A robust Bayesian mixed effects approach for zero inflated and highly skewed longitudinal count data emanating from the zero inflated discrete Weibull distribution

Burger, Divan Aristo; Schall, Robert; Ferreira, Johan; Chen, Ding‐Geng

doi:10.1002/sim.8475

Cited by 9 publications

(20 citation statements)

References 34 publications

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“…The conventional discrete Weibull distribution's key properties, including its dispersion, zero‐inflation, and heavy‐tail indices, can be found in Luyts et al 8 Burger et al 11 present an extension of the DW distribution by adding a zero‐inflation parameter; hence, the so‐called ZIDW distribution. The remainder of this section provides a summary of the ZIDW distribution.…”

Section: Conventional Zidw Distributionmentioning

confidence: 99%

“…The PMF of the ZIDW distribution, using the parameterization of Burger et al 11 for a given count

y_{i}

, is written as:

f (y_{i} | μ, ϕ, π) = π I (y_{i} = 0) + (1 - π) [\exp (- \log (2) {[\frac{y_{i}}{μ}]}^{ϕ}) - \exp (- \log (2) {[\frac{y_{i} + 1}{μ}]}^{ϕ})]

Here,

ϕ

and

π

are, respectively, the shape parameter and zero‐inflation probability, and

μ

is the median of the conventional discrete Weibull distribution. The median of the

y_{i}

under the ZIDW distribution is given by:

M (y_{i}) = λ = {(\frac{\log [\frac{0.5}{1 - π}]}{\log (0.5)})}^{\frac{1}{ϕ}} μ

Recently, Burger et al 11 suggested the use of the log‐link function to model

μ

. Taking an approach similar to that of Preisser et al, 13

λ

's modeling is suggeste...…”

Section: Marginal Zero‐inflated Count Distributionsmentioning

confidence: 99%

“…The ZINB model of Burger et al 20 was applied to the CFU dataset of Dawson et al 25 However, the ZINB distribution also models the bactericidal activity of TB treatments based on the mean count. To model the median count, replacing the ZINB distribution with the ZIDW distribution can be considered 11,26 …”

Section: Motivating Data and Previous Workmentioning

confidence: 99%

“…Limited literature exists on quantile regression modeling of zero‐inflated count data. Examples of quantile regression models for zero‐inflated count data include the two‐part quantile regression model of King and Song 10 and the linear mixed‐effects zero‐inflated discrete Weibull (ZIDW) model of Burger et al 11 demonstrated in a data contamination simulation study that the ZIDW distribution better accommodates excessive skewness and outliers in the data than the ZINB distribution. Therefore, the ZIDW model is more robust to severe skewness in the data than the ZINB model and is convenient to implement since the ZIDW distribution's quantile function is available in closed form.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear mixed‐effects modeling of longitudinal count data: Bayesian inference about median counts based on the marginal zero‐inflated discrete Weibull distribution

Burger

Lesaffre

2021

Statistics in Medicine

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This article proposes a Bayesian regression model for nonlinear zero-inflated longitudinal count data that models the median count as an alternative to the mean count. The nonlinear model generalizes a recently introduced linear mixed-effects model based on the zero-inflated discrete Weibull (ZIDW) distribution. The ZIDW distribution is more robust to severe skewness in the data than conventional zero-inflated count distributions such as the zero-inflated negative binomial (ZINB) distribution. Moreover, the ZIDW distribution is attractive because of its convenience to model the median counts given its closed-form quantile function. The median is a more robust measure of central tendency than the mean when the data, for instance, zero-inflated counts, are right-skewed.In an application of the model we consider a biphasic mixed-effects model consisting of an intercept term and two slope terms. Conventionally, the ZIDW model separately specifies the predictors for the zero-inflation probability and the counting process's median count. In our application, the two latent class interpretations are not clinically plausible. Therefore, we propose a marginal ZIDW model that directly models the biphasic median counts marginally. We also consider the marginal ZINB model to make inferences about the nonlinear mean counts over time. Our simulation study shows that the models have good properties in terms of accuracy and confidence interval coverage. K E Y W O R D SBayesian, marginal zero-inflated discrete Weibull, marginal zero-inflated negative binomial, median counts, nonlinear mixed-effects 5078

