simon_pr: Probabilities of one Simon's Two-Stage Design

Description

Probabilities of frail (i.e., early termination) and success (to reject $H_0$) of one Simon's two-stage design, at given true response rate(s).

Usage

simon_pr(prob, object, ...)
# S3 method for ph2simon
simon_pr(prob, object, ...)
# S3 method for ph2simon4
simon_pr(
  prob,
  object,
  r1 = object@r1,
  n1 = object@n1,
  r = object@r,
  n = object@n,
  ...
)

Value

Function simon_pr() returns simon_pr object.

Arguments

prob: double scalar or vector, true response rate(s) $p$
object: a ph2simon or ph2simon4 object
...: parameters of function ph2simon4(), most importantly type
r1, n1, r, n: (optional) integer scalars, see ph2simon4.

Slots

frail: numeric scalar or vector, probabilities of frail (i.e., early termination) at given true response rate(s) $p$.

reject

numeric scalar or vector, probabilities of success (to reject $H_0$) at given true response rate(s) $p$.

eN

numeric scalar or vector, expected sample size(s) $\textrm{E}(n)$ at given true response rate(s) $p$.

prob

double scalar or vector, true response rate(s) $p$

Details

Given one Simon's two-stage design $(r_1,n_1,r,n)$ and a true 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 $$p_{\textrm{frail}} = \textrm{Pr}(X_1 \leq r_1)$$

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 successfully rejecting $H_0$ is $$\sum_{s_1 = r_1+1}^{n_1} \textrm{Pr}(X_1=s_1)\cdot\textrm{Pr}(X_2 > (r-s_1))$$

The expected sample size is $$\textrm{E}(n) = p_{\textrm{frail}} \cdot n_1 + (1 - p_{\textrm{frail}}) \cdot n$$

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

Examples

Run this code

(x = clinfun::ph2simon(pu = .2, pa = .4, ep1 = .05, ep2 = .1)) 
simon_pr(prob = c(.2, .3, .4), object = x)
simon_pr.ph2simon4(prob = c(.2, .3, .4), r1 = 5L, n1 = 24L, r = 13L, n = 45L) # internal use

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