Handle missing values ​​with brms (2023)

Introduction

Many real-world datasets contain missing values ​​for various reasons. We usually have a few options to deal with these missing values. The easiest solution is to remove all rows from the data set where one or more variables are missing. However, if values ​​are not missing completely at random, this will likely introduce bias in our analysis. So we usually want to impute missing values ​​one way or another. Here, we will consider two very general approaches usingbrms: (1) Assign missing valuesbeforemodel fitting with multiple imputation and (2) imputation of missing values ​​in real timeduringmodel fit1. As a simple example, we will usenhanesdataset, which contains information about the participantsage,IMC(body mass index),hip(hypertensive) andchl(total serum cholesterol). For the purposes of this vignette, we are primarily interested in forecastingIMCwithagemchl.

data("nhanes",package = "rats")head(nhanes)

idade IMC hip chl1 1 NA NA NA2 22,7 1 1873 1 NA 1 1874 3 NA NA NA5 1 20,4 1 1136 3 NA NA 184

Imputation before fitting the model

There are several approaches that allow us to impute missing data before the actual model fitting takes place. From a statistical point of view, multiple imputation is one of the best solutions. Each missing value is not imputed once, butMtimes leading to a totalMfully imputed data sets. The model can then be fitted to each of these datasets separately and the results aggregated into models later. A widely implemented package for multiple imputation israts(Buuren & Groothuis-Oudshoorn, 2010) and will then use it in conjunction withbrms.Here we apply its default settingsrats, which means that all variables will be used to impute missing values ​​in all other variables and the imputation functions are automatically selected based on the characteristics of the variables.

library(rats)HELL BOY<- rats(nhanes,m = 5,print = FALSE)

Now, we havem = 5imputed datasets stored inHELL BOYobject. In practice, we will probably need more than5of them to accurately account for the uncertainty caused by the fault, perhaps even in its area100imputed datasets (Zhou & Reiter, 2010). Of course, this greatly increases the computational load, so we keep itm = 5for the purposes of this vignette. regardless of valueM, we can extract these datasets and then pass them to the actual model fitting function as a list of datasets or passesHELL BOYdirectly. The latter works becausebrmsoffers special support for data derived fromrats. We'll go with the latter approach since it's less typing. It matches the model we are interested inbrmsfor the various imputed data sets is straightforward.

fit_imp1<- brm_multiple(IMC~age*chl,data =HELL BOY,chains = 2)

The return mounted model is a common onebrmsfitobject containing everyone's next plansMsubmodels. While clustering between models is not necessarily straightforward in classical statistics, it is trivial in a Bayesian framework. Here, grouping results from different imputed datasets is achieved by simply combining the posterior designs of the submodels. Likewise, all post-processing methods can be used immediately, without having to worry about grouping.

summary(fit_imp1)

Family: Gaussian Links: mu = identity; sigma = identity Type: bmi ~ age * chl Data: imp (Number of observations: 25) Designs: 10 chains, each with iter = 2000; heating = 1000; skinny = 1; total metathermal draws = 10000 Population Level Effects: Estimate Estimate Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESSIntercept 13.40 8.16 -3.35 29.28 1.08 87 314 .87 314 1.69414 age . 2 43 chl 0.10 0.04 0.01 0.19 1.09 72 253age:chl -0.02 0.02 -0.07 0.02 1.10 67 198Family parameters: Estimate Estimate Error l-95% CI u-95% CI u-95%CI 0.08 4.60 1.26 28 89 The draws were random using (WALNUT) sampling. For each parameter, Bulk_ESS and Tail_ESS are effective measures of sample size, and Rhat is the potential downscaling factor in segregation chains (at convergence, Rhat = 1).

In the summary output, we notice that someAFFORDABLEvalues ​​are greater than\(1.1\)indicating potential convergence problems. For models based on multiple imputed data sets, this is usually afalse positives: Strings from different submodels may not overlap exactly as they match different data. We can see the chains on the right side of it

plot(fit_imp1,variable = "^b",regex = TRUE)

Such non-overlapping chains entail highAFFORDABLEvalues ​​without essentially having a convergence problem. Therefore, we need to investigate the convergence of the submodels separately, which we can do by looking

repetition(fit_imp1$that,2)

b_Intercept b_age b_chl b_age.chl sigma lprior lp__1 1,00 1,00 1,00 1,00 1,00 1,00 12 1,00 1,00 1,00 1,00 1,00 1,001 3 1,01 1,01 1,01 1,00 1. 00 14 1,00 1,00 1,00 1,00 1,01 1,01 15 1,00 1,00 1,00 1,00 1,00 1,00 1

