Dealing with missing data in fitting prediction rule ensembles

Introduction

To deal with missing data in parametric models, multiple imputation is the golden standard (Schafer & Graham, 2002). With GLMs, the models fitted on each imputed dataset can then be pooled. For non-parametric methods and specifically prediction rule ensembles, the jury is still out on how to best deal with missing data. There are several possible approaches:

Listwise deletion. Although the default in pre(), it is certainly the least favorable option.
Single imputation: Perform only a single imputation and fit a prediction rule ensemble on this single dataset. This is likely better than listwise deletion, but likely inferior to multiple imputation. The main advantage is that a PRE can be fit as usual.
Multiple imputation followed by pooling. This approach takes multiple imputed datasets and fits a separate prediction rule ensemble to each of the imputed datasets, which are aggregated to form a final ensemble. The main disadvantage is that this yields an ensemble of ensembles, which will be (much) more complex and thus less interpretable than a single PRE.
Multiple imputation followed by stacking. This approach stacks the multiple imputed datasets and fits a single prediction rule ensemble to this dataset. Rule generation and estimation of the final model can be adjusted to counter the artificial inflation of sample size. The main advantage is that it yields a single PRE, while treating missing data in an optimal manner.

An alternative, would be the Missing-In-Attributes approach for rule generation, followed by mean imputation for the final model. According to Josse et al. (2019), mean imputation and the Missing-In-Attributes approaches perform well from a prediction perspective and are computationally inexpensive. This will be implemented in future versions of package pre.

Below, we provide examples for the four approaches described above.

Example: Predicting wind speed

For the examples, we will be predicting Wind speeds using the airquality dataset (we focus on predicting the wind variable, because it does not have missing values, while variables Ozone and Solar.R do):

head(airquality)
nrow(airquality)
library("mice")
md.pattern(airquality, rotate.names = TRUE)

Listwise Deletion

This option, although not recommended, is the default of function pre():

library("pre")
set.seed(43)
airq.ens <- pre(Wind ~., data = airquality)

## Warning in pre(Wind ~ ., data = airquality): Some observations have missing values and have been removed from the data. New sample size is 111.

airq.ens

## 
## Final ensemble with cv error within 1se of minimum: 
## 
##   lambda =  0.5076221
##   number of terms = 2
##   mean cv error (se) = 9.955139 (1.553628)
## 
##   cv error type : Mean-Squared Error
## 
##          rule   coefficient           description
##   (Intercept)   9.034190233                     1
##         rule8   1.743533723           Ozone <= 45
##         Ozone  -0.006180118  6.75 <= Ozone <= 119

With listwise deletion, only 111 out of 153 observations are retained. We obtain a rather sparse ensemble.

Single Imputation

Here we apply single imputation by replacing missing values with the mean:

imp0 <- airquality
imp0$Solar.R[is.na(imp0$Solar.R)] <- mean(imp0$Solar.R, na.rm=TRUE)
imp0$Ozone[is.na(imp0$Ozone)] <- mean(imp0$Ozone, na.rm=TRUE)
set.seed(43)
airq.ens.imp0 <- pre(Wind ~., data = imp0)
airq.ens.imp0

## 
## Final ensemble with cv error within 1se of minimum: 
## 
##   lambda =  0.2751573
##   number of terms = 5
##   mean cv error (se) = 9.757455 (0.7622478)
## 
##   cv error type : Mean-Squared Error
## 
##          rule  coefficient                 description
##   (Intercept)  10.48717592                           1
##         rule2   1.18133248                 Ozone <= 45
##        rule48  -0.56453456  Temp > 72 & Solar.R <= 258
##        rule28  -0.51357673      Temp > 73 & Ozone > 22
##         Ozone  -0.01910646         7 <= Ozone <= 115.6
##        rule40   0.01440472                  Temp <= 81

We obtain a larger number of rules, and slightly lower cross-validated mean squared error. However, this model cannot really be compared with the listwise deletion model, because they are computed over different sets of observations.

