# Marginal Models For Correlated Ordinal Multinomial Responses

### Description

Solving the generalized estimating equations for correlated ordinal multinomial responses assuming a cumulative link model or an adjacent categories logit model for the marginal probabilities.

### Usage

1 2 3 4 5 | ```
ordLORgee(formula, data, id = id, repeated = NULL,
link = "logit", bstart = NULL, LORstr = "category.exch",
LORem = "3way", LORterm = NULL, add = 0, homogeneous = TRUE,
restricted = FALSE, control = LORgee.control(),
ipfp.ctrl = ipfp.control(), IM = "solve")
``` |

### Arguments

`formula` |
a formula expression as for other regression models for multinomial responses. An intercept term must be included. |

`data` |
an optional data frame containing the variables provided in |

`id` |
a vector that identifies the clusters. |

`repeated` |
an optional vector that identifies the order of observations within each cluster. |

`link` |
a character string that specifies the link function. Options include |

`bstart` |
a vector that includes an initial estimate for the marginal regression parameter vector. |

`LORstr` |
a character string that indicates the marginalized local odds ratios structure. Options include |

`LORem` |
a character string that indicates if the marginalized local odds ratios structure is estimated simultaneously ( |

`LORterm` |
a matrix that satisfies the user-defined local odds ratios structure. It is ignored unless |

`add` |
a positive constant to be added at each cell of the full marginalized contingency table in the presence of zero observed counts. |

`homogeneous` |
a logical that indicates homogeneous score parameters when |

`restricted` |
a logical that indicates monotone score parameters when |

`control` |
a vector that specifies the control variables for the GEE solver. |

`ipfp.ctrl` |
a vector that specifies the control variables for the function |

`IM` |
a character string that indicates the method used for inverting a matrix. Options include |

### Details

The `data`

must be provided in case level or equivalently in ‘long’ format. See details about the ‘long’ format in the function reshape.

A term of the form `offset(expression)`

is allowed in the right hand side of `formula`

.

The default set for the response categories is *\{1,…,J\}*, where *J>2* is the maximum observed response category. If otherwise, the function recodes the observed response categories onto this set.

The *J*-th response category is omitted.

The default set for the `id`

labels is *\{1,…,N\}*, where *N* is the sample size. If otherwise, the function recodes the given labels onto this set.

The argument `repeated`

can be ignored only when `data`

is written in such a way that the *t*-th observation in each cluster is recorded at the *t*-th measurement occasion. If this is not the case, then the user must provide `repeated`

. The suggested set for the levels of `repeated`

is *\{1,…,T\}*, where *T* is the number of observed levels. If otherwise, the function recodes the given levels onto this set.

The variables `id`

and `repeated`

do not need to be pre-sorted. Instead the function reshapes `data`

in an ascending order of `id`

and `repeated`

.

The fitted marginal cumulative link model is

*Pr(Y_{it}≤ j |x_{it})=F(β_{0j} +β^{'} x_{it})*

where *Y_{it}* is the *t*-th multinomial response for cluster *i*, *x_{it}* is the associated covariates vector, *F* is the cumulative distribution function determined by `link`

, *β_{j}* is the *j*-th response category specific intercept and *β* is the marginal regression parameter vector excluding intercepts.

The marginal adjacent categories logit model

*log \frac{Pr(Y_{it}=j |x_{it})}{Pr(Y_{it}=j+1 |x_{it})}=β_{0j} +β^{'} x_{it}*

is fitted if and only if `link="acl"`

. In contrast to a marginal cumulative link model, here the intercepts do not need to be monotone increasing.

The `LORterm`

argument must be an *L* x *J^2* matrix, where *L* is the number of level pairs of `repeated`

. These are ordered as *(1,2), (1,3),…,(1,T), (2,3),…,(T-1,T)* and the rows of `LORterm`

are supposed to preserve this order. Each row is assumed to contain the vectorized form of a probability table that satisfies the desired local odds ratios structure.

### Value

Returns an object of the class `"LORgee"`

. This has components:

`call` |
the matched call. |

`title` |
title for the GEE model. |

`version` |
the current version of the GEE solver. |

`link` |
the marginal link function. |

`local.odds.ratios` |
the marginalized local odds ratios structure variables. |

`terms` |
the |

`contrasts` |
the |

`nobs` |
the number of observations. |

`convergence` |
the values of the convergence variables. |

`coefficients` |
the estimated regression parameter vector of the marginal model. |

`linear.pred` |
the estimated linear predictor of the marginal regression model. The |

`fitted.values` |
the estimated fitted values of the marginal regression model. The |

`residuals` |
the residuals of the marginal regression model. The |

`y` |
the multinomial response variables. |

`id` |
the |

`max.id` |
the number of clusters. |

`clusz` |
the number of observations within each cluster. |

`robust.variance` |
the estimated "robust" covariance matrix. |

`naive.variance` |
the estimated "naive" or "model-based" covariance matrix. |

`xnames` |
the regression coefficients' symbolic names. |

`categories` |
the number of observed response categories. |

`occasions` |
the levels of the |

`LORgee.control` |
the control values for the GEE solver. |

`ipfp.control` |
the control values for the function |

`inverse.method` |
the method used for inverting matrices. |

`adding.constant` |
the value used for |

`pvalue` |
the p-value based on a Wald test that no covariates are statistically significant. |

Generic coef, summary, print, fitted and residuals methods are available. The `pvalue of the Null model`

corresponds to the hypothesis *H_0: β=0* based on the Wald test statistic.

### Author(s)

Anestis Touloumis

### References

Touloumis, A., Agresti, A. and Kateri, M. (2013). GEE for multinomial responses using a local odds ratios parameterization. *Biometrics*, **69**, 633-640.

Touloumis, A. (2015). R Package multgee: A Generalized Estimating Equations Solver for Multinomial Responses. *Journal of Statistical Software*, **64**, 1-14.

### See Also

For a nominal response scale use the function nomLORgee.

### Examples

1 2 3 4 5 |

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