WHO Reference Implementation¶

This page describes how the RCPCHGrowth Python package implements the WHO growth reference, and the reasoning behind some deliberate design decisions that affect the numbers the library returns. It is aimed at developers and researchers who need to understand why the WHO calculations behave as they do, particularly at the edges of the reference range.

Overview¶

For its standalone who reference, the library moved from reusing the UK-WHO tables for the WHO 2006 standard (ages 0-4 years, with Lambda-Mu-Sigma (LMS) values at weekly and monthly intervals) to WHO's daily LMS tables for ages 0-5 years. The uk-who reference remains available and unchanged. Both representations derive from the same WHO standard; using the daily tables removes the need for interpolation in the application layer and aligns calculations with the values distributed in WHO's software.

Historical Context¶

UK-WHO reference (2006)¶

The UK-WHO reference is a hybrid reference combining:

• WHO data for ages 0-4 years
• UK90 data for ages 4-23 years

The WHO dataset used in the UK-WHO reference (published 2006) provided LMS values at discrete time intervals:

• Weekly intervals from birth to 3 months
• Monthly intervals from 3 months to 4 years

This discrete-interval approach meant that when a measurement was taken at an age that fell between published intervals, the library had to interpolate the LMS values. Interpolation introduces:

• Computational complexity
• Potential for rounding errors
• Approximation rather than exact reference values
• Variability depending on interpolation method (the UK-WHO implementation uses cubic interpolation)

WHO daily reference data¶

WHO also distributes LMS values for each day of life from 0-5 years. This dataset:

• Provides a tabulated LMS value for every day in the 0-5 year range
• Eliminates interpolation in the application layer
• Avoids differences caused by independently chosen interpolation methods
• Simplifies calculation logic

In effect this pushes the interpolation step down a layer into the reference data, removing the need for it in the application layer.

Rationale for the Change¶

1. Alignment with WHO's implementation. Using WHO's distributed daily LMS values avoids differences caused by reproducing their interpolation independently.
2. Simplified code. Removing interpolation logic reduces complexity, making the calculation more maintainable and easier to understand.
3. Consistent reference values. Calculations use the same daily tables distributed with WHO's software.
4. Better coverage. The daily LMS values provide a table entry for every day from 0-5 years, whereas the previous discrete intervals required interpolation or approximation at boundaries.
5. Reduced computational cost. Eliminating interpolation calculations may improve performance, particularly for batch processing of measurements.

Implementation Details¶

What Changed¶

• Test fixture. The SDS validation fixture (sds_age_validation_2021_refactored_2026.json) was regenerated from daily WHO values rather than the older weekly/monthly UK-WHO values. The previous fixture was retained as sds_age_validation_2021_deprecated.json. The regenerated fixture is identical to the old one except that it removes 18 cases whose expected values changed under the new implementation (see Numerical differences below).
• Reference data. The LMS tables in rcpchgrowth/data_tables/ were updated to use WHO daily values.
• Calculation logic. This is essentially unchanged. The existing methodology always looked for an exact age match before running any interpolation step. Because there is always a match with daily LMS values, the interpolation step is now always skipped for the under-5s where the WHO standard is used.

Under-2y anthro gold dataset validation¶

As an additional validation exercise, SDS values requested from who_under2_gold_192.csv (generated via anthro_measurements) were compared against SDS recalculated by this package for all 192 under-2 rows:

• Max absolute difference: 1.0980290423567851e-06
• Min signed difference: -1.0980290423567851e-06
• Max signed difference: 7.938643615812424e-07

This is substantially tighter than the accepted tolerance of 1e-3 and supports practical numerical equivalence for the scenarios covered by this under-2 matrix.

WHO Chart Functions: Centile Curve Validation¶

Overview¶

In addition to the SDS-from-measurement tests above, the chart function tests in test_chart_functions.py validate the inverse direction: given a requested SDS, what measurement value is produced? The gold standard for these tests is the centile curve data published directly by the WHO:

• Under 5 years: WHO Child Growth Standards
• Over 5 years: WHO Growth Reference 5-19 years

These published tables list the exact measurement value expected at each centile line for each age point. By testing that measurement_from_sds() reproduces those values within a relative tolerance of 1e-3, we confirm that the chart centile curves rendered by this library match what WHO itself publishes.

Coverage¶

Total parametrized test items: 123,061.

Direction and the WHO extreme-value asymmetry¶

These chart function tests exercise the SDS to measurement direction only. This direction uses the standard inverse-LMS formula throughout:

x = M * (1 + L * S * z)^(1/L)

However, the WHO specifies a different rule when going measurement to SDS for values whose initial LMS z-score is above +3 or below -3. The implementation applies this rule to all WHO 2006 measures under 5 years, and to weight and BMI in the WHO 2007 reference:

• Beyond +3 SDS, the distance between +2 and +3 SDS is used to extrapolate linearly; below -3 SDS, the distance between -2 and -3 SDS is used.
• This correction is documented in the WHO computation guide (PDF).

This correction is deliberately not applied in the SDS to measurement direction. The WHO published centile curve values themselves do not apply this adjustment; they use pure inverse-LMS. Therefore, to remain consistent with the WHO published chart curves, measurement_from_sds() does the same. Commented-out code in global_functions.py preserves the discarded implementation for reference.