Multiple Imputation

We first perform multiple imputation by chained equations, using the predictive mean matching method. We generate 5 imputed datasets:

set.seed(42)
imp <- mice(airquality, m = 5, printFlag = FALSE)

We create a list with imputed datasets:

imp1 <- complete(imp, action = "all", include = FALSE)

To deal with imputed data, we have two options: Stacking the imputed datasets or pooling the resulting ensembles. The former is likely most beneficial for retaining interpretability and therefore an experimental (!) version has been implemented in package pre.

Stacking imputed datasets

Function mi_pre implements a so-called stacking approach (see also Wood et al., 2008), where imputed datasets are combined into one large dataset. In addition to adjustments of the sampling procedures, adjustments to observation weight are made to counter the artificial inflation of sample size. Function mi_pre takes a list of imputed datasets as input data, because it is assumed imputation has already been performed.

Although the option to use the fraction of complete data for computing observation weight is provided through argument compl_frac of function mi_pre, users are not advised to use it. For example, Du et al. (2022) write: “An alternative weight specification, proposed in Wan et al. (2015), is \(o_i = \frac{f_i}{D}\), where \(f_i\) is the number of observed predictors out of the total number of predictors for subject \(i\) […] upweighting subjects with less missingness and downweighting subjects with more missingness can, in some sense, be viewed as making the optimization more like complete-case analysis, which might be problematic for Missing at Random (MAR) and Missing not at Random (MNAR) scenarios.”

We fit a rule ensemble to the imputed data using stacking:

set.seed(42)
airq.ens.sta <- mi_pre(Wind ~ . , data = imp1)

All S3 methods for objects of class pre are also applicable to the ensembles resulting from application of function mi_pre, e.g.:

summary(airq.ens.sta)

## 
## Final ensemble with cv error within 1se of minimum: 
## 
##   lambda =  0.3397143
##   number of terms = 6
##   mean cv error (se) = 9.821413 (1.103229)
## 
##   cv error type : Mean-Squared Error

coefs <- coef(airq.ens.sta)
coefs[coefs$coefficient != 0, ]

##           rule coefficient       description
## 49 (Intercept)  9.44439871                 1
## 6        rule6  1.29750892       Ozone <= 45
## 20      rule23  0.17877849        Temp <= 73
## 2        rule2  0.15142541       Ozone <= 21
## 5        rule5  0.13556989       Ozone <= 47
## 10      rule11  0.03251171       Ozone <= 52
## 50       Ozone -0.01401860 7 <= Ozone <= 118

Computation of partial dependence plots can become computationally prohibitive, especially with multiply-imputed data. To reduce computational load, function mi_mean computes the average imputed dataset, which can then be supplied to the newdata arguments of functions singleplot or pairplot to speed up computation of partial dependence plots:

newdata <- mi_mean(imp1)
singleplot(airq.ens.sta, newdata = newdata, varname = "Ozone")

pairplot(airq.ens.sta, newdata = newdata, varnames = c("Ozone", "Solar.R"))

## Loading required namespace: interp

Pooling

To apply pooling, we create a custom function that fits PREs to several datasets contained in a list:

pre.agg <- function(datasets, ...) {
  result <- list()
  for (i in 1:length(datasets)) {
    result[[i]] <- pre(datasets[[i]], ...)
  }
  result
}

We apply the new function:

set.seed(43)
airq.agg <- pre.agg(imp1, formula = Wind ~ .)

Note that we can used the ellipsis (...) to pass arguments to pre (see ?pre for an overview of arguments that can be specified).

