# Simulation of confidence intervals.

## Setting up the simulations

We use the dataset `bfi` from the package `psych` together with `lavaan` to estimate some realistic factor loadings \(\lambda\) and standard deviations \(\sigma\).

    model <- c("y =~ A1 + A2 + A3 + A4 + A5")
    fit <- lavaan::cfa(model, data = psych::bfi)
    coefs <- lavaan::lavInspect(fit, what = "x")
    lambda <- abs(c(coefs$lambda * sqrt(as.numeric(coefs$psi))))
    sigma <- sqrt(diag(lavaan::lavInspect(fit, what = "x")$theta))

We take the absolute value of the `lambda` vector as the agreement data contains reverse-coded items.

## Comparing confidence intervals coverages and lengths

We compare five confidence intervals, all without transformations. The `adf` interval is the asymptotic distribution-free interval, the `ell` interval is the interval based on elliptical distributions and a kurtosis correction, the `ell_par` is the elliptical interval assuming a parallel model. The same comments hold for `norm` (assuming normal data) and `norm_par` (assuming parallel normal data).

    library("alphaci")
    library("future.apply")
    plan(multisession, workers = availableCores() - 2)
    set.seed(313)

    n_reps <- 10000
    k <- 5
    latent <- \(n) extraDistr::rlaplace(n) / sqrt(2)
    true <- alphaci:::alpha(sigma, lambda)

In this simulation we normal error terms and a Laplace-distributed latent variable. This one has excess kurtosis \(3\), which caries over in large part to the data. `k` is the number of questions ands `n_reps` is the number of simulations.

    success <- \(ci) true <= ci[2] & true >= ci[1]
    len <- \(ci) ci[2] - ci[1]
    simulations <- \(n) {
        results  <- future.apply::future_replicate(n_reps, {
          x <- alphaci:::simulate_congeneric(n, k, sigma, lambda, latent = latent)
          cis <- rbind(adf = alphaci(x, type = "adf"),
            adf_par = alphaci(x, type = "adf", parallel = TRUE),
            ell = alphaci(x, type = "elliptical"),
            ell_par = alphaci(x, type = "elliptical", parallel = TRUE),
            norm = alphaci(x, type = "normal"),
            norm_par = alphaci(x, type = "normal", parallel = TRUE)
          )
          c(cov = apply(cis, 1, success), len = apply(cis, 1, len))
          }, future.seed = TRUE)
      rowMeans(results)
    }

Let’s check out the results when \(n= 10\).

    simulations(10)
    #>      cov.adf  cov.adf_par      cov.ell  cov.ell_par     cov.norm cov.norm_par      len.adf  len.adf_par
    #>    0.8745000    0.7942000    0.9513000    0.9566000    0.9223000    0.9297000    0.7680810   55.3239940
    #>      len.ell  len.ell_par     len.norm len.norm_par
    #>    1.0238478    1.0499270    0.9113473    0.9342908

It appears that the kurtosis corrections work well, at least for small sample size. Let’s see how they perform when \(n\) increases.

    nn <- c(5, 10, 20, 30, 40, 50, 100, 200, 500, 1000, 2000, 5000)
    results <- sapply(nn, simulations)

Plotting the coverages, we find, where `1` is asymptotically distribution-free, `2` is elliptical, `3` is paralell and elliptical, `4` is `normal` and `5` is parallel and normal.

    matplot(nn, t(results[1:5, ]), xlab = "n", ylab = "Coverage", type = "b",
            log = "x")
    abline(h = 0.95, lty = 2)

Hence the kurtosis correction intervals have better coverage than the `adf` interval when \(n\leq 50\) and outperforms the normal theory intervals for all \(n\). If this observation is general remains to be seen.