With `bayesnec`

we have included a function that allows
bootstrapped comparisons of posterior predictions. The main focus here
is to showcase how the user can fit several different `bnec`

model fits and can compare differences in the posterior predictions
across these fits for individual toxicity estimates (e.g. *NEC*,
*N(S)EC*, *NSEC* or *ECx*) or across a range of
predictor values. Below we demonstrate usage of
`compare_posterior`

for objects of class
`bayesnecfit`

and `bayesmanecfit`

. In this example
we compare different types of models and model sets using a single
dataset. However, the intent of this function is to allow comparison
across different datasets that might represent, for example, different
levels of a fixed factor covariate. At this time `bnec`

does
not allow inclusion of an interaction with a fixed factor. Including an
interaction term within each of the non-linear models implemented in
`bayesnec`

is relatively straightforward, and may be
introduced in future releases. However, in many cases the functional
form of the response may change with different levels of a given factor.
The substantial complexity of defining all possible non-linear model
combinations at each factor level means it unlikely this could be
feasibly implemented in `bayesnec`

in the short term. In the
meantime the greatest flexibility in the functional form of individual
model fits can be readily obtained using models fitted independently to
data within each factor level.

To run this vignette, we will also need some additional packages

```
set.seed(333)
library(brms)
library(bayesnec)
data(nec_data)
make_ecx_data <- function(top, bot, ec50, beta, x) {
top + (bot - top) / (1 + exp((ec50 - x) * exp(beta)))
}
x <- seq(0, 10, length = 12)
y <- make_ecx_data(x = x, bot = 0, top = 1, beta = 0.5, ec50 = 5)
set.seed(333)
dat <- data.frame(x = rep(x, 15), y = rnorm(15 * length(x), y, 0.2))
# Fit a set of models
exmp <- bnec(y ~ crf(x, model = "decline"), data = dat, refresh = 0,
open_progress = FALSE)
```

```
class(exmp)
#> [1] "bayesmanecfit" "bnecfit"
```

This call fits all models that are suitable for modelling
Gaussian-distributed response data, excluding all of the hormesis
models, which we are not considering here. Now that we have our example
fit, we can use this to make different model sets, purely for the
purposes of illustrating `compare_posterior`

in
`bayesnec`

and highlighting the rich information that is
contained in the Bayesian posterior draws.

We can pull out the **nec** models and the
**ecx** models separately, to create two alternative model
fits of this data that we can compare to each other, as well as the
original **all** model fit.

Now we have three different averaged model fits, all of class
`bayemanec`

in this case (because they all contain multiple
fits). We can compare their posterior estimates of the ecx10 values
using `compare_posterior`

.

```
post_comp <- compare_posterior(list("all" = exmp, "ecx" = exmp_ecx,
"nec" = exmp_nec),
comparison = "ecx", ecx_val = 10)
names(post_comp)
#> [1] "posterior_list" "posterior_data" "diff_list" "diff_data" "prob_diff"
```

The `compare_posterior`

function outputs several elements
in a named list. This includes the **posterior_data** for
each model in the comparison as a `data.frame`

which we can
use to plot a `geom_density`

plot of the posterior estimates,
so they can be compared visually.

```
ggplot(data = post_comp$posterior_data, mapping = aes(x = value)) +
geom_density(mapping = aes(group = model, colour = model, fill = model),
alpha = 0.3) +
theme_classic()
```

From this you can see that the *EC _{10}* estimates are
very similar for the

`wi`

in
`exmp$mod_stats`

. The The `data.frame`

`diff_data`

can be used to
make a similar plot, but specifically for the differences among
models.

```
ggplot(data = post_comp$diff_data, mapping = aes(x = diff)) +
geom_density(mapping = aes(group = comparison, colour = comparison,
fill = comparison), alpha = 0.3) +
theme_classic()
```

This shows the differences among the three estimates. There is no
difference in the **ecx** and **all**
estimates (the probability density overlaps zero - red shaded curve).
This is because for this example the simulated data are from a smooth
**ecx** type curve and thus those models have high weight.
The **nec** based *EC _{10}* estimates are
greater than the

We can formally test the probability that the toxicity estimate for
one model set is greater than the other using posterior differencing.
This is contained in the compare posterior output as
`prob_diff`

. Here you can see there is 62% chance that
**all** is greater than **ecx**. There is only
a 14% chance that **ecx** is greater than
**nec**, and a 86% chance that the **nec** is
greater than **ecx**.

```
post_comp$prob_diff
#> comparison prob
#> 1 all-ecx 0.6160770
#> 2 all-nec 0.2700338
#> 3 ecx-nec 0.1443930
```

The user can also compare posterior fitted values across the full
range of x values, using `comparison = "fitted"`

.

```
post_comp_fitted <- compare_posterior(list("all" = exmp, "ecx" = exmp_ecx,
"nec" = exmp_nec),
comparison = "fitted")
```

