Decision Boundary for 1 Sample Designs
Calculates the decision boundary for a 1 sample design. This is the critical value at which the decision function will change from 0 (failure) to 1 (success).
decision1S_boundary(prior, n, decision, ...) ## S3 method for class 'betaMix' decision1S_boundary(prior, n, decision, ...) ## S3 method for class 'normMix' decision1S_boundary(prior, n, decision, sigma, eps = 1e-06, ...) ## S3 method for class 'gammaMix' decision1S_boundary(prior, n, decision, eps = 1e-06, ...)
prior |
Prior for analysis. |
n |
Sample size for the experiment. |
decision |
One-sample decision function to use; see |
... |
Optional arguments. |
sigma |
The fixed reference scale. If left unspecified, the default reference scale of the prior is assumed. |
eps |
Support of random variables are determined as the
interval covering |
The specification of the 1 sample design (prior, sample size and decision function, D(y)), uniquely defines the decision boundary
y_c = max_{y}{D(y) = 1},
which is the maximal value of y whenever the decision D(y)
function changes its value from 1 to 0 for a decision function
with lower.tail=TRUE (otherwise the definition is y_c = max_{y}{D(y) = 0}). The decision
function may change at most at a single critical value as only
one-sided decision functions are supported. Here,
y is defined for binary and Poisson endpoints as the sufficient
statistic y = ∑_{i=1}^{n} y_i and for the normal
case as the mean \bar{y} = 1/n ∑_{i=1}^n y_i.
The convention for the critical value y_c depends on whether
a left (lower.tail=TRUE) or right-sided decision function
(lower.tail=FALSE) is used. For lower.tail=TRUE the
critical value y_c is the largest value for which the
decision is 1, D(y ≤q y_c) = 1, while for
lower.tail=FALSE then D(y > y_c) = 1 holds. This is
aligned with the cumulative density function definition within R
(see for example pbinom).
Returns the critical value y_c.
betaMix: Applies for binomial model with a mixture
beta prior. The calculations use exact expressions.
normMix: Applies for the normal model with known
standard deviation σ and a normal mixture prior for the
mean. As a consequence from the assumption of a known standard
deviation, the calculation discards sampling uncertainty of the
second moment. The function decision1S_boundary has an extra
argument eps (defaults to 10^{-6}). The critical value
y_c is searched in the region of probability mass
1-eps for y.
gammaMix: Applies for the Poisson model with a gamma
mixture prior for the rate parameter. The function
decision1S_boundary takes an extra argument eps (defaults to 10^{-6})
which determines the region of probability mass 1-eps where
the boundary is searched for y.
Other design1S: decision1S,
oc1S, pos1S
# non-inferiority example using normal approximation of log-hazard # ratio, see ?decision1S for all details s <- 2 flat_prior <- mixnorm(c(1,0,100), sigma=s) nL <- 233 theta_ni <- 0.4 theta_a <- 0 alpha <- 0.05 beta <- 0.2 za <- qnorm(1-alpha) zb <- qnorm(1-beta) n1 <- round( (s * (za + zb)/(theta_ni - theta_a))^2 ) theta_c <- theta_ni - za * s / sqrt(n1) # double criterion design # statistical significance (like NI design) dec1 <- decision1S(1-alpha, theta_ni, lower.tail=TRUE) # require mean to be at least as good as theta_c dec2 <- decision1S(0.5, theta_c, lower.tail=TRUE) # combination decComb <- decision1S(c(1-alpha, 0.5), c(theta_ni, theta_c), lower.tail=TRUE) # critical value of double criterion design decision1S_boundary(flat_prior, nL, decComb) # ... is limited by the statistical significance ... decision1S_boundary(flat_prior, nL, dec1) # ... or the indecision point (whatever is smaller) decision1S_boundary(flat_prior, nL, dec2)
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