Model 1: Fixed Effects
Our initial model for the logarithm of the load returned on the \(j\)th trip by adult \(i\) will be \(N(\mu_{ij},\sigma_{ij}^2)\) where: \[\mu=\beta_0 + \beta_1 \mathrm{log(IVI)}_{ij} + \beta_2 \mathrm{broodsize}_i + \beta_3 \mathrm{sex}_i\] and \[\log(\sigma_{ij}^2)=\psi_0 + \psi_1 \mathrm{broodsize}_i + \psi_2\mathrm{sex}_i.\] In the framework of generalized linear models, the variance, \(\sigma^2_i\), is equal to the dispersion parameter, which we denote generically with \(\phi\). Hence, the model can then be written as: \(Y_{ij} \sim N(\mu_{ij},\phi_{ij})\) where: \[\mu=\beta_0 + \beta_1 \mathrm{log(IVI)}_{ij} + \beta_2 \mathrm{broodsize}_i + \beta_3 \mathrm{sex}_i\] and \[\log(\phi_{ij})=\psi_0 + \psi_1 \mathrm{broodsize}_i + \psi_2\mathrm{sex}_i.\]
Model
The first step is to define the models for the mean and dispersion components. These models are simply lists with (up to) two named components, fixed and random, which provide the details on the fixed and random effects. Both lists must contain a variable name specifying the basename for the coefficients and a variable formula specifying the effects. The fixed list should also contain an object called priors which specifies the priors for coefficients in the fixed effects portion of the model. Random effects are currently assumed to be normal with mean zero and unknown variances which are assigned the half \(t\)-distribution with ? degrees of freedom and scale ?. The optional parameter link can also be used to specify a link function for either component.
The components of the model for the pied flycatcher data specified above would be generated with:
## Mean model
mymean <- list(fixed=list(name="alpha",
formula=~ log(IVI) + broodsize + sex,
priors=list(c("dnorm",0,.001))))
## Dispersion model
mydisp <- list(fixed=list(name="psi",
formula=~broodsize + sex,
priors=list(c("dnorm",0,.001))),
link="log")These two objects will now be used to generate the JAGS code, data, and initial values for running the model.
Running the Model with dalmatian
The primary function in the package dalmatian automates the creation of the JAGS code, data, and initial values and then passes these to JAGS via the functions in the rjags package. The dalmatian function takes requires several arguments including the data frame for the analysis, the model objects created above, and the name of the JAGS model file. It also requires two lists containing named arguments for the functions jags.model and coda.samples from the rjags package. Descriptions of the arguments for these two functions are available in their own help pages. Any arguments that are not specified in these lists will take the default values given by rjags. The two exceptions are the file argument of jags.model and the n.iter argument of coda.samples which do not have default values and must be specified. The number of parallel chains will be taken from the jags.model list. If this value is not specified then three chains will be run in order that convergence diagnostics can be computed. Note that the chains have been run for a relatively small number of iterations and heavily subsampled to satisfy the size requirements for packages on . Neither of these is recommended in practice.
## Set working directory
## By default uses a system temp directory. You probably want to change this.
workingDir <- tempdir()
## Define list of arguments for jags.model()
jm.args <- list(file=file.path(workingDir,"pied_flycatcher_1_jags.R"),n.adapt=1000)
## Define list of arguments for coda.samples()
cs.args <- list(n.iter=5000,thin=20)
## Run the model using dalmatian
pfmcmc1 <- dalmatian(df=pfdata,
mean.model=mymean,
dispersion.model=mydisp,
jags.model.args=jm.args,
coda.samples.args=cs.args,
rounding=TRUE,
lower="lower",
upper="upper",
residuals = FALSE,
debug=FALSE)
}Results
The function dalmatian returns list containing three objects: 1. jags.model.args The full list of arguments passed to jags.model. This includes the variables in the input list along with the name of the model file, the formatted data, and the initial values. 2. coda.samples.args The full list of arguments passed to coda.samples. This includes the variables in the input list along with the complete object returned by jags.model.args and the character vector including the names of the monitored parameters. 3. coda An object of class mcmc.list and length n.chains containing the samples generated by JAGS.
Results
Inference is based on the output from the MCMC chain which is stored in the coda object. Summaries of the MCMC sampler and the posterior distribution can be constructed from coda with a variety of tools designed to work with objects of the mcmc.list class. For example, we can use the tools in the package coda to generate traceplots, density plots and histograms, numerical summaries of the posterior, and convergence diagnostics. The ggmcmc package can also be used to create fancier looking traceplots, density plots, and plots summarizing the posterior distribution in terms of posterior means and quantiles (credible intervals). However, several wrapper functions exist within the package to automate some of these tasks.
