# sratio: Ordinal Regression with Stopping Ratios In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

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

## Arguments

 `link` Link function applied to the M stopping ratio probabilities. See `Links` for more choices. `parallel` A logical, or formula specifying which terms have equal/unequal coefficients. `reverse` Logical. By default, the stopping 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=j+1|Y<=j+1]) will be used. `zero` Can be an integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2,...,M}. The default value means none are modelled as intercept-only terms. `whitespace` See `CommonVGAMffArguments` for information.

## 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 (see `cratio`) and stopping ratios. Continuation 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.

Simonoff, J. S. (2003). Analyzing Categorical Data, New York, USA: Springer-Verlag.

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/.

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

## Examples

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

### Example output

```Loading required package: stats4

Call:
vglm(formula = cbind(normal, mild, severe) ~ let, family = sratio(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
(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

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   0.9905587   0.9814886
2   0.9203740   0.8538279
3   0.8336534   0.7169229
4   0.7389235   0.5885286
5   0.6415824   0.4749569
6   0.5497802   0.3816118
7   0.4616120   0.3023041
8   0.3974671   0.2500163
```

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