= Solution
Conditioning on marital status produces five <contingency tables>. In each, compare the conditional row <percentages>. The following differences are the <sample proportion> with some bad experiences minus the <sample proportion> without them, in <percentage points>; negative very-safe differences <mean> fewer respondents feeling very safe in the former group.
|| Marital-status stratum
|| Difference: very safe
|| Difference: unsafe
|| Reported <Pearson's chi-squared test> <p-value>
|| <Cramér's V>
|| <Goodman and Kruskal's gamma>
| Single
| $-4.43$
| $+1.82$
| $0.6114$
| $0.048$
| $0.060$
| Married
| $-8.43$
| $+0.72$
| $0.0257$
| $0.092$
| $0.093$
| Living with a partner
| $-13.14$
| $+2.52$
| $0.04885$ (inconsistent; see below)
| $0.144$
| $0.162$
| Divorced
| $-4.01$
| $-5.79$
| $0.3583$
| $0.106$
| $-0.013$
| Widowed
| $-8.96$
| $+13.11$
| $0.3158$
| $0.146$
| $0.220$
The <p-values> test <independence> within the respective strata; the magnitudes and directions come from the conditional <percentages> and association measures. Differences in the table are calculated from the counts before rounding.
Among \b[single] respondents, the very-safe <sample proportion> changes from $33.95\%$ to $29.52\%$, while the unsafe <sample proportion> changes from $36.28\%$ to $38.10\%$. The direction resembles the weak pooled pattern, but $p=0.6114$ supplies little evidence against <independence> in this stratum. This is not evidence that the population association is exactly zero.
Among \b[married] respondents, the very-safe <sample proportion> changes from $41.18\%$ to $32.75\%$ and the fairly-safe <sample proportion> rises from $27.68\%$ to $35.39\%$. Unsafe remains almost unchanged, $31.14\%$ versus $31.87\%$. Thus the within-stratum association again chiefly shifts very safe toward fairly safe. Its $p=0.0257$ is below the $5\%$ <significance level>, while $V=0.092$ still indicates a small association.
For respondents \b[living with a partner], very safe changes from $41.46\%$ to $28.32\%$, fairly safe from $30.08\%$ to $40.71\%$, and unsafe from $28.46\%$ to $30.97\%$. The very-safe difference is larger than in the married stratum. The printed <p-value> $0.04885$ would be just below the conventional $5\%$ <significance level>. However, this particular output is internally inconsistent: recalculating <Pearson's chi-squared test> from its six displayed counts gives $X^2=4.910389$, $p=0.08585$, and $V=0.14425$. The reported $V=0.144$ agrees with the counts, whereas the printed $X^2=6.900389$ would itself imply $p=0.03174$ for two <statistical degrees of freedom>, rather than the printed $0.04885$. \b[Using the displayed counts, this stratum does not reject independence at $5\%$.] The source PDF itself contains these incompatible figures; this is not resolved by rounding.
Among \b[divorced] respondents, very safe changes from $41.18\%$ to $37.17\%$, but unsafe also decreases, from $36.76\%$ to $30.97\%$, as fairly safe increases from $22.06\%$ to $31.86\%$. This is a redistribution toward the middle category, not a simple shift toward greater unsafety. The near-zero gamma, $-0.013$, reflects the lack of a consistent ordinal direction, while $V=0.106$ can still register distributional differences without a direction. The <p-value> $0.3583$ does not reject <independence>.
Among \b[widowed] respondents, very safe changes from $51.43\%$ to $42.47\%$, and unsafe from $14.29\%$ to $27.40\%$. The ordinal association is positive on the chosen ordering, with gamma $0.220$. Its lack of conventional <statistical significance>, $p=0.3158$, does not imply a smaller estimated association than in the married group: the widowed stratum has only $108$ complete observations, versus $857$ in the married stratum, and therefore less precision. All five tables have sufficiently large expected counts for the reported approximations; even the smallest expected count is about $8.10$.
\Image[/past-exam-of-the-digital-social-research-statistics-core-course-of-the-university-of-oxford/2023-marital-status-safety-differences.png]
{title=Derived safety-category differences and unadjusted uncertainty by marital-status stratum}
The plot uses large-sample <Wald confidence intervals> for the <difference of independent sample proportions> within each stratum. They concern the individual category contrasts, whereas the tabulated <Pearson's chi-squared tests> compare the full three-category distributions. The intervals are unadjusted for <multiple hypothesis testing>; the pooled row reuses the stratum observations, so it is not a sixth independent sample.
In the <elaboration model>, the persistence of a generally weak positive ordinal pattern within four strata is closer to \b[partial replication] than to a complete explanation of the pooled association by marital status. The differences in estimated patterns suggest \b[possible specification]: the divorced group's shift toward the middle category differs from the widowed group's shift toward unsafe. They do not by themselves establish population <effect modification>. In particular, <significant and nonsignificant results need not differ significantly>; compare the effects using an <interaction contrast> or a fitted model with the appropriate <interaction terms>, rather than comparing their separate <p-values>.<The Difference Between Significant and Not Significant is not Itself Statistically Significant by Andrew Gelman and Hal Stern>[Printed p. 328, Introduction: comparing significance labels does not test the difference between estimates]{ref}
One additional omnibus check can be derived from the supplied tables without assuming a common effect. Recalculating the five statistics from the counts, including the corrected living-with-partner statistic, the <joint conditional independence test for stratified contingency tables> gives
$$
\sum_{k=1}^5 X_k^2=17.5776085,\qquad \mathrm{df}=5(2-1)(3-1)=10,\qquad p\simeq0.06252.
$$
Under independent-stratum sampling this tests <independence> within every stratum jointly; it does not test whether the association is homogeneous across strata. It does not reject at $5\%$, but this alone does not establish that stratification removed the population association: its ten-degree-of-freedom alternative is less focused than the pooled two-degree-of-freedom test, and nonrejection is not proof of equality.
The <elaboration model> also distinguishes an antecedent <confounder> from an intervening <causal mediator>. These cross-sectional tables do not establish the relevant time ordering or identify either role for marital status. Five separate stratum tests further require caution about <multiple hypothesis testing>: for example a <Bonferroni correction> at familywise $5\%$ would use $0.01$ per test, below all five individual <p-values>. That caution changes the interpretation of selected significance labels, not the observed conditional <percentages>.
\b[The estimated patterns vary across strata, but the tables neither establish a population interaction nor prove that marital status explains away the association.]
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