Concept

Sequential testing — where it appears

Testing repeatedly as data arrives, which multiplies the chances to reject and inflates the error rate unless the thresholds are adjusted. Spending the level across the looks is the repair, and how it is spent is a design choice with real consequences for power.

Named by 5 essays across 5 fields — each of them below, with the objects they name alongside it.

Testing at 0.05 every time the data is looked at. The null is true in every one of these trials and the test is correct every time it is run. Looking once rejects 4.9% of the time, as it should; looking ten times rejects 19.2% of the time. Nothing changed except permission to look.

When the looking happens

A p-value is defined relative to a sampling plan, so the same data means different things under different stopping rules. Testing five times at the nominal level rejects a true null 14% of the time, and no observation in the dataset changed.

sequential · Stopping

What a second positive is worth, prevalence 0.10%. A 90% sensitive, 95% specific test. One positive gives 1.77%. Two independent positives give 24.49%, which is what multiplying the likelihood ratios says. At a correlation of 0.1 between the tests' errors it is 10.16%, and at 0.5 it is 3.16%.

The second test that is not a second opinion

Two positives from a 90/95 test on a one-in-a-thousand condition give a 24.49% chance of disease if the tests are independent. At a correlation of 0.1 between their errors it is 10.16%, and at 0.5 it is 3.16% — barely more than the 1.77% one positive was worth.

screening · Baserate

What a positive test means, sensitivity 90%, specificity 95%. At a prevalence of one in a thousand, 98 of every hundred positives are false. At one in 10, 33 are. The test has not changed.

The base rate was always Bayes

The screening arithmetic everybody finds counter-intuitive is a posterior update with a prior of one in a thousand. Naming it that way turns a famous puzzle into an instance of a rule, and makes the sequential version obvious.

bayes · Baserate

Twenty runs simulating an exactly 95% interval, checked every 250 replications. Each line is one run's running estimate; the dashed band is where the Wilson interval of the running estimate still contains 95%, and a run stops, marked, the first time it leaves the band. 8 of these twenty stop before 10,000 replications. The exact probability of stopping, from the recursion over the count, is 29.54%.

A simulation that stops when it looks settled

A simulation of an interval that covers exactly 95%, checked every 250 replications for a significant departure and stopped when it finds one, flags that correct interval on 29.54% of runs. Stopped instead as soon as its estimate reaches 95%, it reports an interval that covers 94% as meeting its level on 37.21% of runs. Stopped when the estimate stops moving, it reports the right number — and has quietly chosen to run about fifteen hundred replications.

method · Seeds

The bounded error and the unbounded one. How the sequential trace procedure's answer is distributed, against the sample length, for a three-series system with 2 genuine relations. Over-counting — claiming a stationary combination that is a random walk — reads 4.9%, 7.2%, 5.7%, 6.2%, 5.9%, 4.2% across the six lengths, never far from the 5% of a single test. Under-counting reads 69.5%, 40.2%, 14.0%, 0.5%, 0.0%, 0.0%. The procedure is described as a 5% rule and the 5% applies to one of those columns.

The rank is a decision

The sequential procedure's 5% bounds one of its two errors. Over-counting reads between 4.2% and 7.2% at every sample length from fifty observations to three hundred; under-counting reads 69.5% at fifty and 0.0% at three hundred, and nothing in the procedure bounds it.

systems · Rank

Named alongside it

The objects these essays reach for when they reach for this one.

Error rate Posterior Base rate Interim analysis Monte Carlo p-value Predictive value Random walk Screening Statistical power The Bayes factor Binomial proportion

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