Find parameters of a beta distribution to approximate distribution of a p-value derived from a normal test statistic with one-sided significance level=ALPHA and power=1-BETA.
find.ab(n = 1e+05, ALPHA = 0.05, BETA = 0.2, higha = 100)the number of divisions for the numeric integration used to estimate the mean of p-value distribution, H (see details).
one-sided significance level of normal test statistic
type II error for normal test stastic
an upper bound on the beta parameter (see details).
A list with two elements:
numeric value of one of the shape parameters of the beta distribution
numeric value of the other shape parameter of the beta distribution
The cumulative distribution function of the p-value from a normally distributed test statistic with one-sided significance level=ALPHA and power=1-BETA is H(p) = 1-pnorm( qnorm(1-p) - qnorm(1-ALPHA)+qnorm(BETA) ). We approximate this distribution with a beta distribution, B, which has the same mean as H and has B(ALPHA)=1-BETA. If two beta distributions meet both those criteria, we select the one closest to H in terms of integrated square error of the cumulative distribution function. That error is estimated by the sample variance of the differences in the two CDFs evaluated at (0:n)/n. Note that the two beta distributions come from the two roots of the following function: 1-BETA - B(ALPHA) We search for those two roots as the beta parameter within the range (1/higha, higha).
Fay, M.P., and Follmann, D.A. (2002). "Designing Monte Carlo implementations of permutation or bootstrap hypothesis tests" American Statistician, 56: 63-70.
# NOT RUN {
## See first line of Table 1, Fay and Follmann, 2002
find.ab(ALPHA=.05,BETA=.1)
# }
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