We now define print, summary, predict and coef methods to extract results from the fitted ensemble. Again, we can use the ellipsis (...) to pass arguments to the print, summary, predict and coef methods of function pre (see e.g., ?pre:::print.pre for more info):

print.agg <- function(object, ...) {
  result <- list()
  sink("NULL")
  for (i in 1:length(object)) {
    result[[i]] <- print(object[[i]], ...)
  }
  sink()
  print(result)
}
print.agg(airq.agg) ## results suppressed for space considerations

summary.agg <- function(object, ...) {
  for (i in 1:length(object)) summary(object[[i]], ...)
}
summary.agg(airq.agg) ## results suppressed for space considerations

For averaging over predictions, there is only one option for continuous outcomes. For non-continuous outcomes, we can average over the linear predictor, or over the predicted values on the scale of the response. I am not sure which would be more appropriate; the resulting predicted values will not be identical, but highly correlated, though.

predict.agg <- function(object, newdata, ...) {
  rowMeans(sapply(object, predict, newdata = newdata, ...))
}
agg_preds <- predict.agg(airq.agg, newdata = airquality[1:4, ])
agg_preds

##        1        2        3        4 
## 10.42757 10.59272 10.93324 11.00302

Finally, the coef method should return the averaged / aggregated final PRE. That is, it returns:

One averaged intercept;
All rules and linear terms, with their coefficients scaled by the number of datasets;
In presence of identical rules and linear terms, it aggregates those rules and their coefficients into one rule / term, and adds together the scaled coefficients.

Note that linear terms of the same predictors, which obtained different winsorizing points across imputed datasets will be retained seperately and will not be aggregated. Note also that the labels of rules and variables may overlap between different datasets (e.g., the label rule 12 may appear multiple times in the aggregated ensemble, but each rule 12 will have different conditions).

coef.agg <- function(object, ...) {
  coefs <- coef(object[[1]], ...)
  coefs <- coefs[coefs$coefficient != 0, ]
  for (i in 2:length(object)) {
    coefs_tmp <- coef(object[[i]], ...)
    coefs <- rbind(coefs, coefs_tmp[coefs_tmp$coefficient != 0, ])
  }
  ## Divide coefficients by the number of datasets:
  coefs$coefficient <- coefs$coefficient / length(object)
  ## Identify identical rules:
  duplicates <- which(duplicated(coefs$description))
  for (i in duplicates) {
    first_match <- which(coefs$description == coefs$description[i])[1]
    ## Add the coefficients:
    coefs$coefficient[first_match] <- 
      coefs$coefficient[first_match] + coefs$coefficient[i]
  }
  ## Remove duplicates:
  if (length(duplicates) > 0) coefs <- coefs[-duplicates, ]
  ## Check if there are- duplicate linear terms left and repeat:
  duplicates <- which(duplicated(coefs$rule))
  for (i in duplicates) {
    first_match <- which(coefs$rule == coefs$rule[i])[1]
    coefs$coefficient[first_match] <- 
      coefs$coefficient[first_match] + coefs$coefficient[i]
  }
  if (length(duplicates) > 0) coefs <- coefs[-duplicates, ]
  ## Return results:
  coefs
}
coef.agg(airq.agg)

##           rule  coefficient                 description
## 65 (Intercept)  9.433571984                           1
## 29      rule32  0.097328032      Temp > 73 & Ozone > 23
## 3        rule3  0.968746033                 Ozone <= 45
## 7        rule7  0.040507812 Ozone > 14 & Solar.R <= 238
## 66       Ozone -0.006089757         7 <= Ozone <= 115.6
## 6        rule6  0.074835353                 Ozone <= 59
## 25      rule26 -0.067771974      Temp > 63 & Ozone > 14
## 17      rule17 -0.038092316      Temp > 75 & Ozone > 47
## 1        rule1  0.090749273                 Ozone <= 21
## 58      rule62 -0.277510244 Ozone > 45 & Solar.R <= 275
## 36      rule39 -0.046938032 Ozone > 14 & Solar.R <= 201
## 40      rule43 -0.035213431  Temp > 72 & Solar.R <= 255
## 51       rule5  0.031821453                 Ozone <= 52
## 10      rule10  0.070056367                  Temp <= 73
## 19      rule22  0.066962331                 Ozone <= 63
## 14      rule15  0.001044073 Ozone <= 45 & Solar.R > 212

We have obtained a final ensemble of 15 terms.

