = FALSE, reverse = TRUE), data = pneumo))
pchisq(2 * (logLik(fit3) - logLik(fit)),
       df = length(coef(fit3)) - length(coef(fit)), lower.tail = FALSE)
lrtest(fit3, fit)  # More elegant

# A factor() version of fit ----------------------------------
# This is in long format (cf. wide format above)
Nobs <- round(depvar(fit) * c(weights(fit, type = "prior")))
sumNobs <- colSums(Nobs)  # apply(Nobs, 2, sum)

pneumo.long <-
  data.frame(symptoms = ordered(rep(rep(colnames(Nobs), nrow(Nobs)),
                                        times = c(t(Nobs))),
                                levels = colnames(Nobs)),
             let = rep(rep(with(pneumo, let), each = ncol(Nobs)),
                       times = c(t(Nobs))))
with(pneumo.long, table(let, symptoms))  # Should be same as pneumo


(fit.long1 <- vglm(symptoms ~ let, data = pneumo.long, trace = TRUE,
                   cumulative(parallel = TRUE, reverse = TRUE)))
coef(fit.long1, matrix = TRUE)  # Should be as coef(fit, matrix = TRUE)
# Could try using mustart if fit.long1 failed to converge.
mymustart <- matrix(sumNobs / sum(sumNobs),
                    nrow(pneumo.long), ncol(Nobs), byrow = TRUE)
fit.long2 <- vglm(symptoms ~ let, mustart = mymustart,
                  cumulative(parallel = TRUE, reverse = TRUE),
                  data = pneumo.long, trace = TRUE)
coef(fit.long2, matrix = TRUE)  # Should be as coef(fit, matrix = TRUE)

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