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:
