In cbrown5/BenthicLatent: Bayesian latent variable model

Supplement

Calculation of the Watanabe-Aikaike Information Criteria

The WAIC is defined as in Gelman et al. ...

$$WAIC = -2*(\widehat{lpd} - \widehat{p}{{WAIC}})$$
Where the estimate of the log pointwise predictive density (lpd) was calculated:

$$\widehat{lpd} = \sum_{i=1}^{n} ln(\frac{1}{S} \sum_{s=1}^{S}\prod_{j=1}^{J} p(y_{i,j} | \psi^s)$$
Where there are $S$ samples from the posteriors for the parameters, $y_{i,j}$ are the data (counts) and $p(y_{i,j} | \psi^s)$ is the probability density for the count for each species given the parameter estimates for the abundance model taken from posterior draw $s$.

The penalty parameter was calculated:
$$\widehat{p}{{WAIC}} = \sum_{i=1}^{n} var(\sum_{j=1}^{J} ln(p(y_{i,j} | \psi^s)))$$
Where $var$ is the function for the sample variance taken over samples for each observation.
Following Hui et al. we calculate $p(y_{i,j} | \psi^s)$ conditional on the estimates for the latent variables.

Fit of models by habitats

Table S1 Fit of the six latent variable model to each habitat type. r^2 values are calculated on observed versus expected abundances. SLope values of 1 indicate now bias, <1 under-predictions and >1 over-prediction.

load('~/Code/BenthicLatent/data-raw/LV mod estimtes of variance and bias.Rdata')
#Table of estimate of variance and bias for the 6 LV model.  
df <- est_varbias[[6]][,c(3,4,1,2)]
df <- dplyr::arrange(df, meanabund)
knitr::kable(df, digits = 2, col.names = c('Habitat','Mean abundance','r^2','Slope of expected on observed'))