1 Introduction
Hierarchical models, including generalized linear models with mixed random and fixed effects, are increasingly popular. The rapid expansion of applications is largely due to the advancement of the Markov Chain Monte Carlo (MCMC) algorithms and related software (Gilks et al. 1996; Gelman et al. 2003; Lunn et al. 2009). Data cloning is a statistical computing method introduced by Lele et al. (2007). It exploits the computational simplicity of the MCMC algorithms used in the Bayesian statistical framework, but it provides the maximum likelihood point estimates and their standard errors for complex hierarchical models. The use of the data cloning algorithm is especially valuable for complex models, where the number of unknowns increases with sample size (i.e. with latent variables), because inference and prediction procedures are often hard to implement in such situations.
The dclone R package (Sólymos 2010) provides infrastructure for data cloning. Users who are familiar with Bayesian methodology can instantly use the package for maximum likelihood inference and prediction. Developers of R packages can build on the low level functionality provided by the package to implement more specific higher level estimation procedures for users who are not familiar with Bayesian methodology. This paper demonstrates the implementation of the data cloning algorithm, and presents a case study on how to write high level functions for specific modeling problems.
2 Theory of data cloning
Imagine a hypothetical situation where an experiment is repeated by \(k\) different observers, and all \(k\) experiments happen to result in exactly the same set of observations, \(y^{(k)} = \left(y,y,\ldots,y\right)\). The likelihood function based on the combination of the data from these \(k\) experiments is \(L(\theta, y^{\left(k\right)}) = \left[L\left(\theta, y\right)\right]^k\). The location of the maximum of \(L(\theta,y^{(k)})\) exactly equals the location of the maximum of the function \(L\left(\theta, y\right)\), and the Fisher information matrix based on this likelihood is \(k\) times the Fisher information matrix based on \(L\left(\theta, y\right)\).
One can use MCMC methods to calculate the posterior distribution of the model parameters (\(\theta\)) conditional on the data. Under regularity conditions, if \(k\) is large, the posterior distribution corresponding to \(k\) clones of the observations is approximately normal with mean \(\hat{\theta}\) and variance \(1/k\) times the inverse of the Fisher information matrix. When \(k\) is large, the mean of this posterior distribution is the maximum likelihood estimate and \(k\) times the posterior variance is the corresponding asymptotic variance of the maximum likelihood estimate if the parameter space is continuous. When some of the parameters are on the boundaries of their feasible space (Stram and Lee 1994), point estimates can be correct, but currently the Fisher information cannot be estimated correctly by using data cloning. This is an area for further research, but such situations challenge other computing techniques as well.
Data cloning is a computational algorithm to compute maximum likelihood estimates and the inverse of the Fisher information matrix, and is related to simulated annealing (Brooks and Morgan 1995). By using data cloning, the statistical accuracy of the estimator remains a function of the sample size and not of the number of cloned copies. Data cloning does not improve the statistical accuracy of the estimator by artificially increasing the sample size. The data cloning procedure avoids the analytical or numerical evaluation of high dimensional integrals, numerical optimization of the likelihood function, and numerical computation of the curvature of the likelihood function. Interested readers should consult Lele et al. (2007, 2010) for more details and mathematical proofs for the data cloning algorithm.
