boral (version 0.4)

calc.condlogLik: Conditional log-likelihood for an boral model

Description

Calculates the conditional log-likelihood for a set of parameter estimates from an boral model, whereby the latent variables are treated as "fixed effects".

Usage

calc.condlogLik(y, X = NULL, family, trial.size = NULL, lv.coefs, 
	X.coefs = NULL, site.coefs = NULL, lv, cutoffs = NULL, 
     powerparam = NULL)

Arguments

y

The response matrix the boral model was fitted to.

X

The model matrix used in the boral model. Defaults to NULL, in which case it is assumed no model matrix was used.

family

Either a single element, or a vector of length equal to the number of columns in $y$. The former assumes all columns of $y$ come from this distribution. The latter option allows for different distributions for each column of $y$. Elements can be one of "b

trial.size

Either equal to NULL, a single element, or a vector of length equal to the number of columns in $y$. If a single element, then all columns assumed to be binomially distributed will have trial size set to this. If a vector, different trial sizes are allowe

lv.coefs

The column-specific intercept, coefficient estimates relating to the latent variables, and dispersion parameters from the boral model.

X.coefs

The coefficients estimates relating to the model matrix X from the boral model. Defaults to NULL, in which it is assumed there are no covariates in the model.

site.coefs

Row effect estimates for the boral model. Defaults to NULL, in which case it is assumed there are no row effects in the model.

lv

Latent variables "estimates" from the boral model, which the conditional log-likelihood is based on. For boral models with no latent variables, please use calc.logLik.lv0 to calculate the conditiona

cutoffs

Common cutoff estimates from the boral model when any of the columns of $y$ are ordinal responses. Defaults to NULL.

powerparam

Common power parameter from the boral model when any of the columns of $y$ are tweedie responses. Defaults to NULL.

Value

• A list with the following components:
• logLik
  Value of the conditional log-likelihood.
• logLik.comp
  A vector of the log-likelihood values for each row of $y$, such that sum(logLik.comp) = logLik.

Details

For an $n x p$ response matrix $y$, suppose we fit an boral model with $q$ latent varables. If we denote the latent varibles by $\bm{z}_i; i = 1,\ldots,n$, then the conditional log-likelihood is given by (with parameters where appropriate),

$$\log(f) = \sum_{i=1}^n \sum_{j=1}^p \log (f(y_{ij} | \alpha_i, \tau_k, \theta_{0j}, \bm{\theta}_j, \bm{z}_i, \bm{x}_i, \bm{\beta}_j, \phi_j)),$$

where $f(y_{ij}|\cdot)$ is the assumed distribution for column $j$, $\alpha_i$ is the row effect, $\tau_k$ are the common cutoffs for proportional odds regression, $\theta_{0j}$ is the column-specific intercept, $\bm{z}'_i$ are the latent variables, $\bm{\theta}_j$ are the column-specific coefficients relating to the latent variables, $\bm{x}'_i$ is row $i$ of the model matrix,, $\bm{\beta}_j$ are the column-specific coefficients relating to the model matrix of explanatory variables, $\phi_j$ are-column-specific dispersion parameters.

The key difference between this and the marginal likelihood (see calc.marglogLik) is that the conditional log-likelihood treats the latent variables as "fixed effects", while the marginal log-likelihood treats them as "random effects" which require integrating over.

See Also

get.measures for some information criteria based on the conditional log-likelihood; calc.marglogLik for calculation of the marginal log-likelihood; calc.logLik.lv0 to calculate the conditional/marginal log-likelihood for an boral model with no latent variables.

Examples

library(mvabund) ## Load a dataset from the mvabund package
data(spider)
y <- spider$abun
n <- nrow(y); p <- ncol(y); 

## Example 1 - model with 2 latent variables, site effects, 
## 	and no environmental covariates
spider.fit.nb <- boral(y, family = "negative.binomial", num.lv = 2, 
     site.eff = TRUE, save.model = TRUE, calc.ics = FALSE)

## Extract all MCMC samples
fit.mcmc <- as.mcmc(spider.fit.nb$jags.model)[[1]] 

## Find the posterior medians
coef.mat <- matrix(apply(fit.mcmc[,grep("all.params",colnames(fit.mcmc))],
     2,median),nrow=p)
site.coef.median <- apply(fit.mcmc[,grep("site.params", colnames(fit.mcmc))],
     2,median)
lvs.mat <- matrix(apply(fit.mcmc[,grep("lvs",colnames(fit.mcmc))],2,median),nrow=n)

## Caculate the conditional log-likelihood at the posterior median
calc.condlogLik(y, family = "negative.binomial", 
     lv.coefs =  coef.mat, site.coefs = site.coef.median, lv = lvs.mat)


## Example 2 - model with one latent variable, no site effects, 
## 	and environmental covariates
spider.fit.nb2 <- boral(y, X = spider$x, family = "negative.binomial", num.lv = 1, 
     site.eff = FALSE, save.model = TRUE, calc.ics = FALSE)

## Extract all MCMC samples
fit.mcmc <- as.mcmc(spider.fit.nb2$jags.model)[[1]] 

## Find the posterior medians
coef.mat <- matrix(apply(fit.mcmc[,grep("all.params",colnames(fit.mcmc))],
     2,median),nrow=p)
X.coef.mat <- matrix(apply(fit.mcmc[,grep("X.params",colnames(fit.mcmc))],
	2,median),nrow=p)
lvs.mat <- matrix(apply(fit.mcmc[,grep("lvs",colnames(fit.mcmc))],2,median),nrow=n)

## Caculate the log-likelihood at the posterior median
calc.condlogLik(y, X = spider$x, family = "negative.binomial", 
	lv.coefs =  coef.mat, X.coefs = X.coef.mat, lv = lvs.mat)