Missing data and multiple imputation

Correct answer: Those 108 were dropped silently by the fitting function - complete-case analysis is a choice with an assumption behind it, not a default

The printed count is who stayed. The gap of 108 comes from the fitting function skipping any row with a missing covariate, with no warning and no error. 106 is the number who declined randomisation and entered the registry; the two are close but not the same group, because what was dropped is every row missing at least one of the five covariates this model uses, which merely overlaps heavily with the registry. 124 is the event count among the rows that stayed - a Cox model excludes nobody for not having an event, and censored rows still contribute to the risk sets. Whether dropping these rows matters is decided by how much they resemble the rows that stayed, not by how many went.

Correct answer: The standardised mean difference for age is 0.299, well past the usual balance threshold, and those excluded are the older ones

Age at 0.299 is already far from balance: those excluded were about three years older on average. Bilirubin at -0.046 really is close to zero, but a table has to be read across every row - age is imbalanced, albumin is imbalanced, and the two groups differ in observed event rate as well, so letting one row speak for the whole table misses the other two facts. Albumin at -0.222 has its direction reversed: those excluded had lower albumin, not higher, and which group is the minuend decides what the sign means. That is the cell this kind of table is most often misread on. Complete-case analysis does not merely cost sample size, it shifts the cohort's observed event rate too. Almost no paper prints this table, yet it is the only basis for judging whether the exclusions matter.

Correct answer: The within-imputation variance is 0.008878, measuring the uncertainty this estimate would carry with complete data, and what imputation adds on top is comparatively small

The total is the average within-imputation variance plus a factor slightly above 1 times the between-imputation variance. The within part, 0.008878, is the uncertainty the estimate would carry with complete data; the between part, 0.000463, is what imputation adds because the missing values are not known. They differ by more than an order of magnitude, so the total of 0.0094 comes almost entirely from the sample itself. 0.051889 is not a variance at all but the fraction of missing information, the share of the total that the between part accounts for, and reading it as a variance gets the units wrong. One caution: the imputation on this page is deliberately simplified and systematically understates the between-imputation variance, so this fraction demonstrates a mechanism rather than reporting a result about this dataset.

Correct answer: Because imputation brings the analysis back to all 418 rows, so what the discarded rows do carry - age, bilirubin, albumin - returns to the model

The point is that the discarded rows are not blank: age, bilirubin and albumin are all recorded, and only copper or stage is missing. Complete-case analysis throws away a whole row over one missing cell; imputation fills that cell so the row rejoins the 418, and the extra information is where the narrowing comes from. Saying that 310 rows still enter the model is the single-imputation misreading, since the analysed count really does change. The 412 row is a different thing again: it drops copper entirely, which changes the set of covariates and therefore estimates a different quantity. Its interval is narrower not because it is more efficient but because it buys that width by accepting residual confounding, so it cannot be set alongside the other two.

Correct answer: No. Copper and bilirubin have a Spearman correlation of 0.63, so they are not going their separate ways; the closeness came out of the run and was not derivable in advance

0.63 is a moderately strong monotone correlation: when copper is missing, the information it shares with bilirubin goes missing with it, so unrelated to the exposure does not hold in this dataset. The bilirubin row of the comparison table is indeed near zero, but it says something else - the excluded and the retained have similar mean bilirubin, which is neither evidence that the missingness mechanism ignores the exposure nor a guarantee that the adjusted coefficient will not move. 0.26 is the missing fraction, and a fraction on its own settles nothing in either direction: it cannot show the estimate is unbiased, and it cannot guarantee bias the way that option does - the same fraction under a different mechanism gives very different answers, and here the two estimates did come out close. Change the structure - a higher missing rate, or missingness in a variable still strongly related to the outcome after adjustment - and the two approaches can diverge. Closeness is a result, not a premise.

Correct answer: It is the 106 who declined randomisation and entered registry follow-up; the trial laboratories were never drawn for them by protocol, which makes this structural missingness

The block is 106 multiplied by several laboratory columns. These are the ones who declined randomisation and entered registry follow-up, and those laboratories are part of the trial protocol - so the reason for the missingness can be written down, which is what makes the case for treating it as close to MAR. That case comes from knowing how the data were generated, never from the data themselves. 312 is the enrolled count, and their stretch of the map is largely white; reading the red block as the trial arm reverses the mechanism entirely. Platelet count is the exception in the by-group table: it is missing for only a handful of registry rows, which is ordinary scattered missingness and far too little to fill the block. Whether missingness has a structure, and what that structure is, decides whether complete cases, imputation or a sensitivity analysis is the right next step.