Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 41 / 2

Interpret each mechanism for a fixed missingness pattern $r$: the observed and missing response components are the coordinates selected by that pattern, and the covariates $X$ are assumed fully recorded.

For missing completely at random, missingness depends on neither response values nor covariates:For covariate-dependent missing completely at random, response observation may depend on $X$ but not on the response after conditioning on $X$:This is conditional independence $R\perp Y\mid X$ and can still produce a marginal association between response and missingness.

For covariate-dependent missing at random, the observation probability may use observed response components and covariates, but not the values missing under that pattern:For missing not at random, there remains dependence on $Y^m$ after conditioning on $Y^o,X$; the displayed simplification is not valid. For example, refusal could depend on the unreported response itself, even among individuals sharing all recorded covariates. These mechanisms concern the reason for nonobservation, not whether the incomplete records merely look irregular. Their distinction is not generally identifiable from the observed responses alone.

The observed positive counts are $200(0.18)=36$ and $400(0.08)=32$. Thus the responder-only estimate isThis estimates the proportion among the responders; it does not yet address either kind of nonresponse.

Assume that, conditional on the recorded embarrassment category, answering the item is independent of virginity status. Also assume that this category is assessed comparably for both groups and that both response strata have positive observation probabilities. This is covariate-dependent missing completely at random for the item response with embarrassment as the fully observed covariate; it supports MAR standardization over a fully observed covariate.

Use the responder probabilities $0.18$ and $0.08$ within strata, but the stratum sizes in the full group of responders plus item nonresponders: $300$ and $450$. This givesEquivalently, the estimated missing positive counts are $100(0.18)=18$ and $50(0.08)=4$, adding 22 to the 68 observed positives. The marginal estimate changes because the full group contains a larger proportion judged embarrassed. Merely copying the unstratified responder percentage would ignore that difference. The conditional independence assumption cannot be verified using the missing item responses.

The unit and item nonresponse distinction matters: embarrassment was not recorded for the 300 unit nonresponders. Conditional independence of unit response and virginity given embarrassment is therefore not enough unless the embarrassment distribution among unit nonresponders is also supplied.

One sufficient set of additional assumptions is that unit response is independent of the pair consisting of embarrassment and virginity. Equivalently for this estimation purpose, assume that unit nonresponders have the same embarrassment mixture as the other 750 individuals, and the same virginity probabilities within those strata. The former gives an embarrassed fraction $300/750=0.4$, and the latter gives probabilities $0.18$ and $0.08$. Their estimated positive proportion is consequently $0.4(0.18)+0.6(0.08)=0.12$, or 36 of the 300. Combining the groups givesThe item-response assumption from (ii) supplies the first 90; the added unit-response transport assumptions supply the final 36. A weaker sufficient assumption is simply that the unit nonresponders' marginal virginity proportion equals the estimated $12\%$ in the other group, but it is still untestable from these data.

Without any unit-response transport assumption, retain the item assumption and write the unidentified unit proportion as $p_U$. ThenThus the extrapolation is an assumption-dependent estimate, not a quantity determined by the observed table alone.

A sensitivity analysis for unit and item nonresponse asks how much the population estimate and substantive conclusions change when the untestable response assumptions are weakened. Vary plausible virginity probabilities among the missing groups, and vary the unknown embarrassment mixture among unit nonresponders; identify combinations which materially alter the estimate or reverse a conclusion.

For example, let $a,b$ be the positive probabilities among embarrassed and nonembarrassed item nonresponders, let $q$ be the embarrassed fraction among unit nonresponders, and let $c,d$ be their corresponding positive probabilities. The sensitivity estimate isThe nominal assumptions set $a=c=0.18$, $b=d=0.08$ and $q=0.4$, reproducing $12\%$. Different choices model outcome-dependent refusal or a changed mixture. Sampling confidence intervals measure a different uncertainty and should also be reported where appropriate; they do not validate the missingness assumptions.