---
title: "Estimating Non-Linear Models with brms"
author: "Paul Bürkner"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    toc: yes
vignette: >
  %\VignetteIndexEntry{Estimating Non-Linear Models with brms}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
params:
  EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true")
---

```{r, SETTINGS-knitr, include=FALSE}
stopifnot(require(knitr))
options(width = 90)
opts_chunk$set(
  comment = NA,
  message = FALSE,
  warning = FALSE,
  eval = if (isTRUE(exists("params"))) params$EVAL else FALSE,
  dev = "png",
  dpi = 150,
  fig.asp = 0.8,
  fig.width = 5,
  out.width = "60%",
  fig.align = "center"
)
library(brms)
ggplot2::theme_set(theme_default())
```

## Introduction

This vignette provides an introduction on how to fit non-linear multilevel
models with **brms**. Non-linear models are incredibly flexible and powerful,
but require much more care with respect to model specification and priors than
typical generalized linear models. Ignoring group-level effects for the moment,
the predictor term $\eta_n$ of a generalized linear model for observation $n$
can be written as follows:

$$\eta_n = \sum_{i = 1}^K b_i x_{ni}$$

where $b_i$ is the regression coefficient of predictor $i$ and $x_{ni}$ is the
data of predictor $i$ for observation $n$. This also compromises interaction
terms and various other data transformations. However, the structure of $\eta_n$
is always linear in the sense that the regression coefficients $b_i$ are
multiplied by some predictor values and then summed up. This implies that the
hypothetical predictor term

$$\eta_n = b_1 \exp(b_2 x_n)$$ 

would *not* be a *linear* predictor anymore and we could not fit it using
classical techniques of generalized linear models. We thus need a more general
model class, which we will call *non-linear* models. Note that the term
'non-linear' does not say anything about the assumed distribution of the
response variable. In particular it does not mean 'not normally distributed' as
we can apply non-linear predictor terms to all kinds of response distributions
(for more details on response distributions available in **brms** see
`vignette("brms_families")`).

## A Simple Non-Linear Model

We begin with a simple example using simulated data.

```{r}
b <- c(2, 0.75)
x <- rnorm(100)
y <- rnorm(100, mean = b[1] * exp(b[2] * x))
dat1 <- data.frame(x, y)
```

As stated above, we cannot use a generalized linear model to estimate $b$ so we
go ahead an specify a non-linear model.

```{r, results='hide'}
prior1 <- prior(normal(1, 2), nlpar = "b1") +
  prior(normal(0, 2), nlpar = "b2")
fit1 <- brm(bf(y ~ b1 * exp(b2 * x), b1 + b2 ~ 1, nl = TRUE),
            data = dat1, prior = prior1)
```

When looking at the above code, the first thing that becomes obvious is that we
changed the `formula` syntax to display the non-linear formula including
predictors (i.e., `x`) and parameters (i.e., `b1` and `b2`) wrapped in a call to
`bf`. This stands in contrast to classical **R** formulas, where only predictors
are given and parameters are implicit. The argument `b1 + b2 ~ 1` serves two
purposes. First, it provides information, which variables in `formula` are
parameters, and second, it specifies the linear predictor terms for each
parameter. In fact, we should think of non-linear parameters as placeholders for
linear predictor terms rather than as parameters themselves (see also the
following examples). In the present case, we have no further variables to
predict `b1` and `b2` and thus we just fit intercepts that represent our
estimates of $b_1$ and $b_2$ in the model equation above. The formula `b1 + b2 ~
1` is a short form of `b1 ~ 1, b2 ~ 1` that can be used if multiple non-linear
parameters share the same formula. Setting `nl = TRUE` tells **brms** that the
formula should be treated as non-linear.

