Simon_pr: 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)

Value

Simon_pr returns Simon_pr object.

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.

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) $p$

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.

Examples

Run this code

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

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