Control of the False Discovery Proportion for Independently Tested Null Hypotheses

Algorithm 1

Compute the simultaneous UPBs for the false discovery proportion and the UCB for 𝑚₀.

For any given 𝛼 and 𝛾,

(1) Compute 𝑚_0,1−𝛼(𝜆) of (3.15) for some pre-specified 𝜆_𝑖’s, say 𝜆_𝑖=𝑖/1000, for 𝑖=1,…,999.

(2) Compute 𝑚_0,1−𝛼=min_𝑖𝑚_0,1−𝛼(𝜆_𝑖). This 𝑚_0,1−𝛼 is the 1−𝛼 UCB for 𝑚₀.

 If 𝑚_0,1−𝛼 exceeds 𝑚, replace it by 𝑚.

(3) Sort the observed 𝑃-values such that 𝑃₍₁₎≤⋯≤𝑃_(𝑚), and use (3.8) to compute the 1−𝛼

 simultaneous UPBs for the false discovery proportion 𝑄, that is, for 𝑖=1,…,𝑚,

𝑄_1−𝛼(𝑃_(𝑖))=(1/𝑖)(𝑚_0,1−𝛼𝑃_(𝑖)+𝑧̃_1−𝛼(𝑚_0,1−𝛼)𝑚_0,1−𝛼𝑃_(𝑖)(1−𝑃_(𝑖))).

 If 𝑄_1−𝛼(𝑃_(𝑖)) exceeds 1, replace it by 1.

(4) Compute 𝜏=max{𝑃_(𝑖)∶𝑄_1−𝛼(𝑃_(𝑖))≤𝛾},

 reject the hypotheses whose 𝑃-values are no greater than 𝜏,

 which ensures that the false discovery proportion 𝑄 is not exceeding 𝛾 with probability 1−𝛼.