In contrast to generalized linear models, priors on population-level parameters
(i.e., 'fixed effects') are often mandatory to identify a non-linear model.
Thus, **brms** requires the user to explicitly specify these priors. In the
present example, we used a `normal(1, 2)` prior on (the population-level
intercept of) `b1`, while we used a `normal(0, 2)` prior on (the
population-level intercept of) `b2`. Setting priors is a non-trivial task in all
kinds of models, especially in non-linear models, so you should always invest
some time to think of appropriate priors. Quite often, you may be forced to
change your priors after fitting a non-linear model for the first time, when you
observe different MCMC chains converging to different posterior regions. This is
a clear sign of an identification problem and one solution is to set stronger
(i.e., more narrow) priors.

To obtain summaries of the fitted model, we apply

```{r}
summary(fit1)
plot(fit1)
plot(conditional_effects(fit1), points = TRUE)
```

The `summary` method reveals that we were able to recover the true parameter
values pretty nicely. According to the `plot` method, our MCMC chains have
converged well and to the same posterior. The `conditional_effects` method
visualizes the model-implied (non-linear) regression line.

We might be also interested in comparing our non-linear model to a classical
linear model.

```{r, results='hide'}
fit2 <- brm(y ~ x, data = dat1)
```

```{r}
summary(fit2)
```

To investigate and compare model fit, we can apply graphical posterior
predictive checks, which make use of the **bayesplot** package on the backend.

```{r}
pp_check(fit1)
pp_check(fit2)
```

We can also easily compare model fit using leave-one-out cross-validation.

```{r}
loo(fit1, fit2)
```

Since smaller `LOOIC` values indicate better model fit, it is immediately
evident that the non-linear model fits the data better, which is of course not
too surprising since we simulated the data from exactly that model.

## A Real-World Non-Linear model

On his blog, Markus Gesmann predicts the growth of cumulative insurance loss
payments over time, originated from different origin years (see
https://www.magesblog.com/post/2015-11-03-loss-developments-via-growth-curves-and/).
We will use a slightly simplified version of his model for demonstration
purposes here. It looks as follows:

$$cum_{AY, dev} \sim N(\mu_{AY, dev}, \sigma)$$
$$\mu_{AY, dev} = ult_{AY} \left(1 - \exp\left(- \left( \frac{dev}{\theta} \right)^\omega \right) \right)$$

The cumulative insurance payments $cum$ will grow over time, and we model this
dependency using the variable $dev$. Further, $ult_{AY}$ is the (to be
estimated) ultimate loss of accident each year. It constitutes a non-linear
parameter in our framework along with the parameters $\theta$ and $\omega$,
which are responsible for the growth of the cumulative loss and are assumed to
be the same across years. The data is already shipped with brms.

```{r}
data(loss)
head(loss)
```

and translate the proposed model into a non-linear **brms** model.

```{r, results='hide'}
fit_loss <- brm(
  bf(cum ~ ult * (1 - exp(-(dev/theta)^omega)),
     ult ~ 1 + (1|AY), omega ~ 1, theta ~ 1, 
     nl = TRUE),
  data = loss, family = gaussian(),
  prior = c(
    prior(normal(5000, 1000), nlpar = "ult"),
    prior(normal(1, 2), nlpar = "omega"),
    prior(normal(45, 10), nlpar = "theta")
  ),
  control = list(adapt_delta = 0.9)
)
```

We estimate a group-level effect of accident year (variable `AY`) for the
ultimate loss `ult`. This also shows nicely how a non-linear parameter is
actually a placeholder for a linear predictor, which in case of `ult`, contains
only an varying intercept over year. Again, priors on population-level effects
are required and, for the present model, are actually mandatory to ensure
identifiability. We summarize the model using well known methods.

```{r}
summary(fit_loss)
plot(fit_loss, N = 3, ask = FALSE)
conditional_effects(fit_loss)
```

Next, we show marginal effects separately for each year.

```{r}
conditions <- data.frame(AY = unique(loss$AY))
rownames(conditions) <- unique(loss$AY)
me_loss <- conditional_effects(
  fit_loss, conditions = conditions, 
  re_formula = NULL, method = "predict"
)
plot(me_loss, ncol = 5, points = TRUE)
```

It is evident that there is some variation in cumulative loss across accident
years, for instance due to natural disasters happening only in certain years.
Further, we see that the uncertainty in the predicted cumulative loss is larger
for later years with fewer available data points. For a more detailed discussion
of this data set, see Section 4.5 in Gesmann & Morris (2020).

