8.8 - Hypothesis Tests

8.8 - Hypothesis Tests

Problem:

The suggestions dealt with in the previous page are not backed up by appropriate hypothesis tests. Consider hypothesis tests of the form:

\(H_0\colon \Psi = 0\) against \(H_a\colon \Psi \ne 0\)

Univariate Case:

For the univariate case, we may compute the sums of squares for the contrast:

\(SS_{\Psi} = \frac{\hat{\Psi}^2}{\sum_{i=1}^{g}\frac{c^2_i}{n_i}}\)

This sum of squares has only 1 d.f. so that the mean square for the contrast is

\(MS_{\Psi}= SS_{\Psi}\)

Then compute the F-ratio:

\(F = \frac{MS_{\Psi}}{MS_{error}}\)

Reject \(H_{0} \colon \Psi = 0\) at level \(\alpha\) if

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

Multivariate Case

For the multivariate case, the sums of squares for the contrast is replaced by the hypothesis sum of squares and cross-products matrix for the contrast:

\(\mathbf{H}_{\mathbf{\Psi}} = \dfrac{\mathbf{\hat{\Psi}\hat{\Psi}'}}{\sum_{i=1}^{g}\frac{c^2_i}{n_i}}\)

Compute Wilks Lambda

\(\Lambda^* = \dfrac{|\mathbf{E}|}{\mathbf{|H_{\Psi}+E|}}\)

Compute the F-statistic

\(F = \left(\dfrac{1-\Lambda^*_{\mathbf{\Psi}}}{\Lambda^*_{\mathbf{\Psi}}}\right)\left(\dfrac{N-g-p+1}{p}\right)\)

Reject Ho : \(\mathbf{\Psi = 0} \) at level \(α\) if

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

Example 8-9: Pottery Data

The following table gives the results of testing the null hypothesis that each of the contrasts is equal to zero. You should be able to find these numbers in the output by downloading the SAS program here: pottery.sas.

Contrast \(\Lambda _ { \Psi } ^ { * }\) F d.f p
1 0.0284 122.81 5, 18 <0.0001
2 0.9126 0.34 5, 18 0.8788
3 0.4487 4.42 5, 18 0.0084

Conclusions

  1. The mean chemical content of pottery from Ashley Rails and Isle Thorns differs in at least one element from that of Caldicot and Llanedyrn \(\left( \Lambda _ { \Psi } ^ { * } = 0.0284; F = 122. 81; d.f. = 5, 18; p < 0.0001 \right) \).
  2. There is no significant difference in the mean chemical contents between Ashley Rails and Isle Thorns \(\left( \Lambda _ { \Psi } ^ { * } = 0.9126; F = 0.34; d.f. = 5, 18; p = 0.8788 \right) \).
  3. The mean chemical content of pottery from Caldicot differs in at least one element from that of Llanedyrn \(\left( \Lambda _ { \Psi } ^ { * } = 0.4487; F = 4.42; d.f. = 5, 18; p = 0.0084 \right) \).

Once we have rejected the null hypothesis that contrast is equal to zero, we can compute simultaneous or Bonferroni confidence intervals for the contrast:

Simultaneous Confidence Intervals

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.

Note! These standard errors can be obtained directly from the SAS output. Look at the bottom of each page containing the individual ANOVAs.

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 F5,18,0.05 = 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:

Element \(\widehat { \Psi }\) SE\(\widehat { \Psi }\) \(M \times SE \left(\widehat { \Psi }\right)\) Confidence Interval
Al 5.294 0.5972 2.457 2.837, 7.751
Fe -4.640 0.2844 1.170 -5.810, -3.470
Mg -4.065 0.3376 1.389 -5.454, -2.676
Ca -0.175 0.0195 0.080 -0.255, -0.095
Na -0.175 0.0384 0.158 -0.333, -0.017

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.

Element \(\widehat{\Psi}\) SE( \(\widehat{\Psi}\) ) \(M\times SE(\widehat{\Psi})\) Confidence Interval
Al -0.864 1.1199 4.608 -5.472, 3.744
Fe -0.957 0.5333 2.194 -3.151, 1.237
Mg -0.971 0.6331 2.605 -3.576, 1.634
Ca  0.093 0.0366 0.150 -0.057, 0.243
Na -0.201 0.0719 0.296 -0.497, 0.095

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.

Note! Because Contrast 2 is not significant, there is no reason to compute simultaneous confidence intervals for the elements of that contrast.

Bonferroni Confidence Intervals

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:

Element \(\widehat{\Psi}\) SE( \(\widehat{\Psi}\) ) \(t\times SE(\widehat{\Psi})\) Confidence Interval
Al 5.294 0.5972 1.684 3.610, 6.978
Fe -4.640 0.2844 0.802 -5.442, -3.838
Mg -4.0.65 0.3376 0.952 -5.017, -3.113
Ca -0.175 0.0195 0.055 -0.230, -0.120
Na -0.175 0.0384 0.108 -0.283, -0.067

 

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)

Element \(\widehat{\Psi}\) SE( \(\widehat{\Psi}\) ) \(M\times SE(\widehat{\Psi})\) Confidence Interval
Al -0.864 1.1199 3.157 -4.021, 2.293
Fe -0.957 0.5333 1.503 -2.460, 0.546
Mg -0.971 0.6331 1.785 -2.756, 0.814
Ca 0.093 0.0366 0.103 -0.010, 0.196
Na -0.201 0.0719 0.203 -0.404, 0.001

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

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