The asymmetry is therefore intentional and correct:

Discrepancy at the WHO 2006/2007 5-Year Boundary¶

Observation¶

There is a small but reproducible discontinuity in the WHO reference data at the 5-year boundary between the two WHO packages. This has been reported as WorldHealthOrganization/anthro#64 (filed by the RCPCH team; no response from WHO maintainers at the time of writing).

LMS values for boys' height at the boundary:

The difference in the median (M) is ~0.23 cm, which is small but non-trivial for a centile chart.

Root Cause¶

The published WHO 2007 growth reference tables (PDF) actually begin at 61 months, not 60 months. The anthroplus 60-month entry therefore does not appear to represent an empirical data point from the WHO 2007 study - it looks like a backwards-interpolated or extended value added for continuity, and does not precisely match what you would obtain by linearly extrapolating from the anthro 1826-day values.

How This Library Handles It¶

The who_reference() function in who.py uses the constant WHO_2006_REFERENCE_UPPER_THRESHOLD = 1856 / 365.25 (~61 months, 5.079 years) as the cutoff:

• Ages up to and including 1856 days use WHO_CHILD_DATA (from anthro, WHO 2006)
• Ages above 1856 days use WHO_2007_DATA (from anthroplus, WHO 2007)

This means that at exactly 5.0 years (1826.25 days / 60 months) the library uses the anthro WHO 2006 value (M = 109.9593 for boys' height), not the anthroplus 60-month value. This choice aligns with where the WHO 2007 published tables themselves start (61 months) and avoids the ambiguous/interpolated anthroplus row at 60 months.

The practical consequence is that there is no discontinuity in this library's output at exactly 5 years - the transition happens at approximately 61 months, where the WHO 2007 published tables begin.

Numerical differences from the UK-WHO reference¶

Numerical results differ from the UK-WHO reference for some measurements. During the transition, 18 of the SDS validation fixture cases changed by more than the previously accepted 1e-3 tolerance and were removed from the current fixture. These 18 cases are strongly concentrated in preterm and early-infant assessment:

• Chronological ages from approximately 0.016 to 1.97 years (predominantly under 0.5 years)
• Gestations from 27+2 to 44+0 weeks (preterm/late preterm; no full-term cases)
• A female predominance among the affected cases

The differences arise from two things:

• Interpolation method. WHO uses linear interpolation to generate its LMS values, whereas UK-WHO uses cubic (this reflects WHO's published application code).
• Interpolation start point. UK-WHO begins interpolation at 2 weeks of life rather than at 0, which introduces differences around that threshold.

Prof Tim Cole compared the LMS values between the two references and confirmed these are expected behavioural differences between the two reference systems, not bugs. The maximum difference between the SDS derived by each method is 0.011508556081421 - beyond the old UK-WHO test tolerance of 1e-3, but an acceptable excursion given the goal of aligning with the WHO standard.

Backward Compatibility¶

API level. The API is unchanged. Existing code using the library continues to work without modification.

Numerical results. Results differ from the UK-WHO reference for some measurements, particularly in early infancy (0-6 months) and in preterm/late-preterm infants (27-44 weeks gestation). The extent of the difference varies by age and measurement type.

Data Sources & Implementation¶

The WHO reference data in this library is sourced from two WHO R packages, using RCPCH-maintained forks that add inverse-LMS functions and z-score precision control:

WHO anthro package (0-5 years)¶

• Upstream repository: WorldHealthOrganization/anthro
• RCPCH data-generation branch: rcpch/anthro:z-to-measurement
• Coverage used by the library: Birth through 1856 days
• Data precision: Daily LMS values (no interpolation required)
• Measures: Length/height, weight, weight-for-length, BMI, head circumference

WHO anthroplus package (5-19 years)¶

• Upstream repository: WorldHealthOrganization/anthroplus
• RCPCH data-generation branch: rcpch/anthroplus:precision
• Coverage: 5-19 years (61-228 months)
• Data precision: Age-specific LMS values
• Measures: Height, weight (up to 10 years), BMI

RCPCH Modifications¶

Both packages have RCPCH-maintained branches that add:

• A z_precision parameter to control z-score decimal precision
• anthro_measurements / anthroplus_measurements inverse-LMS functions to compute measurements from requested z-scores
• Enhanced extreme-value handling with a correct_extreme parameter

Publication References¶

• UK-WHO Reference: Cole TJ, Freeman JV, Preece MA. British 1990, British 1990r and British 1990sd reference curves for body mass index; and power derived references for weight, height and body mass index in children and adolescents. Eur J Clin Nutr. 1995;49(2):119-126.
• WHO Growth Standards 2006: WHO Multicentre Growth Reference Study Group. WHO Child Growth Standards: Length/height-for-age, weight-for-age, weight-for-length, weight-for-height and body mass index-for-age. Geneva: WHO; 2006. Available: https://www.who.int/tools/child-growth-standards
• WHO Growth Reference 2007 (5-19 years): de Onis M, Onyango AW, Borghi E, et al. Development of a WHO growth reference for school-aged children and adolescents. Bull World Health Organ. 2007;85(9):660-667. Available: https://www.who.int/publications/i/item/9789241563369