The first thing we should do is to check the convergence of the chains. Summary statistics computed from the output in coda are only valid if the chain has converged and is sampling from the correct distribution. The wrapper function convergence() applies three of the convergence diagnostics from the coda package.
## Compute convergence diagnostics
pfconvergence1 <- convergence(pfmcmc1)## Gelman-Rubin diagnostics
pfconvergence1$gelman## Potential scale reduction factors:
##
## Point est. Upper C.I.
## alpha.(Intercept) 1.003 1.015
## alpha.log(IVI) 1.004 1.016
## alpha.broodsizeIncreased 1.005 1.006
## alpha.sexMale 0.999 1.000
## psi.(Intercept) 0.999 0.999
## psi.broodsizeIncreased 1.002 1.009
## psi.sexMale 1.003 1.014## Raftery diagnostics
pfconvergence1$raftery##
## Quantile (q) = 0.025
## Accuracy (r) = +/- 0.005
## Probability (s) = 0.95
##
## You need a sample size of at least 3746 with these values of q, r and s## Effective sample size
pfconvergence1$effectiveSize## alpha.(Intercept) alpha.log(IVI) alpha.broodsizeIncreased
## 777.2150 789.0462 786.6472
## alpha.sexMale psi.(Intercept) psi.broodsizeIncreased
## 846.6271 518.5562 640.6306
## psi.sexMale
## 663.3225Details of these measures can be found in the help files of the coda package. Not surprisingly, these all suggest that the chains need to be run for longer.
Convergence and mixing of the chains can also be examined visually through traceplots. The wrapper function traceplots generates traces using the functions from the ggmcmc package and separates the variables by the model components. One of the advantages of using ggmcmc which is based on ggplot2 is that the plots can be saved and easily reproduced. Here I have thinned the chains further prior to plotting to reduce the size of the vignette. Note that thinning is conducted relative to the original chain, not the already thinned chain. I.e., thinning by 20 selects the output from every 20th iteration of the original chain (every 2nd iteration from the retained samples).
## Generate traceplots
pfraceplots1 <- traceplots(pfmcmc1,show=FALSE,nthin=20)## Fixed effects for mean
pftraceplots1$meanFixed
## Fixed effects for dispersion
pftraceplots1$dispersionFixed
Numerical summaries of the posterior distribution computed from the output in coda can be obtained with the summary function.
## Compute numerical summaries
pfsummary1 <- summary(pfmcmc1)## Print numerical summaries
pfsummary1##
## Iterations = 1020:6000
## Thinning interval = 20
## Number of chains = 3
## Sample size per chain = 250
##
## Posterior Summary Statistics for Each Model Component
##
## Mean Model: Fixed Effects
## Mean Median SD Lower 95% Lower 50% Upper 50% Upper 95%
## (Intercept) -3.31 -3.31 0.07 -3.45 -3.35 -3.26 -3.20
## log(IVI) 0.15 0.15 0.01 0.13 0.14 0.16 0.18
## broodsizeIncreased 0.11 0.11 0.03 0.06 0.09 0.12 0.16
## sexMale -0.07 -0.07 0.02 -0.13 -0.09 -0.06 -0.03
##
## Dispersion Model: Fixed Effects
## Mean Median SD Lower 95% Lower 50% Upper 50% Upper 95%
## (Intercept) -0.70 -0.70 0.06 -0.81 -0.74 -0.66 -0.60
## broodsizeIncreased 0.04 0.05 0.06 -0.07 0.01 0.08 0.16
## sexMale 0.03 0.03 0.06 -0.09 -0.02 0.06 0.12For each component of the model the tables provide statistics including the estimated posterior mean, median, and standard deviation and the end points of the 50% and 95% highest posterior density credible intervals. Note that these summaries are only valid if the previous diagnostics have suggested that the chain has converged and the effective sample size is large enough. Graphical summaries can also be computed with the caterpillar which, like traceplots, is a simple wrapper for the functions in the ggmcmc package.
## Generate caterpillar
pfcaterpillar1 <- caterpillar(pfmcmc1,show = FALSE)## Fixed effects for mean
pfcaterpillar1$meanFixed
## Fixed effects for dispersion
pfcaterpillar1$dispersionFixed