3 The data cloning algorithm
Consider the following Poisson generalized linear mixed model (GLMM) with a random intercept for i.i.d. observations of \(Y_{i}\) counts from \(i = 1,2,\ldots,n\) localities: \[\begin{aligned} \alpha_{i} & \sim \mathrm{normal}\left(0, \sigma^2\right)\\ \lambda_{i} & = \exp\left(\alpha_{i} + \boldsymbol{X}_i^{\mathsf{T}}\boldsymbol{\beta}\right)\\ Y_{i} \mid \lambda_{i} & \sim \mathrm{Poisson}\left(\lambda_{i}\right) \end{aligned}\] The corresponding code for the simulation with \(\boldsymbol{\beta} = \left(1.8, -0.9\right)\), \(\sigma=0.2\), \(x_{i} \sim \mathrm{U}\left(0, 1\right)\) is:
> library(dclone)
> set.seed(1234)
> n <- 50
> beta <- c(1.8, -0.9)
> sigma <- 0.2
> x <- runif(n, min = 0, max = 1)
> X <- model.matrix(~ x)
> alpha <- rnorm(n, mean = 0, sd = sigma)
> lambda <- exp(alpha + drop(X %*% beta))
> Y <- rpois(n, lambda)The first step in the data cloning algorithm is to construct the full Bayesian model of the problem with proper prior distributions for unknown parameters. We use flat normal priors for \(\beta\)s and for \(\log\left(\sigma\right)\). First we use the rjags (Plummer 2010b) and coda (Plummer et al. 2010) R packages and the JAGS (Plummer 2010a) software for model fitting. But the dclone package also supports WinBUGS (Spiegelhalter et al. 2003) and OpenBUGS (Spiegelhalter et al. 2007) via the R packages R2WinBUGS (Sturtz et al. 2005) and BRugs (Thomas et al. 2006), respectively. The corresponding model in the BUGS language is:
> glmm.model <- function() {
+ for (i in 1:n) {
+ Y[i] ~ dpois(lambda[i])
+ lambda[i] <- exp(alpha[i] +
+ inprod(X[i,], beta[1,]))
+ alpha[i] ~ dnorm(0, tau)
+ }
+ for (j in 1:np) {
+ beta[1,j] ~ dnorm(0, 0.001)
+ }
+ log.sigma ~ dnorm(0, 0.001)
+ sigma <- exp(log.sigma)
+ tau <- 1 / pow(sigma, 2)
+ }Note that instead of writing the model into a file, we store it as an R function (see JAGS and WinBUGS documentation for how to correctly specify the model in the BUGS language). Although the BUGS and R syntaxes seem similar, the BUGS model function cannot be evaluated within R. Storing the BUGS model as an R function is handy, because the user does not have to manage different files when modeling. Nevertheless, the model can be supplied in a separate file by giving its name as character.
We also have to define the data as elements of a named list along with
the names of nodes that we want to monitor (we can also set up initial
values, number of burn-in iterations, number of iterations for the
posterior sample, thinning values, etc.; see
dclone package
documentation for details). Now we can do the Bayesian inference by
calling the jags.fit function:
> dat <- list(Y = Y, X = X, n = n,
+ np = ncol(X))
> mod <- jags.fit(dat,
+ c("beta", "sigma"), glmm.model, n.iter = 1000)The output mod is an "mcmc.list" object, which can be explored by
methods such as summary or plot provided by the
coda package.
The dclone package
provides the bugs.fit wrapper function for WinBUGS/OpenBUGS. The BUGS
model needs to be changed to run smoothly in WinBUGS/OpenBUGS:
> glmm.model.bugs <- function() {
+ for (i in 1:n) {
+ Y[i] ~ dpois(lambda[i])
+ lambda[i] <- exp(alpha[i] +
+ inprod(X[i,], beta[1,]))
+ alpha[i] ~ dnorm(0, tau) %_% I(-5, 5)
+ }
+ for (j in 1:np) {
+ beta[1,j] ~ dnorm(0, 0.01) %_% I(-5, 5)
+ }
+ log.sigma ~ dnorm(0, 0.01) %_% I(-5, 5)
+ sigma <- exp(log.sigma)
+ tau <- 1 / pow(sigma, 2)
+ }In the bugs.fit function, the settings besides the data, params,
model, and inits arguments follow the settings in the
bugs/openbugs functions in the
R2WinBUGS package.
This leads to some differences between the arguments of the jags.fit
and the bugs.fit functions. For example bugs.fit uses n.thin
instead of thin, and n.burnin is equivalent to n.adapt + n.update
as compared to jags.fit. The bugs.fit can return the results either
in "mcmc.list" or "bugs" format. The reason for leaving different
arguments for jags.fit and bugs.fit is that the aim of the
dclone package is not to
make the MCMC platforms interchangeable, but to provide data cloning
facility for each. It is easy to adapt an existing BUGS code for data
cloning, but it sometimes can be tricky to adapt a JAGS code to WinBUGS
and vice versa, because of differences between the two dialects
(i.e. truncation, censoring, autoregressive priors, etc., see
Plummer (2010b)).
