DearBeggMonotoneCItheta {selectMeta}

Compute an approximate profile likelihood ratio confidence interval for effect estimate

Description

Under some assumptions on the true underlying p-value density the usual likelihood ratio theory for the finite-dimensional parameter of interest, \theta, holds although we estimate the infinite-dimensional nuisance parameter w, see Murphy and van der Vaart (2000). These functions implement such a confidence interval. To this end we compute the set

\{\theta : \tilde l(\theta, \hat \sigma(\theta), \hat w(\theta)) \ge c\}

where c = - 0.5 \cdot \chi_{1-\alpha}^2(1) and \tilde l is the relative profile log-likelihood function.

The functions DearBeggProfileLL and DearBeggToMinimizeProfile are not intended to be called by the user directly.

Usage

DearBeggMonotoneCItheta(res, lam = 2, conf.level = 0.95, maxiter = 500)
DearBeggProfileLL(z, res0, lam, conf.level = 0.95, maxiter = 500)
DearBeggToMinimizeProfile(vec, theta, y, u, lam)

Arguments

res

Output from function DearBeggMonotone.

lam

Weight of the first entry of w in the likelihood function. Should be the same as used to generate res.

conf.level

Confidence level of confidence interval.

maxiter

Maximum number of iterations of differential evolution algorithm used in computation of confidence limits. Increase this number to get higher accuracy.

z

Variable to maximize over, corresponds to \theta.

res0

Output from DearBeggMonotone, contains initial estimates.

vec

Vector of parameters over which we maximize.

theta

Current \theta.

y

Normally distributed effect sizes.

u

Associated standard errors.

Value

A list with the element

ci.theta

that contains the profile likelihood confidence interval for \theta.

Note

Since we have to numerically find zeros of a suitable function, via uniroot, to get the limits and each iteration involves computation of w(\theta) via a variant of DearBeggMonotone, computation of a confidence interval may take some time (typically seconds to minutes).

Author(s)

Kaspar Rufibach (maintainer), [e-mail],
http://www.kasparrufibach.ch

References

Murphy, S. and van der Vaart, A. (2000). On profile likelihood. J. Amer. Statist. Assoc., 95, 449–485.

Rufibach, K. (2011). Selection Models with Monotone Weight Functions in Meta-Analysis. Biom. J., 53(4), 689–704.

See Also

The estimate under a monotone selection function can be computed using DearBeggMonotone.

Examples

## Not run: 
## compute confidence interval for theta in the education dataset
data(education)
N  <- education$N
y  <- education$theta
u  <- sqrt(2 / N)
lam1 <- 2
res.edu <- DearBeggMonotone(y, u, lam = lam1, maxiter = 1000, 
    CR = 1, trace = FALSE)
r1 <- DearBeggMonotoneCItheta(res.edu, lam = 2, conf.level = 0.95)
res.edu$theta
r1$ci.theta

## compute confidence interval for theta in the passive smoking dataset
data(passive_smoking)
u <- passive_smoking$selnRR
y <- passive_smoking$lnRR
lam1 <- 2
res.toba <- DearBeggMonotone(y, u, lam = lam1, maxiter = 1000, 
    CR = 1, trace = FALSE)
r2 <- DearBeggMonotoneCItheta(res.toba, lam = 2, conf.level = 0.95)
res.toba$theta
r2$ci.theta

## End(Not run)