glmmTMB Balances Speed and Flexibility Among Packages for Zero-inflated Generalized Linear Mixed Modeling

Count data can be analyzed using generalized linear mixed models when observations are correlated in ways that require random effects. However, count data are often zero-inflated, containing more zeros than would be expected from the typical error distributions. We present a new package, glmmTMB, and compare it to other R packages that fit zero-inflated mixed models. The glmmTMB package fits many types of GLMMs and extensions, including models with continuously distributed responses, but here we focus on count responses. glmmTMB is faster than glmmADMB, MCMCglmm, and brms, and more flexible than INLA and mgcv for zero-inflated modeling. One unique feature of glmmTMB (among packages that fit zero-inflated mixed models) is its ability to estimate the Conway-Maxwell-Poisson distribution parameterized by the mean. Overall, its most appealing features for new users may be the combination of speed, flexibility, and its interface’s similarity to lme4.

Introduction

Observed response variables are often in the form of discrete count data, e.g., the number of times that owl nestlings beg for food (Roulin and Bersier 2007), counts of salamanders in streams (Price et al. 2016), or counts of parasite eggs in fecal samples of sheep (Denwood et al. 2008). These counts are often analyzed using generalized linear models (GLMs) and their extensions (Wilson and Grenfell 1997; O’Hara and Kotze 2010). GLMs quantify how expected counts change as a function of predictor variables, e.g., nestlings change their behavior depending on which parent they interact with (Roulin and Bersier 2007), salamander abundance decreases in streams affected by coal mining (Price et al. 2016), and helminth infection intensity in sheep decreases in response to treatment with anthelmintic drugs (Wang et al. 2017). Repeated measurements on the same individual, at the same location, or observations during the same time period are often correlated; this correlation can be accounted for using random effects in generalized linear mixed models (GLMMs; (Bolker et al. 2009; Bolker 2015)).

These types of count data are commonly modeled with GLMs and GLMMs using either Poisson or negative binomial distributions. For the Poisson distribution, the variance is equal to the mean. When data are overdispersed — meaning the variance is larger than the mean — they are often instead modeled using the negative binomial distribution, which can be defined as a mixture of Poisson distributions with Gamma-distributed rates. Overdispersion can also be addressed by adding a random effect with one level for each observation, i.e., a log-normal Poisson distribution (Elston et al. 2001; Hadfield 2010; Harrison 2014, 2015). Ignoring overdispersion causes confidence intervals to be too narrow and inflates the rate of false positives in statistical tests (Rhodes 2015). When data are either over- or underdispersed, they can be modeled with the lesser-known, Conway-Maxwell-Poisson distribution (Shmueli et al. 2005; Barriga and Louzada 2014; Lynch et al. 2014). Depending on the dispersion, the upper tail of the Conway-Maxwell-Poisson distribution can be either longer or shorter than that of the Poisson (Sellers and Shmueli 2010). With two parameters, the Conway-Maxwell-Poisson is a generalization of the Poisson distribution and, depending on the dispersion parameter, it also includes the Bernoulli and geometric distributions as special cases (Sellers and Shmueli 2010). Among other generalizations, the Sichel and Delaporte distributions (Stasinopoulos et al. 2017) provide flexibility in skewness in addition to dispersion.

For these distributions, the expected number of zeros decreases as the mean increases. However, when multiple processes underlie the observed counts, the counts can contain many zeros even if the mean is much greater than zero. For example, an observation of a stream with zero salamanders could be a “structural” zero due to the stream being uninhabitable due to mining waste, or a “sampling” zero due to the combination of a low mean (due to poor ecological suitability and/or low detectability) and sampling variation (Price et al. 2016). Zero-inflated (more broadly zero-altered) GLMs allow us to model count data using a mixture of a Poisson, negative binomial, or Conway-Maxwell-Poisson distribution and a structural zero component. Models that ignore zero-inflation, or attempt to handle it in the same way as simple overdispersion, can yield biased parameter estimates (Harrison 2014).

In this article, we outline the R packages available for fitting models to count data while introducing glmmTMB. We assume that the reader already has a basic understanding of GLMs (Buckley 2015), GLMMs (Bolker 2015), and zero-altered models (Zeileis et al. 2008; Harrison 2014)).

Several R packages are available for fitting zero-inflated models: pscl, INLA, MCMCglmm, glmmADMB, mgcv, brms, and gamlss (Table 1; (Zeileis et al. 2008; HRue et al. 2009; Hadfield 2010; Skaug et al. 2012; Wood et al. 2016; Bürkner 2017; Stasinopoulos et al. 2017)). The widely-used pscl package can fit zero-inflated and hurdle GLMs with predictor variables on the zero-inflation using maximum likelihood estimation (MLE: (Zeileis et al. 2008)). For example, pscl can be used to test the hypothesis that sheep fecal egg counts depend on age and structural zeros depend on genotype. However, pscl cannot model the correlation within sampling units caused by repeated samples; this requires random effects. Omitting random effects and thereby ignoring correlation makes statistical inference anti-conservative (Bolker et al. 2009; Bolker 2015). Several other packages have similar capabilities for fitting zero-inflated GLMs (flexmix, MXM, VGAM: (Grün and Leisch 2008; Lagani et al. 2017; Yee 2017)), but in this paper we focus on packages that can also estimate random effects. One such package is glmmADMB which can fit zero-inflated GLMMs (Skaug et al. 2012). However, glmmADMB cannot fit models where the degree of zero-inflation varies across observational units; thus, it is only appropriate for models where all observational units have an equal probability of producing a structural zero. INLA has the same limitation as glmmADMB (HRue et al. 2009). The mgcv package can only fit zero-inflated GLMMs with predictors of zero-inflation when using a Poisson distribution (Wood et al. 2016). The MCMCglmm and brms packages can fit zero-inflated GLMMs with predictors of zero-inflation, but they are relatively slow (as we will show) because they rely on Markov chain Monte Carlo (MCMC) sampling (Hadfield 2010; Bürkner 2017). gamlss is a flexible package that fits generalized additive models with predictors on all parameters of a distribution; its scope includes several zero-inflated and zero-altered distributions (Stasinopoulos et al. 2017).

The list of features documented here is not exhaustive. It should be appreciated that brms, gamlss and MCMCglmm have additional features that go beyond the scope of zero-inflated GLMMs (Hadfield 2010; Bürkner 2017; Stasinopoulos et al. 2017). We focus on the process of fitting models, largely neglecting questions of statistical frameworks (frequentist vs. Bayesian) or post-fitting procedures such as inference and prediction. For example, having MCMC samples from a fitted model allows a wide range of inferential and predictive procedures.

Here we introduce a new R package, glmmTMB, that estimates GLMs, GLMMs and extensions of GLMMs including zero-inflated and hurdle GLMMs using ML. The ability to fit these types of models quickly and using a single package will make it easier to perform model selection. We focus on zero-inflated counts, but note that there are many other distributions available in glmmTMB, including continuous distributions. We demonstrate the package using two examples without going into details of the reasons why a user would want to fit these models. We use an example of salamander abundance to show how to fit and compare zero-inflated and hurdle GLMMs and then how to extract results from a model. We then use variations of the salamander data to compare how the timings of different packages scale with the number of observations and random effect levels. We use a classic example of owl nestling behavior to compare the timing and parameter estimates from glmmTMB with other R packages.

Table 1: Features of packages that are used for modeling zero-inflated count data. This table only contains packages that can at least fit zero-inflated Poisson GLMMs. lme4 is omitted because it can only estimate zero-inflation when wrapped in an expectation maximization algorithm (Bolker et al. 2013). nbinom2 and nbinom1 are negative binomial distributions in which the variance increases quadratically and linearly (respectively) with the mean. compois is the Conway-Maxwell-Poisson distribution. PIG is the Poisson inverse Gaussian distribution. Notes: 1 We restrict this to mean a single function call, rather than two separate models. 2 Not available with zero-inflation. 3 Conway-Maxwell-Poisson distribution parameterized by the mean. 4 Weights are often used to reduce the influence of some observations over others, e.g., (Gurevitch and Hedges 1999); additionally, glmmTMB’s dispersion formula can be used to model heteroskedasticity. 5 Offsets can be implemented using priors. 6 See vignette("covstruct") for details. 7 Some packages allow for correlation across random effects from different components of the model (e.g., conditional and zero-inflation). 8 Here, we mean that MCMC samples can be obtained from an estimated model (i.e., joint samples from the full distribution, conditional on the data) whether it is estimated using MCMC sampling or MLE. In the case of MLE, flat priors are used for MCMC sampling and chains are initiated at the ML estimate. 9 See vignette("mcmc") for details. 10 This feature is available and widely used e.g., (Chevin et al. 2015), but apparently unsupported by any formal evaluation. 11 We exclude the possibility that the correlation structure of random effects makes a model mathematically equivalent to a multivariate response. 12 Here, we mean generalized additive modeling with automatic smoothness selection. However, a spline can be included in any of these methods using the bs or ns functions from the splines package.

Feature	glmmTMB	glmmADMB	MCMCglmm	brms	INLA	mgcv	gamlss
hurdle models\(^1\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)
predictors of zero-inflation	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)			\(\checkmark\)
predictors of dispersion	\(\checkmark\)			\(\checkmark\)			\(\checkmark\)
zero-truncated distributions	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)
nbinom2 distribution	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)\(^2\)	\(\checkmark\)
nbinom1 distribution	\(\checkmark\)	\(\checkmark\)					\(\checkmark\)\(^2\)
compois distribution	\(\checkmark\)\(^3\)
Delaporte distribution							\(\checkmark\)\(^2\)
Sichel distribution							\(\checkmark\)
geometric distribution			\(\checkmark\)	\(\checkmark\)			\(\checkmark\)\(^2\)
PIG distribution							\(\checkmark\)
weights	\(\checkmark\)\(^4\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
offsets	\(\checkmark\)	\(\checkmark\)	\(^5\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
various RE structures	\(\checkmark\)\(^6\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
RE cor across components \(^7\)			\(\checkmark\)	\(\checkmark\)
MLE	\(\checkmark\)	\(\checkmark\)				\(\checkmark\)	\(\checkmark\)
MCMC samples\(^8\)	\(\checkmark\)\(^9\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\) \(^{10}\)	\(\checkmark\)
multivariate responses \(^{11}\)			\(\checkmark\)	\(\checkmark\)
GAM \(^{12}\)				\(\checkmark\)		\(\checkmark\)	\(\checkmark\)

Implementation of glmmTMB

The design goal of glmmTMB is to extend the flexibility of GLMMs in R while maintaining a familiar interface. To maximize flexibility and speed, glmmTMB’s estimation is done using the TMB package (Kristensen et al. 2016), but users need not be familiar with TMB. We based glmmTMB’s interface (e.g., formula syntax) on the lme4 package — one of the most widely used R packages for fitting GLMMs (Bates et al. 2015). Like lme4, glmmTMB uses MLE and the Laplace approximation to integrate over random effects; unlike lme4, glmmTMB does not have the alternative options of doing restricted maximum likelihood (REML) estimation nor using Gauss-Hermite quadrature to integrate over random effects (Bolker et al. 2009; Bolker 2015). The Laplace approximation may perform poorly when there is little information available on each random effect level (Ogden 2015). REML may be added to glmmTMB in the future. The underlying implementation using TMB is a fundamental difference compared to lme4 and provides glmmTMB with a speed advantage when estimating non-Gaussian models (Figures 1 and 2) and gives it greater flexibility in the classes of distributions it can fit (Table 1).

The flexibility of glmmTMB enables users to fit and compare many varieties of models with assurance that the log-likelihood values are calculated in a consistent way. Comparing likelihoods of models fit by multiple packages must be done carefully because some packages drop constants from the log-likelihood calculations while others do not.

A glmmTMB model has four main components: a conditional model formula, a distribution for the conditional model, a dispersion model formula, and a zero-inflation model formula. Simple GLMs and GLMMs can be fit using the conditional model while leaving the zero-inflation and dispersion formulas at their default values. The mean of the conditional model is specified using a two-sided formula with the response variable on the left and predictors on the right, potentially including random effects and offsets. This formula uses the same syntax as lme4. For example, if salamander counts vary by species (spp) and vary randomly by site, then the formula for the dependence of mean count on species could be

count ~ spp + (1 | site)

The distribution around the mean of the conditional model is specified using the argument family. For the types of count data described in the introduction, the distribution will typically be either Poisson or negative binomial. The Conway-Maxwell-Poisson is a less commonly known distribution for count data that is flexible enough to fit both over- and underdispersed data. Following the standard of glmmTMB described above, the Conway-Maxwell-Poisson distribution (family = compois) is parameterized with the mean (Huang 2017), which differs from the COMPoissonReg package(Sellers et al. 2017). The Poisson, Conway-Maxwell-Poisson, and negative binomial distributions use a log link by default, but other links can be specified as in family = poisson(link = "identity"). glmmTMB provides two parameterizations of the negative binomial which differ in the dependence of the variance (\(\sigma^2\)) on the mean (\(\mu\)). For family = nbinom1, the variance increases linearly with the mean as \(\sigma^2 = \mu(1+\alpha)\), with \(\alpha>0\); for family = nbinom2, the variance increases quadratically with the mean as \(\sigma^2 = \mu(1+\mu/\theta)\), with \(\theta>0\) (Hardin and Hilbe 2007). For the Conway-Maxwell-Poisson distribution, there is no closed form equation for the variance (Huang 2017).

With the default dispersion model (dispformula = ~ 1), the dispersion parameter (e.g., \(\alpha\) or \(\theta\) for the negative binomial distribution) is identical for each observation. Alternatively, the dispersion parameter can vary with fixed effects; in this case, the dispersion model uses a log link. The dispersion model can be used to account for heteroskedasticity. For example, if the response is more variable (relative to the mean) as the year progresses, then a model with either negative binomial distribution might use the one-sided formula dispformula = ~ DOY where DOY is the day of the year. When the same variables are in the conditional and dispersion models, the mean-variance relationship can be manipulated, but this could potentially lead to non-convergence issues. A description of the dispersion parameter for each distribution can be accessed by typing ?sigma.glmmTMB in R.

The zero-inflation model describes the probability of observing an extra (i.e., structural) zero that is not generated by the conditional model. Zero-inflation creates an extra point mass of zeros in the response distribution; the overall distribution is a mixture of the conditional model and zero-inflation model (Lambert 1992; Rhodes 2015). The zero-inflation probability is bounded between zero and one by using a logit link. For example, if salamanders emerged seasonally at each site, such that a structural zero could occur either because a site was contaminated or because it was visited too early in the season, then the model could include the one-sided formula ziformula = ~ DOY. The probability of producing a structural zero can be modeled as equal for all observations with ziformula = ~ 1. In glmmTMB, it is possible to include random effects in the conditional and zero-inflation models, but not the dispersion model.

Installation of glmmTMB

The package is available from The Comprehensive R Archive Network (CRAN) via the command install.packages("glmmTMB"). The current version is 0.2.0. Development versions are available from GitHub and can be installed using devtools (Wickham and Chang 2017). Current details for installing development versions should be accessed on the GitHub page https://github.com/glmmTMB/glmmTMB.

Examples and benchmarks

To illustrate how to use glmmTMB and to compare it to other packages, we applied it to two data sets that are distributed with glmmTMB. Additional code and graphs for these examples can be found in Appendicies A and B.

Salamander data

We demonstrate how to use features of glmmTMB to do model selection and output model results using a data set of the abundance of salamanders (Figure 3). They were observed four times at 23 sites in streams, some of which were impacted by coal mining; multiple species and life stages of salamanders were observed. The data set contains covariates that may affect the habitat suitability of a site and the ability of researchers to capture salamanders that inhabit the site (Price et al. 2015, 2016). Price et al. (2016) analyzed the data using a Bayesian model with an ecological and a sampling component; abundance was estimated with a hurdle Poisson model and then observations were modeled as binomial samples from the abundance.

Model fitting and selection

We fit GLMMs, zero-inflated GLMMs, and hurdle models to the salamander data with Poisson, Conway-Maxwell-Poisson, and negative binomial distributions on the conditional model. For simplicity, we neglected some possible covariates. As a null model, we assumed that counts varied by species (spp) and randomly by site (site). We fit models where the mean count additionally depended on mining status (mined). Our full zero-inflated GLMMs allowed both the conditional and zero-inflation models to differ between mined and unmined sites. Full zero-inflated and hurdle negative binomial GLMMs were fit using the following commands (respectively) :

zinb = glmmTMB(count~spp * mined + (1|site), zi=~spp * mined, 
        data=Salamanders, family=nbinom2)
hnb = glmmTMB(count~spp * mined + (1|site), zi=~spp * mined, 
        data=Salamanders, family=truncated_nbinom2)

As is generally the case for model formulas in R, the * indicates an interaction plus main effects. We used Akaike information criteria (AIC) to compare all models via the AICtab function from the bbmle package (Bolker and Team 2017). For convenience, glmmTMB reports the log-likelihood of unconverged models as NA and version 1.0.19 of bbmle puts these models at the bottom of AIC tables. The code for fitting these models and doing model selection is presented in Appendix A.

Of the models we considered, the most parsimonious was a Conway-Maxwell-Poisson GLMM that allowed counts to vary with species, mining, and their interaction. It did not include zero-inflation, as is common in abundance models (Warton 2005). Two zero-inflated negative binomial and one zero-inflated Conway-Maxwell-Poisson model did not converge. Failed convergence is typically caused by trying to estimate parameters for which the data do not contain information. Models that do not converge should not be considered in model comparison. General model convergence issues are discussed in vignette("troubleshooting",package="glmmTMB").

Model summary

The summary of simple GLMMs from glmmTMB is modeled on the familiar output format of lme4. To demonstrate the extra output from zero-inflation and dispersion models, we present the summary from a more complicated model.

glmmTMB(count ~ mined + (1|site), zi=~mined , disp=~DOY, Salamanders, family=nbinom2)

Following the arguments in the function call above, this model allows the conditional mean to depend on whether or not a site was mined and to vary randomly by site. It allows the number of structural (i.e., extra) zeros to depend on mining. Additionally, it allows the dispersion parameter to depend on the day of the year. This model can be represented by the following set of equations

\[\begin{aligned} \mu &=& E(\texttt{count}|u,\texttt{NSZ}) = \exp( \beta_0 + \beta_\texttt{minedno} + u ),\label{ref:mu}\\ \end{aligned} \tag{1}\]

\[\begin{aligned} u &\sim& N\left(0,\sigma_u^2\right),\label{ref:u}\\ \end{aligned} \tag{2}\]

\[\begin{aligned} \sigma^2 &=& \hbox{Var}(\texttt{count}|u,\texttt{NSZ}) = \mu(1+\mu/\theta),\label{ref:sigma2}\\ \end{aligned} \tag{3}\]

\[\begin{aligned} \hbox{logit}(p) &=& \beta_0^{(\texttt{zi})}+\beta_\texttt{minedno}^{(\texttt{zi})},\label{ref:p_zi}\\ \end{aligned} \tag{4}\]

\[\begin{aligned} \log(\theta) &=& \beta_0^\texttt{(disp)} + \beta_\texttt{DOY}^\texttt{(disp)}\cdot \texttt{DOY}\label{ref:theta}, \end{aligned} \tag{5}\]

where \(u\) is a site specific random effect, NSZ is the event “non-structural zero”, \(p=1-\Pr(\texttt{NSZ})\) is the zero inflation probability, and \(\beta\)’s are regression coefficients with subscript denoting covariate/level (with 0 denoting intercept).

summary(glmmTMB(count~mined+(1|site), zi=~mined , disp=~DOY, Salamanders, family=nbinom2))

#>  Family: nbinom2  ( log )
#> Formula:          count ~ mined + (1 | site)
#> Zero inflation:         ~mined
#> Dispersion:             ~DOY
#> Data: Salamanders
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>     1735     1767     -861     1721      637 
#> 
#> Random effects:
#> 
#> Conditional model:
#>  Groups Name        Variance Std.Dev.
#>  site   (Intercept) 0.134    0.366   
#> Number of obs: 644, groups:  site, 23
#> 
#> Conditional model:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)   -0.540      0.376   -1.44     0.15    
#> minedno        1.424      0.365    3.90 0.000098 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Zero-inflation model:
#>             Estimate Std. Error z value Pr(>|z|)   
#> (Intercept)    0.256      0.487    0.53   0.5988   
#> minedno       -2.244      0.745   -3.01   0.0026 **
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Dispersion model:
#>             Estimate Std. Error z value Pr(>|z|)  
#> (Intercept)   0.0278     0.3177    0.09     0.93  
#> DOY          -0.3947     0.1537   -2.57     0.01 *
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

This summary can be broken down into five sections. The top section is a general overview containing a description of the model specification (Family, Formula, Zero inflation, Dispersion, Data) and resulting information criteria. The information criteria can only be compared to models fitted by packages that, like, glmmTMB, compute the full form of the log-likelihood without dropping constant terms. The second section describes the variability of the Random effects. In this model, we only had random effects in the conditional model (equation (1)), but random effects from the zero-inflation model (equation (4)) could also appear here. The estimated variance 0.134 is \(\sigma_u^2\) in equation (2). The third section describes the coefficients of the Conditional model (\(\beta_0\) and \(\beta_\texttt{minedno}\) in equation (1)) including Wald \(Z\) statistics and \(p\)-values. Apart from the intercept, the estimates are all contrasts as is standard in regression models. This model has a log link as stated in the top line of the summary. The fourth section describes the Zero-inflation model similarly to the Conditional model except that this model has a logit link. The zero-inflation model estimates the probability of an extra zero such that a positive contrast indicates a higher chance of absence (e.g. minedno < 0 means fewer absences in sites unaffected by mining); this is the opposite of the conditional model where a positive contrast indicates a higher abundance (e.g., minedno > 0 means higher abundances in sites unaffected by mining). The estimates in this section correspond to \(\beta_0^{(\texttt{zi})}\) and \(\beta_\texttt{minedno}^{(\texttt{zi})}\) in equation (4). The last section provides estimated coefficients from the Dispersion model (equation (5)), which uses a log link to keep the dispersion parameter \(\theta\) positive. In contrast, a model with the default (simple) dispersion model would report the single dispersion parameter on the natural (rather than log) scale. To interpret the dispersion parameters of any distribution, see ?sigma.glmmTMB.

The current version of the summary function does not display uncertainty estimates for the random effects nor for single dispersion parameters, but confidence intervals can be calculated using the confint function. See ?confint.glmmTMB for details. All confidence intervals produced by the current version of the confint function are Wald intervals, based on the standard errors calculated using the delta method for the parameter on the scale of the internal parameterization (which varies by family).

Additional model output using the predict and simulate functions from glmmTMB is demonstrated in appendix A (Figures 4, 5, 6, and 7).

Timing comparisons

Because glmmTMB is the only package that can fit Conway-Maxwell-Poisson GLMMs, it was not possible to do benchmarking with the most parsimonious model. Therefore, for benchmarking, we used the second best model (\(\Delta\) AIC=0.5 ) which substituted a negative binomial response for the Conway-Maxwell-Poisson response. We measured the time required to fit this model using multiple packages. We performed three sets of timing benchmarks: (1) on simulated data with the same structure as the original salamander data, (2) on the original data replicated to create more observations per random effect level and the same number of random effect levels, (3) on simulated data with increasing numbers of random effect levels and the same number of observations per random effect level. Benchmarks were run in parallel using parLapply on a high performance computing cluster with 12 cores. This performance should match running on a single core. We used default values for all packages except with glmmADMB which required an additional argument (extra.args="-ndi 1000000") to allocate additional memory. By default, brms runs four MCMC chains while MCMCglmm runs one, which greatly affects their estimation time. However, it would be simple to speed up fitting of brms models by running the chains in parallel. For Bayesian methods, the important aspect of timing is sampling efficiency (minimum effective samples per unit time, (Bürkner 2017)), but this is not compatible with the MLE methods, so we limit our presentation of the timings of the Bayesian methods.

Benchmarking showed that fitting the negative binomial model to simulated data with the same structure as the original data was, on average, equally fast in glmmTMB and INLA, 14 times slower with glmmADMB, 29 times slower with lme4, and 190 times slower with brms. mgcv fit the model the fastest, taking 0.03 times as long as glmmTMB. gamlss took 0.24 times as long as glmmTMB. With increasing numbers of observations, the estimation times of all packages appeared to follow power law functions (Figure 1). For simulated data sets with increasing numbers of random effect levels, estimation time increased as a power-law function for all packages except INLA which had estimation times that accelerated (Figure 2). The speed of glmmTMB for models with more random effect levels is due to the sparseness handling by TMB. Benchmarking nuances such as memory usage and how timings scale with model complexity could be investigated in more detail in future studies.

Figure 1: The Salamander data set was replicated by 1, 2, 4, 6, 8, and 10 times to create larger data sets. The time required to fit the same model using functions glmmadmb,glmmTMB,inla, glmer.nb, and gam was recorded. That model can be represented as glmmTMB(count ~ spp * mined + (1 | site), Salamanders, family="nbinom2"). All models had the same number of parameters including random effect levels. Lines represent linear models fit on the log-log scale. With increasing numbers of observations (\(n\)), estimation times increased as a power-law function (\(n^x\)) with exponents (\(x\)) reported next to model names.

Figure 2: Data sets with increasing numbers of levels of the random effect were simulated based on a negative binomial model fit to the salamander data, glmmTMB(count ~ spp * mined + (1 | site), Salamanders, family="nbinom2"). The time required to fit the same model using functions glmmadmb, glmmTMB, inla, glmer.nb, and gam was recorded. Each simulated data set had the same number of observations per random effect level — the same ratio as in the original data. Lines represent linear models fit on the log-log scale.With increasing numbers of random effect levels (\(n\)), estimation times increased as a power-law function (\(n^x\)), except for INLA which accelerates. The exponents (\(x\)) are reported here as labels over the lines.

To further compare R packages for fitting zero-inflated GLMMs, we analyzed counts of begging behavior by owl nestlings. The full analyses can be found in Appendix B. This example previously appeared in (Zuur et al. 2009) and (Bolker et al. 2013) and was originally published by (Roulin and Bersier 2007). We compared the estimates of fixed effects and the amount of time required for fitting the same model in INLA, MCMCglmm, glmmADMB, mgcv, and brms (HRue et al. 2009; Hadfield 2010; Skaug et al. 2012; Wood et al. 2016; Bürkner 2017). For brms and MCMCglmm, we used the default number of iterations, burn-in samples, and thinning. In each package, we fit zero-inflated Poisson models with six fixed effects, one random effect; we also accounted for overdispersion, although sometimes (of necessity) in slightly different ways with different packages (e.g., negative binomial vs. log-normal-Poisson models). We allowed zero-inflation to vary with food treatment and vary randomly with nest. See Appendix B for details of these methods, including code.

Estimates and confidence (or credible) intervals (CI) from brms, mgcv, MCMCglmm, and INLA were nearly identical to those of glmmTMB, when running the Bayesian models with flat priors (Figures 8 and 9).

Conclusions

We have introduced an R package that can quickly estimate a variety of models including GLMs, GLMMs, zero-inflated GLMMs, and hurdle models. By providing this flexibility in a single package, we reduce the need for researchers to learn multiple packages. Another benefit is that models estimated with a single package can be compared using likelihood-based methods including information criteria. Using information criteria to select a model for the salamander data, we found that (among the models we considered) zero-inflation did not improve the fit; we expect that this will be a common result. While glmmTMB allows users to easily fit complicated models, a maximally complex model might not be necessary and might not converge, as we saw here. Other packages have many of the features implemented in glmmTMB, but none have the ability to fit Conway-Maxwell-Poisson GLMMs. Overall, glmmTMB is a very flexible package for modeling count data with zero-inflated GLMMs while still ranking highly in speed comparisons.

Acknowledgements

Thanks to S Price for providing the data on salamanders and for helpful comments on the manuscript. Thanks to P-C Bürkner, J Hadfield, M Bekker-Nielsen Dunbar, and two anonymous reviewers for helpful comments on the manuscript. Thanks to A Zuur for providing the data on owl nestlings. Thanks to G Simpson for advice on capabilities of the mgcv package. This work was supported by grants from the Swiss National Science Foundation (#IZ320Z0_161670) and from the AD Model Builder Foundation to MEB.

1 Appendix A: Salamander example comparing GLMMs, zero-inflated GLMMs, and hurdle models using glmmTMB

In this appendix, we reanalyze counts of salamanders in streams. Repeated samples of salamanders were taken at 23 sites. Some of the sites were affected by mountain top removal coal mining. The data was originally published in (Price et al. 2016) and was acquired from Dryad (Price et al. 2015). These analyses are intended to be a simple demonstration of how to use some features of the glmmTMB package, so we do not attempt to fit all of the models that could be reasonable to try with the covariates that were collected.

Figure 3: Observed numbers of salamanders. Histograms show count data split into separate panels for each salamander species or life stage. Each panel contains two overlaid histograms in which color represents whether the site was affected by mining.

Poisson models

The syntax for fitting GLMMs with glmmTMB is quite similar to using glmer. In the first model, the formula, count ~ spp + (1 | site), says that counts depend on species and vary randomly by site. We also pass it the data frame, Salamanders, and specify a Poisson distribution using the family argument. glmmTMB assumes that we want a log link with the Poisson distribution because that’s the standard.

pm0 = glmmTMB(count~spp + (1|site), Salamanders, family=poisson)
pm1 = glmmTMB(count~spp + mined + (1|site), Salamanders, family=poisson)
pm2 = glmmTMB(count~spp * mined + (1|site), Salamanders, family=poisson)

Conway-Maxwell-Poisson models

To fit Conway-Maxwell-Poisson models, we use family=compois instead of poisson.

cmpm0 = glmmTMB(count~spp + (1|site), Salamanders, family=compois)
cmpm1 = glmmTMB(count~spp + mined + (1|site), Salamanders, family=compois)
cmpm2 = glmmTMB(count~spp * mined + (1|site), Salamanders, family=compois)

Negative binomial models

nbm0 = glmmTMB(count~spp + (1|site), Salamanders, family=nbinom2)
nbm1 = glmmTMB(count~spp + mined + (1|site), Salamanders, family=nbinom2)
nbm2 = glmmTMB(count~spp * mined + (1|site), Salamanders, family=nbinom2)

Unlike the Poisson, the negative binomial distribution has a dispersion parameter. If we expected the counts to become more dispersed (relative to the mean) as the year progresses, then we could use the dispersion formula to model how the dispersion changes with the day of the year (DOY) using disp= ~ DOY.

nbdm0 = glmmTMB(count~spp + (1|site), disp=~DOY, Salamanders, family=nbinom2)
nbdm1 = glmmTMB(count~spp + mined + (1|site), disp=~DOY, Salamanders, family=nbinom2)
nbdm2 = glmmTMB(count~spp * mined + (1|site), disp=~DOY, Salamanders, family=nbinom2)

Zero-inflated models

To fit zero-inflated models, we use the ziformula argument, or glmmTMB will also recognize zi. This is a formula that describes how the probability of an extra zero (i.e., structural zero) will vary with predictors. In this example, we might assume that absences will at least vary by species (spp), so we write zi= ~ spp. This formula only has a right side because the left side is always the probability of having a structural zero in the response that was specified in the first formula. The zero-inflation probability is always modeled with a logit link to keep it between 0 and 1.

Zero-inflation can be used with any of the distributions in glmmTMB, so we compare the same conditional and zero-inflation models with Poisson, Conway-Maxwell-Poisson, and negative binomial distributions.

Warning messages tell us that zicmpm3, zinbm0, and zinbm1 do not converge. The convergence warning refers to vignette("troubleshooting"); this vignette will expand as advice for troubleshooting convergence issues develops. It seems that zicmpm3 is overparameterized, but the problem with zinbm0 and zinbm1 is that they only have mined in the conditional model and not the zero-inflation model, so that they do not agree well with the data. Plotting the data and thinking carefully about the models should help to avoid convergence issues.

zipm0 = glmmTMB(count~spp +(1|site), zi=~spp, Salamanders, family=poisson)
zipm1 = glmmTMB(count~spp + mined +(1|site), zi=~spp, Salamanders, family=poisson)
zipm2 = glmmTMB(count~spp + mined +(1|site), zi=~spp + mined, Salamanders, family=poisson)
zipm3 = glmmTMB(count~spp * mined +(1|site), zi=~spp * mined, Salamanders, family=poisson)

zicmpm0 = glmmTMB(count~spp +(1|site), zi=~spp, Salamanders, family=compois)
zicmpm1 = glmmTMB(count~spp + mined +(1|site), zi=~spp, Salamanders, family=compois)
zicmpm2 = glmmTMB(count~spp + mined +(1|site), zi=~spp + mined, Salamanders, family=compois)
zicmpm3 = glmmTMB(count~spp * mined +(1|site), zi=~spp * mined, Salamanders, family=compois)

#> Warning in fitTMB(TMBStruc): Model convergence problem; non-positive-
#> definite Hessian matrix. See vignette('troubleshooting')

zinbm0 = glmmTMB(count~spp +(1|site), zi=~spp, Salamanders, family=nbinom2)

#> Warning in fitTMB(TMBStruc): Model convergence problem; non-positive-
#> definite Hessian matrix. See vignette('troubleshooting')

zinbm1 = glmmTMB(count~spp + mined +(1|site), zi=~spp, Salamanders, family=nbinom2)

#> Warning in fitTMB(TMBStruc): Model convergence problem; non-positive-
#> definite Hessian matrix. See vignette('troubleshooting')

zinbm2 = glmmTMB(count~spp + mined +(1|site), zi=~spp + mined, Salamanders, family=nbinom2)
zinbm3 = glmmTMB(count~spp * mined +(1|site), zi=~spp * mined, Salamanders, family=nbinom2)

#> Warning in fitTMB(TMBStruc): Model convergence problem; singular
#> convergence (7). See vignette('troubleshooting')

Hurdle models

We can also fit hurdle models in a single model by using a truncated distribution for the conditional model and adding zero-inflation.

hpm0 = glmmTMB(count~spp + (1|site), zi=~spp, Salamanders, family=truncated_poisson)
hpm1 = glmmTMB(count~spp + mined + (1|site), zi=~spp + mined, Salamanders, 
                family=truncated_poisson)
hpm2 = glmmTMB(count~spp * mined + (1|site), zi=~spp + mined, Salamanders, 
                family=truncated_poisson)
hnbm0 = glmmTMB(count~spp + (1|site), zi=~spp, Salamanders, family=truncated_nbinom2)
hnbm1 = glmmTMB(count~spp + mined + (1|site), zi=~spp + mined, Salamanders, 
                family=truncated_nbinom2)
hnbm2 = glmmTMB(count~spp * mined + (1|site), zi=~spp + mined, Salamanders, 
                family=truncated_nbinom2)

Model comparison using AIC

We can use AICtab to compare all the GLMMs, including zero-inflated and hurdle models. Here, to save space, we only output the AICtable for the top four and bottom four models. The most parsimonious model has a Conway-Maxwell-Poisson distribution with effects of species, mining, and their interaction.

AICtab(pm0, pm1, pm2,
       cmpm0, cmpm1, cmpm2, 
       nbm0, nbm1, nbm2, 
       nbdm0, nbdm1, nbdm2, 
       zipm0, zipm1, zipm2, zipm3,
       zicmpm0, zicmpm1, zicmpm2, zicmpm3,
       zinbm0, zinbm1, zinbm2, zinbm3,
       hpm0, hpm1, hpm2, 
       hnbm0, hnbm1, hnbm2)

The top of the table

model	df	dAIC
cmpm2	16	0.00
nbm2	16	0.48
nbdm2	17	1.99
zicmpm2	18	2.09
zinbm3	30	6.80

and the bottom

model	df	dAIC
hpm0	15	314
pm0	8	330
zicmpm3	30	NA
zinbm0	16	NA
zinbm1	17	NA

The log-likelihood of the unconverged models is reported as NA so that these models appear at the end of the AIC table. The negative log-likelihood could be extracted with zinbm1$fit$objective if it was needed.

Plotting model results

There are many decisions to make about marginalizing over or conditioning on the random effects. See discussion at https://cran.r-project.org/web/packages/merTools/vignettes/Using_predictInterval.html.

For demonstration purposes, we plot results from the top zero-inflated model zinbm3.

Quick and dirty plot

It’s easiest to see the pattern by using the predict function. To avoid marginalizing over or conditioning on random effects, we can refit the best model without the random effect of site; however, this is not ideal because it ignores the correlation within sites. We present a more rigorous version next.

The predict function has a parameter zitype that specifies whether you want predictions from the conditional model, the zero-inflation model, or the expected response that combines both parts of the model.

zinbm3FE = glmmTMB(count~spp * mined, zi=~spp * mined, Salamanders, family=nbinom2)
newdata0 = newdata = unique(Salamanders[,c("mined","spp")])
temp = predict(zinbm3FE, newdata, se.fit=TRUE, zitype="response")
newdata$predFE = temp$fit
newdata$predFE.min = temp$fit-1.96*temp$se.fit
newdata$predFE.max = temp$fit+1.96*temp$se.fit

real=ddply(Salamanders, ~site+spp+mined, summarize, m=mean(count))
  
ggplot(newdata, aes(spp, predFE, colour=mined))+geom_point()+
  geom_errorbar(aes(ymin=predFE.min, ymax=predFE.max))+
  geom_point(data=real, aes(x=spp, y=m) )+
  ylab("Average abundance \n including presences and absences")+
  xlab("Species")

Figure 4: Estimated abundance ignoring correlation. Points represent site-specific average counts. Error bars represent the 95% Wald-type confidence intervals for the predicted average count.

Alternative prediction method

We can predict at the population mode, by setting the random effects to zero.

X.cond = model.matrix(lme4::nobars(formula(zinbm3)[-2]), newdata0)
beta.cond = fixef(zinbm3)$cond
pred.cond = X.cond %*% beta.cond

ziformula = zinbm3$modelInfo$allForm$ziformula #$
X.zi = model.matrix(lme4::nobars(ziformula), newdata0)
beta.zi = fixef(zinbm3)$zi
pred.zi = X.zi %*% beta.zi

These are estimates of the linear predictors (i.e., predictions on the link scale: logit(prob) and log(cond)), not the predictions themselves. The easiest thing to do for the point estimates of the unconditional count (ucount) is to transform to the response scale and multiply:

pred.ucount = exp(pred.cond)*(1-plogis(pred.zi))

For the standard errors/confidence intervals, we could use posterior predictive simulations (i.e., draw multivariate normal samples from the parameter for the fixed effects). This conditions on/ignores uncertainty in the random-effect parameters.

library(MASS)
set.seed(101)
pred.condpar.psim = mvrnorm(1000,mu=beta.cond,Sigma=vcov(zinbm3)$cond)
pred.cond.psim = X.cond %*% t(pred.condpar.psim)
pred.zipar.psim = mvrnorm(1000,mu=beta.zi,Sigma=vcov(zinbm3)$zi)
pred.zi.psim = X.zi %*% t(pred.zipar.psim)
pred.ucount.psim = exp(pred.cond.psim)*(1-plogis(pred.zi.psim))
ci.ucount = t(apply(pred.ucount.psim,1,quantile,c(0.025,0.975)))
ci.ucount = data.frame(ci.ucount)
names(ci.ucount) = c("ucount.low","ucount.high")
pred.ucount = data.frame(newdata0, pred.ucount, ci.ucount)

