8  Chapter 8 — How Many Fish Should I Sample?

Applied Epidemiology in Aquaculture — Part III: The Investigator’s Toolkit

Where this sits — Detect (secondary prevention). Surveillance lives or dies on sampling. Too few fish and you miss disease or produce a uselessly wide estimate; too many and you waste money. This chapter makes sampling a calculation, not a guess, and introduces the cost-effective, risk-based schemes that modern Detect programmes run on.

8.1 Overview

Two questions dominate practical sampling. “How precisely do I know the prevalence?” — sampling to estimate. And “Can I be confident the disease is absent?” — sampling to detect (or demonstrate freedom). They need different formulas, both are easy in R, and both depend on choices you must justify. We also cover how to draw the sample (the method matters as much as the size) and how to make surveillance affordable by sampling the right fish, not just more fish.

8.2 8.1 Preparation

pacman::p_load(rio, here, tidyverse, epiR)

8.3 8.2 How to draw the sample

Size is useless if the sample is not representative. Four methods, in rising sophistication:

  • Simple random — every animal equally likely; the gold standard, but hard on a 40,000-fish cage.
  • Systematic — every k-th fish; practical at harvest or grading, valid if there is no hidden periodicity.
  • Stratified — sample within strata (cages, ponds, size classes) and combine; improves precision when strata differ, and lets you report by stratum.
  • Cluster / multi-stage — sample whole units (pens) then fish within them; cheapest in the field, but fish within a pen are correlated, so the effective sample size is smaller than the count (the “design effect”).

The farm reality — fish are housed in pens, and disease clusters by pen — means most aquaculture surveillance is multi-stage: choose pens, then fish within pens.

8.4 8.3 Sampling to estimate a prevalence

To estimate a proportion to within a chosen precision (±d) at 95% confidence:

n = Z² · p(1−p) / d²

# Expected prevalence 20%, want +/- 5%, 95% confidence
epi.sssimpleestb(N = NA, Py = 0.20, epsilon = 0.05,
                 error = "absolute", conf.level = 0.95)   # 246

That returns 246 fish. Two levers matter:

🧮 Epivet Tools — Sample size to estimate a prevalence tools.epivet.cl/ss-estimate — enter expected prevalence, precision and (optionally) population size. Reference model: Ausvet EpiTools.

  • Precision is expensive. Tightening to ±3% raises the requirement to 683 fish — precision scales with 1/d², so halving the margin roughly quadruples the sample.
  • Finite populations help. In a pond of 5,000, the finite-population correction trims 246 down to 235 — a small gain here, a large one when the sample is a big fraction of the population.

Use p = 0.5 when you have no prior estimate; it is the most conservative (largest n).

8.5 8.4 Sampling to detect disease (demonstrating freedom)

Surveillance often asks the opposite: if disease were present at some low level, how many fish must I test to be 95% sure of catching at least one positive? This needs a design prevalence — the minimum prevalence you want to be able to detect — and uses the hypergeometric (finite-population) formula:

# Cage of 20,000; design prevalence 2%; 95% confidence; perfect test
epi.detectsize(N = 20000, prev = 0.02, se = 1, conf.level = 0.95)   # ~148
  • At 2% design prevalence, you need 148 fish; at 5%, only 59. The rarer the disease you want to rule out, the more fish you need — detecting freedom at very low prevalence is genuinely demanding.
  • Imperfect tests cost more. With test sensitivity 0.90, the 148 rises to 165 — you must over-sample to compensate for fish the test will miss. (This is where Chapter 6 meets Chapter 8.)

A negative result then supports probability of freedom, the backbone of the scenario-tree surveillance we build in Chapter 12.

🧮 Epivet Tools — Sample size to detect disease tools.epivet.cl/ss-detect — population, design prevalence and test sensitivity (perfect- and imperfect-test modes). Reference model: Ausvet EpiTools.

8.6 8.5 Risk-based and cost-effective sampling

Testing random fish is rarely the best use of money. Risk-based sampling targets the animals, pens, or sites most likely to be positive — raising the surveillance system’s sensitivity per dollar. For SRS this is concrete: sampling moribund or freshly dead fish with gross lesions, from two or three pens, sharply increases the chance of detection, and a scheme of about five such fish per pen is cost-effective relative to random sampling (Delphino et al., 2021; Laurin et al., 2020). The principle generalises — sample where the signal is, weight by risk, and spend the saved effort on testing more units rather than more fish per unit.

8.7 8.6 Interpretation on the farm

Every surveillance plan embeds three honest choices: the design prevalence (how low a level you care about), the confidence you need, and the test you will use. Make them explicit and the sample size follows; leave them implicit and “we tested 30 fish” means nothing. For a Detect programme the pay-off is direct — a defensible, affordable plan that can either estimate how much disease is present or stand behind a claim of freedom, which is what regulators and trading partners actually require.

8.8 8.7 Exercises

Try it yourself (8.1). Compute the sample size to estimate Streptococcus prevalence in a 10,000-fish tilapia pond to within ±4%, assuming an expected prevalence of 15%. Then recompute with no prior estimate (p = 0.5) and compare.

8.2 You want to be 95% confident of detecting AHPND if it is present in at least 1% of a 200,000-shrimp pond, using a test with Se = 0.85. Roughly how many shrimp must you sample, and why does the low design prevalence make this demanding?

8.3 Explain, using the design effect, why sampling 300 fish from a single cage tells you less than 300 fish spread across ten cages.

8.4 A manager proposes random sampling of apparently healthy fish for SRS surveillance. Using the risk-based principle, suggest a cheaper scheme with higher detection probability, and justify it.

8.9 Summary & what’s next

  • How you sample (random, systematic, stratified, cluster/multi-stage) matters as much as how many.
  • Estimating a prevalence: n = Z²p(1−p)/d²; precision scales with 1/d², so tight margins are costly; finite populations reduce n.
  • Detecting disease needs a design prevalence; rarer targets and imperfect tests both demand larger samples.
  • Risk-based sampling targets the likeliest positives and is the cost-effective backbone of real surveillance.

In Chapter 9 — Is It Really the Cause? we close Part III and the Anticipate arc with the hardest question in epidemiology: separating genuine causes from confounding and bias, so the associations from Chapter 5 can be trusted.

8.10 References & further reading

  • Dohoo I, Martin W, Stryhn H. Veterinary Epidemiologic Research. 2nd ed. VER Inc.; 2009. — sampling and sample size.
  • Delphino MKVC, Mardones FO, et al. Cost-effectiveness of longitudinal surveillance for SRS. J Fish Dis. 2021;44:315–326.
  • Laurin E, Gardner IA, et al. Diagnostic performance and risk-based sampling for P. salmonis. J Fish Dis. 2020;43:1167–1175.
  • Thrusfield M, Christley R. Veterinary Epidemiology. 4th ed. Wiley-Blackwell; 2018. — survey design.

Files for this chapter: none new.