Geoscience Reference
In-Depth Information
assumption that a physical (or biological) meaning can be attributed entirely to a single, individual
pattern (e.g., PC#1) that falls out of a statistical analysis procedure.
Spearman indeed found that the first PC in his analysis explained a substantial percentage—
about 50 to 60 percent—of the variation in the data, but that meant that nearly half of the variation in
the data was still unexplained. Spearman nonetheless ascribed a deep psychometric significance to
the PC#1 of his data and even gave it a name, the
g
factor. Spearman's
g
soon became widely
celebrated as the Holy Grail of psychometric analysis, a unique, mathematically definable measure of
human intelligence. The problem is that
g
was not in fact unique. If one were to adopt different
statistical conventions, one could get a quite different answer using the same dataset.
To explore just how this might work, let us revisit the synthetic example we considered in
chapter 4
, but with a tweak, the details of which we'll return to later. We will first need to draw an
analogy between the seemingly quite disparate problems of characterizing variations in global
temperature (as with the synthetic example from
chapter 4
) and measures of human intelligence (as
with the Spearman analysis discussed above). As different as these two problems might seem, they
potentially lend themselves to strikingly similar data analysis procedures. Imagine that instead of
different potential measures of what we might think of as intelligence (e.g., reading comprehension
and facility with arithmetic), we have different spatial regions (e.g., western and eastern
hemispheres) of temperature variations, and instead of looking for trends across cultures in measures
of intelligence, we are interested in trends across time in temperature.
Let us idealize Spearman's example and imagine that all of the variation in the putative
intelligence data are described by just two principal components. PC#1—Spearman's
g
—would then
reflect a pattern that is similar across all measures of intelligence (from reading comprehension to
arithmetic) and show a clear trend across races and cultures (e.g., high performance in the Western
world and poor performance in those cultures that were thought of as “primitive”). PC#2, by contrast,
would show a more complex pattern of variation across races and cultures, and varying performance
among competing measures of intelligence. Drawing an analogy with the temperature problem, then,
PC #1 would be like the uniform global warming with its clear trend over time for all regions, while
PC#2 would reflect a more complex pattern in space and time that constitutes the regional and
temporal departures from that trend. In both cases, Burt's putative measures of intelligence and our
synthetic temperature example, each of the patterns in the data are important, and one would be in
error to throw out either one and focus exclusively on the single remaining pattern as if it summarizes
all of the important information in the data.
The above situation is portrayed in the right half (panels g-l) of
Figure 9.2
.
PC#1—resolving 55
percent of the variation in the data—shows a uniform warming trend (panels i and j), while PC#2—
resolving the remaining 45 percent of the variation in the data—is a roof-shaped temperature
variation that plays out oppositely in the West (panel k) and East (panel 1). Interestingly, this is
components has reversed, as the percentages of variation they explain has changed. In our original
analysis (left side of figure, panels a-f) PC#1—resolved 60 percent of the variation in the data—and
was associated with the roof-shaped graph and its inverse (panels c and d), while PC#2—resolving
the remaining 40 percent of the variation—was associated with the global warming trend (panels e
and f).
What has allowed for this reversal of fortune for the two patterns? In fact, just a mild change in