Here are the results from the three MCMC platforms:
> mod.wb <- bugs.fit(dat, c("beta", "sigma"),
+ glmm.model.bugs, DIC = FALSE, n.thin = 1)
> mod.ob <- bugs.fit(dat, c("beta", "sigma"),
+ glmm.model.bugs, program = "openbugs",
+ DIC = FALSE, n.thin = 1)> sapply(list(JAGS = mod, WinBUGS = mod.wb,
+ OpenBUGS = mod.ob), coef) JAGS WinBUGS OpenBUGS
beta[1] 1.893 1.910 1.9037
beta[2] -1.050 -1.074 -1.0375
sigma 0.161 0.130 0.0732The idea in the next step of the data cloning algorithm is that instead
of using the likelihood for the observed data, we use the likelihood
corresponding to \(k\) copies (clones) of the data. Actually cloning
(repeating) the data \(k\) times is important if the model includes
unobserved (latent) variables: in this way latent variables are cloned
as well, thus contributing to the likelihood. We can use the rep
function to repeat the data vectors, but it is less convenient for
e.g. matrices or data frames. Thus, there is the dclone generic
function with methods for various R object classes:
> dclone(1:5, 1)[1] 1 2 3 4 5> dclone(1:5, 2) [1] 1 2 3 4 5 1 2 3 4 5
attr(,"n.clones")
[1] 2
attr(,"n.clones")attr(,"method")
[1] "rep"> dclone(matrix(1:4, 2, 2), 2) [,1] [,2]
[1,] 1 3
[2,] 2 4
[3,] 1 3
[4,] 2 4
attr(,"n.clones")
[1] 2
attr(,"n.clones")attr(,"method")
[1] "rep"> dclone(data.frame(a=1:2, b=3:4), 2) a b
1_1 1 3
2_1 2 4
1_2 1 3
2_2 2 4The number of clones can be extracted by the nclones function; it
returns NULL for \(k=1\) and \(k\) otherwise.
The BUGS data specification might contain some elements that we do not
want to clone (e.g. "np", the number of columns of the design matrix
in this case). Thus the dclone method has different behaviour for
lists, than for non list classes (including data frames). We can define
which elements should not be cloned, or which should be multiplied by
\(k\) instead of being cloned \(k\) times.
> dat2 <- dclone(dat, n.clones = 2,
+ multiply = "n", unchanged = "np")
> nclones(dat2)[1] 2
attr(,"method")
Y X n np
"rep" "rep" "multi" NA The "method" attribute of the cloned object stores this information.
There are three different ways of cloning (besides NA standing for
unchanged): "rep" is for (longitudinal) repetitions, "multi" is for
multiplication, and "dim" is repeating the data along an extra
dimension (see later).
Now we do the model fitting with \(k=2\). The "mcmc.list" object
inherits the information about the cloning:
> mod2 <- jags.fit(dat2,
+ c("beta", "sigma"), glmm.model, n.iter = 1000)Similarly, the bugs.fit function takes care of the cloning information
passed through the data argument:
> mod.wb2 <- bugs.fit(dat2, c("beta", "sigma"),
+ glmm.model.bugs, DIC = FALSE, n.thin = 1)
> mod.ob2 <- bugs.fit(dat2, c("beta", "sigma"),
+ glmm.model.bugs, program = "openbugs",
+ DIC = FALSE, n.thin = 1)And here are the results based on \(k=2\) for the three MCMC platforms:
> sapply(list(JAGS = mod2, WinBUGS = mod.wb2,
+ OpenBUGS = mod.ob2), coef) JAGS WinBUGS OpenBUGS
beta[1] 1.918 1.905 1.896
beta[2] -1.114 -1.080 -1.078
sigma 0.207 0.187 0.243For some models, indexing can be more complex, and simple repetitions of
the data ("rep" method) are not appropriate. In case of non
independent data (time series or spatial autoregressive models), cloning
should be done over an extra dimension to ensure that clones are
independent. For this purpose, one can use the dcdim function:
> (obj <- dclone(dcdim(data.matrix(1:5)), 2)) clone.1 clone.2
[1,] 1 1
[2,] 2 2
[3,] 3 3
[4,] 4 4
[5,] 5 5
attr(,"n.clones")
[1] 2
attr(,"n.clones")attr(,"method")
[1] "dim"
attr(,"n.clones")attr(,"method")attr(,"drop")
[1] TRUEIf data cloning consists of repetitions of the data, our BUGS model usually does not need modifications. If we add an extra dimension to the data, the BUGS model and the data specification must reflect the extra dimension, too.
To demonstrate this, we consider a model and data set from Ponciano et al. (2009). They used the single-species population growth data from laboratory experiments of Gause (1934) with Paramecium aurelia. Gause initiated liquid cultures on day 0 at a concentration of two individuals per 0.5 cm\(^3\) of culture media. Then, on days 2–19, he took daily 0.5 cm\(^3\) samples of the microbe cultures and counted the number of cells in each sample. Ponciano et al. (2009) fitted discrete time stochastic models of population dynamics to describe Gause’s data taking into account both process noise and observation error. The Beverton-Holt model incorporates a latent variable component (\(N_t,~t=0,1,\ldots,q\)) to describe an unobserved time series of actual population abundance. The latent variable component contains density dependence (\(\beta\)) and stochastic process noise (\(\sigma^2\)). The model incorporates a Poisson observation component to account for variability caused by sampling: \[\begin{aligned} \mu_{t} & = \log(\lambda) + \log(N_{t-1}) - \log(1 + \beta N_{t-1})\\ \log(N_t) & \sim \mathrm{normal}(\mu_t, \sigma^2)\\ Y_t \mid N_t & \sim \mathrm{Poisson}(N_t) \end{aligned}\] \(\lambda\) is the finite rate of increase in population abundance. The corresponding BUGS model is:
> beverton.holt <- function() {
+ for (j in 1:k) {
+ for(i in 2:(n+1)){
+ Y[(i-1),j] ~ dpois(exp(log.N[i,j]))
+ log.N[i,j] ~ dnorm(mu[i,j], 1 / sigma^2)
+ mu[i,j] <- log(lambda) + log.N[(i-1),j]
+ - log(1 + beta * exp(log.N[(i-1),j]))
+ }
+ log.N[1,j] ~ dnorm(mu0, 1 / sigma^2)
+ }
+ beta ~ dlnorm(-1, 1)
+ sigma ~ dlnorm(0, 1)
+ tmp ~ dlnorm(0, 1)
+ lambda <- tmp + 1
+ mu0 <- log(lambda) + log(2) - log(1 + beta * 2)
+ }Note that besides the indexing for the time series, the model contains
another dimension for the clones. We define the data set by using the
dcdim method for cloning the observations. We include an element
k = 1 that will be multiplied to indicate how many clones (columns)
are in the data, while n (number of observations) remains unchanged:
> paurelia <- c(17, 29, 39, 63, 185, 258, 267,
+ 392, 510, 570, 650, 560, 575, 650, 550,
+ 480, 520, 500)
> bhdat <- list(Y=dcdim(data.matrix(paurelia)),
+ n=length(paurelia), k=1)
> dcbhdat <- dclone(bhdat, n.clones = 5,
+ multiply = "k", unchanged = "n")> bhmod <- jags.fit(dcbhdat,
+ c("lambda","beta","sigma"), beverton.holt,
+ n.iter=1000)> coef(bhmod) beta lambda sigma
0.00218 2.18755 0.12777 Results compare well with estimates in Ponciano et al. (2009) (\(\hat{\beta} = 0.00235\), \(\hat{\lambda} = 2.274\), \(\hat{\sigma} = 0.1274\)).
4 Iterative model fitting
We can use the dc.fit function to iteratively fit the same model with
various \(k\) values as described in Lele et al. (2010). The function takes similar
arguments to dclone and jags.fit (or bugs.fit, if
flavour = "bugs" is used). Because the information in the data
overrides the priors by increasing the number of clones, we can improve
MCMC convergence by making the priors more informative during the
iterative fitting process. We achieve this by modifying the BUGS model
for the Poisson GLMM example:
> glmm.model.up <- function() {
+ for (i in 1:n) {
+ Y[i] ~ dpois(lambda[i])
+ lambda[i] <- exp(alpha[i] +
+ inprod(X[i,], beta[1,]))
+ alpha[i] ~ dnorm(0, 1/sigma^2)
+ }
+ for (j in 1:np) {
+ beta[1,j] ~ dnorm(pr[j,1], pr[j,2])
+ }
+ log.sigma ~ dnorm(pr[(np+1),1], pr[(np+1),2])
+ sigma <- exp(log.sigma)
+ tau <- 1 / pow(sigma, 2)
+ }We also define a function to update the priors. The function returns
values for flat prior specification in the first iteration, and uses the
updated posterior means (via the coef method) and data cloning
standard errors (via the dcsd method) in the rest, because priors that
have large probability mass near the maximum likelihood estimate require
fewer clones to achieve the desired accuracy.
