= Solution
For a specified change, \b[use the log-likelihood-ratio score]. A <log-likelihood ratio> accumulates comparable evidence for the <alternative hypothesis> against the <null hypothesis>, includes the changing expected number of deaths, and yields the standard <CUSUM> designed to detect that alternative. Under the null its expected increment is negative; under the target increase it is positive.
By contrast, raw $O-E$ has zero null drift and does not specify which increase is being sought. Its noise scale changes with expected exposure, since a <Poisson> count with mean $E$ has variance $E$. A reset-at-zero chart with this zero-drift score can wander upwards through chance variation, and its threshold has no direct likelihood-evidence interpretation. An $O-E$ chart can be redesigned with an allowance and appropriately calibrated limits, but unadjusted $O-E$ is not the same as the <Poisson likelihood-ratio CUSUM>.
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