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

Likelihood ratio test — where it appears

A test comparing the largest likelihood a model reaches with and without a restriction, using twice the log of their ratio as the statistic. Under the restriction that statistic is roughly chi-square, with as many degrees of freedom as the restriction removes, which is where its critical values come from.

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

A peak where the recorded data have none. The profile log-likelihood of a selection model in cy, the coefficient that lets the chance of being recorded depend on the outcome itself, for one study of 800 rows whose missingness is at random, with residuals normal; every other parameter is maximised at each fixed value. The model assumes the outcome is normal given the covariates. The curve peaks at cy = 0.35, where the fitted slope is 0.839, and the values of cy within the 95% cut run from −0.13 to 0.75; the likelihood-ratio statistic against cy = 0 is 1.47. The study was drawn with cy = 0.00. With the outcome's law left free, every value of cy fits the recorded rows equally well and this curve would be flat: its curvature is the normal assumption.

The assumption that identifies the mechanism

A selection model estimates how strongly an outcome decides whether it is recorded — the quantity two identical datasets showed no statistic can see — and it does so by assuming the outcome is normal. Where that holds and the outcome does decide, it repairs a slope complete cases put at 0.4318 to 0.5795. Where the missingness is at random and the residual is merely skewed, it reports selection that is not there, moves the slope from 0.5971 to 1.0319, and rejects missingness at random in 72.5% of studies.

missing · Missingness How often a skewed study reports selection that is not there, by the strength of a variable that moves recording only. 120 studies of 800 rows at each strength, missing at random on the regressor with a right-skewed residual, 35% unrecorded. The variable w shifts the chance of being recorded and has no place in the outcome; its share of the recording index's variance is 0.0%, 9.3%, 29.1%, 62.1% at strengths 0, 0.5, 1, 2. The selection model that includes w rejects missingness at random on 75.8%, 27.5%, 7.5%, 5.0% of studies; the same model fitted without w rejects on 78.3%, 85.0%, 100.0%, 100.0%.

A variable that moves recording

A selection model fitted to a skewed outcome that is missing at random reports selection that is not there on three studies in four. Give it a variable that shifts who is recorded and has no place in the outcome — strong enough to carry 29% of the recording index's variance — and the false reports fall to 7.5%, the slope from 1.034 to 0.613 against a truth of 0.6. Leave the same variable out of the model and it does harm instead: false reports reach 100%, and where the outcome really does decide recording, the estimated pull falls from 1.18 to 0.78.

missing · Missingness

Named alongside it

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

Complete-caseMaximum likelihoodMissing at randomMissing not at randomModel misspecificationNon-identifiabilityObservation propensitySelection modelClosed formConfidence intervalMissingness mechanismMonte Carlo

All concepts