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Section: Conventional Zidw Distributionmentioning

confidence: 99%

“…The PMF of the ZIDW distribution, using the parameterization of Burger et al 11 for a given count

y_{i}

, is written as:

f (y_{i} | μ, ϕ, π) = π I (y_{i} = 0) + (1 - π) [\exp (- \log (2) {[\frac{y_{i}}{μ}]}^{ϕ}) - \exp (- \log (2) {[\frac{y_{i} + 1}{μ}]}^{ϕ})]

Here,

ϕ

and

π

are, respectively, the shape parameter and zero‐inflation probability, and

μ

is the median of the conventional discrete Weibull distribution. The median of the

y_{i}

under the ZIDW distribution is given by:

M (y_{i}) = λ = {(\frac{\log [\frac{0.5}{1 - π}]}{\log (0.5)})}^{\frac{1}{ϕ}} μ

Recently, Burger et al 11 suggested the use of the log‐link function to model

μ

. Taking an approach similar to that of Preisser et al, 13

λ

's modeling is suggeste...…”

Section: Marginal Zero‐inflated Count Distributionsmentioning

confidence: 99%

Section: Motivating Data and Previous Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nonlinear mixed‐effects modeling of longitudinal count data: Bayesian inference about median counts based on the marginal zero‐inflated discrete Weibull distribution

Burger

Lesaffre

2021

Statistics in Medicine

Self Cite

View full text Add to dashboard Cite

show abstract

“…Their approach was used to identify potential disparities in outpatient psychiatric service use among white, black, and Latino homeless adults receiving case management. Burger, Schall, Ferreira, and Chen (2020) developed a Bayesian mixed effects approach for highly skewed longitudinal count data that are ZI. In particular, they developed this approach assuming a ZIDW regression model and a matrix generalized half‐ t prior distribution for the covariance matrix of the random effects, instead of the commonly used Wishart prior.…”

Section: Zero Inflation In Correlated Count Settingsmentioning

confidence: 99%

Zero‐inflatedmodeling partII:Zero‐inflatedmodels for complex data structures

Young

Roemmele

Shi

2020

WIREs Computational Stats

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The prequel to this review provided an extensive treatment of classic zero‐inflated count regression models where a univariate discrete distribution is used for the count regression component of the model. The treatment of zero inflation beyond the classic univariate count regression setting has seen a substantial increase in recent years. This second review paper surveys some of this recent literature and focuses on important developments in handling zero inflation for correlated count settings, discrete time series models, spatial models, and multivariate models. We discuss some of the available computational tools for performing estimation in these settings, while again highlighting the diverse data problems that have been addressed using these methods. This article is categorized under: Statistical Models > Multivariate Models Statistical Models > Generalized Linear Models Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory

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Modeling dragonfly population data with a Bayesian bivariate geometric mixed-effects model

Oppen

Milder-Mulderij²,

Brochard³

et al. 2022

Journal of Applied Statistics

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We develop a generalized linear mixed model (GLMM) for bivariate count responses for statistically analyzing dragonfly population data from the Northern Netherlands. The populations of the threatened dragonfly species Aeshna viridis were counted in the years 2015–2018 at 17 different locations (ponds and ditches). Two different widely applied population size measures were used to quantify the population sizes, namely the number of found exoskeletons (‘exuviae’) and the number of spotted egg-laying females were counted. Since both measures (responses) led to many zero counts but also feature very large counts, our GLMM model builds on a zero-inflated bivariate geometric (ZIBGe) distribution, for which we show that it can be easily parameterized in terms of a correlation parameter and its two marginal medians. We model the medians with linear combinations of fixed (environmental covariates) and random (location-specific intercepts) effects. Modeling the medians yields a decreased sensitivity to overly large counts; in particular, in light of growing marginal zero inflation rates. Because of the relatively small sample size ( n = 114) we follow a Bayesian modeling approach and use Metropolis-Hastings Markov Chain Monte Carlo (MCMC) simulations for generating posterior samples.

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A robust Bayesian mixed effects approach for zero inflated and highly skewed longitudinal count data emanating from the zero inflated discrete Weibull distribution

Cited by 9 publications

References 34 publications

Nonlinear mixed‐effects modeling of longitudinal count data: Bayesian inference about median counts based on the marginal zero‐inflated discrete Weibull distribution

Nonlinear mixed‐effects modeling of longitudinal count data: Bayesian inference about median counts based on the marginal zero‐inflated discrete Weibull distribution

Zero‐inflatedmodeling partII:Zero‐inflatedmodels for complex data structures

Modeling dragonfly population data with a Bayesian bivariate geometric mixed-effects model

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