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.
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, ...)
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 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
θ. 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|θ).
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.
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 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.
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.
Other design1S: decision1S_boundary,
decision1S, 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) # 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) designA_nL <- oc1S(flat_prior, nL, decA) designC_n1 <- oc1S(flat_prior, n1, decComb) designC_nL <- oc1S(flat_prior, nL, decComb) # evaluate designs at the key log-HR of positive treatment (HR<1), # the indecision point and the NI margin designA_n1(theta_eval) designA_nL(theta_eval) designC_n1(theta_eval) designC_nL(theta_eval) # 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) # ... or the clinically determined indifference point designC2_nL <- oc1S(flat_prior, nL, dec2) designC1_nL(theta_eval) designC2_nL(theta_eval) # see also ?decision1S_boundary to see which of the two criterions # will drive the decision
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