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Two-stage imputation of longitudinal anthropometric data with cross-reference harmonisation: a simulation study

This simulation study evaluates a reproducible, two-stage method for imputing missing longitudinal weight and height data that combines within-subject linear interpolation with growth reference-based estimation, demonstrating high accuracy and the ability to explicitly handle and audit differing reference standards across data sources.

Original authors: Flavia Alves

Published 2026-06-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Flavia Alves

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to reconstruct a movie of a child's growth, but the film reel is damaged. Some frames are missing (missing data), and worse, some scenes were filmed using two different types of cameras that record color slightly differently (different growth standards like WHO vs. CDC). If you just guess the missing parts or ignore the camera differences, your final movie will be blurry or distorted.

This paper introduces a simple, two-step "film restoration" method to fix these missing growth measurements (weight and height) while keeping a clear record of exactly how every single piece of the puzzle was found.

Here is how the method works, using everyday analogies:

The Problem: Missing Frames and Mixed Cameras

In health research, we track people's weight and height over time. Often, people miss appointments, leaving gaps in the data. Also, some studies use the WHO growth charts (like a "European camera") and others use CDC charts (like an "American camera"). Even if a child is the same size, these two charts might say they are at a different "percentile" (a ranking compared to other kids). If you mix these datasets without noticing, you introduce a hidden error, like trying to edit a movie where the lighting changes randomly from scene to scene.

The Solution: A Two-Stage Restoration Process

The authors propose a "Two-Stage" method to fill in the blanks, prioritizing what we know about the specific person before guessing based on the general population.

Stage 1: The "Connect-the-Dots" Method (Within-Subject Interpolation)

The Analogy: Imagine you have a photo of a child at age 5 and another at age 7, but the photo for age 6 is missing. The easiest, most logical guess is to draw a straight line between the age 5 and age 7 photos. You assume the child grew steadily in between.
What the paper does: If a person has measurements before and after a missing visit, the computer simply draws a straight line between them to fill the gap. It uses only that specific person's data. This is the most accurate way to fill a gap because it respects that specific child's growth pattern.

Stage 2: The "Growth Chart" Method (LMS Centile Imputation)

The Analogy: What if you have no photos at all for a certain year, or the child only has one photo? You can't draw a line. Instead, you look at a standard "growth chart" (like the WHO or CDC charts). You find where that child usually sits on the chart (e.g., the 50th percentile, meaning they are average size), and you assume they stayed on that same "track" until the next time you saw them.
What the paper does: If Stage 1 can't fill a gap, the method looks at the child's known measurements to figure out their "growth track" (centile). It then assumes the child stayed on that track and calculates what their weight/height should be at the missing time, using the specific growth chart (WHO or CDC) that the original data source used.
The "Explicit Parameter" Twist: Crucially, this method forces the researcher to say, "I am using the WHO chart for Source A and the CDC chart for Source B." It doesn't hide this choice. It records exactly which "camera" was used for every single guess.

The "Provenance" Label: The Receipt for Every Guess

The paper emphasizes that every number in the final dataset gets a "label" or a receipt:

Observed: We actually measured this.
Interpolated: We connected the dots (Stage 1).
Growth Reference: We guessed based on the chart (Stage 2).

This is vital because if a researcher wants to be extra careful, they can say, "I will only trust the 'Observed' and 'Interpolated' numbers, and ignore the 'Growth Reference' guesses," or they can analyze them separately to see if the "guesses" change the results.

The Results: How Good is the Restoration?

The authors tested this method using synthetic data (computer-generated fake data that looks like real growth studies). They deliberately hid 20% of the known numbers and tried to guess them back.

The Outcome: The method successfully filled 100% of the gaps.
The Accuracy: The guesses were very close to the truth. For weight, the average error was about 1.78 kg (roughly 3.5% off). For height, it was about 2.84 cm (2% off).
The Verdict: As expected, the "Connect-the-Dots" guesses (Stage 1) were more accurate than the "Growth Chart" guesses (Stage 2). This proves that the two-stage order (try to connect dots first, use charts only if necessary) is the right approach.

Important Limitations (What the Paper Says)

The paper is very honest about what it doesn't do yet:

It's a "Single Guess": The method gives one best guess for a missing number. It doesn't calculate the "uncertainty" or the range of possibilities. If you use these guesses in a final medical study, you might accidentally think you know more than you actually do.
The "Stability" Assumption: Stage 2 assumes a child stays on the same growth track. If a child has a sudden illness or a growth spurt that changes their percentile drastically, this method might not catch that shift. However, because of the "labels," researchers can spot these guesses and treat them with caution.
Synthetic Only: These results are based on computer simulations, not real patients. The method works perfectly in the simulation, but it needs to be tested on real-world messy data before being used for serious medical conclusions.

