cumulative {VGAM}R Documentation

Ordinal Regression with Cumulative Probabilities

Description

Fits a cumulative logit/probit/cloglog/cauchit/... regression model to an ordered (preferably) factor response.

Usage

cumulative(link = "logit", earg = list(),
           parallel = FALSE, reverse = FALSE,
           mv = FALSE, intercept.apply = FALSE)
scumulative(link="logit", earg = list(),
            lscale="loge", escale = list(),
            parallel=FALSE, sparallel=TRUE, reverse=FALSE, iscale = 1)

Arguments

In the following, the response Y is assumed to be a factor with ordered values 1,2,...,J+1. M is the number of linear/additive predictors eta_j; for cumulative() M=J, and for scumulative() M=2J.

link Link function applied to the J cumulative probabilities. See Links for more choices.
lscale Link function applied to the J scaling parameters. See Links for more choices.
earg, escale List. Extra argument for the respective link functions. See earg in Links for general information.
parallel A logical or formula specifying which terms have equal/unequal coefficients.
sparallel For the scaling parameters. A logical, or formula specifying which terms have equal/unequal coefficients. This argument is not applied to the intercept. The scumulative() function requires covariates; for intercept models use cumulative().
reverse Logical. By default, the cumulative probabilities used are P(Y<=1), P(Y<=2), ..., P(Y<=J). If reverse is TRUE, then P(Y>=2), P(Y>=3), ..., P(Y>=J+1) will be used.
This should be set to TRUE for link= golf, polf, nbolf. For these links the cutpoints must be an increasing sequence; if reverse=FALSE for then the cutpoints must be an decreasing sequence.
mv Logical. Multivariate response? If TRUE then the input should be a matrix with values 1,2,...,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 be obtained from Cut.
intercept.apply Logical. Whether the parallel argument should be applied to the intercept term. This should be set to TRUE for link= golf, polf, nbolf.
iscale Numeric. Initial values for the scale parameters.

Details

By default, the non-parallel cumulative logit model is fitted, i.e.,

eta_j = logit(P[Y<=j])

where j=1,2,...,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.

The scaled version of cumulative(), called scumulative(), has J positive scaling factors. They are described in pages 154 and 177 of McCullagh and Nelder (1989); see their equation (5.4) in particular, which they call the generalized rational model.

Value

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

Warning

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

Note

The response should be either a matrix of counts (with row sums that are all positive), or a factor. In both cases, the y slot returned by vglm/vgam/rrvglm is the matrix of counts.

For a nominal (unordered) factor response, the multinomial logit model (multinomial) is more appropriate.

With the logit link, setting parallel=TRUE will fit a proportional odds model. Note that the TRUE here does not apply to the intercept term. In practice, the validity of the proportional odds assumption needs to be checked, e.g., by a likelihood ratio test (LRT). If acceptable on the data, then numerical problems are less likely to occur during the fitting, and there are less parameters. Numerical problems occur when the linear/additive predictors cross, which results in probabilities outside of (0,1); setting parallel=TRUE will help avoid this problem.

Here is an example of the usage of the parallel argument. If there are covariates x1, x2 and x3, then parallel = TRUE ~ x1 + x2 -1 and parallel = FALSE ~ x3 are equivalent. This would constrain the regression coefficients for x1 and x2 to be equal; those of the intercepts and x3 would be different.

In the future, this family function may be renamed to ``cups'' (for cumulative probabilities) or ``cute'' (for cumulative probabilities).

Author(s)

Thomas W. Yee

References

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

Dobson, A. J. (2001) An Introduction to Generalized Linear Models, 2nd 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. and Wild, C. J. (1996) Vector generalized additive models. Journal of the Royal Statistical Society, Series B, Methodological, 58, 481–493.

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

See Also

acat, cratio, sratio, multinomial, pneumo, logit, probit, cloglog, cauchit, golf, polf, nbolf.

Examples

# Fit the proportional odds model, p.179, in McCullagh and Nelder (1989)
data(pneumo)
pneumo = transform(pneumo, let=log(exposure.time))
(fit = vglm(cbind(normal, mild, severe) ~ let,
            cumulative(parallel=TRUE, reverse=TRUE), pneumo))
fit@y   # Sample proportions
weights(fit, type="prior")   # Number of observations
coef(fit, matrix=TRUE)
constraints(fit)   # Constraint matrices

# Check that the model is linear in let ----------------------
fit2 = vgam(cbind(normal, mild, severe) ~ s(let, df=2),
            cumulative(reverse=TRUE), pneumo)
## Not run: 
plot(fit2, se=TRUE, overlay=TRUE, lcol=1:2, scol=1:2)
## End(Not run)

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

# A factor() version of fit ----------------------------------
nobs = round(fit@y * c(weights(fit, type="prior")))
sumnobs = apply(nobs, 2, sum)
mydat = data.frame(
    response = ordered(c(rep("normal", times=sumnobs["normal"]),
                         rep("mild", times=sumnobs["mild"]),
                         rep("severe", times=sumnobs["severe"])),
                       levels = c("normal","mild","severe")),
    LET = c(with(pneumo, rep(let, times=nobs[,"normal"])),
            with(pneumo, rep(let, times=nobs[,"mild"])),
            with(pneumo, rep(let, times=nobs[,"severe"]))))
(fit4 = vglm(response ~ LET, data=mydat,
             cumulative(parallel=TRUE, reverse=TRUE), trace=TRUE))

[Package VGAM version 0.7-7 Index]