= Solution
Let $p_1$ be the alternative death <probability>. The <odds ratio> specifies
$$
\frac{p_1}{1-p_1}=k\frac{p_0}{1-p_0},\qquad p_1=\frac{kp_0}{1-p_0+kp_0},\qquad 1-p_1=\frac{1-p_0}{1-p_0+kp_0}.
$$
Assume $0<p_0<1$ and $k>0$. For observed batch count $y_t$, the common combinatorial factor in the two <binomial likelihoods> cancels, giving the <CUSUM> increment
$$
\begin{aligned}
W_t&=\log\frac{\binom n{y_t}p_1^{y_t}(1-p_1)^{n-y_t}}{\binom n{y_t}p_0^{y_t}(1-p_0)^{n-y_t}}\\
&=y_t\log(p_1/p_0)+(n-y_t)\log[(1-p_1)/(1-p_0)].
\end{aligned}
$$
Therefore
$$
\boxed{W_t=y_t\log k-n\log\{1+(k-1)p_0\}.}
$$
For $k>1$ large death counts add positive evidence of deterioration; for $k<1$ unusually small death counts add evidence for improvement. The score has nonpositive <expectation> under the null and nonnegative <expectation> under the specified alternative, as follows from the nonnegativity of <Kullback-Leibler divergence>. Independent, nonoverlapping batches avoid counting the same patient's outcome repeatedly.
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