5.4 Sampling Strategy: Sample Size, Power, and Worker Selection

Sampling Strategy: Sample Size, Power, and Worker Selection

"How many samples do I need?" is the most frequently asked and least frequently answered question in exposure assessment. Two distinct statistical questions hide inside it, and they have different answers:

1. Whom do I sample so that I do not miss the most exposed workers?
2. How many samples do I need to characterise the exposure distribution of a group with acceptable precision?

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1. Whom to Sample: The Top-Fraction Problem

Suppose a similar exposure group contains many workers and you can only sample a few. You want reasonable confidence that at least one of the workers you pick is among the most exposed.

For random selection from a large group, if the fraction of interest is p (say the top 10%, p = 0.10) and you want confidence C (say 0.90), the probability that a sample of n workers contains none of the top fraction is (1 − p)ⁿ. Setting the complement equal to C:

$1 - (1-p)^n = C \quad \Longrightarrow \quad n = \frac{\ln(1-C)}{\ln(1-p)}$

Worked example. For 90% confidence of including at least one worker from the top 10%:

n = ln(0.10) / ln(0.90) = (−2.3026) / (−0.10536) = 21.85 → 22 workers

The numbers are unforgiving and worth memorising in outline:

The finite-population correction

Real similar exposure groups are small, and sampling without replacement from a small group is far more efficient than the large-group formula implies. If a group has 10 workers, the "top 10%" is a single worker, and sampling 9 of the 10 gives 90% confidence directly. This is a hypergeometric rather than binomial problem, and it is exactly what the NIOSH Occupational Exposure Sampling Strategy Manual tabulated: partial-sampling tables that give the required number of workers as a function of group size.

The practical rule that follows: for groups of six or fewer, sample everyone. The tabulated saving only becomes meaningful for larger groups.

The maximum-risk-employee alternative

Instead of random selection, the industrial hygienist can identify the maximum-risk employee — the worker judged, from process knowledge, proximity to the source, task duration, and work practices, to have the highest exposure — and sample that person.

The two are complementary. Maximum-risk sampling is an efficient screening strategy; only random sampling supports a defensible statement about the exposure distribution of the group as a whole.

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2. How Many Samples: Characterising the Distribution

Once the group is defined, the number of samples determines how precisely you know its geometric mean and geometric standard deviation.

AIHA guidance treats roughly 6 to 10 samples per similar exposure group as the practical minimum for a defensible exposure profile. Fewer than about six makes any estimate of the GSD nearly worthless, and the GSD is what drives the 95th percentile estimate on which the acceptability judgement rests.

Two consequences follow from the mathematics of the lognormal distribution:

• Precision improves only as the square root of n. The standard error of the mean of the log-transformed data is s_y / √n. Halving the width of a confidence interval therefore requires roughly four times as many samples. There is no cheap route to precision.
• A high GSD demands more samples. A homogeneous group (GSD near 1.5) can be characterised with far fewer samples than a heterogeneous one (GSD above 3). A high GSD is usually a symptom of a badly constructed SEG, and the correct response is to split the group, not to keep sampling it.

   +---------------------------------------------------------------+
   |  SAMPLES NEEDED vs GSD (to reach a given precision on X95)     |
   |                                                                |
   |  GSD 1.5  |####                    few samples                 |
   |  GSD 2.0  |########                                            |
   |  GSD 3.0  |################                                    |
   |  GSD 4.0  |############################   split the SEG        |
   +---------------------------------------------------------------+

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3. Statistical Power and the Two Ways to Be Wrong

Exposure decisions carry two error types, and industrial hygiene cares far more about one of them:

Power = 1 − β is the probability of detecting an overexposure that is genuinely present. Under-sampling is not a neutral economy: it directly reduces power and therefore directly increases the probability of leaving an overexposed group unprotected. This asymmetry is why a hygienist should be far more reluctant to declare "acceptable" on three samples than to declare "unacceptable."

Power rises with:

• larger sample size,
• lower variability (a tighter, better-defined SEG),
• a larger true difference between the actual exposure and the OEL.

The last term explains a common field observation: an exposure at 95% of the OEL is very hard to distinguish statistically from one at 105%, while an exposure at 300% of the OEL is obvious from two samples.

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4. Practical Rules

1. Define the SEG first. Sample size questions are unanswerable until the group is defined, and a bad group cannot be rescued by more samples.
2. Sample everyone in groups of six or fewer.
3. Plan 6 to 10 samples per SEG as a baseline, more if the GSD proves high.
4. Use maximum-risk sampling to screen and random sampling to characterise; document which you did, because the inference you may draw differs.
5. Treat a GSD above about 3 as evidence of a heterogeneous group and split it.
6. Never report an "acceptable" judgement from an under-powered data set without stating the uncertainty.