If the data is of a very large dimension, tables of simultaneous or Bonferroni confidence intervals are hard to grasp at a cursory glance. A better approach is to visualize the coverage of the confidence intervals through a profile plot.

Procedure

A profile plot is obtained with the following three step procedure:

Steps

1. Standardize each of the observations by dividing them by their hypothesized means. So the \(i^{th}\) observation of the \(j^{th}\) variable, \(X_{ij}\), is divided by its hypothesized mean for\(j^{th}\) variable \(\mu_0^j\). We will call the result \(Z_{ij}\) as shown below:

   \(Z_{ij} = \dfrac{X_{ij}}{\mu^0_j}\)

2. Compute the sample mean for the \(Z_{ij}\)'s to obtain sample means corresponding to each of the variables j, 1 to p. These sample means, \(\bar{z_j}\), are then plotted against the variable j.

3. Plot either simultaneous or Bonferroni confidence bands for the population mean of the transformed variables,

   Simultaneous \((1 - α) × 100\%\) confidence bands are given by the usual formula, using the z's instead of the usual x's as shown below:

   \(\bar{z}_j \pm \sqrt{\dfrac{p(n-1)}{(n-p)}F_{p,n-p,\alpha}}\sqrt{\dfrac{s^2_{Z_j}}{n}}\)

   The same substitutions are made for the Bonferroni \((1 - α) × 100\%\) confidence band formula:

   \(\bar{z}_j \pm t_{n-1,\frac{\alpha}{2p}}\sqrt{\dfrac{s^2_{Z_j}}{n}}\)