In the case of `comparison = "fitted"`

most of the
elements returned by `compare_posterior`

are class
`data.frame`

, with summary values for the posteriors,
difference values and probabilities returned for each value of the
predictor, for each model or model comparison.

```
head(post_comp_fitted$posterior_data)
#> model x Estimate Q2.5 Q97.5
#> 1 all 0.09090909 1.020778 0.9663590 1.084920
#> 2 all 0.29313544 1.020645 0.9663378 1.083134
#> 3 all 0.49536178 1.020406 0.9663378 1.080866
#> 4 all 0.69758813 1.020050 0.9662984 1.078648
#> 5 all 0.89981447 1.019724 0.9662984 1.076251
#> 6 all 1.10204082 1.019195 0.9662886 1.073099
head(post_comp_fitted$diff_data)
#> comparison diff.Estimate diff.Q2.5 diff.Q97.5 x
#> 1 all-ecx -0.006377007 -0.08644327 0.07743305 0.09090909
#> 2 all-ecx -0.006358821 -0.08530609 0.07573855 0.29313544
#> 3 all-ecx -0.006258424 -0.08354907 0.07444896 0.49536178
#> 4 all-ecx -0.006251275 -0.08176871 0.07254152 0.69758813
#> 5 all-ecx -0.005755167 -0.08009080 0.07157816 0.89981447
#> 6 all-ecx -0.005438621 -0.07845105 0.06951660 1.10204082
```

Using the collated posterior_data we can plot the predicted curves
with confidence bounds for each of the input models. This shows clearly
that the **ecx** model set begins to decline earlier than
the **nec** set, which is flat prior to the
**nec** step point, and then declines more rapidly.

```
ggplot(data = post_comp_fitted$posterior_data) +
geom_line(mapping = aes(x = x, y = Estimate, color = model),
linewidth = 0.5) +
geom_ribbon(mapping = aes(x = x, ymin = Q2.5, ymax = Q97.5, fill = model),
alpha = 0.3)+
labs(x = "Concentration", y = "Posterior estimate") +
theme_classic()
```

We can plot the differences between pairs of models in the list by
plotting `diff.Estimate`

from `diff_data`

and
using colours for the different comparisons. This plot highlights where
the differences among these model sets are the greatest and explains the
differences we see above in the *EC _{10}* toxicity
estimates. The

```
ggplot(data = post_comp_fitted$diff_data) +
geom_line(mapping = aes(x = x, y = diff.Estimate, color = comparison),
linewidth = 0.5) +
geom_ribbon(mapping = aes(x = x, ymin = diff.Q2.5, ymax = diff.Q97.5,
fill = comparison), alpha = 0.3) +
labs(x = "Concentration", y = "Posterior difference") +
theme_classic()
```

And finally we can plot the probability that one model is greater
than the other by plotting `prob`

from
`diff_data`

. The pattern of this plot is identical to the
plot of differences, but the y axis now shows the probability of these
differences. The red line hovers around 0.5 clearly indicating the lack
of significant difference in the **ecx** and
**all** model sets at any point of the predictor axis. The
green and blue curves pass through 0.5 at several points, meaning there
are parts of the curve where there is no significant difference between
the **nec** and the **ecx** or
**all** model set predictions. The greatest probability of
difference among these curves is between values of ~3 and ~4 of the
predictor, where the probability of difference trends towards low values
(**ecx** is higher than **nec**), and at
around 5, where the probability difference is very high
(**nec** is higher than **ecx**). These are
the two points where the **ecx** set deviates most from the
**nec** set.

```
ggplot(data = post_comp_fitted$prob_diff) +
geom_line(mapping = aes(x = x, y = prob, color = comparison),
linewidth = 0.5) +
labs(x = "Concentration", y = "Probability of difference") +
theme_classic()
```

Here we use the difference in the **ecx** and
**nec** models sets purely to showcase the functionality of
`compare_posterior`

and highlight some of the strengths of
using a Bayesian approach. These differences are expected in this case,
because the simulated data we used are from a smooth
**exc** function. The **nec** models are
fitting a break-point model that has a single mean value up to the
**nec**, followed by a relatively sharp decline. When
fitted to these smooth curves the **nec** models do not
really fit the data well and produce inflated (less conservative)
estimates of the *EC _{10}* value. Fortunately, the model
averaging approach solves this issue by yielding a very low weight to
the

The `compare_posterior`

function was actually developed
with the intention of comparing model sets based on different data to
assess statistically the difference in either the toxicity estimates, or
across the entire fitted curve, amongst factors of interest. Examples
would include assessing for changes in toxicity across different WET
tests through time, or comparing different levels of a factor treatment
variable (such as climate change scenarios), to look at how these
interact with concentration to impact toxicity.