Simulating Longer Vectors of Correlated Binary Random Variables via Multinomial Sampling

The ability to simulate correlated binary data is important for sample size calculation and comparison of methods for analysis of clustered and longitudinal data with dichotomous outcomes. One available approach for simulating length n vectors of dichotomous random variables is to sample from the multinomial distribution of all possible length n permutations of zeros and ones. However, the multinomial sampling method has only been implemented in general form (without first making restrictive assumptions) for vectors of length 2 and 3, because specifying the multinomial distribution is very challenging for longer vectors. I overcome this difficulty by presenting an algorithm for simulating correlated binary data via multinomial sampling that can be easily applied to directly compute the multinomial distribution for any n. I demonstrate the approach to simulate vectors of length 4 and 8 in an assessment of power during the planning phases of a study and to assess the choice of working correlation structure in an analysis with generalized estimating equations

GEEQBOX: A MATLAB Toolbox for Generalized Estimating Equations and Quasi-Least Squares

The GEEQBOX toolbox analyzes correlated data via the method of generalized estimating equations (GEE) and quasi-least squares (QLS), an approach based on GEE that overcomes some limitations of GEE that have been noted in the literature. GEEQBOX is currently able to handle correlated data that follows a normal, Bernoulli or Poisson distribution, and that is assumed to have an AR(1), Markov, tri-diagonal, equicorrelated, unstructured or working independence correlation structure. This toolbox is for use with MATLAB.

%QLS SAS Macro: A SAS Macro for Analysis of Correlated Data Using Quasi-Least Squares

Quasi-least squares (QLS) is an alternative computational approach for estimation of the correlation parameter in the framework of generalized estimating equations (GEE). QLS overcomes some limitations of GEE that were discussed in Crowder (1995). In addition, it allows for easier implementation of some correlation structures that are not available for GEE. We describe a user written SAS macro called %QLS, and demonstrate application of our macro using a clinical trial example for the comparison of two treatments for a common toenail infection. %QLS also computes the lower and upper boundaries of the correlation parameter for analysis of longitudinal binary data that were described by Prentice (1988). Furthermore, it displays a warning message if the Prentice constraints are violated. This warning is not provided in existing GEE software packages and other packages that were recently developed for application of QLS (in Stata, MATLAB, and R). %QLS allows for analysis of continuous, binary, or count data with one of the following working correlation structures: the first-order autoregressive, equicorrelated, Markov, or tri-diagonal structures.

%QLS SAS Macro: A SAS macro for Analysis of Longitudinal Data Using Quasi-Least Squares .

Quasi-least squares (QLS) is an alternative computational approach for estimation of the correlation parameter in the framework of generalized estimating equations (GEE). QLS overcomes some limitations of GEE that were discussed in Crowder (Biometrika 82 (1995) 407-410). In addition, it allows for easier implementation of some correlation structures that are not available for GEE. We describe a user written SAS macro called %QLS, and demonstrate application of our macro using a clinical trial example for the comparison of two treatments for a common toenail infection. %QLS also computes the lower and upper boundaries of the correlation parameter for analysis of longitudinal binary data that were described by Prentice (Biometrics 44 (1988), 1033-1048). Furthermore, it displays a warning message if the Prentice constraints are violated; This warning is not provided in existing GEE software packages and other packages that were recently developed for application of QLS (in Stata, Matlab, and R). %QLS allows for analysis of normal, binary, or Poisson data with one of the following working correlation structures: the first-order autoregressive (AR(1)), equicorrelated, Markov, or tri-diagonal structures. Keywords: longitudinal data, generalized estimating equations, quasi-least squares, SAS

Implementation of quasi-least squares With the R package qlspack

Quasi-least squares (QLS) is an alternative method for estimating the correlation parameters within the framework of generalized estimating equations (GEE) that has two main advantages over the moment estimates that are typically applied for GEE: (1) It guarantees a consistent estimate of the correlation parameter and a positive definite estimated correlation matrix, for several correlation structures; and (2) It allows for easier implementation of some correlation structures that have not yet been implemented in the framework of GEE. Furthermore, because QLS is a method in the framework of GEE, existing software can be employed within the QLS algorithm for estimation of the correlation and regression parameters. In this manuscript we describe and demonstrate the user written package qlspack that allows for implementation of QLS in R software. Our package qlspack calls up the geepack package Yan (2002) and Halekoh et al. (2006) to update the estimate of the regression parameter at the current QLS estimate of the correlation parameter; hence, geepack related functions for standard error estimation can be used after implementing qlspack

GEEQBOX: A MATLAB Toolbox for Generalized Estimating Equations and Quasi-Least Squares

The GEEQBOX toolbox analyzes correlated data via the method of generalized estimating equations (GEE) and quasi-least squares (QLS), an approach based on GEE that overcomes some limitations of GEE that have been noted in the literature. GEEQBOX is currently able to handle correlated data that follows a normal, Bernoulli or Poisson distribution, and that is assumed to have an AR(1), Markov, tri-diagonal, equicorrelated, unstructured or working independence correlation structure. This toolbox is for use with MATLAB

Maximum Likelihood Based Analysis of Equally Spaced Longitudinal Count Data with Specified Marginal Means, First-order Antedependence, and Linear Conditional Expectations

This manuscript implements a maximum likelihood based approach that is appropriate for equally spaced longitudinal count data with over-dispersion, so that the variance of the outcome variable is larger than expected for the assumed Poisson distribution. We implement the proposed method in the analysis of two data sets and make comparisons with the semi-parametric generalized estimating equations (GEE) approach that incorrectly ignores the over-dispersion. The simulations demonstrate that the proposed method has better small sample efficiency than GEE. We also provide code in R that can be used to recreate the analysis results that we provide in this manuscript

On the designation of the patterned associations for longitudinal Bernoulli data: weight matrix versus true correlation structure?

Due to potential violation of standard constraints for the correlation for binary data, it has been argued recently that the working correlation matrix should be viewed as a weight matrix that should not be confused with the true correlation structure. We propose two arguments to support our view to the contrary for the first-order autoregressive AR(1) correlation matrix. First, we prove that the standard constraints are not unduly restrictive for the AR(1) structure that is plausible for longitudinal data; furthermore, for the logit link function the upper boundary value only depends on the regression parameter and the change in covariate values between successive measurements. In addition, for given marginal means and parameter $\alpha$, we provide a general proof that satisfaction of the standard constraints for consecutive marginal means will guarantee the existence of a compatible multivariate distribution with an AR(1) structure. The relative laxity of the standard constraints for the AR(1) structure coupled with the existence of a simple model that yields data with an AR(1) structure bolsters our view that for the AR(1) structure at least, it is appropriate to view this model as a correlation structure versus a weight matrix