= Solution
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 $s_c(p,z)=\Pr(A=1\mid C=c,P=p,Z=z)$. Within $P>1$, <Bayes' theorem> gives
$$
\boxed{\operatorname{logit}\Pr(C=1\mid p,z,A=1)
=\operatorname{logit}\Pr(C=1\mid p,z)+\log\frac{s_1(p,z)}{s_0(p,z)}.}
$$
If 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.
\b[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.
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