# 7.2.5 - Profile Plots

7.2.5 - Profile PlotsProfile plots provide another useful graphical summary of the data. These are only meaningful if all variables have the same units of measurement. They are not meaningful if the variables have different units of measurement. For example, some variables may be measured in grams while other variables are measured in centimeters. In this case, profile plots should not be constructed.

- In the traditional profile plot, the samples mean for each group are plotted against the variables.
- For the bank notes, it is preferable to subtract off the government specifications before carrying out the analyses.

#### Using Minitab

At this time Minitab does not support this procedure.

#### Analysis

The results are shown below:

From this plot, we can see that the bottom and top margins of the counterfeit notes are larger than the corresponding mean for the genuine notes. Likewise, the left and right margins are also wider for the counterfeit notes than the genuine notes. However, the diagonal and length measurement for the counterfeit notes appear to be smaller than the genuine notes. Please note, however, this plot does not show which results are significant. The significance is from the previous simultaneous or Bonferroni confidence intervals.

One of the things to look for in these plots is if the line segments joining the dots are parallel to one another. In this case, they are not even close to being parallel for any pair of variables.

### Profile Analysis

Profile Analysis is used to test the null hypothesis that these line segments are indeed parallel. You should test the hypothesis that the line segments in the profile plot are parallel to one another only if the variables have the same units of measurement. We might expect parallel segments if all of the measurements for the counterfeit notes are consistently larger than the measurements for the genuine notes by some constant.

Use the following procedure to test this null hypothesis:

**Step 1**: For each group, we create a new random vector \(Y_{ij}\) corresponding to the \(j^{th}\) observation from population *i*. The elements in this vector are the differences between the values of the successive variables as shown below:

\( \mathbf{Y}_{ij} = \left(\begin{array}{c}X_{ij2}-X_{ij1}\\X_{ij3}-X_{ij2}\\\vdots \\X_{ijp}-X_{ij,p-1}\end{array}\right)\)

**Step 2**: Apply the two-sample Hotelling's T-square to the data \(\mathbf{Y}_{ij}\) to test the null hypothesis that the means of the \(\mathbf{Y}_{ij}\)'s for population 1 are the same as the means of the \(\mathbf{Y}_{ij}\)'s for population 2:

\(H_0\colon \boldsymbol{\mu}_{\mathbf{Y}_1} = \boldsymbol{\mu}_{\mathbf{Y}_2}\)