Concept

Conditional independence — where it appears

Two quantities carrying no information about each other once a third is fixed. It is what licenses multiplying likelihoods, it is much weaker than independence, and two tests sharing a specimen or a mechanism fail it in the direction that overstates the evidence.

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

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

One distribution, two effects. Three causal structures fitted to one covariance matrix over a treatment, a covariate and an outcome. Each reproduces it exactly — the largest entry-wise disagreement across all three is 4.4e-16 — so no sample of any size distinguishes them. The regression of the outcome on the treatment and the covariate returns 0.500 in all three, to within 4.4e-16, because that coefficient is a function of the covariance and of nothing else. The effect the three worlds hold is 0.500, 0.848 and 0.848: adjusting is exactly right in the first and off by −0.348 in the other two. The arithmetic cannot see the difference and the difference is the whole question.

The two worlds that look the same

Three causal structures were fitted to one covariance matrix and agree with it to 4.4·10⁻¹⁶. The regression returns 0.5000 under all three; the effect they hold is 0.5000, 0.8481 and 0.8481. What separates structures is a missing edge, and the signature of one is a correlation of exactly zero.

collider · Conditioning

Named alongside it

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

Base rate Causal diagram Causal effect Covariance matrix Fisher transform Identification Likelihood ratio Markov equivalence Null hypothesis Observational equivalence Partial correlation Posterior

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