Cohen’s d for TimePoint×Condition in Pooled Linear Mixed Models

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Summary

The issue involves a failure to correctly map effect size calculations to specific interaction terms within a pooled Linear Mixed Model (LMM) derived from multiple imputation. The user is attempting to calculate Cohen’s d for Estimated Marginal Means (EMMs) across multiple time points and conditions, but the current implementation fails to isolate the effect sizes for specific TimePoint $\times$ Condition interactions, instead returning a generalized or incorrectly scoped set of comparisons.

Root Cause

The technical failure stems from three primary architectural mismatches in the R implementation:

  • Scope Mismatch: The eff_size function in the emmeans package requires a specific contrast object or a subset of the emmeans object to map the effect size to the correct comparison.
  • Imputation Pooling Conflict: When using mice and lmer, the standard sigma (residual standard deviation) and edf (effective degrees of freedom) from a single model iteration do not accurately represent the pooled variance of the entire imputed dataset.
  • Interaction Complexity: The user is performing a factorial design (TimePoint * Condition). Calculating Cohen’s d for a simple main effect is trivial, but calculating it for an interaction-driven comparison requires the function to know which specific pairs (e.g., Condition 1 vs Condition 2 at Month 3) are being evaluated.

Why This Happens in Real Systems

In complex statistical modeling and production-grade data pipelines, this happens due to:

  • Implicit vs. Explicit Context: Functions often assume the user wants all possible pairwise comparisons unless a specific contrast matrix is provided.
  • Data Heterogeneity: In imputed datasets, variance is not a single scalar but a distribution across multiple datasets. Applying a single sigma from one model to the entire pooled result is a mathematical fallacy.
  • API Abstraction Leaks: High-level wrappers like emmeans abstract the underlying math, which leads users to believe that passing a model object is sufficient, when in reality, the underlying residuals must be mathematically compatible with the specific marginal means being compared.

Real-World Impact

  • Statistical Invalidity: Reporting Cohen’s d using a single-model sigma instead of a pooled estimate leads to inflated or deflated effect sizes, potentially causing Type I or Type II errors.
  • Misleading Clinical Conclusions: In medical research (implied by the RCT_id variable), an incorrect effect size can lead to the false conclusion that a treatment is more effective than it actually is.
  • Automated Pipeline Failure: In automated reporting systems, if the eff_size function returns a matrix of all permutations instead of the requested subset, downstream data parsers will crash or ingest incorrect values.

Example or Code

# Correct approach to isolate specific comparisons and use pooled parameters

# 1. Define the specific contrasts you want (e.g., Condition 1 vs 2 at each timepoint)
# Instead of applying eff_size to the whole emm object, 
# we subset or define the comparison first.

# Assuming 'mm' is the emmeans object: mm <- emmeans(model, ~ Condition | TimePoint)

# 2. Calculate the pooled residuals and degrees of freedom 
# Note: In a real production environment, sigma should be derived from 
# the pooled model results, not a single iteration.
pooled_res <- pool(model)
pooled_sigma <- sd(residuals(model)) # Simplified for demonstration
pooled_df <- df.residual(model)

# 3. Target specific comparisons to avoid the "all comparisons" output
# We use the emmeans object restricted to the interaction of interest
specific_comparisons <- contrast(mm, method = "pairwise", by = "TimePoint")

# 4. Apply effect size to the targeted contrasts
final_effect_sizes <- eff_size(specific_comparisons, sigma = pooled_sigma, edf = pooled_df)

summary(final_effect_sizes)

How Senior Engineers Fix It

  • Explicit Contrast Definition: Instead of relying on default behavior, senior engineers explicitly define a contrast matrix to ensure only the relevant comparisons (Condition A vs B at $T_n$) are calculated.
  • Variance Propagation: They ensure that the sigma parameter passed to the effect size function is the pooled standard deviation derived from the $M$ imputed datasets, rather than a single-model estimate.
  • Unit Testing Statistical Outputs: They implement checks to ensure the dimensions of the eff_size output matrix match the expected number of time points, preventing “dimension mismatch” errors in reporting.

Why Juniors Miss It

  • Over-reliance on Defaults: Juniors often assume that if a function accepts a model object, it “just knows” which specific interaction the user is interested in.
  • Ignoring the “Pooled” Aspect: They often treat imputed data as a single dataset, forgetting that multiple imputation requires specific pooling rules (like Rubin’s Rules) for all parameters, including standard deviations.
  • Confusion between Means and Contrasts: They fail to distinguish between the Estimated Marginal Means (the point estimates) and the Contrasts (the differences between those points), which is where the Cohen’s d calculation actually lives.

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