# cratio: Ordinal Regression with Continuation Ratios In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2``` ```cratio(link = "logitlink", parallel = FALSE, reverse = FALSE, zero = NULL, whitespace = FALSE) ```

## Arguments

 `link` Link function applied to the M continuation ratio probabilities. See `Links` for more choices. `parallel` A logical, or formula specifying which terms have equal/unequal coefficients. `reverse` Logical. By default, the continuation ratios used are eta_j = logit(P[Y>j|Y>=j]) for j=1,…,M. If `reverse` is `TRUE`, then eta_j=logit(P[Y

## Details

In this help file the response Y is assumed to be a factor with ordered values 1,2,…,M+1, so that M is the number of linear/additive predictors eta_j.

There are a number of definitions for the continuation ratio in the literature. To make life easier, in the VGAM package, we use continuation ratios and stopping ratios (see `sratio`). Stopping ratios deal with quantities such as `logitlink(P[Y=j|Y>=j])`.

## 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 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.

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

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.

Thomas W. Yee

## References

Agresti, A. (2013). Categorical Data Analysis, 3rd ed. Hoboken, NJ, USA: Wiley.

McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.

Yee, T. W. (2010). The VGAM package for categorical data analysis. Journal of Statistical Software, 32, 1–34. https://www.jstatsoft.org/v32/i10/.

`sratio`, `acat`, `cumulative`, `multinomial`, `margeff`, `pneumo`, `logitlink`, `probitlink`, `clogloglink`, `cauchitlink`.

## Examples

 ```1 2 3 4 5 6 7 8``` ```pneumo <- transform(pneumo, let = log(exposure.time)) (fit <- vglm(cbind(normal, mild, severe) ~ let, cratio(parallel = TRUE), data = pneumo)) coef(fit, matrix = TRUE) constraints(fit) predict(fit) predict(fit, untransform = TRUE) margeff(fit) ```

### Example output

```Loading required package: stats4

Call:
vglm(formula = cbind(normal, mild, severe) ~ let, family = cratio(parallel = TRUE),
data = pneumo)

Coefficients:
(Intercept):1 (Intercept):2           let
-8.733797     -8.051302      2.321359

Degrees of Freedom: 16 Total; 13 Residual
Residual deviance: 7.626763
Log-likelihood: -26.39023
logit(P[Y>1|Y>=1]) logit(P[Y>2|Y>=2])
(Intercept)          -8.733797          -8.051302
let                   2.321359           2.321359
\$`(Intercept)`
[,1] [,2]
[1,]    1    0
[2,]    0    1

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

logit(P[Y>1|Y>=1]) logit(P[Y>2|Y>=2])
1         -4.6531774         -3.9706824
2         -2.4474398         -1.7649448
3         -1.6117442         -0.9292491
4         -1.0403809         -0.3578859
5         -0.5822388          0.1002563
6         -0.1997827          0.4827124
7          0.1538548          0.8363499
8          0.4160301          1.0985252
P[Y>1|Y>=1] P[Y>2|Y>=2]
1 0.009441281  0.01851142
2 0.079625970  0.14617212
3 0.166346597  0.28307708
4 0.261076501  0.41147144
5 0.358417612  0.52504310
6 0.450219786  0.61838815
7 0.538388014  0.69769591
8 0.602532898  0.74998366
, , 1

normal        mild        severe
(Intercept)  0.08167972 -0.07878663 -0.0028930983
let         -0.02170968  0.02090961  0.0008000745

, , 2

normal       mild      severe
(Intercept)  0.6400622 -0.4664909 -0.17357136
let         -0.1701224  0.1221861  0.04793632

, , 3

normal       mild     severe
(Intercept)  1.2111629 -0.5965056 -0.6146573
let         -0.3219154  0.1524215  0.1694939

, , 4

normal       mild    severe
(Intercept)  1.6848854 -0.4825758 -1.202310
let         -0.4478263  0.1167953  0.331031

, , 5

normal        mild     severe
(Intercept)  2.0083753 -0.23426941 -1.7741059
let         -0.5338068  0.04605304  0.4877538

, , 6

normal        mild     severe
(Intercept)  2.1618063  0.03043788 -2.1922442
let         -0.5745873 -0.02736296  0.6019503

, , 7

normal        mild     severe
(Intercept)  2.170579  0.25808932 -2.4286681
let         -0.576919 -0.08919657  0.6661155

, , 8

normal       mild     severe
(Intercept)  2.0916309  0.3866929 -2.4783238
let         -0.5559354 -0.1232739  0.6792092
```

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