> upfun <- function(x) {
+ if (missing(x)) {
+ np <- ncol(X)
+ return(cbind(rep(0, np+1),
+ rep(0.001, np+1)))
+ } else {
+ ncl <- nclones(x)
+ if (is.null(ncl))
+ ncl <- 1
+ par <- coef(x)
+ se <- dcsd(x)
+ log.sigma <- mcmcapply(x[,"sigma"], log)
+ par[length(par)] <- mean(log.sigma)
+ se[length(se)] <- sd(log.sigma) * sqrt(ncl)
+ return(cbind(par, se))
+ }
+ }Finally, we define prior specifications as part of the data ("pr"),
and provide the updating function in the dc.fit call:
> updat <- list(Y = Y, X = X, n = n,
+ np = ncol(X), pr = upfun())
> k <- c(1, 5, 10, 20)
> dcmod <- dc.fit(updat, c("beta", "sigma"),
+ glmm.model.up, n.clones = k, n.iter = 1000,
+ multiply = "n", unchanged = "np",
+ update = "pr", updatefun = upfun)> summary(dcmod)Iterations = 1001:2000
Thinning interval = 1
Number of chains = 3
Sample size per chain = 1000
Number of clones = 20
1. Empirical mean and standard deviation for each
variable, plus standard error of the mean:
Mean SD DC SD Naive SE
beta[1] 1.894 0.0368 0.164 0.000671
beta[2] -1.082 0.0734 0.328 0.001341
sigma 0.278 0.0256 0.114 0.000467
Time-series SE R hat
beta[1] 0.00259 1.01
beta[2] 0.00546 1.01
sigma 0.00194 1.04
2. Quantiles for each variable:
2.5% 25% 50% 75% 97.5%
beta[1] 1.823 1.869 1.89 1.920 1.964
beta[2] -1.230 -1.133 -1.08 -1.029 -0.943
sigma 0.226 0.260 0.28 0.296 0.323The summary contains data cloning standard errors (DC SD) and
\(\hat{R}\) values for MCMC chain convergence (Gelman and Rubin 1992).
5 Diagnostics
We can see how the increase in the number of clones affects our
inferences on single nodes by using the dctable function. This
function retrieves the information stored during the iterative fitting
process (or can be used to compare more than one fitted model). Only the
last MCMC object is returned by dc.fit, but descriptive statistics of
the posterior distribution are stored in each step
(Figure 1). The asymptotic convergence can be visually
evaluated by plotting the posterior variances scaled by the variance for
the model at \(k=1\) (or the smallest \(k\)). If scaled variances are
decreasing at a \(1/k\) rate and have reached a lower bound (say \(<0.05\)),
the data cloning algorithm has converged. If scaled variances are not
decreasing at the proper rate, that might indicate identifiability
issues (Lele et al. 2010). On the log scale, this graph should show an
approximately linear decrease of log(scaled variance) vs. \(\log(k)\) for
each parameter (Figure 2).
> dct <- dctable(dcmod)
> plot(dct)
> plot(dct, type="log.var")
Lele et al. (2010) introduced diagnostic measures for checking the convergence of
the data cloning algorithm which are based on the joint posterior
distribution and not only on single parameters. These include
calculating the largest eigenvalue of the posterior variance covariance
matrix (lambdamax.diag), or calculating the mean squared error and
another correlation-like fit statistic (\(r^2\)) based on a \(\chi^2\)
approximation (chisq.diag with a plot method). The maximum
eigenvalue reflects the degeneracy of the posterior distribution, while
the two fit measures reflect the adequacy of the normal approximation.
All three statistics should converge to zero as \(k\) increases. If this
happens, different prior specifications are no longer influencing the
results (Lele et al. 2007, 2010).
