# cumulative: Ordinal Regression with Cumulative Probabilities In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2``` ```cumulative(link = "logitlink", parallel = FALSE, reverse = FALSE, multiple.responses = FALSE, whitespace = FALSE) ```

## Arguments

 `link` Link function applied to the J cumulative probabilities. See `Links` for more choices, e.g., for the cumulative `probitlink`/`clogloglink`/`cauchitlink`/... 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.
 `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) are used. This should be set to `TRUE` for `link=` `gordlink`, `pordlink`, `nbordlink`. For these links the cutpoints must be an increasing sequence; if `reverse = FALSE` for then the cutpoints must be an decreasing sequence. `multiple.responses` Logical. Multiple responses? 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., multiple responses. A suitable matrix can be obtained from `Cut`.
 `whitespace` See `CommonVGAMffArguments` for information.

## Details

In this help file the response Y is assumed to be a factor with ordered values 1,2,…,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<=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 (`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`.

## 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.

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.

Dobson, A. J. and Barnett, A. (2008). An Introduction to Generalized Linear Models, 3rd ed. Boca Raton, FL, USA: 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. https://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.

`propodds`, `R2latvar`, `ordsup`, `prplot`, `margeff`, `acat`, `cratio`, `sratio`, `multinomial`, `pneumo`, `Links`, `hdeff.vglm`, `logitlink`, `probitlink`, `clogloglink`, `cauchitlink`, `gordlink`, `pordlink`, `nbordlink`, `logistic1`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50``` ```# 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 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) # 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) ```

### Example output

```Loading required package: stats4

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
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
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
[,1]
1   98
2   54
3   43
4   48
5   51
6   38
7   28
8   11
logit(P[Y>=2]) logit(P[Y>=3])
(Intercept)      -9.676093     -10.581725
let               2.596807       2.596807
\$`(Intercept)`
[,1] [,2]
[1,]    1    0
[2,]    0    1

\$let
[,1]
[1,]    1
[2,]    1

1 2 3 4 5 6 7 8
1 1 1 1 1 1 1 3
1 2 3 4 5 6 7 8
1 1 1 1 1 1 1 3
 0.5142885

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
 0.7058849
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
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
VGLM    linear loop  1 :  deviance = 428.98474
VGLM    linear loop  2 :  deviance = 411.73439
VGLM    linear loop  3 :  deviance = 408.69065
VGLM    linear loop  4 :  deviance = 408.54867
VGLM    linear loop  5 :  deviance = 408.54833
VGLM    linear loop  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
logit(P[Y>=2]) logit(P[Y>=3])
(Intercept)      -9.676092     -10.581724
let               2.596806       2.596806
VGLM    linear loop  1 :  deviance = 428.98474
VGLM    linear loop  2 :  deviance = 411.73439
VGLM    linear loop  3 :  deviance = 408.69065
VGLM    linear loop  4 :  deviance = 408.54867
VGLM    linear loop  5 :  deviance = 408.54833
VGLM    linear loop  6 :  deviance = 408.54833
logit(P[Y>=2]) logit(P[Y>=3])
(Intercept)      -9.676092     -10.581724
let               2.596806       2.596806
```

VGAM documentation built on Jan. 16, 2021, 5:21 p.m.