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Let the annual count be $Y_t=y_t$, with null mean ${\lambda }_{0t}$ and alternative mean $\theta {\lambda }_{0t}$. Dividing the two probability mass functions for the Poisson distributions cancels $y_t!$ and the common power of ${\lambda }_{0t}$:Consequently the score for the Poisson likelihood-ratio CUSUM isAt $\theta =2$, this becomesUnder the stated Poisson model this identity is exact, rather than an additional approximation. The approximation in a practical application concerns the mortality model and the use of an estimated expected count.

For $\theta >1$, the expected scores verify the direction of evidence:The first inequality follows from $\log \theta <\theta -1$; the second bracket is zero at one and has positive derivative $\log \theta$ above one.

For a specified change, 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.