After we have assessed the assumptions, our next step is to proceed with the MANOVA.

Analysis

The concentrations of the chemical elements depend on the site where the pottery sample was obtained \(\left( \Lambda ^ { \star } = 0.0123 ; F = 13.09 ; \mathrm { d } . \mathrm { f } = 15,50 ; p < 0.0001 \right)\). It was found, therefore, that there are differences in the concentrations of at least one element between at least one pair of sites.

 Question: How do the chemical constituents differ among sites?

Analysis

Results from the profile plots are summarized as follows:

• The sample sites appear to be paired: Ashley Rails with Isle Thorns and Caldicot with Llanedyrn.
• Ashley Rails and Isle Thorns appear to have higher aluminum concentrations than Caldicot and Llanedyrn.
• Caldicot and Llanedyrn appear to have higher iron and magnesium concentrations than Ashley Rails and Isle Thorns.
• Calcium and sodium concentrations do not appear to vary much among the sites.

Note: These results are not backed up by appropriate hypotheses tests.  Hypotheses need to be formed to answer specific questions about the data. These should be considered only if significant differences among group mean vectors are detected in the MANOVA.

Specific Questions

• Which chemical elements vary significantly across sites?
• How do the sites differ?
  • Is the mean chemical constituency of pottery from Ashley Rails and Isle Thorns different from that of Llanedyrn and Caldicot?
  • Is the mean chemical constituency of pottery from Ashley Rails equal to that of Isle Thorns?
  • Is the mean chemical constituency of pottery from Llanedyrn equal to that of Caldicot?

Analysis of Individual Chemical Elements

A naive approach to assessing the significance of individual variables (chemical elements) would be to carry out individual ANOVAs to test:

\(H_0\colon \mu_{1k} = \mu_{2k} = \dots = \mu_{gk}\)

for chemical k. Reject \(H_0 \) at level \(\alpha\) if

\(F > F_{g-1, N-g, \alpha}\)

 Problem: If we're going to repeat this analysis for each of the p variables, this does not control for the experiment-wise error rate.

Just as we can apply a Bonferroni correction to obtain confidence intervals, we can also apply a Bonferroni correction to assess the effects of group membership on the population means of the individual variables.

Bonferroni Correction: Reject \(H_0 \) at level \(\alpha\) if

\(F > F_{g-1, N-g, \alpha/p}\)

or, equivalently, if the p-value is less than \(α/p\).

Analysis

Here, p = 5 variables, g = 4 groups, and a total of N = 26 observations. So, for an α = 0.05 level test, we reject

\(H_0\colon \mu_{1k} = \mu_{2k} = \dots = \mu_{gk}\)

if

\(F > F_{3,22,0.01} = 4.82\)

or equivalently, if the p-value reported by SAS is less than 0.05/5 = 0.01. The results of the individual ANOVAs are summarized in the following table. All tests are carried out with 3, 22 degrees freedom (the d.f. should always be noted when reporting these results).

Because all of the F-statistics exceed the critical value of 4.82, or equivalently, because the SAS p-values all fall below 0.01, we can see that all tests are significant at the 0.05 level under the Bonferroni correction.

Conclusion: The means for all chemical elements differ significantly among the sites. For each element, the means for that element are different for at least one pair of sites.