Add WHO reference

[eatyourpeas]

WHO reference is included already for the smaller children in the US and UK references.
The WHO reference though exists from birth through to adulthood and should be implement as its own standard.

[eatyourpeas]

Just reopening this issue as I have spent some time with the 2007 reference and seeing now a few issues that are unresolved. Adding @statist7 and @fredho-42

The WHO standard was introduced in 2006 for the under 5s, with the over 5s added in 2007.

The 2006 standard

Runs from birth to 5 years (0-1856 days or just over 5 y). Helpfully an L, M and S value are provided for
every day of life (so no interpolation is needed) and cover:

• length/height
• weight
• head circumference
• BMI

There is an R library published on the WHO Github called Anthro

---

The 2007 reference

Runs from 61 months (just over 5 years) and covers:

• height from 5-19y
• weight to 10 y
• BMI 5-19y

L, M and S values are offered at monthly intervals and is available as an R package also on the WHO GitHub as AnthroPlus.

---

Difference between WHO standard implementation and UK-WHO reference

There are some important differences, when introspecting the published packages with how things are currently implemented at RCPCH

1. Interpolation: RCPCH uses cubic interpolation. WHO AnthroPlus in the >5y uses linear interpolation:

if (any(is_diff_age_pos)) {
    adjust_param <- function(x) {
      x_name <- as.character(substitute(x))
      x[is_diff_age_pos] + diff_age[is_diff_age_pos] *
        (match_upp_age[[x_name]][is_diff_age_pos] - x[is_diff_age_pos])
    }
    m[is_diff_age_pos] <- adjust_param(m)
    l[is_diff_age_pos] <- adjust_param(l)
    s[is_diff_age_pos] <- adjust_param(s)
  }

2. SDS Adjustment: RCPCH does not adjust for extreme SDS, but WHO applies a correction to SDS values outside +/-3 SD

compute_zscore_adjusted <- function(y, m, l, s) {
  stopifnot(is.numeric(y), is.numeric(m), is.numeric(l), is.numeric(s))
  calc_sd <- function(sd) m * ((1 + l * s * sd)^(1 / l))

  zscore <- compute_zscore(y, m, l, s)
  SD3pos <- calc_sd(3)
  SD3neg <- calc_sd(-3)
  SD23pos <- SD3pos - calc_sd(2)
  SD23neg <- calc_sd(-2) - SD3neg

  # a type-safe way to use ifelse
  not_zscore_na <- !is.na(zscore)
  zscore_gt_3 <- not_zscore_na & zscore > 3
  zscore[zscore_gt_3] <- (3 + ((y - SD3pos) / SD23pos))[zscore_gt_3]
  zscore_lt_3 <- not_zscore_na & zscore < -3
  zscore[zscore_lt_3] <- (-3 + ((y - SD3neg) / SD23neg))[zscore_lt_3]

  zscore
}

This is applied to all measurement methods in the under 5s but only to weight and BMI in the over 5s. There is documentation on the WHO website about the methodology

Charting

The UK-WHO publish centile charts with centiles each 0.667 SD apart for all measurement methods:

centile (rounded for convenience)	SDS
0.4th	-2.6667
2nd	-2.0000
9th	-1.3333
25th	-0.6667
50th	0
75th	0.6667
91st	1.3333
98th	2.000
99.6th	2.6667

The WHO charts use different intervals

centile	SDS
0.1	-3.0903
1	-2.33
3	-1.88
5	-1.645
10	-1.28
15	-1.036
25	-0.67
50	0
75	0.67
85	1.036
95	1.28
99	2.33
99.9	3.0903

The key point here is that the 0.1st and 99.9th centiles are outside the 3 SD threshold documented above. Oddly though here if we use the WHO published L, M and S values to replicate the centile lines but do not apply the SD adjustment documented above we can replicate the published centiles. However, if we do apply the SD adjustment we get slightly different lines.

Implications

This leaves us with 2 questions that need addressing before we can reliably publish the WHO reference:

1. when implementing the WHO standard as part of the UK-WHO reference, should we apply the SD adjustment for values outside +/- 3 SD?
2. why does the WHO publish that an adjustment is applied to all values outside +/-3 SD (and implement this in their R package) yet seemingly not apply this to their own centile lines?

[statist7]

Some comments and a question.

It is the 2007 reference not the 2007 standard, in contrast to the 2006 standard. And weight in the 2007 reference is for age 5-10y.

The centile-SDS tabulations are potentially misleading. For UK-WHO the centiles are defined to be exactly two-thirds of an SDS apart, i.e. 0.6667 rounded to four decimal places, while the corresponding centile labels are rounded for convenience. Providing the SDS for the rounded centiles (e.g. the 2nd centile) should be discouraged as it may give people the wrong idea!

Are the WHO SDS values exactly as tabulated, with precision varying between two and four decimal places? If so, I suggest padding them with zeroes to four decimal places to make the discrepancy explicit. And if not, do please give them to full precision.

Concerning Question 1, UK-WHO is a mix of UK90 and WHO (as the name implies). The ±3 SDS adjustment is an invention of WHO and has never applied to UK90. So logically it should not be applied to the WHO in UK-WHO either. This needs stating explicitly somewhere, to avoid confusion when researchers compare extreme z-scores from anthro and dGC.

Concerning Question 2, the descriptions of the two WHO packages, anthro and anthroplus, make clear that they are designed to calculate z-scores, but there is no suggestion that they do the reverse, i.e. calculate centiles for a given SDS. So there will be no smoking gun in the packages to confirm the implication of Question 2 that WHO does not apply the 3 SD adjustment to centiles outside ±3 SDS.

[eatyourpeas]

This is really helpful, thank you @statist7
I have updated the 2007 reference to be a reference. The weight range for that is already in there.
I have updated the table to reflect 4dp (do please check i have that right)
The anthro and anthroplus packages seem to round all their results to 2 decimal places:
apply_zscore_and_growthstandards in anthro:

y <- merged_df[["measure"]]
  m <- merged_df[["m"]]
  l <- merged_df[["l"]]
  s <- merged_df[["s"]]
  zscore <- zscore_fun(y, m, l, s)
  round(zscore, digits = 2L)

and anthroplus_zscores in anthroplus:

  zhfa <- round(zhfa, digits = 2L)
  zwfa <- round(zwfa, digits = 2L)
  zbfa <- round(zbfa, digits = 2L)

In effect this means I can only run tests to match this to 2 decimal places also.

For chart construction you are right there is no reverse function to generate measurements from SDS in either package. This means I am not sure what precision they have used to create the published centile line values against age. I know that for the 0.1st and 99.9th centiles they have used +/- 3.0903 (without the adjustment) as my tests pass using a tolerance of 1e-3. The other values I am less certain but again, the same tests pass for the same tolerance. I will experiment with different levels of precision and update the table.