manova. samples of romano-british pottery were taken at four sites in the united kingdom. a chemical...

MANOVA

Samples of Romano-British pottery were taken at four sites in the United Kingdom. A chemical analysis of the pottery was performed to measure the percentage of five metal oxides present in each sample. The purpose of the analysis was to determine if different sites produced pottery with different chemical compositions. Since we have five different measures of chemical composition, we should perform a multivariate analysis of variance (MANOVA) analysis to determine if there is a significant difference between sites considering all five variables simultaneously. The dependent variables (Y-variables) for the MANOVA are the five chemical variables; the predictor variable (X-variable) is 'Site'.

• Twenty-six samples of Romano-British pottery were found at four different kiln sites in Wales, Gwent and the New Forest. The sites are Llanederyn (L), Caldicot (C), Island Thorns (I), and Ashley Rails (A). The other variables are the percentage of oxides of various metals measured by atomic absorption spectrophotometry. The data were collected to see if different sites contained pottery of different chemical compositions.

• Al: Percentage of aluminum oxide in sample • Fe: Percentage of iron oxide in sample • Mg: Percentage of magnesium oxide in sample • Ca: Percentage of calcium oxide in sample • Na: Percentage of sodium oxide in sample • Site: Site where pottery sample was collected

Between-Subjects Factors

5

2

5

14

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siteN

Multivariate Testsc

.989 319.930a 5.000 18.000 .000

.011 319.930a 5.000 18.000 .000

88.869 319.930a 5.000 18.000 .000

88.869 319.930a 5.000 18.000 .000

1.554 4.298 15.000 60.000 .000

.012 13.089 15.000 50.091 .000

35.439 39.376 15.000 50.000 .000

34.161 136.644b 5.000 20.000 .000

Pillai's Trace

Wilks' Lambda

Hotelling's Trace

Roy's Largest Root

Pillai's Trace

Wilks' Lambda

Hotelling's Trace

Roy's Largest Root

EffectIntercept

site

Value F Hypothesis df Error df Sig.

Exact statistica.

The statistic is an upper bound on F that yields a lower bound on the significance level.b.

Design: Intercept+sitec.

Tests of Between-Subjects Effects

175.610a 3 58.537 26.669 .000

134.222b 3 44.741 89.883 .000

103.351c 3 34.450 49.120 .000

.205d 3 .068 29.157 .000

.258e 3 .086 9.503 .000

3676.822 1 3676.822 1675.154 .000

231.962 1 231.962 466.006 .000

102.149 1 102.149 145.647 .000

.341 1 .341 145.505 .000

.167 1 .167 18.430 .000

175.610 3 58.537 26.669 .000

134.222 3 44.741 89.883 .000

103.351 3 34.450 49.120 .000

.205 3 .068 29.157 .000

.258 3 .086 9.503 .000

48.288 22 2.195

10.951 22 .498

15.430 22 .701

.051 22 .002

.199 22 .009

5684.600 26

664.140 26

375.381 26

.815 26

1.110 26

223.898 25

145.172 25

118.780 25

.256 25

.458 25

Dependent Variableal

fe

mg

ca

na

al

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mg

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SourceCorrected Model

Intercept

site

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .784 (Adjusted R Squared = .755)a.

R Squared = .925 (Adjusted R Squared = .914)b.

R Squared = .870 (Adjusted R Squared = .852)c.

R Squared = .799 (Adjusted R Squared = .772)d.

R Squared = .564 (Adjusted R Squared = .505)e.

The MANOVA analysis shows that there is a significant difference between the sites when considering all five measures simultaneously and when considering each individually. This is convincing evidence that a difference exists; however, it does not tell us which group(s) were different from the others. To determine this, we must rely on post-hoc tests of each chemical measure across the different sites.

Multiple Comparisons

LSD

5.6200* 1.23953 .000 3.0494 8.1906

-.8600 .93700 .369 -2.8032 1.0832

4.7557* .77186 .000 3.1550 6.3564

-5.6200* 1.23953 .000 -8.1906 -3.0494

-6.4800* 1.23953 .000 -9.0506 -3.9094

-.8643 1.11993 .448 -3.1869 1.4583

.8600 .93700 .369 -1.0832 2.8032

6.4800* 1.23953 .000 3.9094 9.0506

5.6157* .77186 .000 4.0150 7.2164

-4.7557* .77186 .000 -6.3564 -3.1550

.8643 1.11993 .448 -1.4583 3.1869

-5.6157* .77186 .000 -7.2164 -4.0150

-3.9030* .59028 .000 -5.1272 -2.6788

-.2000 .44621 .658 -1.1254 .7254

-4.8601* .36757 .000 -5.6224 -4.0978

3.9030* .59028 .000 2.6788 5.1272

3.7030* .59028 .000 2.4788 4.9272

-.9571 .53333 .086 -2.0632 .1489

.2000 .44621 .658 -.7254 1.1254

-3.7030* .59028 .000 -4.9272 -2.4788

-4.6601* .36757 .000 -5.4224 -3.8978

4.8601* .36757 .000 4.0978 5.6224

.9571 .53333 .086 -.1489 2.0632

4.6601* .36757 .000 3.8978 5.4224

-3.2490* .70067 .000 -4.7021 -1.7959

-.0680 .52966 .899 -1.1664 1.0304

-4.2204* .43631 .000 -5.1253 -3.3156

3.2490* .70067 .000 1.7959 4.7021

3.1810* .70067 .000 1.7279 4.6341

-.9714 .63306 .139 -2.2843 .3415

.0680 .52966 .899 -1.0304 1.1664

-3.1810* .70067 .000 -4.6341 -1.7279

-4.1524* .43631 .000 -5.0573 -3.2476

4.2204* .43631 .000 3.3156 5.1253

.9714 .63306 .139 -.3415 2.2843

4.1524* .43631 .000 3.2476 5.0573

-.2430* .04047 .000 -.3269 -.1591

.0260 .03060 .405 -.0375 .0895

-.1501* .02520 .000 -.2024 -.0979

.2430* .04047 .000 .1591 .3269

.2690* .04047 .000 .1851 .3529

.0929* .03657 .019 .0170 .1687

-.0260 .03060 .405 -.0895 .0375

-.2690* .04047 .000 -.3529 -.1851

-.1761* .02520 .000 -.2284 -.1239

.1501* .02520 .000 .0979 .2024

-.0929* .03657 .019 -.1687 -.0170

.1761* .02520 .000 .1239 .2284

-.0020 .07963 .980 -.1671 .1631

-.0060 .06020 .922 -.1308 .1188

-.2027* .04959 .000 -.3056 -.0999

.0020 .07963 .980 -.1631 .1671

-.0040 .07963 .960 -.1691 .1611

-.2007* .07195 .011 -.3499 -.0515

.0060 .06020 .922 -.1188 .1308

.0040 .07963 .960 -.1611 .1691

-.1967* .04959 .001 -.2996 -.0939

.2027* .04959 .000 .0999 .3056

.2007* .07195 .011 .0515 .3499

.1967* .04959 .001 .0939 .2996

(J) siteC

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(I) siteA

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Dependent Variableal

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mg

ca

na

MeanDifference

(I-J) Std. Error Sig. Lower Bound Upper Bound

95% Confidence Interval

Based on observed means.

The mean difference is significant at the .05 level.*.

manova. samples of romano-british pottery were taken at four sites in the united kingdom. a chemical...

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