The convergence of each of the submodels appears good. Consequently, we can proceed to post-processing and interpretation of the results. For example, we could investigate the combined effect ofagemchl.

conditional_effect(fit_imp1,"age:cl")

In summary, the advantages of multiple imputation are obvious: you can apply it to all types of models, since the model fitting functions do not need to know that the datasets have been imputed beforehand. We also do not need to worry about clustering between submodels when using fully Bayesian methods. The only downside is the time it takes to set up the model. Estimation of Bayesian models is already quite slow with a single data set and only gets worse when working with multiple imputations.

Billing when placing the model

Imputation during model fitting is generally considered more complex than imputation before model fitting because everything has to be considered in one step. This remains true when missing values ​​are imputedbrms, but probably to a slightly lesser extent. Think it over againnhanesdata for forecastingIMCwithage, echl. Fromageit doesn't contain missing values, we just have to be extra carefulIMCmchl.We need to tell the model two things. (1) Which variables contain missing values ​​and how they should be predicted, and (2) which of these imputed variables should be used as predictors. Inbrmswe can do it like this:

bform<- friend(IMC| m()~age* m(chl))+ friend(ch| m()~age)+ set_rescor(FALSE)fit_imp2<- brm(bforma,data =nhanes)

The model has become multivariate, as we not only predictIMCbut alsochl(edvignette ( "brms_multivariate" )for details on its multivariate syntaxbrms). We ensure that errors in both variables are modeled rather than excluded by addition| mi()on the left side of the types2. We wrotemi(chl)on the right side of the formula forIMCto ensure that the estimated missing valueschlwill be used in the forecastIMC. The summary is a bit more confusing as we get coefficients for both response variables, but other than that we can interpret the coefficients in the usual way.

summary(fit_imp2)

Family: MV (Gaussian, Gaussian) Links: mu = identity; sigma = identity mu = identity; sigma = identity Type: bmi | mi() ~ age * mi(chl) chl | mi() ~ age Data: nhanes (Number of observations: 25) Designs: 4 chains, each with iter = 2000; heating = 1000; skinny = 1; Total draws after warm = 4000 Population Level Effects: Estimate Estimate Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESSbmi_Intercept 13.90 8.81 -3.24 31.08 1.00 17413_19 1.31 19 0 ,26 1.00 2937 2826BMI_AGE 2.76 5.61 -8.42 13.59 1.00 1510 1614CHL_AGE 28.51 13.58 1.95 56.23 1.00 2846 2473BMI_MICHL 0 .10 0.05 0.01 0 .19 11,003 miles: -0.08 0.02 1.00 14 30 1815Family Specific Parameters: Estimate Estimate Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESSsigma_bmi 3.40 0.81 2.21 5.30 1.00 1510 2211sigma_chl 40.57 7.76 28.83 58.99 1, 00 2221 2780 samples used NUTSmplings. For each parameter, Bulk_ESS and Tail_ESS are effective measures of sample size, and Rhat is the potential downscaling factor in segregation chains (at convergence, Rhat = 1).

conditional_effect(fit_imp2,"age:cl",resp = "bmi")

The results look quite similar to those obtained by multiple calculations, but be aware that this may not be the case in general. In multiple imputation, the default is to impute all variables based on all other variables, while in the one-step approach, we must explicitly specify the variables used in the imputation. Thus, arguably, multiple imputation is easier to implement. An obvious advantage of the "one-step" approach is that the model only needs to be adjusted once and notMtimes. Also, within itbrmsframework, we can use multilevel structure and complex non-linear relationships to impute missing values, which is not so easily achieved in standard multiple imputation software. On the other hand, it is not currently possible to impute discrete variables becauseStan(the engine behindbrms) does not allow the estimation of discrete parameters.

nhanes$com<- reprint(nrow(nhanes),2)

bform<- friend(IMC| m()~age* m(chl))+ friend(ch| m(com)~age)+ set_rescor(FALSE)fit_imp3<- brm(bforma,data =nhanes)

bibliographical references

Buuren, S.V. & Groothuis-Oudshoorn, K. (2010). mice: Multivariate imputation with chained equations in R.Journal of Statistical Software, 1-68. doi.org/10.18637/jss.v045.i03

Zhou, X. & Reiter, J.P. (2010). A note on Bayesian inference after multiple imputation.The American statistician, 64(2),159-163. doi.org/10.1198/tast.2010.09109