These measures and multivariate \(\hat{R}\) values for MCMC chain
convergence (Brooks and Gelman 1997) are calculated during the iterations by
dc.fit as well, and can be retrieved by the function dcdiag:
> dcdiag(dcmod) n.clones lambda.max ms.error r.squared r.hat
1 1 0.11538 0.1282 0.02103 1.66
2 5 0.02225 0.0229 0.00277 1.02
3 10 0.01145 0.0383 0.00612 1.01
4 20 0.00643 0.0241 0.00173 1.03The data cloning algorithm requires that MCMC chains are properly mixed and the posterior distribution is nearly degenerate multivariate normal. These requirements have been satisfied in the case of the Poisson GLMM model. \(\hat{R}\) values show better mixing properties of the MCMC chains with higher \(k\) values, and in this example it is expected, because we have used informative priors near the maximum likelihood estimates for the cases \(k>1\).
The functions dctable and dcdiag can be used to determine the number
of clones required for a particular model and data set. Also, these
diagnostic functions can alert the modeller when the model contains
non-identifiable parameters. Lele et al. (2010) gives several examples; here we
consider the normal-normal mixture: \[\begin{aligned}
\mu_{i} & \sim \mathrm{normal}(\gamma, \tau^2)\\
Y_{i} \mid \mu_i & \sim \mathrm{normal}(\mu_i, \sigma^2)
\end{aligned}\] where the parameters (\(\gamma, \sigma^2+\tau^2\)) are
known to be identifiable, but (\(\gamma, \sigma^2, \tau^2\)) are not.
We simulate random observations under this model (\(\gamma=2.5, \sigma=0.2, \tau=0.5\)) and fit the corresponding BUGS model:
> gamma <- 2.5
> sigma <- 0.2
> tau <- 0.5
> set.seed(2345)
> mu <- rnorm(n, gamma, tau)
> Y <- rnorm(n, mu, sigma)
> nn.model <- function() {
+ for (i in 1:n) {
+ Y[i] ~ dnorm(mu[i], prec1)
+ mu[i] ~ dnorm(gamma, prec2)
+ }
+ gamma ~ dnorm(0, 0.001)
+ log.sigma ~ dnorm(0, 0.001)
+ sigma <- exp(log.sigma)
+ prec1 <- 1 / pow(sigma, 2)
+ log.tau ~ dnorm(0, 0.001)
+ tau <- exp(log.tau)
+ prec2 <- 1 / pow(tau, 2)
+ }
> nndat <- list(Y = Y, n = n)> nnmod <- dc.fit(nndat, c("gamma","sigma","tau"),
+ nn.model, n.clones=c(1,10,20,30,40,50),
+ n.iter=1000, multiply="n")> dcdiag(nnmod) n.clones lambda.max ms.error r.squared r.hat
1 1 0.0312 0.508 0.02985 1.18
2 10 0.0364 0.275 0.00355 2.06
3 20 1.2617 1.111 0.13714 50.15
4 30 0.1530 0.753 0.10267 12.91
5 40 1.7972 0.232 0.03770 92.87
6 50 1.8634 0.241 0.04003 15.72> vars <- mcmcapply(nnmod[,c("sigma","tau")],
+ array)^2
> sigma^2 + tau^2[1] 0.29> summary(rowSums(vars)) Min. 1st Qu. Median Mean 3rd Qu. Max.
0.21 0.23 2.87 3.00 6.04 6.84 The high r.hat and the variable lambda.max and fit statistic values
that are not converging to zero indicate possible problems with
identifiability.
6 Inference and prediction
We can explore the results with methods defined for "mcmc.list"
objects (many such methods are available in the
coda package,
e.g. summary, plot, etc.). The
dclone package adds a few
more methods: coef returns the mean of the posterior, dcsd the data
cloning standard errors. Any function returning a scalar statistic can
be passed via the mcmcapply function:
> coef(dcmod)beta[1] beta[2] sigma
1.894 -1.082 0.278 > dcsd(dcmod)beta[1] beta[2] sigma
0.164 0.328 0.114 > mcmcapply(dcmod, sd) * sqrt(nclones(dcmod))beta[1] beta[2] sigma
0.164 0.328 0.114 The asymptotic multivariate normality can be used to get Wald-type
confidence intervals for the estimates based on the inverse of the
Fisher information matrix. The vcov method returns the inverse Fisher
information matrix, the confint method calculates confidence intervals
assuming multivariate normality for MCMC objects with \(k > 1\):
> confint(dcmod) 2.5 % 97.5 %
beta[1] 1.5718 2.217
beta[2] -1.7253 -0.438
sigma 0.0534 0.502> vcov(dcmod) beta[1] beta[2] sigma
beta[1] 0.02705 -0.04604 -0.00291
beta[2] -0.04604 0.10783 -0.00156
sigma -0.00291 -0.00156 0.01308Confidence intervals can also be obtained via parametric bootstrap or based on profile likelihood (Ponciano et al. 2009), but these are not currently available in the dclone package and often require substantial user intervention.
