Week 2, Session 2 — Multiple imputation with mice

Course 3 — #courses

R. Heller

Note

Inference lab using the five-step template: Hypothesis → Visualise → Assumptions → Conduct → Conclude.

Learning objectives

• Run multiple imputation by chained equations with mice.
• Diagnose convergence with trace and density plots.
• Pool regression coefficients across imputations using Rubin’s rules.

Prerequisites

Session 1 of this week (MCAR/MAR/MNAR).

Background

Multiple imputation (MI) replaces each missing value with several plausible values drawn from a predictive distribution, producing m completed datasets. Each completed dataset is analysed separately; the results are pooled using Rubin’s rules so that the standard errors reflect both the within-imputation uncertainty (ordinary sampling variance) and the between-imputation uncertainty (the variance of the imputations themselves).

mice implements MI by chained equations: for each incomplete variable, it fits a conditional model given the others and draws imputations from the posterior predictive distribution of that model. The procedure iterates until the imputations stabilise. The standard diagnostics are trace plots (should mix and not drift) and density plots comparing observed and imputed values (should overlap but need not match exactly).

The inclusion of the outcome in the imputation model is not optional. Omitting it biases the imputations toward the null and attenuates the estimated effect in the pooled analysis.

Setup

library(tidyverse)
library(mice)
set.seed(42)
theme_set(theme_minimal(base_size = 12))

1. Hypothesis

In the mice::nhanes teaching dataset, the cholesterol outcome (chl) is related to age and BMI (bmi). We will estimate that relationship under MI.

2. Visualise

data(nhanes, package = "mice")
md.pattern(nhanes, plot = FALSE)

   age hyp bmi chl   
13   1   1   1   1  0
3    1   1   1   0  1
1    1   1   0   1  1
1    1   0   0   1  2
7    1   0   0   0  3
     0   8   9  10 27

nhanes |>
  mutate(missing_chl = is.na(chl)) |>
  ggplot(aes(bmi, chl, colour = missing_chl)) +
  geom_point(size = 2) +
  labs(x = "BMI", y = "Cholesterol", colour = "chl missing?")

3. Assumptions

MAR conditional on the variables in the imputation model; a compatible imputation model (predictive mean matching by default); enough imputations (here m = 10) to stabilise the pooled standard errors.

4. Conduct

imp <- mice(nhanes, m = 10, method = "pmm",
            printFlag = FALSE, seed = 42)
imp

Class: mids
Number of multiple imputations:  10 
Imputation methods:
  age   bmi   hyp   chl 
   "" "pmm" "pmm" "pmm" 
PredictorMatrix:
    age bmi hyp chl
age   0   1   1   1
bmi   1   0   1   1
hyp   1   1   0   1
chl   1   1   1   0

plot(imp)

densityplot(imp)

fit   <- with(imp, lm(chl ~ age + bmi))
pooled <- pool(fit)
summary(pooled, conf.int = TRUE)

         term   estimate std.error  statistic        df    p.value       2.5 %
1 (Intercept) -22.553452 60.422948 -0.3732597 17.319419 0.71348446 -149.855865
2         age  35.321205 11.255699  3.1380730  9.947218 0.01061075   10.223895
3         bmi   5.711016  2.036136  2.8048302 14.980225 0.01334229    1.370596
     97.5 %    conf.low conf.high
1 104.74896 -149.855865 104.74896
2  60.41852   10.223895  60.41852
3  10.05144    1.370596  10.05144

5. Concluding statement

Multiple imputation of the nhanes dataset (m = 10, predictive mean matching) gave a pooled coefficient for BMI on cholesterol of 5.71 (see pooled table above), with standard errors reflecting both within- and between-imputation uncertainty through Rubin’s rules.

Students should leave this session able to say out loud that “MI accounts for missing-data uncertainty” does not mean “MI fixes MNAR”. It does not.

Common pitfalls

• Imputing the outcome separately from the covariates with incompatible models.
• Using m = 5 when the fraction of missing information is high.
• Pooling means or SDs by hand instead of pool().
• Dropping the outcome from the imputation model because it “feels like cheating”.

Further reading

• van Buuren S, Groothuis-Oudshoorn K (2011), mice: Multivariate Imputation by Chained Equations in R.
• van Buuren S (2018), Flexible Imputation of Missing Data, ch. 4–6.
• White IR, Royston P, Wood AM (2011), Multiple imputation using chained equations: issues and guidance for practice.

Session info

sessionInfo()

R version 4.4.1 (2024-06-14)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.4 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0

locale:
 [1] LC_CTYPE=C.UTF-8       LC_NUMERIC=C           LC_TIME=C.UTF-8       
 [4] LC_COLLATE=C.UTF-8     LC_MONETARY=C.UTF-8    LC_MESSAGES=C.UTF-8   
 [7] LC_PAPER=C.UTF-8       LC_NAME=C              LC_ADDRESS=C          
[10] LC_TELEPHONE=C         LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C   

time zone: UTC
tzcode source: system (glibc)

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] mice_3.19.0     lubridate_1.9.5 forcats_1.0.1   stringr_1.6.0  
 [5] dplyr_1.2.1     purrr_1.2.2     readr_2.2.0     tidyr_1.3.2    
 [9] tibble_3.3.1    ggplot2_4.0.3   tidyverse_2.0.0

loaded via a namespace (and not attached):
 [1] gtable_0.3.6       shape_1.4.6.1      xfun_0.57          htmlwidgets_1.6.4 
 [5] lattice_0.22-6     tzdb_0.5.0         vctrs_0.7.3        tools_4.4.1       
 [9] Rdpack_2.6.6       generics_0.1.4     pan_1.9            pkgconfig_2.0.3   
[13] jomo_2.7-6         Matrix_1.7-0       RColorBrewer_1.1-3 S7_0.2.2          
[17] lifecycle_1.0.5    compiler_4.4.1     farver_2.1.2       codetools_0.2-20  
[21] htmltools_0.5.9    yaml_2.3.12        glmnet_5.0         pillar_1.11.1     
[25] nloptr_2.2.1       MASS_7.3-60.2      reformulas_0.4.4   iterators_1.0.14  
[29] rpart_4.1.23       boot_1.3-30        foreach_1.5.2      mitml_0.4-5       
[33] nlme_3.1-164       tidyselect_1.2.1   digest_0.6.39      stringi_1.8.7     
[37] labeling_0.4.3     splines_4.4.1      fastmap_1.2.0      grid_4.4.1        
[41] cli_3.6.6          magrittr_2.0.5     survival_3.6-4     broom_1.0.12      
[45] withr_3.0.2        scales_1.4.0       backports_1.5.1    timechange_0.4.0  
[49] rmarkdown_2.31     otel_0.2.0         nnet_7.3-19        lme4_2.0-1        
[53] hms_1.1.4          evaluate_1.0.5     knitr_1.51         rbibutils_2.4.1   
[57] rlang_1.2.0        Rcpp_1.1.1-1.1     glue_1.8.1         minqa_1.2.8       
[61] jsonlite_2.0.0     R6_2.6.1