ctree_stacked: Conditional Inference Tree on Stacked Multiply Imputed Data

In ctreeMI: Conditional Inference Trees with Stacked Multiple Imputation

Conditional Inference Tree on Stacked Multiply Imputed Data

Description

Fits a conditional inference tree (ctree) on stacked multiply imputed datasets using the Stack / M rescaling procedure described in Sherlock et al. (2026). Multiply imputed datasets are concatenated vertically ("stacked"), and the significance threshold used for node-level pruning is divided by the number of imputations M to counteract the artificially inflated sample size. This yields a single, coherent, interpretable tree that incorporates imputation variability without requiring the pooling of structurally different trees.

Usage

ctree_stacked(formula, data, m = NULL, alpha = 0.05, verbose = TRUE, ...)

Arguments

formula	A model formula, passed to ctree.
data	A mids object from mice, a list of imputed data frames, or a single complete data frame. If a single complete data frame is supplied, the function falls back to a standard ctree call with a warning.
m	Integer. Number of imputations to use. Defaults to all available datasets. Ignored when data is a plain data frame.
alpha	Numeric. Nominal significance threshold for node-level splitting (default 0.05). The corrected threshold actually applied is alpha / m. Must be strictly between 0 and 1.
verbose	Logical. If TRUE (default), prints a message summarizing the stacking and correction applied.
...	Additional arguments passed to ctree_control.

Details

Methodological background

Conditional inference trees (Hothorn, Hornik & Zeileis, 2006) use permutation-based significance tests to select splits, providing built-in protection against spurious partitioning. When data are multiply imputed, pooling trees fitted to separate imputations is infeasible: structurally different trees define different subgroups, making the targets of inference incomparable across imputations.

Rodgers et al. (2021) proposed stacking the M imputed datasets and fitting a single tree to the combined data. This produces one coherent, interpretable tree. The complication is that stacking inflates the nominal sample size by M, causing test statistics at each node to be similarly inflated.

Sherlock et al. (2026) proposed and validated the Stack / M correction: use a significance threshold of alpha / M. Monte Carlo simulations under MCAR confirmed sub-nominal (conservative) type-I error and acceptable power, making this approach well-suited for exploratory analyses where interpretability is prioritised.

Value

An object of class c("ctreeMI", "constparty", "party"). All partykit methods (plot, predict, etc.) work on this object. An additional ctreeMI_info attribute carries:

m

Number of imputations used.

n_original

Rows in one imputed dataset.

n_stacked

Total rows in the stacked dataset (M \times n).

alpha_nominal

Nominal alpha supplied by the user.

alpha_applied

Corrected alpha applied (alpha / m).

formula

The model formula.

call

The matched call.

Node-level sample sizes reported by print() and plot() reflect the stacked dataset. Divide by M to obtain effective per-node counts in the original data.

References

Sherlock, P., Mansolf, M., Hofheimer, J., Hockett, C. W., O'Connor, T. G., Roubinov, D., Graff, J. C., Lai, J.-S., Bush, N. R., Wright, R. J., & Chiu, Y.-H. M. (2026). Beyond linear risk: A machine learning approach to understanding perinatal depression in context. Multivariate Behavioral Research, 1–16. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00273171.2026.2661244")}

Hothorn, T., Hornik, K., & Zeileis, A. (2006). Unbiased recursive partitioning: A conditional inference framework. Journal of Computational and Graphical Statistics, 15(3), 651–674. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1198/106186006X133933")}

Hothorn, T., & Zeileis, A. (2015). partykit: A modular toolkit for recursive partitioning in R. Journal of Machine Learning Research, 16, 3905–3909.

Rodgers, J., Khoo, S.-T., & Ludtke, O. (2021). Handling missing data in structural equation models using multiple imputation and stacking. Structural Equation Modeling, 28(6), 915–930.

See Also

ctree, ctree_control, mice, stack_imputations, rescale_alpha, print.ctreeMI, summary.ctreeMI

Examples

  library(mice)

  # Introduce missingness into the airquality dataset
  set.seed(42)
  aq <- airquality
  aq$Ozone[sample(nrow(aq), 20)]   <- NA
  aq$Solar.R[sample(nrow(aq), 15)] <- NA

  # Impute (M = 20)
  imp <- mice(aq, m = 20, printFlag = FALSE)

  # Fit ctree with Stack/M correction
  fit <- ctree_stacked(Ozone ~ Solar.R + Wind + Temp + Month,
                       data  = imp,
                       alpha = 0.05)
  print(fit)
  plot(fit)


# Example using a list of data frames (no mice required)
set.seed(1)
make_df <- function(i) {
  set.seed(i)
  n  <- 100
  x1 <- rnorm(n)
  y  <- x1 + rnorm(n)
  data.frame(y = y, x1 = x1)
}
imp_list <- lapply(1:10, make_df)
fit <- ctree_stacked(y ~ x1, data = imp_list, alpha = 0.05, verbose = FALSE)
print(fit)

ctreeMI documentation built on July 26, 2026, 1:06 a.m.