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Within the displayed sequential response selection model, $\theta =0$ makes every attempt's nonresponse probability independent of $Y$, and hence gives the outcome-independence needed for complete-case analysis. The estimate $\hat{\theta }=0.38$ suggests that larger outcomes are associated with greater odds of an unsuccessful attempt. Per one-unit outcome increase, the fitted odds ratio is $e^{0.38}\approx 1.46$, with the reported interval transformed to $(e^{-0.06},e^{0.82})\approx (0.94,2.27)$.

The confidence interval for $\theta$ includes zero, so there is no rejection of outcome-independent missingness at the five-percent level within this model. This does not establish that the missing at random assumption is true: the interval also allows appreciable outcome dependence, and inference depends on the chosen selection model. The intercepts describe response levels, not independence from the outcome. Nor can the respondents' mean and standard deviation establish the missing outcomes' distribution. There are $300-211=89$ unobserved outcomes, about $29.7\%$, so conclusions about the full trial depend materially on the missingness assumptions.

Under missing completely at random, the observed second-year outcomes form a random subsample, so the direct pooled complete-case analysis estimate isIt is valid under missing completely at random because dropout no longer changes the marginal distribution of second-year use. Pooling is simpler than reporting separate conditional probabilities, but MCAR alone does not make this estimate more efficient than the estimate in the previous part. The fully observed first-year variable carries useful outcome information and can improve precision even when dropout is completely random.

To make the qualification explicit, write $X=Y_1$, $m(X)=P(Y_2=1\mid X)$, $\theta =E\{m(X)\}$ and $\rho =P(R=1)$ for the constant response probability. For $N$ patients the leading variance of the pooled estimator is $\mathrm{Var}(Y_2)/(N\rho )$. The standardization estimator in the preceding part has leading varianceIndeed its first-order centered contribution is $m(X)-\theta +R\{Y_2-m(X)\}/\rho$: the two terms have zero covariance, and their variances give that expression. The law of total variance therefore yieldsThe inequality is strict when there is dropout and first-year use predicts second-year use, as the distinct conditional rates suggest here. Thus the requested universal efficiency claim needs qualification. In the unrestricted joint binary-outcome model, maximizing the observed-data likelihood under either missing at random or missing completely at random gives the same standardization estimate $0.384889$: the all-patient first-year proportion and the two observed conditional second-year proportions maximize its factored outcome likelihood. Restricting the distinct missingness parameters to a common response probability affects their factor, not this estimate. The pooled value is the simple valid MCAR answer; retaining the first-year information gives the efficient MCAR answer and does not require replacing the previous estimate.

A complete-case analysis changes the mix of people contributing to the logistic regression. Let $A=1$ mean that both required variables are observed, and write $s_c(p,z)=\mathrm{Pr}(A=1\mid C=c,P=p,Z=z)$. Within $P>1$, Bayes' theorem givesIf completeness depends differently on the two responses, and that selection also varies with partner count or the other predictors, the extra term changes the fitted regression coefficients. For example, people with high counts and poor condom use can be under-represented, weakening or otherwise distorting the apparent association. Dropping people with missing $P$ also excludes some whose membership of the $P>1$ target group is unknown.

The fitted conditional association among complete responders need not equal the target-population association. No direction of selection bias follows without specifying the response mechanism. Selection on predictors alone need not bias a correctly specified conditional logistic regression; it is the outcome-related selection described here that removes that reassurance.

A complete-case analysis estimates each success probability among clinic attenders. If attendance depends on the unobserved outcome even after conditioning on treatment, the observed success proportions differ systematically from those in the randomized groups. This outcome-dependent dropout causes selection bias and can bias their difference.

Suppose outcome collection stops at the first success or after $m$ unsuccessful attempts, and let $A$ count unsuccessful attempts. Conditional on the outcome $y$, put $q_j(y)=\mathbb{P}(A\ge j\mid A\ge j-1,Y=y)$. For $k<m$, the joint density of $A=k$ and an observed $Y=y$ isFor $A=m$, the unobserved outcome is integrated out, giving $\int f_{\beta }(y){\prod }_{j=1}^m{q}_j(y)dy$. These probabilities sum to one by telescoping and integration. Dependence of $q_j$ on $y$ models missing not at random. If all $q_j$ are independent of $y$ conditional on the regression predictors, the outcome density among responders remains $f_{\beta }$, as required for complete-case analysis of that conditional model.