Ordinal Regression with Cumulative Probabilities

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

models, regression
cumulative(link = "logit", parallel = FALSE, reverse = FALSE,
           multiple.responses = FALSE, whitespace = FALSE)

Link function applied to the \(J\) cumulative probabilities. See Links for more choices, e.g., for the cumulative probit/cloglog/cauchit/… models.


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 parallel = TRUE then it does not apply to the intercept.


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 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.


Logical. Multiple responses? 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., multiple responses. A suitable matrix can be obtained from Cut.


See CommonVGAMffArguments for information.


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.


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


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 = "logit")). 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.


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


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.

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.

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, prplot, margeff, acat, cratio, sratio, multinomial, pneumo, Links, hdeff.vglm, logit, probit, cloglog, cauchit, golf, polf, nbolf, logistic1.

  • cumulative
# 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)
# }
Documentation reproduced from package VGAM, version 1.0-4, License: GPL-3

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