Ordinal Regression with Cumulative Probabilities

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

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 $J$ cumulative probabilities. See Links for more choices, e.g., for the cumulative probitlink/clogloglink/... models.

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

The partial proportional odds model can be fitted by assigning this argument something like parallel = TRUE ~ -1 + x3 + x5 so that there is one regression coefficient for x3 and x5. Equivalently, setting parallel = FALSE ~ 1 + x2 + x4 means $M$ regression coefficients for the intercept and x2 and x4. It is important that the intercept is never parallel. See CommonVGAMffArguments for more information.

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.

ynames

See multinomial for information.

multiple.responses

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.

Thresh

Character. The choices concern constraint matrices applied to the intercepts. They can be constrained to be equally-spaced (equidistant) etc. See CommonVGAMffArguments and constraints for general information. Basically, the choice is pasted to the end of "CM." and that function is called. This means users can easily write their own CM.-type function.

If equally-spaced then the direction and the reference level are controlled by Trev and Tref, and the constraint matrix will be $M$ by 2, with the second column corresponding to the distance between the thresholds.

If "symm1" then the fitted intercepts are symmetric about the median ($M$ odd) intercept. If $M$ is even then the median is the mean of the two most inner and adjacent intercepts. For this, CM.symm1 is used to construct the appropriate constraint matrix.

If "symm0" then the median intercept is 0 by definition and the symmetry occurs about the origin. Thus the intercepts comprise pairs that differ by sign only. The appropriate constraint matrix is as with "symm1" but with the first column deleted. The choices "symm1" and "symm0" are effectively equivalent to "symmetric" and "symmetric2" respectively in ordinal.

For "qnorm" then CM.qnorm uses the qnorm((1:M)/(M+1)) quantiles of the standard normal.

Trev, Tref

Support arguments for Thresh for equally-spaced intercepts. The logical argument Trev is applied first to give the direction (i.e., ascending or descending) before row Tref (ultimately numeric) of the first (intercept) constraint matrix is set to the reference level. See constraints for information.

whitespace

See CommonVGAMffArguments for information.

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.

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.

Author

Thomas W. Yee

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.

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.

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))
#> 
#> Call:
#> vglm(formula = cbind(normal, mild, severe) ~ let, family = cumulative(parallel = TRUE, 
#>     reverse = TRUE), data = pneumo)
#> 
#> 
#> Coefficients:
#> (Intercept):1 (Intercept):2           let 
#>     -9.676093    -10.581725      2.596807 
#> 
#> Degrees of Freedom: 16 Total; 13 Residual
#> Residual deviance: 5.026826 
#> Log-likelihood: -25.09026 
depvar(fit)  # Sample proportions (good technique)
#>      normal       mild     severe
#> 1 1.0000000 0.00000000 0.00000000
#> 2 0.9444444 0.03703704 0.01851852
#> 3 0.7906977 0.13953488 0.06976744
#> 4 0.7291667 0.10416667 0.16666667
#> 5 0.6274510 0.19607843 0.17647059
#> 6 0.6052632 0.18421053 0.21052632
#> 7 0.4285714 0.21428571 0.35714286
#> 8 0.3636364 0.18181818 0.45454545
fit@y        # Sample proportions (bad technique)
#>      normal       mild     severe
#> 1 1.0000000 0.00000000 0.00000000
#> 2 0.9444444 0.03703704 0.01851852
#> 3 0.7906977 0.13953488 0.06976744
#> 4 0.7291667 0.10416667 0.16666667
#> 5 0.6274510 0.19607843 0.17647059
#> 6 0.6052632 0.18421053 0.21052632
#> 7 0.4285714 0.21428571 0.35714286
#> 8 0.3636364 0.18181818 0.45454545
weights(fit, type = "prior")  # Number of observations
#>   [,1]
#> 1   98
#> 2   54
#> 3   43
#> 4   48
#> 5   51
#> 6   38
#> 7   28
#> 8   11
coef(fit, matrix = TRUE)
#>             logitlink(P[Y>=2]) logitlink(P[Y>=3])
#> (Intercept)          -9.676093         -10.581725
#> let                   2.596807           2.596807
constraints(fit)  # Constraint matrices
#> $`(Intercept)`
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> $let
#>      [,1]
#> [1,]    1
#> [2,]    1
#> 
apply(fitted(fit), 1, which.max)  # Classification
#> 1 2 3 4 5 6 7 8 
#> 1 1 1 1 1 1 1 3 
apply(predict(fit, newdata = pneumo, type = "response"),
      1, which.max)  # Classification
#> 1 2 3 4 5 6 7 8 
#> 1 1 1 1 1 1 1 3 
R2latvar(fit)
#> [1] 0.5142885

