7.1.5 - Profile Plots

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.


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


  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:

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

Example 7-5: Women's Health Survey (Profile Plots) Section

Using SAS

The profile plots are computed using the SAS program.

Download the SAS Program: nutrient6.sas

View the video below to see how to create a profile plot using the SAS statistical software application.

Using Minitab

View the video below to see how to create a profile plot using the Minitab statistical software application.


From this plot, the results are immediately clear. We can easily see that the confidence intervals for calcium and iron fall below 1 indicating that the average daily intakes for these two nutrients are below recommended levels. The protein confidence interval falls above the value 1 indicating that the average daily intake for protein exceeds the recommended level. The confidence intervals for vitamin A and C both contain 1 showing no significant evidence against the null hypothesis and suggesting that they meet the recommended intake of these two vitamins.