Normal Distribution
nothing is called 95% until it has been counted
Concept

Odds ratio — where it appears

The ratio of the odds of an outcome in one group to the odds in another, usually analysed on the log scale. It is the contrast logistic regression estimates, and for a rare outcome it is close to the ratio of the two risks.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Also named here as risk difference — the same set of essays touches all of them, so they are one junction rather than several.

One odds ratio in every stratum, and a different one marginally. Five strata with baseline risks from 5% to 85%, a treatment allocated by a coin in each, and a conditional odds ratio of exactly 2.5 throughout. The marginal odds ratio is 1.789. Nothing is confounded; the odds ratio is simply not a weighted average of odds ratios.

The change that is not confounding

Five strata, a treatment allocated by a coin in every one, and an odds ratio of exactly 2.5 in all five. The odds ratio computed on the pooled table is 1.789. Nothing is confounded — an odds ratio is not a weighted average of odds ratios, and the risk difference, on the same table, is exactly its own stratum value.

reversal · Simpson How often reporting whichever contrast is most significant rejects when the arms do not differ, by the split. 20,000 trials a point. With both arms at 0.05, 0.1, 0.3: at shares 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7 the chosen contrast rejects 6.5%, 5.6%, 5.0%, 5.0%, 5.1%, 5.1%, 5.2%, 5.2%, 6.3%; 6.0%, 5.4%, 5.3%, 4.9%, 4.8%, 4.9%, 5.2%, 5.4%, 5.8%; 5.8%, 5.6%, 5.2%, 5.1%, 5.2%, 5.1%, 5.3%, 5.4%, 5.7%.

A contrast chosen after the data

A two-arm trial with a binary outcome that reports whichever of the risk difference, the log risk ratio and the log odds ratio comes out most significant has tested three statistics that ask one question. At an equal split the choice is no choice at all: the difference has the largest z in every one of twenty thousand trials, so the selected test is the difference's, at its 5%. At a 30–70 split the choice is real and costs a point and a half of false positives, 6.5% at the rarest outcome. Pre-registering the difference and the odds ratio as a pair with Bonferroni's correction — the cautious alternative — costs eight to twelve points of power at every split.

allocation · Allocation

Named alongside it

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

Risk differenceAggregationAllocationBonferroniCollapsibilityConfoundingMultiple comparisonsRandomisationRelative riskRisk ratioSpecification searchStatistical power

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