Simultaneous \((1 - α) × 100\%\) Confidence Intervals for the Elements of \(\Psi\) are obtained as follows:

\(\hat{\Psi}_j \pm \sqrt{\dfrac{p(N-g)}{N-g-p+1}F_{p, N-g-p+1}}SE(\hat{\Psi}_j)\)

where

\(SE(\hat{\Psi}_j) = \sqrt{\left(\sum\limits_{i=1}^{g}\dfrac{c^2_i}{n_i}\right)\dfrac{e_{jj}}{N-g}}\)

where \(e_{jj}\) is the \( \left( j, j \right)^{th}\) element of the error sum of squares and cross products matrix, and is equal to the error sums of squares for the analysis of variance of variable j.

Contrast 1

Recall that we have p = 5 chemical constituents, g = 4 sites, and a total of N = 26 observations. From the F-table, we have F_5,18,0.05 = 2.77. Then our multiplier 

\begin{align} M &= \sqrt{\frac{p(N-g)}{N-g-p+1}F_{5,18}}\\[10pt] &= \sqrt{\frac{5(26-4)}{26-4-5+1}\times 2.77}\\[10pt] &= 4.114 \end{align}

Simultaneous 95% Confidence Intervals are computed in the following table. The elements of the estimated contrast together with their standard errors are found at the bottom of each page, giving the results of the individual ANOVAs. For example, the estimated contrast from aluminum is 5.294 with a standard error of 0.5972. The fourth column is obtained by multiplying the standard errors by M = 4.114. So, for example, 0.5972 × 4.114 = 2.457. Finally, the confidence interval for aluminum is 5.294 plus/minus 2.457:

Conclusion

Pottery from Ashley Rails and Isle Thorns have higher aluminum and lower iron, magnesium, calcium, and sodium concentrations than pottery from Caldicot and Llanedyrn.

Contrast 3

Simultaneous 95% Confidence Intervals for Contrast 3 are obtained similarly to those for Contrast 1.

Conclusion

All of the above confidence intervals cover zero. Therefore, the significant difference between Caldicot and Llanedyrn appears to be due to the combined contributions of the various variables.

Bonferroni \((1 - α) × 100\%\) Confidence Intervals for the Elements of Ψ are obtained as follows:

\(\hat{\Psi}_j \pm t_{N-g, \frac{\alpha}{2p}}SE(\hat{\Psi}_j)\)

where

\(SE(\hat{\Psi}_j) = \sqrt{\left(\sum\limits_{i=1}^{g}\dfrac{c^2_i}{n_i}\right)\dfrac{e_{jj}}{N-g}}\)

and \(e_{jj}\) is the \( \left( j, j \right)^{th}\) element of the error sum of squares and cross products matrix and is equal to the error sums of squares for the analysis of variance of variable j.

Contrast 1

Here we have a \(t_{22,0.005} = 2.819\). The Bonferroni 95% Confidence Intervals are:

Conclusion

Pottery from Ashley Rails and Isle Thorns have higher aluminum and lower iron, magnesium, calcium, and sodium concentrations than pottery from Caldicot and Llanedyrn.

Contrast 3

Bonferroni 95% Confidence Intervals (note: the "M" multiplier below should be the t-value 2.819)

All resulting intervals cover 0 so there are no significant results.