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 comprises 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")`

).

We begin with a simple example using simulated data.

```
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.

```
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

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: y ~ b1 * exp(b2 * x)
b1 ~ 1
b2 ~ 1
Data: dat1 (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
b1_Intercept 1.82 0.11 1.60 2.04 1.00 1587 1837
b2_Intercept 0.75 0.04 0.68 0.82 1.00 1574 1875
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 1.04 0.08 0.91 1.20 1.00 2326 2109
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

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.

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: y ~ x
Data: dat1 (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 2.40 0.13 2.14 2.65 1.00 4305 2619
x 1.80 0.13 1.55 2.05 1.00 3844 2789
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 1.31 0.10 1.14 1.52 1.00 3479 2854
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

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

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

```
Output of model 'fit1':
Computed from 4000 by 100 log-likelihood matrix
Estimate SE
elpd_loo -147.2 6.7
p_loo 3.0 0.8
looic 294.4 13.3
------
Monte Carlo SE of elpd_loo is 0.0.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 98 98.0% 1521
(0.5, 0.7] (ok) 2 2.0% 963
(0.7, 1] (bad) 0 0.0% <NA>
(1, Inf) (very bad) 0 0.0% <NA>
All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.
Output of model 'fit2':
Computed from 4000 by 100 log-likelihood matrix
Estimate SE
elpd_loo -172.4 14.6
p_loo 6.4 4.3
looic 344.8 29.2
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 99 99.0% 2700
(0.5, 0.7] (ok) 0 0.0% <NA>
(0.7, 1] (bad) 1 1.0% 27
(1, Inf) (very bad) 0 0.0% <NA>
See help('pareto-k-diagnostic') for details.
Model comparisons:
elpd_diff se_diff
fit1 0.0 0.0
fit2 -25.2 15.0
```

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.

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.

```
AY dev cum premium
1 1991 6 357.848 10000
2 1991 18 1124.788 10000
3 1991 30 1735.330 10000
4 1991 42 2182.708 10000
5 1991 54 2745.596 10000
6 1991 66 3319.994 10000
```

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

```
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.

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: cum ~ ult * (1 - exp(-(dev/theta)^omega))
ult ~ 1 + (1 | AY)
omega ~ 1
theta ~ 1
Data: loss (Number of observations: 55)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Group-Level Effects:
~AY (Number of levels: 10)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(ult_Intercept) 740.53 228.84 430.68 1299.73 1.00 1238 1833
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
ult_Intercept 5282.42 291.01 4722.34 5858.59 1.00 1004 1677
omega_Intercept 1.33 0.05 1.24 1.43 1.00 2591 2302
theta_Intercept 46.23 2.15 42.42 50.86 1.00 2423 2022
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 140.22 16.11 113.57 176.18 1.00 2125 1968
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

Next, we show marginal effects separately for each year.

```
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).

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.

```
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.

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

```
Family: bernoulli
Links: mu = logit
Formula: answer ~ ability
Data: dat_ir (Number of observations: 300)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 0.93 0.13 0.67 1.19 1.00 3312 2606
ability 0.48 0.15 0.19 0.77 1.00 2847 2409
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

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

```
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.

```
Family: bernoulli
Links: mu = identity
Formula: answer ~ 0.33 + 0.67 * inv_logit(eta)
eta ~ ability
Data: dat_ir (Number of observations: 300)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
eta_Intercept 0.27 0.17 -0.07 0.60 1.00 3362 2632
eta_ability 0.68 0.22 0.28 1.13 1.00 3153 2688
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

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

```
Output of model 'fit_ir1':
Computed from 4000 by 300 log-likelihood matrix
Estimate SE
elpd_loo -178.8 7.4
p_loo 2.0 0.2
looic 357.7 14.7
------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.
Output of model 'fit_ir2':
Computed from 4000 by 300 log-likelihood matrix
Estimate SE
elpd_loo -178.6 7.4
p_loo 2.1 0.2
looic 357.2 14.8
------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.
Model comparisons:
elpd_diff se_diff
fit_ir2 0.0 0.0
fit_ir1 -0.2 0.4
```

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.

```
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).

```
Family: bernoulli
Links: mu = identity
Formula: answer ~ guess + (1 - guess) * inv_logit(eta)
eta ~ 0 + ability
guess ~ 1
Data: dat_ir (Number of observations: 300)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
eta_ability 0.78 0.25 0.33 1.29 1.00 3026 2470
guess_Intercept 0.42 0.05 0.32 0.51 1.00 2967 2548
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

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.

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