# Define Custom Response Distributions with brms

#### Paul Bürkner

#### 2020-02-21

## Introduction

The brms package comes with a lot of built-in response distributions – usually called families in R – to specify among others linear, count data, survival, response times, or ordinal models (see `help(brmsfamily)` for an overview). Despite supporting over two dozen families, there is still a long list of distributions, which are not natively supported. The present vignette will explain how to specify such custom families in brms. By doing that, users can benefit from the modeling flexibility and post-processing options of brms even when using self-defined response distributions.

## A Case Study

As a case study, we will use the `cbpp` data of the lme4 package, which describes the development of the CBPP disease of cattle in Africa. The data set contains four variables: `period` (the time period), `herd` (a factor identifying the cattle herd), `incidence` (number of new disease cases for a given herd and time period), as well as `size` (the herd size at the beginning of a given time period).

    data("cbpp", package = "lme4")
    head(cbpp)

      herd incidence size period
    1    1         2   14      1
    2    1         3   12      2
    3    1         4    9      3
    4    1         0    5      4
    5    2         3   22      1
    6    2         1   18      2

In a first step, we will be predicting `incidence` using a simple binomial model, which will serve as our baseline model. For observed number of events \(y\) (`incidence` in our case) and total number of trials \(T\) (`size`), the probability mass function of the binomial distribution is defined as

\[ P(y | T, p) = \binom{T}{y} p^{y} (1 - p)^{N-y} \]

where \(p\) is the event probability. In the classical binomial model, we will directly predict \(p\) on the logit-scale, which means that for each observation \(i\) we compute the success probability \(p_i\) as

\[ p_i = \frac{\exp(\eta_i)}{1 + \exp(\eta_i)} \]

where \(\eta_i\) is the linear predictor term of observation \(i\) (see `vignette("brms_overview")` for more details on linear predictors in brms). Predicting `incidence` by `period` and a varying intercept of `herd` is straight forward in brms:

    fit1 <- brm(incidence | trials(size) ~ period + (1|herd),
                data = cbpp, family = binomial())

In the summary output, we see that the incidence probability varies substantially over herds, but reduces over the cource of the time as indicated by the negative coefficients of `period`.

    summary(fit1)

     Family: binomial
      Links: mu = logit
    Formula: incidence | trials(size) ~ period + (1 | herd)
       Data: cbpp (Number of observations: 56)
    Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
             total post-warmup samples = 4000

    Group-Level Effects:
    ~herd (Number of levels: 15)
                  Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sd(Intercept)     0.77      0.24     0.40     1.32 1.00     1425     1567

    Population-Level Effects:
              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    Intercept    -1.42      0.27    -1.99    -0.94 1.00     1924     1779
    period2      -1.01      0.31    -1.61    -0.40 1.00     5138     2962
    period3      -1.15      0.33    -1.82    -0.51 1.00     4403     2796
    period4      -1.62      0.44    -2.52    -0.82 1.00     5166     2606

    Samples were drawn 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 drawback of the binomial model is that – after taking into account the linear predictor – its variance is fixed to \(\text{Var}(y_i) = T_i p_i (1 - p_i)\). All variance exceeding this value cannot be not taken into account by the model. There are multiple ways of dealing with this so called overdispersion and the solution described below will serve as an illustrative example of how to define custom families in brms.

## The Beta-Binomial Distribution

The beta-binomial model is a generalization of the binomial model with an additional parameter to account for overdispersion. In the beta-binomial model, we do not predict the binomial probability \(p_i\) directly, but assume it to be beta distributed with hyperparameters \(\alpha > 0\) and \(\beta > 0\):

\[ p_i \sim \text{Beta}(\alpha_i, \beta_i) \]

The \(\alpha\) and \(\beta\) parameters are both hard to interprete and generally not recommended for use in regression models. Thus, we will apply a different parameterization with parameters \(\mu \in [0, 1]\) and \(\phi > 0\), which we will call \(\text{Beta2}\):

\[ \text{Beta2}(\mu, \phi) = \text{Beta}(\mu \phi, (1-\mu) \phi) \]

The parameters \(\mu\) and \(\phi\) specify the mean and precision parameter, respectively. By defining

\[ \mu_i = \frac{\exp(\eta_i)}{1 + \exp(\eta_i)} \]

we still predict the expected probability by means of our transformed linear predictor (as in the original binomial model), but account for potential overdispersion via the parameter \(\phi\).

