From weight and height to a z-score

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What this lesson covers

• What a z-score says
• Three z-scores, three questions
• The LMS method
• The lookup key has three parts
• Computing it
• Flagging: two rules, two answers
• The standard deviation is already a finding
• Save the z-scores with their inputs
• What comes next

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What a z-score says

• A z-score answers one question: how far is this child from the median of a healthy reference population, in standard deviations?

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Three z-scores, three questions

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Three z-scores, three questions

• Weight-for-age is the one to be careful with — It mixes the two deficits, so a low WAZ does not say whether the child…

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The LMS method

• L — the Box-Cox power that normalises a skewed distribution
• M — the median
• S — the coefficient of variation

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The LMS method — Example

z = ((weight / M)^L - 1) / (L * S)

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The LMS method — In Python

import pandas as pd

reference = pd.read_csv("who-2006-weight-for-lenhei.csv")
print(reference.head(3))
print(f"{len(reference)} rows: sex x standard x lenhei in 0.1 cm steps")

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The LMS method — In R

reference <- readr::read_csv("who-2006-weight-for-lenhei.csv")
nrow(reference)

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The lookup key has three parts — In Python

def lookup_key(row):
    lying = row["measured_lying"] == "true"
    standard = "L" if row["age_months"] < 24 else "H"

    lenhei = row["height_cm"]
    if standard == "L" and not lying:
        lenhei += 0.7
    elif standard == "H" and lying:
        lenhei -= 0.7

    return row["sex"], standard, round(lenhei, 1)

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The lookup key has three parts — In R

lookup_key <- function(sex, age, height, lying) {
  standard <- if (age < 24) "L" else "H"
  lenhei <- height +
    if (standard == "L" && !lying) 0.7 else if (standard == "H" && lying) -0.7 else 0
  list(sex = sex, lorh = standard, lenhei = round(lenhei, 1))
}

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The lookup key has three parts

• The standard follows the age — not the position
• The adjustment goes on the measurement, not the standard — You are converting the measurement into the one the standard…
• The rounding is to one decimal, on both sides, in one place — A float computed from an adjustment is not the same float…

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Computing it — In Python

reference["key"] = list(zip(reference["sex"], reference["lorh"],
                            reference["lenhei"].round(1)))
lms = reference.set_index("key")[["l", "m", "s"]]

def whz(row):
    key = lookup_key(row)
    if key not in lms.index or pd.isna(row["weight_kg"]):
        return None
    l, m, s = lms.loc[key]
    return (((row["weight_kg"] / m) ** l) - 1) / (l * s)

smart["whz"] = smart.apply(whz, axis=1)
print(f"{smart['whz'].notna().sum()} of {len(smart)} computable")

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Computing it — In R

# In R, use the official package rather than the table:
# anthro::anthro_zscores(sex = ..., age = ..., weight = ..., lenhei = ..., measure = ...)

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Computing it

• Missing weight or age — 32 children, and they are missing data.
• Height outside the reference range — the length standard runs 45.0 to 110.0 cm and the height standard 65.0 to…
• A measurement in the wrong unit — four heights recorded in metres, which the joining course's lab found by…
• Report the three separately — "66 children excluded" hides that some were data errors you could have fixed

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Flagging: two rules, two answers

• WHO flags — fixed bounds, exclude z outside -5 to +5. Absolute, comparable across surveys.
• SMART flags — relative, exclude observations more than 3 SD from the survey's own mean. Adapts to the survey, and…

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Flagging: two rules, two answers — In Python

who_flagged = smart["whz"].between(-5, 5)

mean, sd = smart["whz"].mean(), smart["whz"].std()
smart_flagged = smart["whz"].between(mean - 3 * sd, mean + 3 * sd)

print(f"WHO flags keep {who_flagged.sum()}, SMART flags keep {smart_flagged.sum()}")

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Flagging: two rules, two answers — In R

smart <- smart |>
  mutate(who_ok = between(whz, -5, 5),
         smart_ok = between(whz, mean(whz, na.rm = TRUE) - 3 * sd(whz, na.rm = TRUE),
                            mean(whz, na.rm = TRUE) + 3 * sd(whz, na.rm = TRUE)))

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The standard deviation is already a finding — In Python

analysable = smart.loc[who_flagged, "whz"]
print(f"n = {len(analysable)}, mean = {analysable.mean():.2f}, "
      f"sd = {analysable.std():.2f}")

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The standard deviation is already a finding — In R

smart |> filter(who_ok) |> summarise(n = n(), mean = mean(whz), sd = sd(whz))

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Save the z-scores with their inputs — In Python

smart[["child_id", "cluster", "team", "age_months", "sex", "weight_kg",
       "height_cm", "measured_lying", "oedema", "whz"]].to_csv(
    "outputs/smart_with_zscores.csv", index=False)

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Save the z-scores with their inputs — In R

readr::write_csv(smart, here::here("outputs", "smart_with_zscores.csv"))

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What comes next

• You have a z-score for every analysable child.

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Where this goes next

Read the full lesson, with runnable code Back to the lesson