# Compute an approximate profile likelihood ratio confidence interval for effect estimate 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 [REMOVE_ME]$$ \{\theta : \tilde l(\theta, \hat \sigma(\theta), \hat w(\theta)) \ge c\} [REMOVE_ME_2]$$ 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. ## 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. ```r 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. ## Returns A list with the element - **`ci.theta`**: that contains the profile likelihood confidence interval for $\theta$. ## 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. ## Author(s) Kaspar Rufibach (maintainer), kaspar.rufibach@gmail.com , [http://www.kasparrufibach.ch](http://www.kasparrufibach.ch) ## 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). ## See Also The estimate under a monotone selection function can be computed using DearBeggMonotone . ## Examples ```r ## 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) ```

DearBeggMonotoneCItheta function