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

The value that is not there

A missing value is missing conditional on something, and which something decides everything that follows. Dropping the incomplete rows leaves a regression slope exactly right when the chance of being observed depends on the regressor, however strongly, and wrong by a quarter of itself when it depends on the outcome — and the data cannot tell those two cases apart. Filling the gaps in is not free either: one filled value is not an observation, and the arithmetic that makes several of them into one is two corrections rather than the one everybody quotes.
Three mechanisms leave the slope alone; one does not. The bias of the complete-case slope under each of four missingness rules, counted over 4000 studies of 200 rows at 35.0% missing, with the closed form printed beside each count. Missingness that depends on nothing, on the regressor, or on the second covariate leaves the slope exactly where it was — the closed forms are zero to machine precision and the counts are -0.0005, -0.0005 and -0.0011 against standard errors of about 0.0018. Missingness that depends on the outcome moves it by -0.1635, which is 27.3% of the slope being estimated. The same share of rows is lost in every case.

Three mechanisms and one dataset

Four rules for which outcomes go missing, each calibrated to lose the same 35% of the rows and each leaning on what it reads with the same coefficient. Three leave the fitted slope exactly where it was, and the one that reads the outcome moves it by 0.163531.

 Unrepresentative in every respect but the one that matters. Three properties of the complete cases as the chance of being observed leans harder on the regressor, in closed form, at 35.0% of outcomes missing throughout. The mean of the regressor among the rows kept climbs from 0.0000 to 0.5528 against a population mean of zero, and the mean of the outcome from 0.0000 to 0.3980 above its own. The bias in the fitted slope is exactly zero at every one of the ten settings, because selection acting on the regressor alone leaves the conditional law of the outcome given the regressor untouched and least squares conditions on exactly that. The sample is wrong about almost everything and right about the one quantity being estimated.

Dropping the incomplete rows

Push the missingness until the rows that survive have a covariate mean of 0.543905 against a population zero and a variance of 0.5041 against one, and the fitted slope is still exactly right. Where the rule reads the outcome instead, the same sweep takes coverage to 2.42% at eight hundred rows.

 A filled value is not an observation. What a 95% interval for the slope actually covers after each way of handling 35.0% missing outcomes, counted over 4000 studies of 200 rows. Dropping the incomplete rows covers 95.93%. Filling with the observed mean covers 13.85%, because the estimate itself has moved. Filling with a fitted value covers 80.85% against a closed prediction of 79.73%: the estimate is right and the reported standard error is short by a factor of 0.6567 against a predicted 0.6500, because the residual sum of squares is divided by the whole sample's degrees of freedom. Adding residual noise recovers the spread and covers 85.78% against a predicted 84.62%, since the interval still ignores the variance of having imputed at all.

One imputation is not an observation

Three ways of filling a missing outcome, under a mechanism that makes dropping the rows beyond reproach. Filling with the observed mean covers 13.85%, filling with a fitted value covers 80.85%, adding noise covers 85.78%, and the thing all three were meant to improve on covers 95.93%.

 The extra 1/m, and the correction nobody quotes. What a pooled 95% interval covers against the number of imputations, counted over 2000 studies of 200 rows at 35.0% of outcomes missing. Rubin's rules — total variance W̄ + (1 + 1/m)B, read against a t distribution on (m − 1)(1 + W̄/((1 + 1/m)B))² degrees of freedom — cover 94.10% at two imputations and reach their promise by 5, at 95.25%. Dropping the (1 + 1/m) factor takes two imputations to 93.10%; using a normal quantile instead of the degrees-of-freedom correction takes it to 92.55%; dropping both takes it to 91.45%. The median degrees of freedom at two imputations is 12.95, which is why the second correction is the larger.

The variance between imputations

Pooling several filled datasets covers 94.10% at two imputations and reaches its promise at five, where a single fill covered 85.78%. The correction everybody quotes is the smaller of the two doing the work — 1.00 ± 0.22 points against 1.55 ± 0.28.

 The damage does not stay in the term that was left out. Where each coefficient lands when the model that fills the missing outcomes and the model that analyses them disagree, over 1500 studies of 200 rows at 35.0% missing and 20 imputations. An imputer that omits a covariate the analysis fits attenuates that covariate's coefficient by exactly the missing fraction — -0.1405 counted against a closed -0.1400 — and pushes the coefficient it did impute on the other way by exactly the product of the omitted coefficient, the covariates' correlation and the missing fraction: 0.0402 counted against 0.0420. Both closed forms come out of the same two-by-two solve. Matching models leave both alone, and so does an imputer that knows more than the analysis.

An imputation model the analysis does not contain

A model that fills the gaps without a covariate the analysis fits attenuates that covariate's coefficient by exactly the missing share, 0.4 to 0.26, and moves the one it did carry by exactly γρf, 0.6 to 0.642. The reverse case is supposed to inflate the interval, and at four strengths of the extra knowledge it does not.

 An estimate reported as a function of an assumption. What the slope really is, against a shift in the outcomes nobody saw — line from the closed form, dots counted over 2000 studies of 200 rows at 35.0% missing. Every point on this line produces exactly the same observed data, and the complete-case estimate is the flat line at 0.5996 regardless. The truth moves at -0.2845 per unit of shift, which is a function of the missingness model and the missing fraction and of nothing that can be estimated: across the swept range the true slope runs from 0.8845 to 0.3155, a span of 0.5691 against a value of 0.60 in the world where the shift is zero. Reporting the line is the honest form of the answer.

The mechanism the data cannot see

Two worlds produce identical covariates, identical patterns of what is recorded and identical recorded outcomes, to the last bit. Their true slopes are 0.6 and 0.315452, and the truth moves at 0.284548 per unit of an assumption nothing in the data can inform.

 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.

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