Summary

This paper offers a simple, transparent tool to fix missing growth data. It prioritizes using a person's own history first, then falls back to standard growth charts if needed, while loudly announcing which "standard" was used for every single calculation. It's a way to make sure that when we combine data from different sources, we aren't accidentally mixing apples and oranges without knowing it.

Technical Summary: Two-Stage Imputation of Longitudinal Anthropometric Data

Problem Statement
Longitudinal health datasets frequently suffer from missing weight and height measurements due to non-standardized collection or missed assessments. Naïve handling of this missingness (e.g., complete-case analysis) can introduce bias. Furthermore, combining data from different sources often involves indexing measurements against disparate growth reference standards (e.g., WHO vs. CDC). Because these standards assign different centiles to the same physical measurement, combining them without explicit harmonization introduces systematic, often unstated, differences. Imputation in growing individuals is particularly complex because plausible values depend jointly on age and sex, precluding imputation from a single fixed distribution.

Methodology
The paper proposes a reproducible, two-stage imputation method implemented in base statistical software (R) with no external package dependencies. The process is designed to prioritize individual longitudinal data before resorting to population-level references, while explicitly tracking the source of every value.

Stage 1: Within-Subject Linear Interpolation
For each subject, observed values are ordered by visit date. Missing interior values (gaps between two observed measurements) are filled using linear interpolation. No extrapolation is performed; values before the first or after the last observation remain missing for Stage 2. This stage relies solely on the subject's own data.
Stage 2: Growth-Reference (LMS) Centile Imputation
Remaining missing values are imputed using age- and sex-specific growth references via the LMS method (which models skewness $L$ , median $M$ , and coefficient of variation $S$ ).
1. For observed measurements, the LMS standard-deviation score ( $z$ ) and corresponding centile are calculated.
2. These centiles are carried forward and backward across visits for the subject, assuming approximate centile stability over time.
3. If a subject has no observed values for a measure, the 50th centile is assigned by default.
4. The expected measurement value is derived by reading the reference curve at the specific visit age and sex.
5. Different reference tables can be supplied per data source (e.g., WHO-style for one source, CDC-style for another), making the standard an explicit, auditable parameter.
Pre-processing and Provenance
Before imputation, implausible values are flagged using age-banded robust z-scores and set to missing if they exceed a configurable threshold (default: absolute robust z-score > 3). Every final value is labeled by its provenance: Observed, Interpolated (Stage 1), or Growth Reference (Stage 2). This labeling enables downstream stratification or sensitivity analysis.

Evaluation and Results
The method was evaluated using a synthetic dataset ( $n=60$ subjects, 288 visits) with 30% missingness and four gross outliers. A "mask-and-recover" design was used, where 20% of observed values were hidden and re-imputed to assess accuracy.

Completeness: The method achieved 100% data completeness.
Accuracy: The mean absolute error (MAE) for recovered weight was 1.78 kg (3.5% MAPE), and for height, 2.84 cm (2.0% MAPE). Bias was negligible (−0.07 kg for weight; +0.17 cm for height).
Stage Comparison: As hypothesized, values recovered via within-subject interpolation were more accurate than those derived from growth references (Weight MAE: 1.47 kg vs. 2.06 kg; Height MAE: 2.60 cm vs. 2.99 cm), validating the two-stage ordering.

Key Contributions and Significance
The paper identifies four primary innovations:

Explicit Harmonization: It treats the choice of growth reference standard as an explicit, source-specific parameter rather than a hidden assumption, rendering the harmonization process auditable and reversible in sensitivity analyses.
Prioritized Data Usage: The two-stage design prioritizes within-subject longitudinal information before falling back to population references, reflecting the higher reliability of individual trends.
Provenance Tracking: By labeling every imputed value by its source, analysts can stratify results or exclude specific imputation types (e.g., excluding growth-reference imputations for subjects with rapid physiological changes).
Accessibility and Reproducibility: The method is implemented in base software without package dependencies, lowering the barrier to reuse and ensuring full reproducibility on synthetic data.

Limitations and Future Work
The authors explicitly note that the method relies on the assumption of centile stability within a subject, which may not hold for individuals undergoing rapid or pathological changes. Additionally, the method produces single-point predictions; using these as observed data in downstream inferential models understates uncertainty. The authors state that the necessary next steps are:

Application to empirical data to validate recovery accuracy in real-world settings.
Propagation of imputation uncertainty into downstream models (e.g., via multiple imputation).
Comparison of alternative reference standards head-to-head.

The paper concludes that while the current results are based on synthetic data to demonstrate behavior and reproducibility, the approach offers a simple, dependency-free, and transparent tool for addressing missing anthropometry in combined datasets.