Comparing accuracy and sparsity

We compare performance using 10-fold cross validation. We evaluate predictive accuracy and the number of selected rules. We only evaluate accuracy for observations that have no missing values.

k <- 10
set.seed(43)
fold_ids <- sample(1:k, size = nrow(airquality), replace = TRUE)

observed <- c()
for (i in 1:k) {
  ## Separate training and test data
  test <- airquality[fold_ids == i, ]
  test <- test[!is.na(test$Ozone), ]
  test <- test[!is.na(test$Solar.R), ]
  observed <- c(observed, test$Wind)
}  

preds <- data.frame(observed)
preds$LWD <- preds$SI <- preds$MI <- preds$observed
nterms <- matrix(nrow = k, ncol = 3)
colnames(nterms) <- c("LWD", "SI", "MI")
row <- 1

for (i in 1:k) {

  if (i > 1) row <- row + nrow(test)
  
  ## Separate training and test data
  train <- airquality[fold_ids != i, ]
  test <- airquality[fold_ids == i, ]
  test <- test[!is.na(test$Ozone), ]
  test <- test[!is.na(test$Solar.R), ]

  ## Fit and evaluate listwise deletion
  premod <- pre(Wind ~ ., data = train)
  preds$LWD[row:(row+nrow(test)-1)] <- predict(premod, newdata = test)
  tmp <- print(premod)
  nterms[i, "LWD"] <- nrow(tmp) - 1
  
  ## Fit and evaluate single imputation
  imp0 <- train
  imp0$Solar.R[is.na(imp0$Solar.R)] <- mean(imp0$Solar.R, na.rm=TRUE)
  imp0$Ozone[is.na(imp0$Ozone)] <- mean(imp0$Ozone, na.rm=TRUE)
  premod.imp0 <- pre(Wind ~., data = imp0)
  imp1 <- test
  imp1$Solar.R[is.na(imp1$Solar.R)] <- mean(imp0$Solar.R, na.rm=TRUE)
  imp1$Ozone[is.na(imp1$Ozone)] <- mean(imp0$Ozone, na.rm=TRUE)
  preds$SI[row:(row+nrow(test)-1)] <- predict(premod.imp0, newdata = imp1)  
  tmp <- print(premod.imp0)
  nterms[i, "SI"] <- nrow(tmp) - 1
  
  ## Perform multiple imputation
  imp <- mice(train, m = 5)
  imp1 <- complete(imp, action = "all", include = FALSE)
  airq.agg <- pre.agg(imp1, formula = Wind ~ .)
  preds$MI[row:(row+nrow(test)-1)] <- predict.agg(airq.agg, newdata = test)
  nterms[i, "MI"] <- nrow(coef.agg(airq.agg)) - 1

}

sapply(preds, function(x) mean((preds$observed - x)^2)) ## MSE

##  observed        MI        SI       LWD 
##  0.000000  9.462657 10.087872  9.824913

sapply(preds, function(x) sd((preds$observed - x)^2)/sqrt(nrow(preds))) ## SE of MSE

## observed       MI       SI      LWD 
## 0.000000 1.472118 1.538254 1.499660

var(preds$observed) ## benchmark: Predict mean for all

## [1] 12.65732

Interestingly, we see that all three methods yield similar predictions and accuracy, and explain about 20% of variance in the response. Multiple imputation performed best, followed by listwise deletion, followed by single imputation. Taking into account the standard errors, however, these differences are not significant. Also, this simple evaluation on only a single dataset should not be taken too seriously. The better performance of multiple imputation does come at the cost of increased complexity:

boxplot(nterms, main = "Number of selected terms \nper missing-data method",
        cex.main = .8)

In line with findings of Josse et al. (2019), we expect MIA to work better for rules than mean imputation. In future versions of package pre, we plan to implement MIA (for the rules) and combine it with mean imputation (for the linear terms).

