Thompson-Howarth

Overview

The Thompson-Howarth method is a quality control technique widely used in mining, geochemistry, assay laboratories, and analytical testing to monitor analytical precision. It evaluates the difference between paired measurements, such as duplicate samples, field duplicates, laboratory duplicates, check assays and repeat analyses.

The objective is to determine whether the observed differences between paired results are consistent with expected analytical precision. Unlike Shewhart, CUSUM, or EWMA methods which monitor process performance over time, Thompson-Howarth analysis focuses on the agreement between duplicate measurements, and on how precision changes with concentration.

Pair statistics

For each duplicate pair of an original result X1 and a duplicate result X2:

Pair mean: M = (X1 + X2) / 2    Absolute difference: D = |X1 − X2|

How Cloud QC calculates it

Cloud QC follows Thompson and Howarth’s partition method, separately for each check stage:

  1. For each pair: the mean M = (X₁ + X₂) / 2 and the difference |X₁ − X₂|, or |X₁ − X₂| / √2 (chosen by “Y axis statistic to use”).
  2. The pairs are sorted by M and split into consecutive groups of n pairs, the “Partition window” (default 11, as Thompson and Howarth recommend).
  3. For each group: the mean of the pair means (or their median, with “Use Median of Means”) and the median difference. The median keeps a single bad pair from moving the group.
  4. A straight line is fitted through the group points: median difference = intercept + slope × concentration. Robust regression is the default (“Slope/intercept by robust regression (recommended)”), otherwise ordinary least squares. A zero or negative intercept can be forced to 0.01.
  5. The fitted line is in the units of the plotted statistic, so its intercept and slope are scaled to one standard deviation: × 1.04836 for the median of |X₁ − X₂|, × 1.4826 for the median of |X₁ − X₂| / √2, and × 0.7071 with “Compute for nugget” (the root mean square of |X₁ − X₂|). The factors are from Stanley (2003) and Stanley and Lawie (2008): 1.4826 turns a median absolute value into a standard deviation, and 1/√2 turns the difference of two results into one result’s.
  6. The scaled intercept estimates the precision at zero concentration (σ₀) and the scaled slope the part that grows with concentration (k): σc = σ₀ + k × c.

Precision and practical detection limit

Precision (%) at concentration c = (2 × σ₀ / c + 2 × k) × 100

Practical Detection Limit (PDL) = 2 × σ₀ / (1 − 2 × k)

The precision is the “Precision v Concentration” curve for each check stage; the PDL is the concentration at which the precision is 100%. The curve is drawn only when a check stage has at least five times the partition window in pairs (55 for a window of 11) and a positive intercept.

The “Thompson Howarth” view shows each pair, the group points and the fitted lines (ordinary and robust) with 95% confidence bands; the panel heading gives the number of pairs and the intercept. “Compute for nugget” follows Stanley (2006) for variables with a nugget effect: the squared differences are averaged in each group and the square root taken.

Worked example

Ten duplicate assay pairs:

Pair Original Duplicate Mean (M) Difference (D) RD %
1 1.20 1.18 1.19 0.02 1.68
2 0.85 0.88 0.87 0.03 3.45
3 2.15 2.10 2.13 0.05 2.35
4 1.60 1.68 1.64 0.08 4.88
5 3.20 3.08 3.14 0.12 3.82
6 0.95 0.90 0.93 0.05 5.41
7 2.50 2.42 2.46 0.08 3.25
8 1.75 1.60 1.68 0.15 8.93
9 4.10 4.00 4.05 0.10 2.47
10 5.00 4.20 4.60 0.80 17.39

For illustration, with a partition window of 5 (the app’s smallest is 9), the ten pairs, sorted by pair mean, form two groups:

  • Group 1: pairs 2, 6, 1, 4 and 8. Mean of means 1.259, median difference 0.05.
  • Group 2: pairs 3, 7, 5, 9 and 10. Mean of means 3.275, median difference 0.10.

The line through the two group points has slope (0.10 − 0.05) / (3.275 − 1.259) = 0.025 and intercept 0.05 − 0.025 × 1.259 = 0.019. Scaled to one standard deviation (× 1.04836 for |X₁ − X₂|): σ₀ = 0.0197 and k = 0.0260.

Precision at 1.0 = (2 × 0.0197 / 1.0 + 2 × 0.0260) × 100 = 9.1%    at 3.0 = 6.5%    PDL = 2 × 0.0197 / (1 − 2 × 0.0260) = 0.04

Pair 10’s large difference (0.80) does not move its group’s median (0.10); this is why Thompson and Howarth use medians. Ten pairs are far fewer than the app needs to draw a curve: the example only shows the arithmetic.

Interpretation

  • Most duplicate pairs exhibit small differences relative to their mean value; pairs 1–9 demonstrate good agreement.
  • Pair 10 shows a substantially larger difference than the remaining pairs and should be investigated for sampling variability, sample preparation issues, analytical error or heterogeneous mineralisation.
  • The Thompson-Howarth method allows laboratories to determine whether observed differences are proportional to concentration and consistent with expected analytical performance.

Advantages and limitations

Advantage Description
Industry standard Widely used in mining and geochemical QA/QC
Precision focused Directly measures duplicate agreement
Concentration sensitive Accounts for increasing variance at higher grades
Easy visualisation Simple Thompson-Howarth plots
Practical interpretation Identifies analytical or sampling issues quickly
Limitation Description
Requires duplicate samples Cannot be applied to single observations
Focuses on precision only Does not assess analytical bias
Influenced by heterogeneous material Large natural variability can increase differences
Requires historical data Precision envelopes must be established from representative data

Summary

The Thompson-Howarth method is a duplicate precision monitoring technique used extensively in mining, geochemical, and laboratory QA/QC programmes. By comparing duplicate results through pair means and absolute differences, the method quantifies analytical precision and shows how precision changes with concentration. Graphical presentation using Thompson-Howarth plots provides a straightforward and effective means of assessing laboratory performance and identifying potential sampling or analytical issues.