17.9.11.2 Algorithms (PSS: Two-Sample Poisson Rate Test)

Power

One-sided power:\(H_0:\lambda_1\le \lambda_2\)

\[Power =1-F\left(\frac{\lambda_2-\lambda_1+z_{\alpha }\sqrt{\frac{\lambda_c}{n}\left(\frac{1}{T_1}+\frac{1}{T_2}\right)}}{\sqrt{\frac{\lambda_1}{nT_1}+\frac{\lambda_2}{nT_2}}}\right)\]

One-sided power:\(H_0:\lambda_1\ge \lambda_2\)

\(Power =F\left(\frac{\lambda_2-\lambda_1-z_{\alpha }\sqrt{\frac{\lambda_c}{n}\left(\frac{1}{T_1}+\frac{1}{T_2}\right)}}{\sqrt{\frac{\lambda_1}{nT_1}+\frac{\lambda_2}{nT_2}}}\right)\)

Two-sided power \(H_0: \lambda_1=\lambda_2\!\)

\[Power =1-F\left(\frac{\lambda_2-\lambda_1+z_{\frac{\alpha}{2} }\sqrt{\frac{\lambda_c}{n}\left(\frac{1}{T_1}+\frac{1}{T_2}\right)}}{\sqrt{\frac{\lambda_1}{nT_1}+\frac{\lambda_2}{nT_2}}}\right)+F\left(\frac{\lambda_2-\lambda_1-z_{\frac{\alpha}{2} }\sqrt{\frac{\lambda_c}{n}\left(\frac{1}{T_1}+\frac{1}{T_2}\right)}}{\sqrt{\frac{\lambda_1}{nT_1}+\frac{\lambda_2}{nT_2}}}\right)\]

\(n\): The sample size of each group

\(\lambda_1\): The rate of 1st population

\(\lambda_2\): The rate of 2nd population

\[\lambda_c =\frac{T_1\lambda_{1}+T_2\lambda_{2}}{T_1+T_2}\]

\(z_{\alpha }\): The \(\alpha\)-level upper critical value of Normal distribution

\(z_{\frac{\alpha}{2} }\): The \(\frac{\alpha}{2}\)-level two-side critical value of Normal distribution

\(F\): The cumulative distribution function of the standard normal distribution

\(T_1\): Exposure of 1st group

\(T_2\): Exposure of 2nd group

Sample Size

Origin uses an iterative algorithm with the power equation. At each iteration, the power is evaluated for a candidate sample size. The iteration stops when the computed power reaches the target value. The reported sample size is then the smallest integer greater than or equal to this value (i.e., the nearest larger integer).