These predicted counts should be close to the median counts, so we plot them together to compare.

real.count = ddply(Salamanders, ~spp+mined, summarize, m=median(count), mu=mean(count))
ggplot(pred.ucount, aes(x=spp, y=pred.ucount, colour=mined))+geom_point(shape=1, size=2)+
  geom_errorbar(aes(ymin=ucount.low, ymax=ucount.high))+
  geom_point(data=real.count,  aes(x=spp, y=m, colour=mined), shape=0, size=2)+
  geom_point(data=real.count,  aes(x=spp, y=mu, colour=mined), shape=5, size=2)+
  ylab("Abundance \n including presences and absences")+
  xlab("Species")

Figure 5: Estimated abundance at mode. Circles represent predicted unconditional counts at the mode (i.e., site effect = 0) and error bars represent the 95% confidence intervals for that mode. Squares represent the observed median and diamonds represent observed means calculated across samples and sites. In this highly skewed data, the mode is closer to the mean than the median.

Simulating from a fitted model

We could also examine the distribution of simulated values from the best fitted model. For this we use the function simulate.glmmTMB. This function works for zero-inflated and hurdle models as well as less complex models.

sims=simulate(nbm2, seed = 1, nsim = 1000)

This function returns a list of vectors. The list has one element for each simulation (nsim) and the vectors are the same shape as our response variable.

simdatlist=lapply(sims, function(count){ 
  cbind(count, Salamanders[,c('site', 'mined', 'spp')])
})
simdatsums=lapply(simdatlist, function(x){
  ddply(x, ~spp+mined, summarize, 
        absence=mean(count==0),
        mu=mean(count))
})
ssd=do.call(rbind, simdatsums)

Then we can plot them with the observations summarized in the same way.

real = ddply(Salamanders, ~spp+mined, summarize, 
             absence=mean(count==0),
             mu=mean(count))
ggplot(ssd, aes(x=absence, color=mined))+
  geom_density(adjust=4)+
  facet_wrap(~spp)+
  geom_point(data=real, aes(x=absence, y=1, color=mined), size=2)+
  xlab("Probability that salamanders are not observed")+ylab(NULL)

Figure 6: Simulated zero counts. Each panel represents a different species or life stage of a species. Densities are values from 1000 data sets simulated from our best fit model. Points represent the observed data.

We can see that this model does a good job of capturing the observed zero counts.

ggplot(ssd, aes(x=mu, color=mined))+
  geom_density(adjust=4)+
  facet_wrap(~spp)+
  geom_point(data=real, aes(x=mu, y=.5, color=mined), size=2)+
  xlab("Abundance including presences and absences")+ylab(NULL)

Figure 7: Simulated unconditional abundances. Each panel represents a different species or life stage of a species. Densities are values from 1000 data sets simulated from our best fit model. Points represent the observed data.

2 Appendix B: Compare zero-inflated mixed models across R packages

In this appendix, we analyze counts of begging behavior by owl nestlings. This example previously appeared in similar forms (Zuur et al. 2009) and (Bolker et al. 2013); the data were originally published by (Roulin and Bersier 2007). The response variable is the number of calls from chicks (NCalls) in a nest. Two changes from the example published in (Bolker et al. 2013) are that (1) we use an observation-level random effect to account for overdispersion; (2) instead of assuming that the number of calls is strictly proportional to brood size (i.e., using an offset of log(brood size)), we fit the model with log(brood size) as a predictor, equivalent to assuming that \(\textrm{calls} \propto (\textrm{brood size})^\gamma\), with \(\gamma\) not necessarily equal to 1.

Since nests were repeatedly measured, Nest is included as a random effect; observation-level random effects are incorporated to allow for overdispersion (Elston et al. 2001; Hadfield 2010). Covariates of interest include the sex of the parent visiting the nest (SexParent), whether the chicks were satiated or not (FoodTreatment), and the timing of the parent’s arrival (ArrivalTime).

Preliminaries

Load packages

library(glmmTMB)
library(glmmADMB)
library(MCMCglmm)
library(brms)
library(INLA)
library(mgcv)
library(broom) #for tidy devtools::install_github("bbolker/broom")
library(plyr)
library(dplyr) #tidyverse
library(ggplot2); theme_set(theme_bw())
library(ggstance)#for position_dodgev

Data organization and helper functions (hidden)

data(Owls,package="glmmTMB")
Owls = plyr::rename(Owls, c(SiblingNegotiation="NCalls"))
Owls = transform(Owls,
                 ArrivalTime=scale(ArrivalTime, center=TRUE, scale=FALSE),
                 obs=factor(seq(nrow(Owls))))

Constant zero-inflation

Here we fit the model with zero-inflation assumed to be constant across the data set, i.e., zero-inflation is independent of the predictor variables.

glmmTMB

fixef1 = NCalls~(FoodTreatment + ArrivalTime) * SexParent + logBroodSize
form1 =  update(fixef1, . ~ . + (1|Nest)+(1|obs))
time.tmb = tfun(m1.tmb <<- glmmTMB(form1,
                                  ziformula=~1, data = Owls,
                                  family=poisson))

glmmADMB

With the additions to the model (logBroodSize as a covariate and observation-level random effects), we were unsuccessful in fitting the model with glmmADMB. Some variants (e.g., with observation-level random effects, but with logBroodSize as an offset) were possible by modifying control settings (i.e. admb.opts=admbControl(shess=FALSE,noinit=FALSE)), but even when successful these fits were very slow (>10 minutes).

MCMCglmm

Code for this example was modified from (Bolker et al. 2013); a more complete description appears in the supplementary material for that paper.

fixef2 = NCalls~trait-1+ # intercept terms for both count and binary terms
    # other fixed-effect terms only apply to count term
    at.level(trait,1):logBroodSize+
    at.level(trait,1):((FoodTreatment+ArrivalTime)*SexParent)
nfix = 8
# residual variances independent for count and binary terms;
#    fixed to 1 for binary term
# random-effects variances independent for count and binary terms;
#    fixed very small (1e-6) for binary term
prior_overdisp  = list(R=list(V=diag(c(1,1)),nu=0.002,fix=2),   
                       G=list(list(V=diag(c(1,1e-6)),nu=0.002,fix=2)))
prior_overdisp_broodoff = c(prior_overdisp,
                            list(B=list(mu=rep(0,nfix),
                                        V=diag(rep(1e4,nfix)))))
set.seed(101)
time.MCMCglmm = tfun(m1.MCMCglmm <<- MCMCglmm(fixef2,
                                   rcov=~idh(trait):units,
                                   random=~idh(trait):Nest,
                                   prior=prior_overdisp_broodoff,
                                   data=Owls,
                                   nitt=103000,
                                   thin=100,
                                   family="zipoisson",
                                   verbose=FALSE))