## Advanced Item-Response Models

As a third example, we want to show how to model more advanced item-response
models using the non-linear model framework of **brms**. For simplicity, suppose
we have a single forced choice item with three alternatives of which only one is
correct. Our response variable is whether a person answers the item correctly
(1) or not (0). Person are assumed to vary in their ability to answer the item
correctly. However, every person has a 33% chance of getting the item right just
by guessing. We thus simulate some data to reflect this situation.

```{r}
inv_logit <- function(x) 1 / (1 + exp(-x))
ability <- rnorm(300)
p <- 0.33 + 0.67 * inv_logit(ability)
answer <- ifelse(runif(300, 0, 1) < p, 1, 0)
dat_ir <- data.frame(ability, answer)
```

The most basic item-response model is equivalent to a simple logistic regression
model.

```{r, results='hide'}
fit_ir1 <- brm(answer ~ ability, data = dat_ir, family = bernoulli())
```

However, this model completely ignores the guessing probability and will thus
likely come to biased estimates and predictions.

```{r}
summary(fit_ir1)
plot(conditional_effects(fit_ir1), points = TRUE)
```

A more sophisticated approach incorporating the guessing probability looks as
follows:

```{r, results='hide'}
fit_ir2 <- brm(
  bf(answer ~ 0.33 + 0.67 * inv_logit(eta),
     eta ~ ability, nl = TRUE),
  data = dat_ir, family = bernoulli("identity"), 
  prior = prior(normal(0, 5), nlpar = "eta")
)
```

It is very important to set the link function of the `bernoulli` family to
`identity` or else we will apply two link functions. This is because our
non-linear predictor term already contains the desired link function (`0.33 +
0.67 * inv_logit`), but the `bernoulli` family applies the default `logit` link
on top of it. This will of course lead to strange and uninterpretable results.
Thus, please make sure that you set the link function to `identity`, whenever
your non-linear predictor term already contains the desired link function.

```{r}
summary(fit_ir2)
plot(conditional_effects(fit_ir2), points = TRUE)
```

Comparing model fit via leave-one-out cross-validation 

```{r}
loo(fit_ir1, fit_ir2)
```

shows that both model fit the data equally well, but remember that predictions
of the first model might still be misleading as they may well be below the
guessing probability for low ability values. Now, suppose that we don't know the
guessing probability and want to estimate it from the data. This can easily be
done changing the previous model just a bit.

```{r, results='hide'}
fit_ir3 <- brm(
  bf(answer ~ guess + (1 - guess) * inv_logit(eta), 
    eta ~ 0 + ability, guess ~ 1, nl = TRUE),
  data = dat_ir, family = bernoulli("identity"), 
  prior = c(
    prior(normal(0, 5), nlpar = "eta"),
    prior(beta(1, 1), nlpar = "guess", lb = 0, ub = 1)
  )
)
```

Here, we model the guessing probability as a non-linear parameter making sure
that it cannot exceed the interval $[0, 1]$. We did not estimate an intercept
for `eta`, as this will lead to a bias in the estimated guessing parameter (try
it out; this is an excellent example of how careful one has to be in non-linear
models).

```{r}
summary(fit_ir3)
plot(fit_ir3)
plot(conditional_effects(fit_ir3), points = TRUE)
```

The results show that we are able to recover the simulated model parameters with
this non-linear model. Of course, real item-response data have multiple items so
that accounting for item and person variability (e.g., using a multilevel model
with varying intercepts) becomes necessary as we have multiple observations per
item and person. Luckily, this can all be done within the non-linear framework
of **brms** and I hope that this vignette serves as a good starting point.

## References 

Gesmann M. & Morris J. (2020). Hierarchical Compartmental Reserving Models.
*CAS Research Papers*.
