| cumulative {VGAM} | R Documentation |
Ordinal Regression with Cumulative Probabilities
Description
Fits a cumulative link regression model to a (preferably ordered) factor response.
Usage
cumulative(link = "logitlink", parallel = FALSE,
reverse = FALSE, multiple.responses = FALSE,
ynames = FALSE, Thresh = NULL, Trev = reverse,
Tref = if (Trev) "M" else 1, whitespace = FALSE)
Arguments
link |
Link function applied to the |
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.
If The partial proportional odds model can be
fitted by assigning this argument something like
|
reverse |
Logical.
By default, the cumulative probabilities used are
|
ynames |
See |
multiple.responses |
Logical.
Multiple responses?
If |
Thresh |
Character.
The choices concern constraint matrices applied
to the intercepts.
They can be constrained to be
equally-spaced (equidistant)
etc.
See If equally-spaced then the direction and the
reference level are controlled
by If If For |
Trev, Tref |
Support arguments for |
whitespace |
See |
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
(clogloglink) 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.
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.
Boersch-Supan (2021) looks at sparse data and
the numerical problems that result;
see sratio.
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.
The formula must contain an intercept term.
Other VGAM family functions for an ordinal response
include
acat,
cratio,
sratio.
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 x2, x3 and
x4, then
parallel = TRUE ~ x2 + x3 -1 and
parallel = FALSE ~ x4 are equivalent.
This would constrain the regression coefficients
for x2 and x3 to be equal;
those of the intercepts and x4 would be different.
If the data is inputted in long format
(not wide format, as in pneumo below)
and the self-starting initial values are not
good enough then try using
mustart,
coefstart and/or
etatstart.
See the example below.
To fit the proportional odds model one can use the
VGAM family function propodds.
Note that propodds(reverse) is equivalent to
cumulative(parallel = TRUE, reverse = reverse)
(which is equivalent to
cumulative(parallel =
TRUE, reverse = reverse, link = "logitlink")).
It is for convenience only. A call to
cumulative() is preferred since it reminds the user
that a parallelism assumption is made, as well as
being a lot more flexible.
Category specific effects may be modelled using
the xij-facility; see
vglm.control and fill1.
With most Threshold choices,
the first few fitted regression coefficients
need care in their interpretation. For example,
some values could be the distance away from
the median intercept. Typing something
like constraints(fit)[[1]] gives the
constraint matrix of the intercept term.
Author(s)
Thomas W. Yee
References
Agresti, A. (2013). Categorical Data Analysis, 3rd ed. Hoboken, NJ, USA: Wiley.
Agresti, A. (2010). Analysis of Ordinal Categorical Data, 2nd ed. Hoboken, NJ, USA: Wiley.
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.
Tutz, G. (2012). Regression for Categorical Data, Cambridge: Cambridge University Press.
Tutz, G. and Berger, M. (2022). Sparser ordinal regression models based on parametric and additive location-shift approaches. International Statistical Review, 90, 306–327. doi:10.1111/insr.12484.
Yee, T. W. (2010). The VGAM package for categorical data analysis. Journal of Statistical Software, 32, 1–34. doi:10.18637/jss.v032.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.
See Also
propodds,
constraints,
CM.ones,
CM.equid,
R2latvar,
ordsup,
prplot,
margeff,
acat,
cratio,
sratio,
multinomial,
CommonVGAMffArguments,
pneumo,
budworm,
Links,
hdeff.vglm,
logitlink,
probitlink,
clogloglink,
cauchitlink,
logistic1.
Examples
# Proportional odds model (p.179) of McCullagh and Nelder (1989)
pneumo <- transform(pneumo, let = log(exposure.time))
(fit <- vglm(cbind(normal, mild, severe) ~ let,
cumulative(parallel = TRUE, reverse = TRUE), 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
R2latvar(fit)
# Check that the model is linear in let ----------------------
fit2 <- vgam(cbind(normal, mild, severe) ~ s(let, df = 2),
cumulative(reverse = TRUE), data = 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))
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) # cf. 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) # cf. coef(fit, matrix = TRUE)