= Solution
The prose names a second year that differs from the table label. The calculations use the four supplied counts, interpreting the first and second rows as the two displayed periods; no numerical conclusion depends on resolving that label discrepancy.
The first <generalized linear model> treats the four counts as independent <Poisson random variables>, with a <log link> and both row and column <regression factors>. The row-by-column interaction makes the <log-linear model> saturated. With $r,c\in\{0,1\}$, write
$$
\log\mu_{rc}=\alpha+\beta r+\gamma c+\delta rc.
$$
Its four <maximum-likelihood fitted values> equal the observed counts. Thus $\widehat\alpha=\log20$, $\widehat\beta=\log(12/20)$, $\widehat\gamma=\log(9/20)$, and
$$
\widehat\delta=\log\frac{20\cdot11}{9\cdot12}=0.711496.
$$
The interaction is the <log odds ratio> measuring association of row and column. Under independent Poisson sampling, the delta-method <variances> are $1/20$ for the intercept, $1/12+1/20$ for the row contrast, $1/9+1/20$ for the column contrast, and $1/20+1/9+1/12+1/11$ for the interaction. Their square roots reproduce the printed <standard errors>. Although the software labels the standardized ratios as t values, these are asymptotic normal <Wald statistics>, with known <Poisson> dispersion one, not exact finite-df Student tests.
The <statistical saturated model> has zero <residual deviance> and zero <residual degrees of freedom>; this is a perfect interpolation, not evidence of predictive adequacy. The null deviance $5.016056$ compares the four observed counts with an intercept-only model, whose four fitted means are $52/4=13$. <Fisher scoring> is the numerical score/information iteration used to fit the model; the iteration count is a convergence report, not a test statistic.
Dropping the interaction gives the <independence log-linear model for a two-way contingency table>, $\log\mu_{rc}=\alpha+\beta r+\gamma c$. Its <Poisson regression margin-matching score equations> match the row and column totals, so
$$
\widehat\mu_{rc}=\frac{r_r c_c}{N},\qquad \widehat\mu=\begin{pmatrix}17.846154&11.153846\\14.153846&8.846154\end{pmatrix},\qquad N=52.
$$
Here $r_r$ and $c_c$ denote the observed row and column totals. In particular the fitted row multiplier is $23/29$, the column multiplier is $20/32$, and the intercept is $\log(29\cdot32/52)=2.881788$. These produce the second coefficient table. Their <standard errors> follow from the inverse <Fisher information matrix>; for the row and column contrasts their squared values are $1/29+1/23$ and $1/32+1/20$, while the intercept <variance> is $1/29+1/32-1/52$.
The <Poisson deviance> relative to the saturated fit is
$$
G^2=2\sum_{r,c}y_{rc}\log\frac{y_{rc}}{\widehat\mu_{rc}}=1.527855.
$$
The linear terms cancel because fitted and observed totals agree. The additive model has three coefficients, leaving one <degree of freedom>. <Wilks theorem> calibrates this <likelihood-ratio test> approximately by $\chi^2_1$, giving $p\simeq0.2164$. The interaction <Wald test> instead gives $p\simeq0.2192$; the two are only asymptotically equivalent.
<Fisher's exact test> conditions on both margins under independence. If $X$ is the upper-left count, then
$$
P(X=x)=\frac{\binom{32}{x}\binom{20}{29-x}}{\binom{52}{29}},\qquad9\le x\le29.
$$
Its usual two-sided <p-value> sums the probabilities of all tables no more probable than the observed $X=20$. That sum is $0.259724$, reproducing the output without an asymptotic approximation. It tests the same absence of row-column association, but conditioning and discreteness explain the numerical difference from the <likelihood-ratio test>.
In ordinary language, the male proportion among the cases is $20/29\simeq69.0\%$ in the first row and $12/23\simeq52.2\%$ in the second. The estimated case-sex <odds ratio> is $2.037$, but its approximate 95-percent <confidence interval> is wide, $(0.655,6.338)$. \b[These small samples do not give convincing evidence that the sex composition of cases changed between the two periods.] Failure to reject is not evidence that the compositions are exactly equal. Nor do these counts establish a male-to-female disease-incidence ratio or an incidence trend: population denominators and comparable ascertainment would be needed for those interpretations.
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