Session info

In case you obtained different results, these results were obtained using the following:

sessionInfo()

## R version 4.3.1 (2023-06-16 ucrt)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 19045)
## 
## Matrix products: default
## 
## 
## locale:
## [1] LC_COLLATE=C                       LC_CTYPE=Dutch_Netherlands.utf8   
## [3] LC_MONETARY=Dutch_Netherlands.utf8 LC_NUMERIC=C                      
## [5] LC_TIME=Dutch_Netherlands.utf8    
## 
## time zone: Europe/Amsterdam
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] pre_1.0.7   mice_3.16.0
## 
## loaded via a namespace (and not attached):
##  [1] sass_0.4.7         utf8_1.2.3         generics_0.1.3     tidyr_1.3.0       
##  [5] shape_1.4.6        stringi_1.7.12     lattice_0.21-8     inum_1.0-5        
##  [9] plotmo_3.6.2       lme4_1.1-35.1      digest_0.6.31      magrittr_2.0.3    
## [13] mitml_0.4-5        evaluate_0.23      grid_4.3.1         iterators_1.0.14  
## [17] mvtnorm_1.2-3      fastmap_1.1.1      foreach_1.5.2      jomo_2.7-6        
## [21] jsonlite_1.8.4     glmnet_4.1-8       Matrix_1.6-2       nnet_7.3-19       
## [25] backports_1.4.1    Formula_1.2-5      survival_3.5-5     purrr_1.0.1       
## [29] fansi_1.0.4        codetools_0.2-19   jquerylib_0.1.4    cli_3.6.0         
## [33] rlang_1.1.1        splines_4.3.1      plotrix_3.8-4      cachem_1.0.8      
## [37] yaml_2.3.7         pan_1.9            tools_4.3.1        deldir_1.0-9      
## [41] MatrixModels_0.5-3 earth_5.3.2        nloptr_2.0.3       minqa_1.2.6       
## [45] dplyr_1.1.2        interp_1.1-4       boot_1.3-28.1      broom_1.0.5       
## [49] rpart_4.1.19       vctrs_0.6.3        R6_2.5.1           lifecycle_1.0.4   
## [53] stringr_1.5.1      libcoin_1.0-10     MASS_7.3-60        partykit_1.2-20   
## [57] pkgconfig_2.0.3    pillar_1.9.0       bslib_0.5.1        TeachingDemos_2.12
## [61] glue_1.6.2         Rcpp_1.0.10        highr_0.10         xfun_0.41         
## [65] tibble_3.2.1       tidyselect_1.2.0   rstudioapi_0.15.0  knitr_1.45        
## [69] htmltools_0.5.7    nlme_3.1-162       rmarkdown_2.25     compiler_4.3.1

References

Du, J., Boss, J., Han, P., Beesley, L. J., Kleinsasser, M., Goutman, S.A., … & Mukherjee, B. (2022). Variable selection with multiply-imputed datasets: choosing between stacked and grouped methods. Journal of Computational and Graphical Statistics, 31(4), 1063-1075.

Josse, J., Prost, N., Scornet, E., & Varoquaux, G. (2019). On the consistency of supervised learning with missing values. arXiv preprint arXiv:1902.06931. https://arxiv.org/abs/1902.06931

Miles, A. (2016). Obtaining predictions from models fit to multiply imputed data. Sociological Methods & Research, 45(1), 175-185.

Schafer, J. L., & Graham, J. W. (2002). Missing data: our view of the state of the art. Psychological Methods, 7(2), 147.

Wood, A. M., White, I. R., & Royston, P. (2008). How should variable selection be performed with multiply imputed data? Statistics in Medicine, 27(17), 3227-3246.