Simon_pr {VisualizeSimon2Stage}

Simon_pr: Probabilities of a Simon's Two-Stage Design

Description

Probability of frail (i.e., early termination), fail (to reject the null) and success (to reject the null) of a Simon's two-stage design, at given true response rate(s).

Usage

Simon_pr(prob, n1, n, r1, r)

Arguments

prob

double vector, true response rate(s) p

n1, n

positive integer scalars, Stage-1 sample size n_1 and total sample size n

r1, r

non-negative integer scalars, number of response in Stage-1 r_1 and overall r required exclusively, i.e., passing Stage-1 indicates observing >r_1 responses, and rejecting H_0 indicates observing >r responses.

Details

Given the Simon's two-stage design (n_1, r_1, n, r), for a response rate p, we have the number of Stage-1 positive responses X_1 \sim \textrm{Binom}(n_1, p) and the number of Stage-2 positive responses X_2 \sim \textrm{Binom}(n-n_1, p). Obviously X_1 and X_2 are independent.

The probability of early termination is \textrm{Pr}(X_1 \leq r_1).

The probability of failure to reject H_0 is

\sum_{s_1 = r_1+1}^{n_1} \textrm{Pr}(X_1=s_1)\cdot\textrm{Pr}(X_2 \leq (r-s_1))

The probability of rejecting H_0 is

\sum_{s_1 = r_1+1}^{n_1} \textrm{Pr}(X_1=s_1)\cdot\textrm{Pr}(X_2 > (r-s_1))

Parameters nomenclature of n1, n, r1 and r follows that of PASS and function ph2simon.

Value

Simon_pr returns Simon_pr object.

Slots

.Data

ncol-3 double matrix, probability of frail (i.e., early termination), fail (to reject the null) and success (to reject the null), at each response rate p given in ⁠@prob⁠

eN

numeric vector, expected sample size(s) \textrm{E}(N) for each of response rate(s) p

prob

double vector, response rate(s) p

Examples

Simon_pr(prob = c(.2, .4), n1 = 15L, r1 = 3L, n = 24L, r = 7L)