Generating Correlated and/or Overdispersed Count Data: A SAS Implementation

Analysis of longitudinal count data has, for long, been done using a generalized linear mixed model (GLMM), in its Poisson-normal version, to account for correlation by specifying normal random effects. Univariate counts are often handled with the negativebinomial (NEGBIN) model taking into account overdispersion by use of gamma random effects. Inherently though, longitudinal count data commonly exhibit both features of correlation and overdispersion simultaneously, necessitating analysis methodology that can account for both. The introduction of the combined model (CM) by Molenberghs, Verbeke, and Demétrio (2007) and Molenberghs, Verbeke, Demétrio, and Vieira (2010) serves this purpose, not only for count data but for the general exponential family of distributions. Here, a Poisson model is specified as the parent distribution of the data with a normally distributed random effect at the subject or cluster level and/or a gamma distribution at observation level. The GLMM and NEGBIN model are special cases. Data can be simulated from (1) the general CM, with random effects, or, (2) its marginal version directly. This paper discusses an implementation of (1) in SAS software (SAS Inc. 2011). One needs to reflect on the mean of both the combined (hierarchical) and marginal models in order to generate correlated and/or overdispersed counts. A pre-specification of the desired marginal mean (in terms of covariates and marginal parameters), a marginal variance-covariance structure and the hierarchical mean (in terms of covariates and regression parameters) is required. The implied hierarchical parameters, the variance-covariance matrix of the random effects, and the variance-covariance matrix of the overdispersion part are then derived from which correlated Poisson data are generated. Sample calls of the SAS macro are presented as well as output

Medical conditions at enrollment do not impact efficacy and safety of the adjuvanted recombinant zoster vaccine:a pooled post-hoc analysis of two parallel randomized trials

In two pivotal efficacy studies (ZOE-50; ZOE-70), the adjuvanted recombinant zoster vaccine (RZV) demonstrated >90% efficacy against herpes zoster (HZ). Adults aged ≥50 or ≥70 years (ZOE-50 [NCT01165177]; ZOE-70 [NCT01165229]) were randomized to receive 2 doses of RZV or placebo 2 months apart. Vaccine efficacy and safety were evaluated post-hoc in the pooled (ZOE-50/70) population according to the number and type of selected medical conditions present at enrollment. At enrollment, 82.3% of RZV and 82.7% of placebo recipients reported ≥1 of the 15 selected medical conditions. Efficacy against HZ ranged from 84.5% (95% Confidence Interval [CI]: 46.4–97.1) in participants with respiratory disorders to 97.0% (95%CI: 82.3–99.9) in those with coronary heart disease. Moreover, efficacy remained >90% irrespective of the number of selected medical conditions reported by a participant. As indicated by the similarity of the point estimates, this post-hoc analysis suggests that RZV efficacy remains high in all selected medical conditions, as well as with increasing number of medical conditions. No safety concern was identified by the type or number of medical conditions present at enrollment

Pseudo-likelihood methodology for hierarchical count data

© 2014 Taylor & Francis Group, LLC. Generalized Estimating Equations (GEE) are a widespread tool for modeling correlated data, based on properly formulating a marginal regression function, combined with working assumptions about the correlation function. Should interest be placed in addition on the correlation function, then, apart from second-order GEE, pseudo-likelihood (PL) also provides an attractive alternative, especially in its pairwise form, where the covariance between each pair of the response vector is modeled as well. An elegant PL approach is formulated in this paper, based on a flexible bivariate Poisson model. The performance of the PL-method is studied, relative to GEE, using simulations. Data on repeated counts of epileptic seizures in a two-arm clinical trial are analyzed. A macro has been developed by the authors and made available on their web pages.peerreview_statement: The publishing and review policy for this title is described in its Aims & Scope. aims_and_scope_url: http://www.tandfonline.com/action/journalInformation?show=aimsScope&journalCode=lsta20status: publishe

Generating Correlated and/or Overdispersed Count Data: A SAS

Analysis of longitudinal count data has, for long, been done using a generalized linear mixed model (GLMM), in its Poisson-normal version, to account for correlation by specifying normal random effects. Univariate counts are often handled with the negativebinomial (NEGBIN) model taking into account overdispersion by use of gamma random effects. Inherently though, longitudinal count data commonly exhibit both features of correlation and overdispersion simultaneously, necessitating analysis methodology that can account for both. The introduction of the combined model (CM) by Molenberghs, Verbeke, and Demétrio (2007) and Molenberghs, Verbeke, Demétrio, and Vieira (2010) serves this purpose, not only for count data but for the general exponential family of distributions. Here, a Poisson model is specified as the parent distribution of the data with a normally distributed random effect at the subject or cluster level and/or a gamma distribution at observation level. The GLMM and NEGBIN model are special cases. Data can be simulated from (1) the general CM, with random effects, or, (2) its marginal version directly. This paper discusses an implementation of (1) in SAS software (SAS Inc. 2011). One needs to reflect on the mean of both the combined (hierarchical) and marginal models in order to generate correlated and/or overdispersed counts. A pre-specification of the desired marginal mean (in terms of covariates and marginal parameters), a marginal variance-covariance structure and the hierarchical mean (in terms of covariates and regression parameters) is required. The implied hierarchical parameters, the variance-covariance matrix of the random effects, and the variance-covariance matrix of the overdispersion part are then derived from which correlated Poisson data are generated. Sample calls of the SAS macro are presented as well as output

Second-order generalized estimating equations for correlated count data

© 2015, Springer-Verlag Berlin Heidelberg. Generalized estimating equations have been widely used in the analysis of correlated count data. Solving these equations yields consistent parameter estimates while the variance of the estimates is obtained from a sandwich estimator, thereby ensuring that, even with misspecification of the so-called working correlation matrix, one can draw valid inferences on the marginal mean parameters. That they allow misspecification of the working correlation structure, though, implies a limitation of these equations should scientific interest also be in the covariance or correlation structure. We propose herein an extension of these estimating equations such that, by incorporating the bivariate Poisson distribution, the variance-covariance matrix of the response vector can be properly modelled, which would permit inference thereon. A sandwich estimator is used for the standard errors, ensuring sound inference on the parameters estimated. Two applications are presented.status: publishe

Medical conditions at enrollment do not impact efficacy and safety of the adjuvanted recombinant zoster vaccine : a pooled post-hoc analysis of two parallel randomized trials

In two pivotal efficacy studies (ZOE-50; ZOE-70), the adjuvanted recombinant zoster vaccine (RZV) demonstrated >90% efficacy against herpes zoster (HZ). Adults aged >= 50 or >= 70 years (ZOE-50 [NCT01165177]; ZOE-70 [NCT01165229]) were randomized to receive 2 doses of RZV or placebo 2 months apart. Vaccine efficacy and safety were evaluated post-hoc in the pooled (ZOE-50/70) population according to the number and type of selected medical conditions present at enrollment. At enrollment, 82.3% of RZV and 82.7% of placebo recipients reported >= 1 of the 15 selected medical conditions. Efficacy against HZ ranged from 84.5% (95% Confidence Interval [CI]: 46.4-97.1) in participants with respiratory disorders to 97.0% (95%CI: 82.3-99.9) in those with coronary heart disease. Moreover, efficacy remained >90% irrespective of the number of selected medical conditions reported by a participant. As indicated by the similarity of the point estimates, this post-hoc analysis suggests that RZV efficacy remains high in all selected medical conditions, as well as with increasing number of medical conditions. No safety concern was identified by the type or number of medical conditions present at enrollment