Operating Characteristics for 1 Sample Design

The oc1S function defines a 1 sample design (prior, sample size, decision function) for the calculation of the frequency at which the decision is evaluated to 1 conditional on assuming known parameters. A function is returned which performs the actual operating characteristics calculations.

Usage

oc1S(prior, n, decision, ...)

# S3 method for class 'betaMix'
oc1S(prior, n, decision, ...)

# S3 method for class 'normMix'
oc1S(prior, n, decision, sigma, eps = 1e-06, ...)

# S3 method for class 'gammaMix'
oc1S(prior, n, decision, eps = 1e-06, ...)

Arguments

prior: Prior for analysis.
n: Sample size for the experiment.
decision: One-sample decision function to use; see decision1S.
...: 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 1-eps probability mass. Defaults to $10^{-6}$.

Value

Returns a function with one argument theta which calculates the frequency at which the decision function is evaluated to 1 for the defined 1 sample design. Note that the returned function takes vectors arguments.

Details

The oc1S function defines a 1 sample design and returns a function which calculates its operating characteristics. This is the frequency with which the decision function is evaluated to 1 under the assumption of a given true distribution of the data defined by a known parameter $\theta$. The 1 sample design is defined by the prior, the sample size and the decision function, $D(y)$. These uniquely define the decision boundary, see decision1S_boundary.

When calling the oc1S function, then internally the critical value $y_c$ (using decision1S_boundary) is calculated and a function is returns which can be used to calculated the desired frequency which is evaluated as

$$ F(y_c|\theta). $$

Methods (by class)

oc1S(betaMix): Applies for binomial model with a mixture beta prior. The calculations use exact expressions.
oc1S(normMix): Applies for the normal model with known standard deviation $\sigma$ 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 oc1S 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 $y$.
oc1S(gammaMix): Applies for the Poisson model with a gamma mixture prior for the rate parameter. The function oc1S takes an extra argument eps (defaults to $10^{-6}$) which determines the region of probability mass 1-eps where the boundary is searched for $y$.

Examples


# 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)

# standard NI design
decA <- decision1S(1 - alpha, theta_ni, lower.tail=TRUE)

# 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)

theta_eval  <- c(theta_a, theta_c, theta_ni)

# evaluate different designs at two sample sizes
designA_n1  <- oc1S(flat_prior, n1, decA)
#> Using default prior reference scale 2
designA_nL  <- oc1S(flat_prior, nL, decA)
#> Using default prior reference scale 2
designC_n1  <- oc1S(flat_prior, n1, decComb)
#> Using default prior reference scale 2
designC_nL  <- oc1S(flat_prior, nL, decComb)
#> Using default prior reference scale 2

# evaluate designs at the key log-HR of positive treatment (HR<1),
# the indecision point and the NI margin

designA_n1(theta_eval)
#> [1] 0.80084529 0.49980774 0.04995032
designA_nL(theta_eval)
#> [1] 0.92039728 0.64489439 0.04997268
designC_n1(theta_eval)
#> [1] 0.80084529 0.49980774 0.04995032
designC_nL(theta_eval)
#> [1] 0.84991646 0.49995959 0.02185859

# to understand further the dual criterion design it is useful to
# evaluate the criterions separatley:
# statistical significance criterion to warrant NI...
designC1_nL <- oc1S(flat_prior, nL, dec1)
#> Using default prior reference scale 2
# ... or the clinically determined indifference point
designC2_nL <- oc1S(flat_prior, nL, dec2)
#> Using default prior reference scale 2

designC1_nL(theta_eval)
#> [1] 0.92039728 0.64489439 0.04997268
designC2_nL(theta_eval)
#> [1] 0.84991646 0.49995959 0.02185859

# see also ?decision1S_boundary to see which of the two criterions
# will drive the decision