We adjusted the number of samples until the effective sample size of the most poorly sampled parameter (the zero-inflation parameter, in this case) was greater than 500; with 100,000 samples after a burn-in of 3,000, we achieved a minimum effective sample size of 568 (for the zero-inflation parameter).

brms

time.brms = tfun(m1.brms <<- brm(form1, data = Owls,
                                 iter=1000,
                                 family="zero_inflated_poisson"))

#> Compiling the C++ model

#> Start sampling

One of the known advantages of Hamiltonian Monte Carlo, as implemented in Stan (on which brms is built), is that it achieves high effective sample size per MCMC step; we were able to cut down the number of samples considerably from the default of 2000 and still achieve a minimum effective sample size of approximately 500 (the minimum effective sample size achieved was 460).

We resample to estimate the time required for sampling only (i.e., not including model compilation

time.brms2 = tfun(m1.brms2 <<- update(m1.brms))

#> Start sampling

INLA

form2 = update(fixef1, . ~ . + f(Nest, model="iid") + f(obs, model="iid"))
time.inla = tfun(m1.inla <<-
       inla(form2,
            family= "zeroinflatedpoisson1",
            data=Owls))

mgcv

To the best of our knowledge, mgcv::gam is currently unable to fit models with the combination of zero-inflation and overdispersion (the ziplss() famiy that handles zero-inflation handles only a ZIP case rather than zero-inflated negative binomial or other extensions, and observation-level random effects cannot be fit with gam’s random-effect approach). To address this issue, we fitted a GAM with zero-inflated Poisson responses and used a quasi-likelihood approach: i.e., estimating overdispersion as [sum of squared Pearson residuals/residual degrees of freedom] and inflating the parameter standard errors by the square root of the overdispersion.

form3 = update(fixef1, . ~ . + s(Nest, bs="re"))
time.mgcv = tfun(m1.gam <<- gam(list(form3, ~ 1),
                                data = Owls,
                                family = ziplss(), method = "REML"))

Comparing the results

Timings:

	time (sec)
mgcv	0.2
glmmTMB	2.5
INLA	4.2
MCMCglmm	53.0
brms (no compilation)	55.2
brms (with compilation)	115.6

The deterministic methods (gam, glmmTMB and inla) were all fast; gam was fastest, because we fitted a simpler model (see above). The stochastic methods (MCMCglmm and brm) were about an order of magnitude slower.

Figure 8: Estimated fixed-effect coefficients: Estimates are from similar zero-inflated Poisson models fit using functions glmmTMB, MCMCglmm, brm, inla, and gam.

Because we ran brms with flat priors, the estimates are very close to the ML estimates of glmmTMB. The most different results are from gam, because we used a quasi-likelihood model with a slightly different implicit variance scaling.

Complex zero-inflation

Here we fit the model with zero-inflation depending on some of the predictor variables. We can no longer use glmmADMB or INLA (INLA allows the zero-inflation probabilities to depend on covariates in hurdle models — “type 0” in the INLA documentation — but not for zero-inflated models).

glmmTMB

form0 = update(fixef1, . ~ . + (1|Nest))
ziform = ~FoodTreatment+(1|Nest)
time.tmb_czi = tfun(m1.tmb_czi <<- glmmTMB(form0,
                                           ziformula=ziform,
                                           data = Owls, family=nbinom2))

Attempting to fit the glmmTMB model with a log Normal-Poisson model (i.e., a Poisson model with observation-level random effects) and covariate-dependent zero-inflation led to convergence failure, so we substituted a similar model (a negative-binomial model without observation-level random effects).

MCMCglmm

fixef3 = NCalls~trait-1+ # intercept terms for both count and binary terms
    # fixed-effect terms for count term
    at.level(trait,1):logBroodSize+
    at.level(trait,1):((FoodTreatment+ArrivalTime)*SexParent)+
    # fixed-effect terms for binary term
    at.level(trait,2):FoodTreatment
nfix = 9
# residual variances independent for count and binary terms;
#    fixed to 1 for binary term
# random-effects variances now allow estimated variance for binary term
#   as well
prior_overdisp_czi  = list(R=list(V=diag(c(1,1)),nu=0.002,fix=2),   
                       G=list(list(V=diag(c(1,1)),nu=0.002)))
prior_overdisp_broodoff_czi = c(prior_overdisp_czi,
                              list(B=list(mu=rep(0,nfix),
                                          V=diag(rep(1e4,nfix)))))
set.seed(101)
time.mcmc_czi=tfun(m1.mcmc_czi <<- MCMCglmm(fixef3,
                                   rcov=~idh(trait):units,
                                   random=~idh(trait):Nest,
                                   prior=prior_overdisp_broodoff_czi,
                                   data=Owls,
                                   nitt=153000,
                                   family="zipoisson",
                                   verbose=FALSE))
## warning message suppressed ...

Minimum effective sample size: 884

brms

time.brms_czi = tfun(m1.brms_czi <<- brm(brmsformula(form1, zi=ziform),
                                 data = Owls,
                                 family="zero_inflated_poisson"))

#> Compiling the C++ model

#> Start sampling

time.brms_czi2 = tfun(m1.brms_czi2 <<- update(m1.brms_czi))

#> Start sampling

mgcv

time.mgcv_czi = tfun(m1.gam_czi <<- gam(list(form3,
                                             ~FoodTreatment+ s(Nest, bs = "re")),
                                        data = Owls, family = ziplss(), method = "REML"))

Comparison

Timings:

	time (sec)
mgcv	0.3
TMB	3.9
brms (no compilation)	53.1
MCMCglmm	88.5
brms (with compilation)	174.4

Figure 9: Estimated fixed-effect coefficients: Estimates are from the same zero-inflated Poisson model with predictors on zero-inflation fit using functions glmmTMB, MCMCglmm, brm, and gam.

Package versions:

#>     brms glmmADMB  glmmTMB     INLA MCMCglmm     mgcv 
#>   1.10.2    0.8.5    0.2.0  17.6.20     2.25   1.8.20

CRAN packages used

glmmTMB, pscl, MCMCglmm, mgcv, brms, gamlss, flexmix, MXM, VGAM, TMB, COMPoissonReg package, devtools

CRAN Task Views implied by cited packages

Bayesian, Cluster, Distributions, Econometrics, Environmetrics, ExtremeValue, GraphicalModels, MetaAnalysis, MixedModels, Phylogenetics, Psychometrics, Spatial, Survival

Note

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