So we want to go to Analyze and then go to Dimension Reduction. And first of all,
notice that name there, dimension reduction. The key here, reduction, we're
trying to reduce a certain number of variables or items to a smaller number
of factors or components. And we can refer to these as dimensions, so if we
have one factor that's a one dimension(al) solution, two factors is a
two dimension(al) solution, and so on.
Let's go ahead and select Factor. And then here we want to move all our variables over
to the right. Go ahead and select Descriptives, and then we'll select Univariate
descriptives, to get some univariate descriptive statistics on each of our
variables. And I also want to select KMO and Bartlett's test of sphericity.
Then we'll click Continue. And then next we go to Extraction and notice here, by default,
the method is principal components. And that's what I had mentioned that we're going
to run here today. So that's good, we want to leave that selected. But if you are
looking for an alternative procedure, you can find a number of them here. Now
We're just going to do principal components, which I said earlier, is the most
commonly used method of analysis. OK we'll go ahead and leave these defaults, we'll have the
Unrotated factor solution displayed, and then I also want to display a Scree plot,
which I'll tell you about more in a few minutes. And then let's leave this
Extraction default option selected. So notice that the extraction is, based on
eigenvalue, where eigenvalues greater than 1 will be retained or extracted.
And I will go into that in detail in just a few moments. So go ahead and click
Continue.
OK Rotations, let's go ahead and look at that. Now I'll go and select Varimax, and we'll see
what happens when we run the analysis. But notice here we have five different
options. And Rotation is one of the things that can get a little bit
confusing sometimes in factor analysis. The first thing to note is that there's
two key types of rotation, there's Orthogonal rotation, and there's Oblique
rotation. Now orthogonal rotation means that your factors or components, if
there is more than one, if there's two or more factors or components, they will be
uncorrelated. In fact that rotational solution forces them to be uncorrelated.
Now oblique, on the other hand, they're rotated in such a way where they're
allowed to be correlated. So you'll get solutions where the factors typically
will be correlated to some degree.
Sometimes they'll be highly correlated, sometimes they'll be barely correlated
at all,
it depends on the solution. But the oblique rotation allows for the
correlation. Now of these rotation procedures in SPSS, Varimax, Quartimax
and Equamax are all different types of orthogonal, or uncorrelated rotations,
whereas Direct Oblimin and Promax are oblique, or correlated rotations. We'll go
and select Varimax in this case. OK go ahead and click Continue. And then that
looks good, so go ahead and click OK. And then here we have
our analysis, and our first table we'll look at here is the KMO and Bartlett's
test. This is sometimes reported, so I want to be sure that you understand what it
is here.
Bartlett's test the sphericity, that's what we're going to be focusing
on. And Bartlett's test of sphericity, notice first of all, that it is significant,
it's less than .05. And it approximates a chi-square distribution,
so we can consider it chi-square distributed. And what this is testing is,
it's actually testing whether this correlation matrix, are these variables,
so item 1 with 2, item 1 with 3, item 2 with 3, and so on, this entire
triangle are these variables, are they correlated significantly different than
zero. But unlike the correlation matrix, it doesn't test each individual
correlation separately, but what it does is, in one overall test, it assesses whether
these 10 correlations, taken as a group,
do they significantly differ from zero. And more precisely, for those who are
familiar with matrix algebra, it's testing whether this correlation matrix is
significantly different than an identity matrix. An identity matrix just has ones
along the main diagonal and zeros in all other places. So in other words, it's a
matrix where variables are not correlated whatsoever with each other,
but as always, a variable correlates 1.0 with itself. So it has 1s on the main
diagonal, 0 everywhere else. And the fact that this is significant, and it's
extremely significant, the p-value is very small, it gives us confidence that
our variables are significantly correlated. So once again that's testing
whether the variables, as a set,
does this matrix, does this group of variables, differ significantly from all
zeros here, and it definitely does.
So that's what that test measures.
OK next we have our commonalities, and I'm going to skip over that for a minute,
we'll get back to that soon though. Let's go and look at the total variance explained.