multgee: GEE Solver for Correlated Nominal or Ordinal Multinomial Responses

Installation

You can install the release version of multgee:

install.packages("multgee")

The source code for the release version of multgee is available on CRAN at:

• https://CRAN.R-project.org/package=multgee

Or you can install the development version of multgee:

# install.packages('devtools')
devtools::install_github("AnestisTouloumis/multgee")

The source code for the development version of multgee is available on github at:

• https://github.com/AnestisTouloumis/multgee

To use multgee, you should load the package as follows:

library("multgee")
#> Loading required package: gnm

Usage

This package provides a generalized estimating equations (GEE) solver for fitting marginal regression models with correlated nominal or ordinal multinomial responses based on a local odds ratios parameterization for the association structure (see Touloumis, Agresti and Kateri, 2013).

There are two core functions to fit GEE models for correlated multinomial responses:

• ordLORgee for an ordinal response scale. Options for the marginal model include cumulative link models or an adjacent categories logit model,
• nomLORgee for a nominal response scale. Currently, the only option is a marginal baseline category logit model.

The main arguments in both functions are:

• an optional data frame (data),
• a model formula (formula),
• a cluster identifier variable (id),
• an optional vector that identifies the order of the observations within each cluster (repeated).

The association structure among the correlated multinomial responses is expressed via marginalized local odds ratios (Touloumis et al., 2013). The estimating procedure for the local odds ratios can be summarized as follows: For each level pair of the repeated variable, the available responses are aggregated across clusters to form a square marginalized contingency table. Treating these tables as independent, an RC-G(1) type model is fitted in order to estimate the marginalized local odds ratios. The LORstr argument determines the form of the marginalized local odds ratios structure. Since the general RC-G(1) model is closely related to the family of association models, one can instead fit an association model to each of the marginalized contingency tables by setting LORem = "2way" in the core functions.

There are also five useful utility functions:

• confint for obtaining Wald–type confidence intervals for the regression parameters using the standard errors of the sandwich (method = "robust") or of the model–based (method = "naive") covariance matrix. The default option is the sandwich covariance matrix (method = "robust"),
• waldts for assessing the goodness-of-fit of two nested GEE models based on a Wald test statistic,
• vcov for obtaining the sandwich (method = "robust") or model–based (method = "naive") covariance matrix of the regression parameters,
• intrinsic.pars for assessing whether the underlying association structure does not change dramatically across the level pairs of repeated,
• gee_criteria for reporting commonly used criteria to select variables and/or association structure for GEE models.

Example

The following R code replicates the GEE analysis presented in Touloumis et al. (2013).

data("arthritis")
intrinsic.pars(y, arthritis, id, time, rscale = "ordinal")
#> [1] 0.6517843 0.9097341 0.9022272

The intrinsic parameters do not differ much. This suggests that the uniform local odds ratios structure might be a good approximation for the association pattern.

fitord <- ordLORgee(formula = y ~ factor(time) + factor(trt) + factor(baseline),
    data = arthritis, id = id, repeated = time)
summary(fitord)
#> GEE FOR ORDINAL MULTINOMIAL RESPONSES 
#> version 1.6.0 modified 2017-07-10 
#> 
#> Link : Cumulative logit 
#> 
#> Local Odds Ratios:
#> Structure:         category.exch
#> Model:             3way
#> 
#> call:
#> ordLORgee(formula = y ~ factor(time) + factor(trt) + factor(baseline), 
#>     data = arthritis, id = id, repeated = time)
#> 
#> Summary of residuals:
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -0.5176739 -0.2380475 -0.0737290 -0.0001443 -0.0066846  0.9933154 
#> 
#> Number of Iterations: 6 
#> 
#> Coefficients:
#>                   Estimate   san.se   san.z Pr(>|san.z|)    
#> beta10            -1.84003  0.38735 -4.7504      < 2e-16 ***
#> beta20             0.27712  0.34841  0.7954      0.42639    
#> beta30             2.24779  0.36509  6.1568      < 2e-16 ***
#> beta40             4.54824  0.41994 10.8307      < 2e-16 ***
#> factor(time)3     -0.00079  0.12178 -0.0065      0.99485    
#> factor(time)5     -0.36050  0.11413 -3.1586      0.00159 ** 
#> factor(trt)2      -0.50463  0.16725 -3.0173      0.00255 ** 
#> factor(baseline)2 -0.70291  0.37861 -1.8565      0.06338 .  
#> factor(baseline)3 -1.27558  0.35066 -3.6376      0.00028 ***
#> factor(baseline)4 -2.65579  0.41039 -6.4714      < 2e-16 ***
#> factor(baseline)5 -3.99555  0.53246 -7.5040      < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Local Odds Ratios Estimates:
#>        [,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]  [,8]  [,9] [,10] [,11] [,12]
#>  [1,] 0.000 0.000 0.000 0.000 1.919 1.919 1.919 1.919 2.484 2.484 2.484 2.484
#>  [2,] 0.000 0.000 0.000 0.000 1.919 1.919 1.919 1.919 2.484 2.484 2.484 2.484
#>  [3,] 0.000 0.000 0.000 0.000 1.919 1.919 1.919 1.919 2.484 2.484 2.484 2.484
#>  [4,] 0.000 0.000 0.000 0.000 1.919 1.919 1.919 1.919 2.484 2.484 2.484 2.484
#>  [5,] 1.919 1.919 1.919 1.919 0.000 0.000 0.000 0.000 2.465 2.465 2.465 2.465
#>  [6,] 1.919 1.919 1.919 1.919 0.000 0.000 0.000 0.000 2.465 2.465 2.465 2.465
#>  [7,] 1.919 1.919 1.919 1.919 0.000 0.000 0.000 0.000 2.465 2.465 2.465 2.465
#>  [8,] 1.919 1.919 1.919 1.919 0.000 0.000 0.000 0.000 2.465 2.465 2.465 2.465
#>  [9,] 2.484 2.484 2.484 2.484 2.465 2.465 2.465 2.465 0.000 0.000 0.000 0.000
#> [10,] 2.484 2.484 2.484 2.484 2.465 2.465 2.465 2.465 0.000 0.000 0.000 0.000
#> [11,] 2.484 2.484 2.484 2.484 2.465 2.465 2.465 2.465 0.000 0.000 0.000 0.000
#> [12,] 2.484 2.484 2.484 2.484 2.465 2.465 2.465 2.465 0.000 0.000 0.000 0.000
#> 
#> p-value of Null model: < 0.0001

