what we want to do is retain the number
of components that are above what's
known as the scree, or where this plot
tends to not drop much, when it tends to,
I wouldn't say flatline, but taper off
very gradually. Notice how these 4
points here, these 4 eigenvalues,
the rate of change, or the slope here, is
quite minimal as we move across. But this
value,
there's a big drop from component 1 to
component 2, and then from component 2
all the way through 5,
there's not much of a change anymore. So
according to the scree plot, we interpret
the number of components above where
they tend to not change much anymore. And
where this name comes from, the scree
plot, scree is a geological term which
indicates the rubble or the stones that
fall from a cliff. So if you think of a
cliff, you're driving along the road,
you're going to see a lot of stones,
smaller sized rocks and some bigger
rocks, but they're all collected along
the side of the mountain, right? Well this
is the scree or the rubble that is
collected off the cliff.
So that's where this name comes from. So
we want to retain the number of
components above the scree. So the scree
plot would indicate to us here that we
want to retain one component. Suppose
there was a component right here as well.
Well notice that it still drops quite a
bit from here to here, and then it flat
lines. So if we had a component here as 
well, then we would retain two components 
in
that case. But since we only have one
before the drop off, then we're just
going to retain the one component.
OK now it's nice when these two
solutions agree, both the eigenvalue
greater than one and the scree plot both
indicated one component, but that's
definitely not always the case. So it's
nice when it does work out. I find that
the more components you retain, so if you
have a factor analysis or principal
components analysis or you have like 50
variables, for example, that you're
analyzing, you're probably going to get a
solution with 3, 4, 5, 6
components, something like that. Well, in
that case, in my experience, they're not
typically going to agree that often. So
as you add components the likelihood of
these two rules of thumb agreeing
completely, decreases. It certainly can
happen, they can agree, without question,
but the likelihood tends to decrease. One
of the interesting things about these two
rules are that, the eigenvalue greater
than one rule has been around for a very
long time, as has the scree plot. 
The eigenvalue
greater than one rule was published by
Kaiser in 1960, so that's quite a long
time, and it's still one of the primary
methods for extraction, for determining
the number of components, used today. And
the scree plot, the key publication for
that was in 1966 by Raymond Cattell. So
this came out in 66, the publication
anyway, and the publication for this came
out in 1960 by Kaiser, so that's quite a long
time ago, and they're still the two
primary methods that are used for factor
extraction.
Now that being said, for those who are
interested in a more advanced look at
factor analysis, there are better methods
that can be used, such as parallel
analysis. But, unfortunately, they're not
output in SPSS. You can go ahead and run
a parallel analysis if you search on the
web, and you can use syntax for SPSS to
run it, or you can use, some websites have
it all ready, where you just input the 
number of
variables you have, your sample size, and
so on, and you can get out the solution
for the parallel analysis. But that's
really beyond the scope of this video. If
I get a chance, I'll try to make a video
on how to run and interpret a parallel
analysis as well.
But for now, these are the two most
commonly used methods of extraction.
OK, so to review, in our example here, we
have one component. And next we'll go
ahead and look at our Component Matrix
and we'll also look at our Rotated
Component Matrix here. And let's start
with this one.
Notice it says Rotated Component Matrix
only one component was extracted the
solution cannot be rotated. And that's a
very important point to make, and that is,
when you have a one component solution
in principal components analysis, then
there is no rotation. Rotation only comes
into play when there are two or more
components. So with one component there
is no rotation, and that's why we got
this output, and the reason why we got this
output, if you recall, when we did our
factor analysis in SPSS,
under rotation, we asked for Varimax.
So basically SPSS is telling us, we can't
do Varimax rotation because there's
only one component, and rotation doesn't
come into play in that case. So as one
measure of the effectiveness of our
solution, we noted the total variance
explained by our one component. I had said 
that 63% of the variance was
pretty good in practice. That's one way
to look at it. That's the overall
variance that the component accounts for.
Now here on the Component Matrix