These methods are handy when we make predictions. We can use the maximum likelihood estimates and the variance-covariance matrix defined as a multivariate normal node in the BUGS model. For the Poisson mixed model example, the BUGS model for prediction will look like:
> glmm.pred <- function() {
+ for (i in 1:n) {
+ Y[i] ~ dpois(lambda[i])
+ lambda[i] <- exp(mu[i])
+ mu[i] <- alpha[i] +
+ inprod(X[i,], beta[1,])
+ alpha[i] ~ dnorm(0, tau)
+ }
+ tmp[1:(np+1)] ~ dmnorm(param[], prec[,])
+ beta[1,1:np] <- tmp[1:np]
+ sigma <- tmp[(np+1)]
+ tau <- 1 / pow(sigma, 2)
+ }Now we add the estimates and the precision matrix prec to the data
(the make.symmetric function prevents some problems related to matrix
symmetry and numerical precision), and define \(\boldsymbol{X}\) for the
predictions (now we simply use the observed values of the covariates).
Then do the modeling as usual by sampling the node "lambda":
> prec <- make.symmetric(solve(vcov(dcmod)))
> prdat <- list(X = X, n = nrow(X), np = ncol(X),
+ param = coef(dcmod), prec = prec)
> prmod <- jags.fit(prdat, "lambda", glmm.pred,
+ n.iter = 1000)7 Writing high level functions
Suppose we want to provide a user friendly function to fit the Poisson
mixed model with random intercept. We are now modeling the observed
abundances (count based on point counts) of the Ovenbird (Seiurus
aurocapilla) as a function of ecological site characteristics
(upland/lowland, uplow) and percentage of total human disturbance
around the sites (thd in the ovenbird data set). Data were collected
from 182 sites in the Northern Boreal region of Alberta, Canada, between
2003 and 2008. Data were collected by the Alberta Biodiversity
Monitoring Institute and are available at http://www.abmi.ca.
Our goal is to determine the effect of human disturbance on Ovenbird abundance, by controlling for site characteristics. But we know that other factors not taken into account, e.g. the amount of deciduous forest, might influence the abundance as well (Hobson and Bayne 2002). So the random intercept will account for this unexplained environmental variability. The Poisson error component will account for random deviations from expected abundances (\(\lambda_{i}\)) and observed counts (\(Y_{i}\)) represent a realization of this quantity.
Here is the high level function for fitting the Poisson mixed model
built on data cloning with a simple print, summary and predict
method:
> glmmPois <- function(formula,
+ data = parent.frame(), n.clones, ...) {
+ lhs <- formula[[2]]
+ Y <- eval(lhs, data)
+ formula[[2]] <- NULL
+ rhs <- model.frame(formula, data)
+ X <- model.matrix(attr(rhs, "terms"), rhs)
+ dat <- list(n = length(Y), Y = Y,
+ X = X, np = ncol(X))
+ dcdat <- dclone(dat, n.clones,
+ multiply = "n", unchanged = "np")
+ mod <- jags.fit(dcdat, c("beta", "sigma"),
+ glmm.model, ...)
+ coefs <- coef(mod)
+ names(coefs) <- c(colnames(X),
+ "sigma")
+ rval <- list(coefficients = coefs,
+ call = match.call(),
+ mcmc = mod, y = Y, x = rhs,
+ model = X, formula = formula)
+ class(rval) <- "glmmPois"
+ rval
+ }
> print.glmmPois <- function(x, ...) {
+ cat("glmmPois model\n\n")
+ print(format(coef(x), digits = 4),
+ print.gap = 2, quote = FALSE)
+ cat("\n")
+ invisible(x)
+ }
> summary.glmmPois <- function(object, ...) {
+ x <- cbind("Estimate" = coef(object),
+ "Std. Error" = dcsd(object$mcmc),
+ confint(object$mcmc))
+ cat("Call:", deparse(object$call,
+ width.cutoff = getOption("width")),
+ "\n", sep="\n")
+ cat("glmmPois model\n\n")
+ printCoefmat(x, ...)
