Imputation before model fitting
There are many approaches allowing us to impute missing data before the actual model fitting takes place. From a statistical perspective, multiple imputation is one of the best solutions. Each missing value is not imputed once but m times leading to a total of m fully imputed data sets. The model can then be fitted to each of those data sets separately and results are pooled across models, afterwards. One widely applied package for multiple imputation is mice (Buuren & Groothuis-Oudshoorn, 2010) and we will use it in the following in combination with brms. Here, we apply the default settings of mice, which means that all variables will be used to impute missing values in all other variables and imputation functions automatically chosen based on the variables’ characteristics.
Now, we have m = 5 imputed data sets stored within the imp object. In practice, we will likely need more than 5 of those to accurately account for the uncertainty induced by the missingness, perhaps even in the area of 100 imputed data sets (Zhou & Reiter, 2010). Of course, this increases the computational burden by a lot and so we stick to m = 5 for the purpose of this vignette. Regardless of the value of m, we can either extract those data sets and then pass them to the actual model fitting function as a list of data frames, or pass imp directly. The latter works because brms offers special support for data imputed by mice. We will go with the latter approach, since it is less typing. Fitting our model of interest with brms to the multiple imputed data sets is straightforward.
The returned fitted model is an ordinary brmsfit object containing the posterior samples of all m submodels. While pooling across models is not necessarily straightforward in classical statistics, it is trivial in a Bayesian framework. Here, pooling results of multiple imputed data sets is simply achieved by combining the posterior samples of the submodels. Accordingly, all post-processing methods can be used out of the box without having to worry about pooling at all.
Family: gaussian
Links: mu = identity; sigma = identity
Formula: bmi ~ age * chl
Data: imp (Number of observations: 25)
Samples: 10 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup samples = 10000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 15.96 7.76 1.03 31.53 1.10 67 418
age 0.54 4.66 -8.60 9.67 1.06 125 1257
chl 0.08 0.04 -0.00 0.17 1.16 41 211
age:chl -0.02 0.02 -0.06 0.03 1.08 89 744
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 3.21 0.71 2.11 4.84 1.41 21 106
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).In the summary output, we notice that some Rhat values are higher than \(1.1\) indicating possible convergence problems. For models based on multiple imputed data sets, this is often a false positive: Chains of different submodels may not overlay each other exactly, since there were fitted to different data. We can see the chains on the right-hand side of
Such non-overlaying chains imply high Rhat values without there actually being any convergence issue. Accordingly, we have to investigate the convergence of the submodels separately, which we can do by looking at
b_Intercept b_age b_chl b_age.chl sigma lp__
1 1.00 1.00 1.00 1.00 1.00 1.00
2 1.00 1.00 1.00 1.00 1.00 1.00
3 1.01 1.01 1.01 1.01 1.00 1.00
4 1.00 1.00 1.00 1.00 1.00 1.01
5 1.00 1.00 1.00 1.00 1.01 1.01The convergence of each of the submodels looks good. Accordingly, we can proceed with further post-processing and interpretation of the results. For instance, we could investigate the combined effect of age and chl.
To summarize, the advantages of multiple imputation are obvious: One can apply it to all kinds of models, since model fitting functions do not need to know that the data sets were imputed, beforehand. Also, we do not need to worry about pooling across submodels when using fully Bayesian methods. The only drawback is the amount of time required for model fitting. Estimating Bayesian models is already quite slow with just a single data set and it only gets worse when working with multiple imputation.
Compatibility with other multiple imputation packages
brms offers built-in support for mice mainly because I use the latter in some of my own research projects. Nevertheless, brm_multiple supports all kinds of multiple imputation packages as it also accepts a list of data frames as input for its data argument. Thus, you just need to extract the imputed data frames in the form of a list, which can then be passed to brm_multiple. Most multiple imputation packages have some built-in functionality for this task. When using the mi package, for instance, you simply need to call the mi::complete function to get the desired output.
