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
- 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:
\(Z_{ij} = \dfrac{X_{ij}}{\mu^0_j}\)
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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.
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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.
Analysis
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.