17.8.6.2 Algorithm (PSS: Two-Proportion Test)


Power

One-sided power:H_0:p_1\ge p_2

Power =F(\frac{-(p_2-p_1)-z_{\alpha }\sqrt{\frac{(2p_c(1-p_c)}{n} }}{\sqrt{\frac{p_2(1-p_2)}{n}+\frac{p_1(1-p_1)}{n}}})

One-sided power:H_0:p_1\le p_2

Power =1-F(\frac{-(p_2-p_1)+z_{\alpha }\sqrt{\frac{(2p_c(1-p_c)}{n} }}{\sqrt{\frac{p_2(1-p_2)}{n}+\frac{p_1(1-p_1)}{n}}})

Two-sided power H_0: p_1=p_2\!

Power =1-F(\frac{-(p_2-p_1)+z_{\frac{\alpha}{2} }\sqrt{\frac{(2p_c(1-p_c)}{n} }}{\sqrt{\frac{p_2(1-p_2)}{n}+\frac{p_1(1-p_1)}{n}}})+F(\frac{-(p_2-p_1)-z_{\frac{\alpha}{2}}\sqrt{\frac{(2p_c(1-p_c)}{n} }}{\sqrt{\frac{p_2(1-p_2)}{n}+\frac{p_1(1-p_1)}{n}}})

n: The sample size of each group

p_1:The proportion of 1st population

p_2:The proportion of 2nd population

p_c =\frac{p_{1}+p_{2}}{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

Sample Size

Origin uses an iterative algorithm with the power equation. At each iteration,the power for a trial sample size are evaluated and iteration stops when the power evaluated reaches the values which corresponding to an integer sample size, and which is nearest to, yet greater than, the target value.