Community estuary bacteria monitoring near an outfall
A community estuary group samples E. coli at four sites: two upstream and two downstream of a wastewater treatment plant outfall. Volunteers collect one surface sample per site per month for 18 months, giving 72 results. They treat each result as independent and find downstream mean 310 cfu/100 mL vs upstream 180 cfu/100 mL; a t-test gives p = 0.03. The laboratory detection limit is 10 cfu/100 mL and 11 samples are reported as '<10'. Sampling times vary with tide and rainfall, and the two downstream sites are 150 m apart. The group wants to tell the regional council that the plant is causing elevated bacteria and that the result is statistically significant.
1. Identify the main statistical problems with the group's analysis and explain how each affects the p-value or effect estimate.
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The 72 results are not 72 independent observations: each site is sampled repeatedly, and the two downstream sites are close together. Treating them as independent is pseudoreplication and makes the p-value too small. Tide and rainfall affect E. coli, so sampling at different times confounds site with conditions. The 11 '<10' results are censored, not true zeros; substituting zero or the limit biases the mean and variance. A t-test assumes independence and roughly normal, equal-variance data, which is doubtful for counts. The two upstream sites may differ from each other, and the downstream sites may share a plume. The effect estimate is also sensitive to the detection-limit handling. The p = 0.03 should not be read as strong evidence.
2. Propose a more appropriate analysis and a sampling design for the next 12 months. Be explicit about the unit of replication and how you would treat '<10'.
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Use the site as the unit of replication and pair upstream and downstream samples by date where possible. Fit a mixed model with site random effects and fixed effects for position, rainfall, tide, and season; or use a randomisation test that permutes site labels, respecting the few sites. Treat '<10' with a censored regression (e.g., Tobit) or report a range of assumptions and sensitivity. Increase the number of sites and sample at standardised tide and rainfall strata; take composite samples. Report the estimated downstream–upstream difference with a confidence interval and the between-site variation. A p-value from a model that respects clustering is more defensible, but with four sites the evidence is still limited.
3. The council asks whether the result proves the plant is the cause. Write a short response that distinguishes association, contribution, and proof.
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Association, not proof. The data may be consistent with a plant contribution, but bacteria can come from birds, pets, runoff, boats, or other sources. To argue contribution, you need a gradient with distance from the outfall, repeated before/after or up/down comparisons, microbial source tracking, flow data, and control for rainfall and tide. Proof of causation is not usually available from observational monitoring. Say what the result supports: elevated downstream counts and a plausible plant influence, with uncertainty. Recommend further sampling and source tracking before asserting cause.