# Check that the model is linear in let ----------------------
fit2 <- vgam(cbind(normal, mild, severe) ~ s(let, df = 2),
             cumulative(reverse = TRUE), data = pneumo)
if (FALSE) { # \dontrun{
 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), pneumo))
#> 
#> Call:
#> vglm(formula = cbind(normal, mild, severe) ~ let, family = cumulative(parallel = FALSE, 
#>     reverse = TRUE), data = pneumo)
#> 
#> 
#> Coefficients:
#> (Intercept):1 (Intercept):2         let:1         let:2 
#>     -9.593308    -11.104791      2.571300      2.743550 
#> 
#> Degrees of Freedom: 16 Total; 12 Residual
#> Residual deviance: 4.884404 
#> Log-likelihood: -25.01905 
pchisq(2 * (logLik(fit3) - logLik(fit)), df =
       length(coef(fit3)) - length(coef(fit)), lower.tail = FALSE)
#> [1] 0.7058849
lrtest(fit3, fit)  # More elegant
#> Likelihood ratio test
#> 
#> Model 1: cbind(normal, mild, severe) ~ let
#> Model 2: cbind(normal, mild, severe) ~ let
#>   #Df  LogLik Df  Chisq Pr(>Chisq)
#> 1  12 -25.019                     
#> 2  13 -25.090  1 0.1424     0.7059

# 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
#>                   symptoms
#> let                normal mild severe
#>   1.75785791755237     98    0      0
#>   2.70805020110221     51    2      1
#>   3.06805293513362     34    6      3
#>   3.31418600467253     35    5      8
#>   3.51154543883102     32   10      9
#>   3.67630067190708     23    7      8
#>   3.8286413964891      12    6     10
#>   3.94158180766969      4    2      5


(fit.long1 <- vglm(symptoms ~ let, data = pneumo.long, trace = TRUE,
                   cumulative(parallel = TRUE, reverse = TRUE)))
#> Iteration 1: deviance = 428.98474
#> Iteration 2: deviance = 411.73439
#> Iteration 3: deviance = 408.69065
#> Iteration 4: deviance = 408.54867
#> Iteration 5: deviance = 408.54833
#> Iteration 6: deviance = 408.54833
#> 
#> Call:
#> vglm(formula = symptoms ~ let, family = cumulative(parallel = TRUE, 
#>     reverse = TRUE), data = pneumo.long, trace = TRUE)
#> 
#> 
#> Coefficients:
#> (Intercept):1 (Intercept):2           let 
#>     -9.676092    -10.581724      2.596806 
#> 
#> Degrees of Freedom: 742 Total; 739 Residual
#> Residual deviance: 408.5483 
#> Log-likelihood: -204.2742 
coef(fit.long1, matrix = TRUE)  # cf. coef(fit, matrix = TRUE)
#>             logitlink(P[Y>=2]) logitlink(P[Y>=3])
#> (Intercept)          -9.676092         -10.581724
#> let                   2.596806           2.596806
# 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)
#> Iteration 1: deviance = 428.98474
#> Iteration 2: deviance = 411.73439
#> Iteration 3: deviance = 408.69065
#> Iteration 4: deviance = 408.54867
#> Iteration 5: deviance = 408.54833
#> Iteration 6: deviance = 408.54833
coef(fit.long2, matrix = TRUE)  # cf. coef(fit, matrix = TRUE)
#>             logitlink(P[Y>=2]) logitlink(P[Y>=3])
#> (Intercept)          -9.676092         -10.581724
#> let                   2.596806           2.596806