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):