## Fitting Custom Family Models

The beta-binomial distribution is not natively supported in brms and so we will have to define it ourselves using the `custom_family` function. This function requires the family’s name, the names of its parameters (`mu` and `phi` in our case), corresponding link functions (only applied if parameters are prediced), their theoretical lower and upper bounds (only applied if parameters are not predicted), information on whether the distribuion is discrete or continuous, and finally, whether additional non-parameter variables need to be passed to the distribution. For our beta-binomial example, this results in the following custom family:

    beta_binomial2 <- custom_family(
      "beta_binomial2", dpars = c("mu", "phi"),
      links = c("logit", "log"), lb = c(NA, 0),
      type = "int", vars = "vint1[n]"
    )

The name `vint1` for the variable containing the number of trials is not chosen arbitrarily as we will see below. Next, we have to provide the relevant Stan functions if the distribution is not defined in Stan itself. For the `beta_binomial2` distribution, this is straight forward since the ordinal `beta_binomial` distribution is already implemented.

    stan_funs <- "
      real beta_binomial2_lpmf(int y, real mu, real phi, int T) {
        return beta_binomial_lpmf(y | T, mu * phi, (1 - mu) * phi);
      }
      int beta_binomial2_rng(real mu, real phi, int T) {
        return beta_binomial_rng(T, mu * phi, (1 - mu) * phi);
      }
    "

For the model fitting, we will only need `beta_binomial2_lpmf`, but `beta_binomial2_rng` will come in handy when it comes to post-processing. We define:

    stanvars <- stanvar(scode = stan_funs, block = "functions")

To provide information about the number of trials (an integer variable), we are going to use the addition argument `vint()`, which can only be used in custom families. Simiarily, if we needed to include additional vectors of real data, we would use `vreal()`. Actually, for this particular example, we could more elegantly apply the addition argument `trials()` instead of `vint()`as in the basic binomial model. However, since the present vignette is ment to give a general overview of the topic, we will go with the more general method.

We now have all components together to fit our custom beta-binomial model:

    fit2 <- brm(
      incidence | vint(size) ~ period + (1|herd), data = cbpp,
      family = beta_binomial2, stanvars = stanvars
    )

The summary output reveals that the uncertainty in the coefficients of `period` is somewhat larger than in the basic binomial model, which is the result of including the overdispersion parameter `phi` in the model. Aparat from that, the results looks pretty similar.

    summary(fit2)

     Family: beta_binomial2
      Links: mu = logit; phi = identity
    Formula: incidence | vint(size) ~ period + (1 | herd)
       Data: cbpp (Number of observations: 56)
    Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
             total post-warmup samples = 4000

    Group-Level Effects:
    ~herd (Number of levels: 15)
                  Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sd(Intercept)     0.40      0.26     0.02     0.97 1.00      802     1448

    Population-Level Effects:
              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    Intercept    -1.36      0.26    -1.89    -0.87 1.00     2999     2219
    period2      -1.01      0.41    -1.87    -0.22 1.00     3361     2442
    period3      -1.27      0.46    -2.21    -0.40 1.00     3637     2826
    period4      -1.55      0.52    -2.63    -0.60 1.00     3765     2622

    Family Specific Parameters:
        Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    phi    18.13     16.52     5.38    60.12 1.00     1279     1023

    Samples were drawn 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).

## Post-Processing Custom Family Models

Some post-proecssing methods such as `summary` or `plot` work out of the box for custom family models. However, there are three particularily important methods, which require additional input by the user. These are `pp_expect`, `posterior_predict` and `log_lik` computing predicted mean values, predicted response values, and log-likelihood values, respectively. They are not only relevant for their own sake, but also provide the basis of many other post-processing methods. For instance, we may be interested in comparing the fit of the binomial model with that of the beta-binomial model by means of approximate leave-one-out cross-validation implemented in method `loo`, which in turn requires `log_lik` to be working.

The `log_lik` function of a family should be named `log_lik_<family-name>` and have the two arguments `i` (indicating observations) and `draws`. You don’t have to worry too much about how `draws` is created. Instead, all you need to know is that parameters are stored in slot `dpars` and data are stored in slot `data`. Generally, parameters take on the form of a \(S \times N\) matrix (with \(S =\) number of posterior samples and \(N =\) number of observations) if they are predicted (as is `mu` in our example) and a vector of size \(N\) if the are not predicted (as is `phi`).

