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Correlation & Spearman

 
  • Dec 11, 2025

Correlation & Spearman Rank Correlation

1. Rough guideline for correlation magnitude

Here, r or ρ means “some correlation coefficient”.

  • r < 0.1 → almost none / very weak
  • r ≈ 0.3 → weak to modest
  • r ≈ 0.5 → clearly noticeable correlation
  • r ≥ 0.7 → fairly strong correlation
  • r ≥ 0.9 → very strong correlation

These cutoffs are only rules of thumb; different books or fields may use slightly different labels.

2. What is Spearman rank correlation?

2.1 Basic idea

Spearman rank correlation (usually denoted ρ or ρ_s) measures the association between ranks, not the raw values.

Steps (conceptual):

  1. Sort the X values from smallest to largest and assign ranks: 1, 2, 3, …
  2. Do the same for Y values and assign ranks.
  3. Compute the ordinary Pearson correlation between the two sets of ranks.
  4. That Pearson correlation of the ranks is the Spearman ρ_s.

Advantages:

  • Much less sensitive to outliers.
  • More robust when the distribution is skewed.
  • Great when you care about ordering (“who is higher/lower”) rather than exact numeric values.

3. Spearman correlations in VALID.DAT (Exercises 11.72–11.75)

Data: VALID.DAT, n = 173 subjects.

  • DR (diet record): short-term record of what was actually eaten.
  • FFQ (food frequency questionnaire): questionnaire-based estimate of usual intake.

We compare DR vs FFQ using Spearman rank correlation to see how well
they agree on “who eats more” vs “who eats less” of each nutrient.

3.1 11.72 Alcohol intake (alco_dr vs alco_ffq)

  • X = alco_dr (alcohol intake from DR)
  • Y = alco_ffq (alcohol intake from FFQ)

Spearman correlation:

  • ρ_s ≈ 0.899

Significance:

  • Corresponding t statistic ≈ 26.8
  • Degrees of freedom df = 171
  • p-value ≈ 4.3 × 10^(-63) (essentially 0)

Interpretation:

  • ρ_s ≈ 0.90 → very strong positive rank correlation.
  • Subjects who drink a lot according to the DR almost always rank high according to the FFQ as well.
  • For alcohol, DR and FFQ agree on the ranking of subjects almost perfectly.

3.2 11.73 Total fat intake (tfat_dr vs tfat_ffq)

  • X = tfat_dr (total fat from DR)
  • Y = tfat_ffq (total fat from FFQ)

Spearman correlation:

  • ρ_s ≈ 0.371

Significance:

  • t ≈ 5.22, df = 171
  • p ≈ 5.1 × 10^(-7) (p < 0.0001)

Interpretation:

  • ρ_s ≈ 0.37 → weak to modest positive correlation.
  • Statistically highly significant (p < 0.0001).
  • DR and FFQ are reasonably consistent about “high-fat eaters vs low-fat eaters,”
    but far from perfect, and clearly weaker than alcohol (ρ_s ≈ 0.90).

3.3 11.74 Saturated fat intake (sfat_dr vs sfat_ffq)

  • X = sfat_dr (saturated fat from DR)
  • Y = sfat_ffq (saturated fat from FFQ)

Spearman correlation:

  • ρ_s ≈ 0.422

Significance:

  • t ≈ 6.09, df = 171
  • p ≈ 7.3 × 10^(-9) (p < 0.0001)

Interpretation:

  • ρ_s ≈ 0.42 → a bit stronger than for total fat.
  • DR and FFQ show a reasonably good agreement on who eats more saturated fat.
  • Again, statistically very significant.

3.4 11.75 Total calorie intake (cal_dr vs cal_ffq)

  • X = cal_dr (total calories from DR)
  • Y = cal_ffq (total calories from FFQ)

Spearman correlation:

  • ρ_s ≈ 0.340

Significance:

  • t ≈ 4.72, df = 171
  • p ≈ 4.8 × 10^(-6) (p < 0.0001)

Interpretation:

  • ρ_s ≈ 0.34 → weak to modest positive correlation.
  • DR and FFQ do a decent job at distinguishing “high-calorie eaters” from “low-calorie eaters,”
    but the agreement is not very strong.
  • Definitely weaker than alcohol (ρ_s ≈ 0.90).

4. Are ρ_s = 0.34, 0.37, 0.42 “real” correlations?

4.1 Effect size view

Using the rough thresholds above:

  • ρ_s ≈ 0.34 (calories)
  • ρ_s ≈ 0.37 (total fat)
  • ρ_s ≈ 0.42 (saturated fat)

All of these fall in the weak to modest range.

So:

  • The measurements are not useless.
  • They provide some ordering information about who eats more or less.
  • But they are far from very strong or perfect agreement (especially compared with alcohol ρ_s ≈ 0.90).

Typical wording in a paper:

  • “There was a modest positive correlation between DR and FFQ for total fat intake (ρ_s ≈ 0.37).”

4.2 Statistical significance vs size of correlation

Important distinction:

  • p-value answers:
    “Is the true correlation exactly 0 or not?”

  • Correlation magnitude (e.g., 0.34, 0.37, 0.42) answers:
    “How strong is the relationship in practice?”

In these problems:

  • p-values are extremely small (all < 0.0001) →
    we can confidently say the true correlation is not 0.

  • But the effect sizes (around 0.3–0.4) are modest →
    the relationship is real, but not very strong.

5. How do we get p-values from ρ_s?

All p-values above are calculated from the sample Spearman correlation ρ_s.

5.1 Testing structure

  • Null hypothesis H0: ρ_s = 0 (no rank correlation in the population)
  • Alternative hypothesis H1: ρ_s ≠ 0

We first compute ρ_s from the data (say 0.37), then ask:

“If the true ρ_s were 0,
how likely is it to observe a sample correlation as extreme as 0.37 (or more)
just by random chance?”

That probability is the p-value.

5.2 Practical calculation (large n, t-approximation)

When sample size n is large (here n = 173), we can approximate:

  1. Compute ρ_s from the data.

  2. Convert ρ_s to a t statistic:

    t = ρ_s * sqrt( (n - 2) / (1 - ρ_s^2) )

    with degrees of freedom df = n - 2.

  3. Use the t distribution with df = n - 2 to find the two-sided p-value.

Example (total fat, 11.73):

  • ρ_s ≈ 0.371, n = 173
  • t ≈ 0.371 * sqrt( 171 / (1 - 0.371^2) ) ≈ 5.22
  • For df = 171, t = 5.22 gives p ≈ 5.1 × 10^(-7).

This p is tiny, so we reject H0: ρ_s = 0.

5.3 One-line summary

To get the p-value, we go:

data → ranks → ρ_s → t statistic → t distribution → p-value.

So yes, the p-value is based on the sample Spearman correlation.

6. Exercise 11.76 – Parametric vs nonparametric (final verdict)

Characteristics of dietary intake data in VALID.DAT:

  • Likely skewed and not normally distributed.
  • Contains zeros and potential outliers (especially for alcohol and some nutrients).
  • For validation, we mostly care about rank ordering of subjects (who eats more/less).

Therefore:

  • A nonparametric method like Spearman rank correlation is more appropriate and robust.
  • It does not require normality and is less affected by extreme values.

In practice, a good strategy is:

  • Use Spearman to assess rank agreement,
  • Optionally apply transformations (e.g., log) and also report Pearson correlations,
  • Compare both to get a fuller picture.
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