Supplementary Table S3
Abouchami et al.
F-test statistics of the Loa and Kea regression trends.
df
SS
MS
F
df
LOA trend
Regression
1
0.96061 0.96061
Residual
67 0.20784 0.00310
KEA trend
Regression
1
1.03775 1.03775
Residual
135
0.29179 0.00216
1.44 67, 135
Total
202 0.49963 0.00247
Pooled
204 0.61175 0.00300
LOA-KEA
Difference
between slopes
1
0.11212 0.11212
45.3
1, 204
F5%crit. F1%crit.
1.38
3.89
1.51
6.76
P (%)
3.79
<10-4
Table S3. F-test statistics for the 208Pb/204Pb- 206Pb/204Pb variations, with the Loa and Kea
trend volcanoes taken as two populations. Calculations were performed using the Free
Statistics Software (http://www.wessa.net) and XLSTAT (http://www.xlstat.com). df:
degree of freedom; SS: Sum of Squares; MS: residual Mean Square; Fx crit.: critical F
value at x significance level; P: probability level of the null hypothesis that the two
populations have the same variance about the regressions.
Notes:
We first note that the subdivision of the dataset into Loa and Kea trend is based upon
geographical location of the volcanoes rather than the Pb isotopic compositions of the
samples. The aim here is to see whether the Pb isotopic compositions of the samples
from the two trends are derived from two statistically distinct populations or not.
The F-test is designed to evaluate if the standard deviations of two populations are
equal, and is simply the ratio of two population variances44. The F-ratio is always taken
such that F is greater or equal to unity. If two populations are different from one another,
the confidence level for this conclusion depends on the number of degrees of freedom
(samples) and the magnitude of the F-value. The usefulness of this F-test is limited by the
fact that two population variances may be different from each other even if their mean
values are identical or vice-versa.
For multivariate data, such as the 208Pb/204Pb and 206Pb/204Pb of the samples analyzed
here, the F-test can also be applied to the variances about the best-fit regression lines
through the two populations to see whether the residual variances are statistically
different from one another. Here we ask the question: Can the Loa and Kea regression
lines be regarded as the same or do they scatter differently about their respective
regression lines?
One can also compare the residual variances of the individual regression lines of two
populations with the variance of a “global” regression line by pooling all the data
together. If the variances of the individual regressions are lower than the variance of the
global regression of the pooled data, the two populations will be significantly different
from each other, even if their individual variances are identical.
In this case, we ask: Are the slopes of the Loa and Kea regression lines significantly
different from the single Loa-Kea pooled slope? In other words, are the data better
considered as two separate regressions lines rather than a single one?
Table 3 shows the results of the F-test performed on the Loa and Kea trend volcanoes
in
208
Pb/204Pb vs.
206
Pb/204Pb space, as described above. The F value for Loa and Kea
regressions is 1.44, that is, larger than the critical F-value at the 5% significance level
(1.38). The Loa-Kea F-value corresponds to a null hypothesis probability level of 3.8%
(Table 3), indicating that the residual variances from the Loa and Kea regression lines are
different at better than 95% confidence.
Comparison of the individual regressions with a single pooled regression through all
the data results in an F-value of 45. The latter is much greater than the critical F-value at
the 1% (F = 6.76) and even 0.1% (F = 11.45) significance levels. The null hypothesis
probability for F = 45 is less than 10-4%, demonstrating that there is an overwhelming
difference between the Loa and Kea trend volcanoes in
208
Pb/204Pb- 206Pb/204Pb space. In
other words, the F-test provides strong confirmation of the visual impression that the Loa
and Kea trend volcanoes should be treated as two separate populations rather than a
single pooled population.
In addition to the various F-tests outlined above, we have calculated regression slopes
of the Pb isotope arrays in
208
Pb/204Pb-206Pb/204Pb space for the individual volcanoes of
both trends (Supplementary Table 2). These slopes differ widely from one another and
are, particularly in the case of negative slopes, interpreted as local mixing lines. The
results of these calculations are shown in Supplementary Fig. 1.
44. Snedecor, J. H., Statistical Methods, 8th edn (Iowa State Univ. Press, Ames,
1993).