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.
ordLORgee(formula = formula(data), data = parent.frame(), 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")Returns an object of the class "LORgee". This has components:
the matched call.
title for the GEE model.
the current version of the GEE solver.
the marginal link function.
the marginalized local odds ratios structure variables.
the terms structure describing the model.
the contrasts used for the factors.
the number of observations.
the values of the convergence variables.
the estimated regression parameter vector of the marginal model.
the estimated linear predictor of the marginal regression model. The \(j\)-th column corresponds to the \(j\)-th response category.
the estimated fitted values of the marginal regression model. The \(j\)-th column corresponds to the \(j\)-th response category.
the residuals of the marginal regression model. The \(j\)-th column corresponds to the \(j\)-th response category.
the multinomial response variables.
the id variable.
the number of clusters.
the number of observations within each cluster.
the estimated sandwich (robust) covariance matrix.
the estimated model-based (naive) covariance matrix.
the regression coefficients' symbolic names.
the number of observed response categories.
the levels of the repeated variable.
the control values for the GEE solver.
the control values for the function ipfp.
the method used for inverting matrices.
the value used for add.
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 \(H_0: \beta=0\) based on
the Wald test statistic.
a formula expression as for other regression models for multinomial responses. An intercept term must be included.
an optional data frame containing the variables provided in
formula, id and repeated.
a vector that identifies the clusters.
an optional vector that identifies the order of observations within each cluster.
a character string that specifies the link function. Options
include "logit", "probit", "cauchit", "cloglog"
or "acl".
a vector that includes an initial estimate for the marginal regression parameter vector.
a character string that indicates the marginalized local odds
ratios structure. Options include "independence", "uniform",
"category.exch", "time.exch", "RC" or "fixed".
a character string that indicates if the marginalized local
odds ratios structure is estimated simultaneously ("3way") or
independently at each level pair of repeated ("2way").
a matrix that satisfies the user-defined local odds ratios
structure. It is ignored unless LORstr="fixed".
a positive constant to be added at each cell of the full marginalized contingency table in the presence of zero observed counts.
a logical that indicates homogeneous score parameters
when LORstr="time.exch" or "RC".
a logical that indicates monotone score parameters when
LORstr="time.exch" or "RC".
a vector that specifies the control variables for the GEE solver.
a vector that specifies the control variables for the
function ipfp.
a character string that indicates the method used for inverting a
matrix. Options include "solve", "qr.solve" or
"cholesky".
Anestis Touloumis
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 \(\{1,\ldots,J\}\), 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 \(\{1,\ldots,N\}\), 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 \(\{1,\ldots,T\}\), 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 $$Pr(Y_{it}\le j
|x_{it})=F(\beta_{j0} +\beta^{'} x_{it})$$ 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, \(\beta_{j0}\) is the \(j\)-th
response category specific intercept and \(\beta\) is the marginal
regression parameter vector excluding intercepts.
The marginal adjacent categories logit model $$log \frac{Pr(Y_{it}=j
|x_{it})}{Pr(Y_{it}=j+1 |x_{it})}=\beta_{j0} +\beta^{'} x_{it}$$ 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 formulae are easier to read from either the Vignette or the Reference Manual (both available here).
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 \((1,2), (1,3),\ldots,(1,T), (2,3),\ldots,(T-1,T)\) 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.
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.
For a nominal response scale use the function nomLORgee.
data(arthritis)
intrinsic.pars(y, arthritis, id, time)
fitmod <- ordLORgee(formula = y ~ factor(time) + factor(trt) + factor(baseline),
data = arthritis, id = id, repeated = time, LORstr = "uniform")
summary(fitmod)
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