+ cat("\n")
+ invisible(x)
+ }
> predict.glmmPois <- function(object,
+ newdata = NULL, type = c("mu", "lambda", "Y"),
+ level = 0.95, ...){
+ prec <- solve(vcov(object$mcmc))
+ prec <- make.symmetric(prec)
+ param <- coef(object)
+ if (is.null(newdata)) {
+ X <- object$model
+ } else {
+ rhs <- model.frame(object$formula, newdata)
+ X <- model.matrix(attr(rhs, "terms"), rhs)
+ }
+ type <- match.arg(type)
+ prdat <- list(n = nrow(X), X = X,
+ np = ncol(X), param = param, prec = prec)
+ prval <- jags.fit(prdat, type, glmm.pred, ...)
+ a <- (1 - level)/2
+ a <- c(a, 1 - a)
+ rval <- list(fit = coef(prval),
+ ci.fit = quantile(prval, probs = a))
+ rval
+ }Note that the functions glmm.model and glmm.pred containing the BUGS
code are used within these R functions. This implementation works fine,
but is not adequate when building a contributed R package, because
functions such as dnorm and inprod are not valid R objects, etc. For
R packages, the best way is to represent the BUGS model as a character
vector with lines as elements, and put that inside the R function. The
custommodel function of the
dclone package can be
used to create such character vectors and pass them to other
dclone functions via the
model argument.
Now we fit the model for the ovenbird data set to estimate the effect
of human disturbance on Ovenbird abundance. We fit the model using the
function glmmPois:
> data(ovenbird)
> obmod <- glmmPois(count ~ uplow + thd,
+ ovenbird, n.clones = 5, n.update = 1000,
+ n.iter = 1000)Then print the object and inspect the summary,
> obmodglmmPois model
(Intercept) uplowlowland thd
2.00312 -1.34242 -0.01647
sigma
1.19318 > summary(obmod)Call:
glmmPois(formula = count ~ uplow + thd, data = ovenbird,
n.clones = 5, n.update = 1000, n.iter = 1000)
glmmPois model
Estimate Std. Error 2.5 % 97.5 %
(Intercept) 2.00312 0.13767 1.73328 2.27
uplowlowland -1.34242 0.21503 -1.76387 -0.92
thd -0.01647 0.00569 -0.02763 -0.01
sigma 1.19318 0.09523 1.00653 1.38Finally predict abundances as a function of disturbance (0–100%) by controlling for site characteristics (Figure 3):
> thd <- seq(0, 100, len = 101)
> ndata <- data.frame(uplow = rep("lowland",
+ length(thd)), thd = thd)
> levels(ndata$uplow) <- levels(ovenbird$uplow)
> obpred <- predict(obmod, ndata, "lambda")
Ovenbird abundance was significantly higher in upland sites, and human disturbance had a significant negative effect on expected Ovenbird abundance. Unexplained variation (\(\sigma^2=\)1.425 \(\pm\) 0.102 SE) was substantial, thus the choice of the Poisson mixed model makes sense for this data set.
8 Summary
The data cloning algorithm is especially useful for complex models for which other likelihood based computational methods fail. The algorithm also can numerically reveal potential identifiability issues related to hierarchical models. The dclone package supports established MCMC software and provides low level functions to help implementing high level estimating procedures to get maximum likelihood inferences and predictions for more specialized problems based on the data cloning algorithm.
9 Acknowledgements
Subhash Lele, Khurram Nadeem and Gabor Grothendieck have provided invaluable suggestions and feedback on this work. Comments of Martyn Plummer and two anonymous reviewers greatly improved the quality of the paper. Funding was provided by the Alberta Biodiversity Monitoring Institute and the Natural Sciences and Engineering Research Council.
CRAN packages used
dclone, rjags, coda, R2WinBUGS, BRugs
CRAN Task Views implied by cited packages
Bayesian, Cluster, GraphicalModels, HighPerformanceComputing, MixedModels, Optimization
Note
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