The 95% Wald confidence intervals for the regression parameters are

confint(fitord)
#>                        2.5 %      97.5 %
#> beta10            -2.5992134 -1.08084855
#> beta20            -0.4057572  0.95999854
#> beta30             1.5322296  2.96335073
#> beta40             3.7251701  5.37130863
#> factor(time)3     -0.2394781  0.23790571
#> factor(time)5     -0.5841988 -0.13680277
#> factor(trt)2      -0.8324304 -0.17683500
#> factor(baseline)2 -1.4449754  0.03916401
#> factor(baseline)3 -1.9628588 -0.58829522
#> factor(baseline)4 -3.4601391 -1.85143755
#> factor(baseline)5 -5.0391534 -2.95195213

To illustrate model comparison, consider another model with age and sex as additional covariates:

fitord1 <- update(fitord, formula = . ~ . + age + factor(sex))
waldts(fitord, fitord1)
#> Goodness of Fit based on the Wald test 
#> 
#> Model under H_0: y ~ factor(time) + factor(trt) + factor(baseline)
#> Model under H_1: y ~ factor(time) + factor(trt) + factor(baseline) + age + factor(sex)
#> 
#> Wald Statistic = 3.8313, df = 2, p-value = 0.1472
gee_criteria(fitord, fitord1)
#>              QIC      CIC     RJC   QICu Parameters
#> fitord  1268.444 1268.375 11.0347 1.2779         11
#> fitord1 1279.759 1279.582 13.0885 0.9964         13

According to the Wald test, there is no evidence of no difference between the two models. The QICu criterion suggest that fitord should be preferred over fitord1.

Getting help

The statistical methods implemented in multgee are described in Touloumis et al. (2013). A detailed description of the functionality of multgee can be found in Touloumis (2015). Note that an updated version of this paper also serves as a vignette:

browseVignettes("multgee")

How to cite

To cite 'multgee' in publications, please use:

  Touloumis A. (2015). "R Package multgee: A Generalized Estimating
  Equations Solver for Multinomial Responses." _Journal of Statistical
  Software_, *64*(8), 1-14.
  <https://www.jstatsoft.org/index.php/jss/article/view/v064i08>.

A BibTeX entry for LaTeX users is

  @Article{,
    title = {{R} Package {multgee}: A Generalized Estimating Equations Solver for Multinomial Responses},
    author = {{Touloumis A.}},
    journal = {Journal of Statistical Software},
    year = {2015},
    volume = {64},
    number = {8},
    pages = {1-14},
    url = {https://www.jstatsoft.org/index.php/jss/article/view/v064i08},
  }

To cite the methodology implemented in 'multgee' in publications,
please use:

  Touloumis A., Agresti A., Kateri M. (2013). "GEE for multinomial
  responses using a local odds ratios parameterization." _Biometrics_,
  *69*(3), 633-640. <https://doi.org/10.1111/biom.12054>.

A BibTeX entry for LaTeX users is

  @Article{,
    title = {GEE for multinomial responses using a local odds ratios parameterization},
    author = {{Touloumis A.} and {Agresti A.} and {Kateri M.}},
    journal = {Biometrics},
    year = {2013},
    volume = {69},
    number = {3},
    pages = {633-640},
    url = {https://doi.org/10.1111/biom.12054},
  }

References

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

Touloumis, A., Agresti, A. and Kateri, M. (2013) GEE for Multinomial Responses Using a Local Odds Ratios Parameterization. Biometrics, 69, 633–640.