Supplementary Code: Translated CES and Epistemic Deduction for Non-Compensatory Composite Indicators
Note for Double-Blind Peer Reviewers
This file and the accompanying `Supplementary_Code.r` script have been provided via the journal's Editorial Manager strictly for reviewer verification purposes for submission to Quality & Quantity. To preserve strict double-blind anonymity during the peer-review process, this code is hosted locally within the submission system. Upon acceptance, a permanent, publicly accessible repository with a citable DOI will be generated and linked in the final publication.
In the interim, reviewers are highly encouraged to run the Minimal Working Example (MWE) via the provided `MWE.r` script to verify the Monte Carlo rank-stability claims, the Epistemic Shielding Trap corrections, and the new Empirical Proof of Concept detailed in the manuscript.
Overview
Recent literature on multidimensional social progress has strongly favored "strong dual necessity" frameworks, utilizing the Constant Elasticity of Substitution (CES) function to prevent severe deficits in objective or subjective well-being from being masked by excellence. However, operationalizing this requires resolving scale incomparability, which is conventionally done via destructive Z-score standardization or relative Min-Max normalization—inherently destroying the absolute policy zero-point.
This computational engine (Online Resource 1 for the accompanying manuscript) provides a rigorous mathematical alternative that preserves the absolute normative zero-point by:
Aggregating negative, threshold-anchored scores using a Global Normative-Anchor Non-Homothetic (Quasi-Homothetic/Stone-Geary) Translated CES with Baseline Subtraction. This bypasses the `NaN` domain collapse of standard CES functions while mathematically guaranteeing the preservation of the Leontief bottleneck.
Deriving domain-safeguard translation buffers ($C_{\text{global}}$) via the Normative Absolute Floor of the indicators, guaranteeing $X_d > 0$ algebraically without ad hoc censoring artifacts.
Applying a certainty-adjusted pre-screening mechanism to explicitly penalize statistical ignorance exclusively at the bottleneck dimension without violating the geometry of the aggregator.
(Note: Functions 1 & 2 in the R script calculate a diagnostic for L-statistic cardinality bias in lower-tail aggregation. While the main manuscript focuses strictly on the dimension-to-index CES aggregation pipeline, these functions are provided here as a Supplementary Pre-processing Module for practitioners who require complete indicator-to-dimension replication. To avoid the"adequacy inversion" artifact, the shrinkage estimator intentionally bypasses shrinkage on the absolute score, delegating uncertainty penalization to the CES pipeline. To prevent single-indicator positive shock vulnerabilities while strictly respecting formative measurement theory, the lower-tail aggregation departs from strict OWA anonymity and natively implements a Lower-Tail Trimmed Mean (20% Trim), alongside a strict `NA` guard to enforce the manuscript's mandate for rigorous pre-imputation.)
Important Methodological Note: The Epistemic Shielding Trap
$C_{\text{global}}$ acts strictly as a normative/computational buffer against domain collapse, not a normative policy threshold. To address the Epistemic Shielding Trap (where noisy dimensions inadvertently shield regions from Leontief bottlenecks if buffers are derived from empirical variance/noise), do not adjust the inner CES weights, as this violates strong dual necessity by granting compensability via omission.
Instead, calculate the standard Translated CES index using uniform/normative weights, and apply the certainty-adjusted pre-screening to the input space prior to aggregation:
$$Y_{\text{final}} = \max\left(L_{\text{min}}, Y_{\text{CES}}(X^*)\right)$$
where the CES aggregation is natively evaluated on the pre-screened space $X^*$ to ensure mathematical consistency and avoid cross-space normative distortions. The deduction is defined as the exact geometric shift between the unadjusted baseline $Y_{\text{CES}}(X)$ and the pre-screened aggregate $Y_{\text{CES}}(X^*)$.
Note on Missing Data & Leontief Capture
The framework assumes inputs are rigorously imputed prior to aggregation. To satisfy the Max-Min ambiguity-averse framework for idiosyncratic missingness, practitioners must utilize Pessimistic Single Imputation (drawing from the lower tail of the posterior predictive distribution, e.g., the 10th percentile) rather than standard Multiple Imputation (MI) pooling. Standard MI pooling (Rubin's Rules) violates the axioms of Max-Min Expected Utility (MEU) in strictly non-linear, non-compensatory spaces, as pooling averages the worst-case scenarios across imputations. The supplementary code now includes a `pessimistic_impute()` helper function to facilitate this.
Warning on Joint Worst-Case Collapse: If a severely deprived region suffers from simultaneous MNAR missingness across multiple dimensions, applying the 10th percentile independently to each dimension assumes a perfect positive correlation in the worst-case errors. This will crash the index to the absolute floor. To resolve this mathematically, the supplementary code now includes a `pessimistic_impute_multivariate()` function utilizing `TruncatedNormal::mvrandn` to approximate the deterministic conditional expectation of a multivariate truncated normal distribution (via large-sample averaging), preserving the MEU ambiguity premium while preventing artificial floor-crashing.
Computational Efficiency Warning
The `null_expectation_sd` function runs Monte Carlo simulations to establish the baseline. Do not call this function inside a loop over thousands of regions. Pre-compute and cache the null baselines per dimension outside the region loop, and pass them to `lower_tail_owa` via the `precomputed_null_exp` argument to avoid catastrophic $O(N \times 10000)$ time complexity.
Furthermore, the `monte_carlo_rank_stability` validation function utilizes the `matrixStats` package for vectorized $O(N)$ execution. Due to C-level vectorization, this simulation completes in < 5 seconds on modern hardware.
Minimal Working Example (MWE)
To execute the MWE, please run the accompanying `MWE.r` script in your R environment. Ensure your working directory is set to the folder containing both `Supplementary_Code.r` and `MWE.r`.
Note: Step 11 of the MWE requires the `ggplot2` and `ggrepel` packages to generate the empirical scatterplot demonstrating the Epistemic Shielding Trap using subnational DHS/MICS parameters.