We could define the complete log-likelihood function R directly, or we can expose the self-defined Stan functions and apply them. The latter approach is usually more convenient, but the former is more stable and the only option when implementing custom families in other R packages building upon brms. For the purpose of the present vignette, we will go with the latter approach

    expose_functions(fit2, vectorize = TRUE)

and define the required `log_lik` functions with a few lines of code.

    log_lik_beta_binomial2 <- function(i, draws) {
      mu <- draws$dpars$mu[, i]
      phi <- draws$dpars$phi
      trials <- draws$data$vint1[i]
      y <- draws$data$Y[i]
      beta_binomial2_lpmf(y, mu, phi, trials)
    }

With that being done, all of the post-processing methods requiring `log_lik` will work as well. For instance, model comparison can simply be performed via

    loo(fit1, fit2)

    Output of model 'fit1':

    Computed from 4000 by 56 log-likelihood matrix

             Estimate   SE
    elpd_loo   -100.2 10.4
    p_loo        22.4  4.5
    looic       200.4 20.7
    ------
    Monte Carlo SE of elpd_loo is NA.

    Pareto k diagnostic values:
                             Count Pct.    Min. n_eff
    (-Inf, 0.5]   (good)     42    75.0%   831
     (0.5, 0.7]   (ok)       11    19.6%   203
       (0.7, 1]   (bad)       2     3.6%   31
       (1, Inf)   (very bad)  1     1.8%   15
    See help('pareto-k-diagnostic') for details.

    Output of model 'fit2':

    Computed from 4000 by 56 log-likelihood matrix

             Estimate   SE
    elpd_loo    -95.0  8.3
    p_loo        11.1  2.0
    looic       189.9 16.6
    ------
    Monte Carlo SE of elpd_loo is 0.1.

    Pareto k diagnostic values:
                             Count Pct.    Min. n_eff
    (-Inf, 0.5]   (good)     49    87.5%   676
     (0.5, 0.7]   (ok)        7    12.5%   182
       (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.

    Model comparisons:
         elpd_diff se_diff
    fit2  0.0       0.0
    fit1 -5.2       4.4

Since larger `ELPD` values indicate better fit, we see that the beta-binomial model fits somewhat better, although the corresponding standard error reveals that the difference is not that substantial.

Next, we will define the function necessary for the `posterior_predict` method:

    posterior_predict_beta_binomial2 <- function(i, draws, ...) {
      mu <- draws$dpars$mu[, i]
      phi <- draws$dpars$phi
      trials <- draws$data$vint1[i]
      beta_binomial2_rng(mu, phi, trials)
    }

The `posterior_predict` function looks pretty similar to the corresponding `log_lik` function, except that we are now creating random samples of the response instead of log-liklihood values. Again, we are using an exposed Stan function for convenience. Make sure to add a `...` argument to your `posterior_predict` function even if you are not using it, since some families require additional arguments. With `posterior_predict` to be working, we can engage for instance in posterior-predictive checking:

    pp_check(fit2)

When defining the `pp_expect` function, you have to keep in mind that it has only a `draws` argument and should compute the mean response values for all observations at once. Since the mean of the beta-binomial distribution is \(\text{E}(y) = \mu T\) definition of the corresponding `pp_expect` function is not too complicated, but we need to get the dimension of parameters and data in line.

    pp_expect_beta_binomial2 <- function(draws) {
      mu <- draws$dpars$mu
      trials <- draws$data$vint1
      trials <- matrix(trials, nrow = nrow(mu), ncol = ncol(mu), byrow = TRUE)
      mu * trials
    }

A post-processing method relying directly on `pp_expect` is `conditional_effects`, which allows to visualize effects of predictors.

    conditional_effects(fit2, conditions = data.frame(size = 1))

For ease of interpretation we have set `size` to 1 so that the y-axis of the above plot indicates probabilities.

## Turning a Custom Family into a Native Family

Family functions built natively into brms are saver to use and more convenient, as they require much less user input. If you think that your custom family is general enough to be useful to other users, please feel free to open an issue on GitHub so that we can discuss all the details. Provided that we agree it makes sense to implement your family natively in brms, the following steps are required (`foo` is a placeholder for the family name):

- In `family-lists.R`, add function `.family_foo` which should contain basic information about your family (you will find lots of examples for other families there).
- In `families.R`, add family function `foo` which should be a simple wrapper around `.brmsfamily`.
- In `stan-likelihood.R`, add function `stan_llh_foo` which provides the likelihood of the family in Stan language.
- If necessary, add self-defined Stan functions in separate files under `inst/chunks`.
- Add functions `posterior_predict_foo`, `pp_expect_foo` and `log_lik_foo` to `posterior_predict.R`, `pp_expect.R` and `log_lik.R`, respectively.
- If necessary, add distribution functions to `distributions.R`.