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A complete-case analysis uses only observations for which every variable required by the analysis is observed. It can be biased when completeness selects observations according to their outcomes or other relevant variables.

Let $R$ indicate observation of a regression response $Y$, with fully observed predictors $X$. If $Y\perp R\mid X$ and $\mathbb{P}(R=1\mid X)>0$ on the target support, the observed conditional density equals the full conditional density. Thus a correctly specified regression model can be fitted to observed responses under conditional missing at random. This does not justify the unadjusted complete-case mean when observation rates and outcome means both vary with $X$, nor does it imply missing completely at random.

When a response $Y$ is missing at random conditional on a fully observed category $X$, with positive response probability in every occupied category, estimate its conditional probabilities using respondents within each category and average them with the category frequencies from all patients. For a binary response, $\hat{P}(Y=1)={\sum }_x\hat{P}(X=x)\hat{P}(Y=1\mid X=x,R=1)$. The weighting uses the full cohort rather than only complete cases.

With a fully observed covariate $X$ taking finitely many values of positive probability, response $Y$ of finite variance, and an independent response indicator with probability $\rho$, let $m(X)=E(Y\mid X)$. A covariate-standardized mean has asymptotic variance $[\mathrm{Var}(m)+E\{\mathrm{Var}(Y\mid X)\}/\rho ]/N$, whereas the complete-case mean has variance $\mathrm{Var}(Y)/(N\rho )$. The difference is $(1-\rho )\mathrm{Var}(m)/(N\rho )$. Thus missing completely at random does not justify discarding predictive observed covariates on efficiency grounds.

A frequency table gives category counts and percentages. A full-sample percentage divides by 1784, whereas a valid percentage divides by the number observed for that variable. A cumulative percentage adds categories in the displayed order. Differences between the two percentage columns therefore reflect missing values, while small discrepancies from 100% can arise from rounding error.

The age groups contain 329 respondents aged 18–24, 320 aged 25–34, 371 aged 35–44, 323 aged 45–54 and 441 aged 55+, approximately 18.44%, 17.94%, 20.80%, 18.11% and 24.72%. The oldest category is the largest, but it is open-ended and spans a different width from the younger categories, so its larger count does not establish a higher frequency per single year of age. The ordered cumulative 57.18% means that about that proportion is younger than 45. The displayed final 100.01% is compatible with rounding, not an extra respondent. There are no reported missing ages.

Blocking has valid counts 133 Not know, 1299 No and 268 Yes. Dividing by all respondents gives approximately 7.46%, 72.81% and 15.02%; dividing by the 1700 valid responses gives about 7.82%, 76.41% and 15.76%. The remaining 84 observations, 4.71% of the full sample, are missing. The distinction matters: 15.02% is not the same denominator-based statement as 15.76%. Not knowing that blocking was possible is a supplied answer category, not one of those 84 missing values. It should not be deleted or merged merely because its label contains “know”.

Blocking is a nominal categorical variable in this form. Yes and No identify a binary behaviour, while Not know additionally records awareness. There is no automatic substantive ordinal scale Not know $<$ No $<$ Yes. Consequently its cumulative percentage is tied to the printing order and should not be read as a universal low-to-high blocking scale. This point will limit interpretation of the later ordinal coefficients.

There are 478 observed Not-populist classifications and 1214 Populist classifications. These represent 26.79% and 68.05% of the full sample, or 28.25% and 71.75% of the 1692 classified respondents. The 92 missing classifications account for 5.16%. These percentages describe the particular classification and sample; the output does not supply the instrument used to define populism or establish prevalence in a wider population. Adding missing cases to either substantive group would invent their classifications.

The later table uses only 1629 observations with both blocking and populism observed. It is smaller than either marginal valid total, as expected under complete-case analysis. The missingness overlap from marginal and complete-case counts givesThus 63 have blocking missing but populism observed, 71 have populism missing but blocking observed, and 21 have both missing. These totals reconcile the denominators without reconstructing individual records. They do not establish a missing-data mechanism such as missing completely at random; selection into complete cases could affect later associations.

The ordinary least squares fit estimates the conditional mean of the continuous safety response as a linear function of the income coding:The regression coefficient for income is positive. Since larger response values mean greater safety, one additional coded income unit corresponds to an estimated increase of $0.2369$ safety-scale units in the fitted mean. Following the question's stated units, a unit is a £10,000 income step. The observed coding also uses income bands with open end categories, so this is a linear trend per coded band step; it is not evidence of a known constant effect for every extra pound, especially within the open top band. The output's abbreviated variable label does not override the stated units.

The intercept is the fitted mean at income code zero. It is estimated as $6.5933$, but zero is outside the observed income codes, so that value is an extrapolated model baseline, not an observed estimate for a particular income category. Its small p-value tests a zero intercept; it is not the main test of the income association.

For the income coefficient, the standard error is $0.0604$. The reported test statistic and two-sided test giveproviding strong evidence against the null hypothesis of zero linear slope under the stated model assumptions. A $95\%$ Student t confidence interval, using $1727$ residual statistical degrees of freedom, isThis is an interval for the population linear coefficient under the model, not an interval containing $95\%$ of individual safety scores. The large-sample estimate is precise enough to detect a positive association, but that does not establish substantial individual predictive power.

The coefficient of determination is $R^2=0.008828$: the fitted line accounts for about $0.883\%$ of the in-sample variation about the response sample mean. The adjusted coefficient of determination is about $0.825\%$. Thus income alone explains very little of the observed variation, despite the small slope p-value. The residual standard error of $3.997$ response units describes substantial scatter about the fitted line. This is an estimated residual standard deviation, rather than the standard error of the slope or a complete individual prediction interval.

The overall F-test has $F=15.38$ on $1,1727$ statistical degrees of freedom and $p=9.129\times 10^{-5}$. With one predictor and an intercept, it tests the same zero-slope hypothesis as the coefficient Student's t-test; $F=t^2$, up to rounding. It is not an additional independent confirmation. In this simple linear regression, the Pearson correlation coefficient has the slope's sign and magnitude $\sqrt{R^2}\simeq 0.0940$, another description of the weak linear association.

The regression uses $1729$ complete observations and estimates two coefficients, giving $1729-2=1727$ residual statistical degrees of freedom. The deletion of $88$ observations is complete-case analysis; extending its conclusions to the full target population requires suitable assumptions about the missing-data mechanism. Classical reported standard errors also rely on an adequate linear conditional mean, independent errors and homoscedasticity; a normal linear model gives the finite-sample Student t-distribution and F-test calibration, while large-sample approximations can justify similar tests under weaker distributional assumptions. The printed summary alone does not establish these assumptions. Regression diagnostics and, where appropriate, heteroskedasticity-consistent standard errors address different parts of that assessment.

Finally, this unadjusted observational study estimates an association. Other covariates, such as age or experience status, may be related to both income and reported safety. Without additional design or causal identification assumptions, the positive fitted slope does not establish that increasing income would cause an increase in safety.

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.