Fitting models with zoid

Overdispersion or not?

One optional feature of the model is to include overdispersion in the calculations of the 3 probabilities for each cell. Including overdispersion is generally only advised with replicated data or shared information – and may fit heterogeneous datasets with lots of 0s better than the model without overdispersion. For the cod diet data, we’ll include overdispersion because of poor MCMC sampling without it.

Model selection

Next, we can test some hypotheses about how the data are structured. We’ll create the following models (1) a model with all observations as replicate samples, (2) a model with only seasonal effects, (3) a model with only differences by year, and (4) a model with both year and season. Both year and season are treated as factors here – but continuous covariates can also be included.

Note that for fitting, the data_matrix should be a matrix, but the design_matrix should be a data frame.

design_matrix$Season = as.factor(design_matrix$Season)
design_matrix$Year = as.factor(design_matrix$Year)
design_matrix$y = 1 # dummy variable

set.seed(123)
fit_1 <- fit_zoid(data_matrix = as.matrix(data_matrix),
                      overdispersion = TRUE,
                      chains=4,
                      iter=4000)

fit_2 <- fit_zoid(formula = y ~ Season, 
                      design_matrix = design_matrix, 
                      data_matrix = as.matrix(data_matrix),
                      overdispersion = TRUE,
                      chains=4,
                      iter=4000)

fit_3 <- fit_zoid(formula = y ~ Year, 
                      design_matrix = design_matrix, 
                      data_matrix = as.matrix(data_matrix),
                      overdispersion = TRUE,
                      chains=4,
                      iter=4000)

fit_4 <- fit_zoid(formula = y ~ Year + Season, 
                      design_matrix = design_matrix, 
                      data_matrix = as.matrix(data_matrix),
                      overdispersion = TRUE,
                      chains=4,
                      iter=4000)

To compare models, we could use criteria like LOOIC in the loo package – this is accessible by calling

loo_1 = loo::loo(fit_1$model)
loo_2 = loo::loo(fit_2$model)
loo_3 = loo::loo(fit_3$model)
loo_4 = loo::loo(fit_4$model)

For our example, the LOOIC from the model with Season and Year is lowest (2584.9) indicating the most support over the base model (2879.7), model with Season only (2892.4), and model with Year only (2637.2). Two words of caution for this application are (1) the standard errors of the LOOIC estimates are all in the 120-160 range, so many of the models are within +/- 1 SE of each other and (2) the Pareto-k diagnostic values fall into the ‘bad’ category for about 20% of the data points. There’s a couple solutions for this, including more MCMC sampling, and using PSIS smooth sampling

Summarizing estimates

We include several helper functions for processing output into more manageable data frames. First, we can extract the predicted point estimates (and uncertainty intervals) around proportions,

fit_1 <- fit_zoid(data_matrix = as.matrix(data_matrix), 
                      chains=1,
                      iter=mcmc_iter,
                      overdispersion = TRUE, refresh=0)

fitted_vals = get_fitted(fit_1)
head(fitted_vals)

##   obs group       mean     median         lo         hi
## 1   1     1 0.03297343 0.03293401 0.03073215 0.03537796
## 2   2     1 0.03297343 0.03293401 0.03073215 0.03537796
## 3   3     1 0.03297343 0.03293401 0.03073215 0.03537796
## 4   4     1 0.03297343 0.03293401 0.03073215 0.03537796
## 5   5     1 0.03297343 0.03293401 0.03073215 0.03537796
## 6   6     1 0.03297343 0.03293401 0.03073215 0.03537796

Second, we can return all parameters (including betas for coefficients and \(\phi\), the overdispersion term)

fitted_vals = get_pars(fit_1)
head(fitted_vals$betas)

##   m cov         par        mean       median          lo         hi
## 1 1   1 (Intercept) -0.01228533 -0.012246645 -0.08396062 0.06038681
## 2 2   1 (Intercept) -0.02500844 -0.021716494 -0.07386585 0.02544750
## 3 3   1 (Intercept) -0.01173933 -0.015288713 -0.05752490 0.04729847
## 4 4   1 (Intercept) -0.01271186 -0.013370911 -0.05564834 0.03140008
## 5 5   1 (Intercept) -0.00943012 -0.017199185 -0.04218768 0.04559571
## 6 6   1 (Intercept) -0.00284256 -0.002717758 -0.04878939 0.04051401

Random effects

For models with grouping variables or factor predictor variables, either fixed or random effects may be included. Both need to be included via the formula interface, and random effects may be included in the typical form (lme4, glmmTMB),

formula <- ~ (1|group)
formula <- ~ (1|group) + (1|year) + (1|season)

For now, only random intercepts are allowed; random slopes will be included eventually.