we'll also pull up our Scree plot here. These two tables here, the Total Variance
Explained and Scree Plot, both deal with what's known as our factor extraction
methods. If you recall when we went through SPSS, the options, we left the
eigenvalue greater than one rule option selected as the default, but we also
selected that a Scree plot be output in our analysis. And these are two of the
most commonly used procedures for deciding how many components or factors
to retain; how many do you want to keep in our solution. So we'll go ahead and
take a look at these in turn here. Here for our Total Variance Explained table,
notice first of all that we have 5 components in our rows here. And you may
be wondering, well wait a second, I thought factor analysis, the whole
purpose of it, was to reduce our number of variables into a smaller number of
components? And if you are thinking that, you're correct, that is our purpose here.
But, as just a matter of definition, it's always the case that the number of
variables we input in our analysis, will always be equal to the number of
components shown here. So we have five variables input in our analysis,
therefore we have 5 rows or 5 components shown here. Now here in our
Initial Eigenvalues table, notice that we have these various eigenvalues. So the
first one is 3.136 and everything after that is less than 1. Now if you recall
our first rule was eigenvalue greater than one rule. So that was, keep the
number of factors or components that have eigenvalues greater than one. All
other components with eigenvalues less than one, such as these here, we do not
keep.
If you look at the
Extraction Sums of Squared Loadings section of this table, notice that there's only one
value here now. And what this means is this is how many components SPSS
retained or kept, based on the rule. So since only one component had an eigenvalue
greater than one, we only have one component in our solution here. So the
results of this rule tells us, or indicates, that we want to have one
component. So in other words, we reduce those 5 variables down to one
component. Or that one component, from this perspective, does a pretty good job
at explaining the relationships between SWLS1 through SWLS5. And, as a side note,
there are different ways of assessing how good of a job this component did at
explaining those relationships, but for now, using this rule, we know that we have
one component, that's all we're going to keep.
One way to assess how good of a job
this analysis did at explaining the relationships between those variables, is
to look at the percent of variance accounted for by the component. And in
this example, our one component solution accounted for 62.72% of the variance,
or about 63% of variance, which is pretty good in
practice. I typically see solutions between 40% and 60% of the
variance, in the 40s through 60s, in that range. I don't typically see many
solutions with variance higher than 70, and a solution below 40 is not very
strong. But that's typically the range that I'll see them in, so I would say
that 63% is pretty good in practice. Now an interesting thing
here, recall that we had 5 components. If you add up these
eigenvalues they will equal to 5, within rounding error. So the sum of the
eigenvalues is always equal to the number of components, or put another way,
the number of original variables in your analysis. So if I had 10 variables in my
analysis here, then these values would sum up to 10. And in fact would be 10
rows in this table. Now since I have 5 variables, I'm going to have 5 components
output in my initial solution,
and the eigenvalues will sum to 5. And the reason why that's good to know is that
if you divide the eigenvalue for our retained component the 3.136/5 you will
get exactly .6272 or 62.72% when converted to a percentage.
So the percent of variance accounted for is literally the magnitude of the eigen-
value divided by the sum of the eigenvalues, or 5 in this case.
OK, so in summary, our eigenvalue greater than one rule indicated that one
component should be retained.
Next let's look at the Scree plot. So here our Scree plot, notice first of all
that on the X-axis, the component number is plotted, so this is the first
component, second component, third, and so on. And on the Y-axis we have our
eigenvalue plotted. And in fact if you think about it, this graph is really just
plotting, notice this first value, 3.136, that is right here. Component 2 is somewhere
between .6 and .7, and if you look here, here we go component 2, .625. Notice component
3 drops off just a little, it is .534 Component 4 is .463, and then
component 5 is .231. So this Scree plot is literally just these eigenvalues
plotted from left to right.
Now the rule of thumb for interpreting the Scree plot is as follows: