Methods in Psychology Week 09 Outline for this week This week well continue with our exploration of exploratory factor analysis Well begin with a consideration of the common factor model and what its constituents are ID: 532258
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Slide1
1
CLP 6529, Applied Multivariate
Methods in Psychology
Week 09Slide2
Outline for this week
This week, we’ll continue with our exploration of exploratory factor analysisWe’ll begin with a consideration of the common factor model, and what its constituents areThen, we’ll explore some example data sets and implementation in SPSS
2Slide3
Comparing EFA and PCA
So, while PCA tries to find dimensions that explain all of the variance in a set of measured variables, EFA tries to find the dimensionality of reliable, common, shared variance only.Variance in a measured variable that is not shared with any other variable is treated as unique variance.
3Slide4
Preview to EFAh
2 = communality = proportion of variance that can be accounted for by common factors
4Slide5
Preview to EFAh
2 = communality = proportion of variance that can be accounted for by common factorsu2 = uniqueness = 1 - h2
5Slide6
Preview to EFAh
2 = communality = proportion of variance that can be accounted for by common factorsu2 = uniqueness = 1 - h2Reliability of a measure is given as rxt
2
6Slide7
Preview to EFAh
2 = communality = proportion of variance that can be accounted for by common factorsu2 = uniqueness = 1 - h2Reliability of a measure is given as rxt
2s2 = rxt2 – h2 = reliable systematic variance in a measured variable that is not due to common factors
7Slide8
Preview to EFAh
2 = communality = proportion of variance that can be accounted for by common factorsu2 = uniqueness = 1 - h2Reliability of a measure is given as rxt
2s2 = rxt2 – h2 = reliable systematic variance in a measured variable that is not due to common factorsError in a measure e2 = 1 - rxt2
8Slide9
Preview to EFAh
2 = communality = proportion of variance that can be accounted for by common factorsu2 = uniqueness = 1 - h2Reliability of a measure is given as rxt
2s2 = rxt2 – h2 = reliable systematic variance in a measured variable that is not due to common factorsError in a measure e2 = 1 - rxt2
u
2
= s
2
+ e
2
(uniqueness is both systematic variance that is not shared with other measures, and unsystematic or “error”/”noise” variance)
9Slide10
Preview to EFAVariables are estimated from common factors by multiplying each factor by the appropriate weight and summing across factors
Common factors correlate zero with unique factorsUnique factors correlate zero with each other
10Slide11
Preview to EFA
Letter
Sets
Letter
Series
Number
Series
Spot-A-
Word
Verbal
Meaning
Vocabu-
lary
Identical
Pictures
Number
Compar.
Finding
As
EFA
There are two key differences. FIRST, we now try to separate reliable variance (defined as variance shared by two or more measures) from unique variance. SECOND, we generally always expect a “truncated” model (fewer factors than variables) because parsimony is our chief goal.
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What is a factor?
How can I tell, in a mess of correlations like this? I haven’t even been told the variable names!
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What is a factor?
Now, still without knowing the variable names, the correlation matrix has been reordered so that “conceptually close” variables are adjacent to each other. Since the above- and below-diagonal segments are symmetrical, I’ll just focus on the lower half.
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What is a factor?
The “eyeball” method of factor analysis. Can we see pockets of local covariation? Can we find clusters of variables that are more closely related to one another than they are to other variables?
14Slide15
What is a factor?
Well, there does seem to be some slight patterning in the data…this is clearest for the first three variables. But what ARE these clusters?
15Slide16
What is a factor?
The first three variables are reasoning…it makes sense they hang together.
The next three are “locus of control” variables. It does look like “chance” control is reasonably strongly related to the other two.
Vocabulary and years of education do kind of relate (some people would call this an ‘acquired cognitive reserve’ factor)…but vocabulary is also quite related to two of the reasoning measures.
16
Reasoning Control Reserve Slide17
Factor analysis: Some terms
“Indeterminacy in exploratory factor analysis”There is no a priori criterion against which to test a factor solutionThe “number of factors problem” means there is no absolute way to determine how many factors really underly a set of variablesThe “number of rotations” problem means we never know exactly how a matrix ought to be rotated to best describe data (more to come)
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Factor analysis: Some termsThe correlation matrix among measured variables is called the
observed correlation matrixFactor analysis gives you sigma (S), which is the reproduced correlation matrix
Rotation: This is used to produce an increase in the interpretability of factors without altering the mathematical meaning of those factors. This can be achieved in several ways, but generally involves maximizing high loadings and minimizing low loadings.
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Factor analysis: Unrotated, orthogonal solution
A large set of measures is shown here in terms of their loadings on two factors, factor 1 and factor 2. The right angle between the two axes means that the two factors are uncorrelated. This first solution is
orthogonal
(uncorrelated factors)
and
unrotated
(in that the axes have not been moved to maximize fit to the data. This is a typical principal components kind of solution,
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-.5
.5
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-.5
.5
.9
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Factor analysis:
Rotated
, orthogonal solution
A large set of measures is shown here in terms of their loadings on two factors, factor 1 and factor 2. The right angle between the two axes means that the two factors are uncorrelated. This second solution is orthogonal (uncorrelated factors)
and
rotated because you can see the axes have been moved to try to get as close to the data points as possible. However, because of the orthogonality constraint, this just isn’t slicing through the points like we’d like a regression line to slice!
-.9
-.5
.5
.9
-.9
-.5
.5
.9
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Factor analysis:
Rotated, oblique
solution
A large set of measures is shown here in terms of their loadings on two factors, factor 1 and factor 2. The oblique angle between the two axes means that the two factors are correlated. This third solution is oblique (correlated factors)
and
rotated. With the orthogonality constraint gone, these lines beautifully slice through the data points. You can see that most measures will have near-zero loadings on one of the factors, and another loading (from low to high) on a single factor.
.9
-.9
-.5
.5
-.9
-.5
.5
.9
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Factor analysis: Rotated, oblique solution—
Simple structure
You can see that most measures will have near-zero loadings on one of the factors, and another loading (from low to high) on a single factor. As we will soon see, this is called “rotating for simple structure”, or “Thurstonian rotation”. This will generally be our preferred solution.
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-.5
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-.5
.5
.9
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Factor analysis: Some terms
Factor intercorrelation matrix: In oblique solutions, we’re going to want to look at the relationships among factors. In some programs, this is called phi f or psi
yFactor structure: In both orthogonal and oblique solutions, this is the matrix of correlations when measured variables are related to their factor. They give the total relationship between each measure and its factors, and are the preferred/only loadings to report in an orthogonal solution
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Factor analysis: Some
terms
Factor pattern
: In oblique solutions, this is the matrix of standardized regression coefficients when measured variables are regressed on their factor. They give the
unique
relationship between measures and factors, and are the
preferred
loadings to report in an
oblique
solution
Loadings
: The generic term used to reflect the path coefficients that describe the relationships between measures and factors. In some programs, the matrix of loadings is called lambda
l
, and we also know this as the
eigenmatrix
or set of eigenvectors.
24Slide25
Factor analysis: Some terms
Factor scores: For some applications, it is desirable to have an estimate of individual subjects’ scores on factors. (Say you have three measures of inductive reasoning, but what you really want is a global “inductive reasoning score” to use in a subsequent analysis). Factor scores are these point estimates of individuals scores on the factors (if we could measure the unobserved factors). The program can also give you the matrix of factor scoring coefficients used to multiple the original measures by (i.e., weight) to get factor scores. Factor scoring coefficients are linear
recombinations of the loadings.
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Factor analysis: Key points
Generally, the correlations among measures should be “high” to sustain a factor. There is no hard and fast rule about how highOne rule says that rs should be modest...above absolute value of 0.30. Singularity (r = 1) is a problem, however (non “positive definite” matrices)
As a correlational analysis, FA is very sensitive to outliers and missing data problems. You will commonly use listwise deletion, but if missing data is spotty, you could lose many cases this way
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Factor analysis: Key pointsSample sizes...Stevens offers this:
n = 50 = very poorn = 100 = poorn = 200 = fairn
= 300 = goodn = 500 = very goodn = 1000 = excellent
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Factor analysis: Key points
Multivariate normality is not a critical assumption, exceptas it influences correlation sizesinferential tests for the # of factors are used
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Factor analysis: Key points
Thus, the communality of a variable is the variance accounted for by the factorsh2 = 1 = u2A variables communality is the sum of its squared loadings across all the factors on which it loads
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FA: Summary of extraction rules
30Slide31
EFA for categorical (ordinal items)Strictly speaking, the tools provided by SPSS may not be adequate…OLS estimators are used for clearly non-normal data
Mplus has appropriate estimatorsYou all have free access to Mplus now (apps.ufl.edu) Consequently, we will look at this a bit in a week or two
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FA: RotationFactor analysis, I’ve already tried to argue, will be very hard to interpret without rotation. It does not alter the mathematic quality of the solution...only interpretability
Indeed, all orthogonal solutions on some data set with the same number of factors are mathematically equivalent
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FA: Orthogonal RotationThe most common orthogonal rotation is “
varimax”, in part because it is mathematically easiest to do and easy to understand. It maximizes the “variance of loadings” within and across factors, so as to maximize the pattern of “highs and lows” in the loadings.Varimax is inappropriate when the theoretical expectation includes a ‘general factor’, because the tendency to produce such a factor is minimized.
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FA: Orthogonal RotationQuartimax rotation maximizes the variance of loadings within
variablesEquamax tries to maximize the variance of loadings both within variables and within-and-between factorsIn practice, similar results will come from each
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FA: Oblique rotationIn an oblique rotation, there are two general approaches: A ‘direct
oblimin’ and a promaxDirect oblimin is much less commonly used. For this, you need to set a criterion (delta, gamma). If you set it at “0” (default), it will produce the most correlated factors it can. If you set it at “-4”, it will produce orthogonal factors. Thus, it gives you control over the degree of correlation allowed. This is very hard to justify.
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FA: Oblique rotation
Much more commonly used is “promax”, which was relatively recently added to SPSSPromax first starts with an orthogonal, varimax rotation. This, it rotates that solution further to allow correlations between factors, and to achieve Thurstone’s
“simple structure” (i.e., as many loadings as possible to be maximally close to either 1 or 0, and as few loadings as possible to fall in between).1/0 logic helps in interpretation. A “1” would mean “yes, definitely loads on this factor”, and a “0” would mean “definitely does not load”
36Slide37
FA: Oblique rotationPromax is short for “Procrustean maximization”.
Procrustes ????????????????????????
37Slide38
Procrustes
Procrustes (
proh
-KRUS-
teez
)
Procrustes was a host who adjusted his guests to their bed. Procrustes, whose name means "he who stretches", was arguably the most interesting of Theseus's challenges on the way to becoming a hero. He kept a house by the side of the road where he offered hospitality to passing strangers, who were invited in for a pleasant meal and a night's rest in his very special bed. Procrustes described it as having the unique property that its length exactly matched whomsoever lay down upon it. What Procrustes didn't volunteer was the method by which this "one-size-fits-all" was achieved, namely as soon as the guest lay down Procrustes went to work upon him, stretching him on the rack if he was too short for the bed and chopping off his legs if he was too long. Theseus turned the tables on Procrustes, fatally adjusting him to fit his own bed.
38Slide39
Procrustes
You design the bedYou specify the pattern of factor loadings that you expect to see, and the program tells you how well it fitsThis means that we expect some variables to load on some factors and not others—by theoryIn a promax rotation, we try to stick as closely as possible to this principle. We don’t actually tell the program what should load on what, but we do—as much as possible—try to have variables load on one factor at a time. Clear loadings.
39Slide40
Procrustes
40Slide41
ProcrustesThe Procrustes rotation is the fundamental building block for confirmatory factor analysis
Exploratory factor analysis: Throw the data at the computer, it investigates the correlations, and it tells you the pattern of factors (structure) that it findsConfirmatory factor analysis: You tell the computer what you want the factor structure to be. The computer tests that structure, and tells you how well it fits (very procrustean)
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EFA vs. CFAAs a summary, the analyses we do in SPSS will be exploratory. We’ll see what the structure in the data is
The analyses we’ll do in AMOS are confirmatory. We’ll tell the computer what the factors should be
42Slide43
Back to Promax
In promax, we don’t actually specify the pattern of 1s and 0s.But, the computer has the attainment of a pattern of 1s and 0s as its goalIt never quite reaches that goal, but it gives you
much more interpretable loadings
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PromaxWe will generally prefer promax
because it gives factors that are similar to visual rotation, tend to be invariant, easier to calculate, and if orthogonality is true, it will “show up”
44Slide45
More differentiation from PCAFactor analysis differs from PCA in that
communality values replace ones in the R-matrix diagonal before extractionThis is done to try to tell the program how much of the variance is truly shared and reliable (that is the variance that we want to factor), versus how much is unique and error (we want to throw that away)
45Slide46
Prior communality estimates
This process of throwing away unique and error variance is called “providing prior communality estimates” to the program. SPSS calls them “initial communality estimates”.You do this because you are trying to dimensionalize only reliable, common variance. You want to throw away unreliable, uncommon variance. This is the core fundamental critical important assumption of the common factor model.
46Slide47
Prior communality estimatesSo, how do we get these prior communality estimates?
squared multiple correlations (SMC): Each variable is treated as a DV, and predicted from all the other variables in a multiple regression. The resulting Rsquared is the squared multiple correlation, and it is our estimate of how much variance each variable shares with ever other variable.
47Slide48
Prior communality estimates
User specified values:
use this if you have some a priori knowledge about the reliable, shared variance in a measure...communalities from previous, trusted factor analyses; the published reliability of a measure
maximum absolute correlation (MAC)
: The problem with SMC is that it is in no way accounting for specific or local correlations with other measures of the same factor. The MAC criterion has the benefit in that it will tend to give you the best possible correlation each measure has
48Slide49
CommunalitiesOne way in which we judge the quality of a FA is also to look at the “extraction” communalities...which provide a nice index of how much variance is explained in each measured variable.
49Slide50
Loadings and communalities
For
unrotated
factors
only
, the sum of squared loadings for a factor is equal to the eigenvalue. Once loadings are rotated, the sum of squared loadings is called SSL and is no longer equal to the eigenvalue. HOWEVER, it is often (for orthogonal solutions) very
very
close.
Meanwhile, within a measure, the sum of squared loadings across factors gives an estimate of its “extracted” communality. Again, the sum of squared communalities is pretty close to the final extracted cumulative eigenvalues of the solution.
50Slide51
Return to the number of factors problem
In exploratory factor analysis, our GOAL is a “truncated” solution. We want the most parsimonious solution we can get that explains a lot of variance in the measures. So, again, we have to grapple with how many factors to extract.**Note...this almost always takes a lot of runs. You can almost never just do this once!!!!!!!The familiar criteria (Kaiser’s rule, scree plot) still apply.
There is a maximum likelihood test (Bartlett’s, Lawley), but strict multivariate assumptions apply. Also, the larger your sample, the more likely you are to find more factors needed. Ugh
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Return to the number of factors problemThis is a strong place for the role of theory. In the best situation, you know going in how many factors you expect
(and, if you do, why aren’t you doing this analysis in Mplus or AMOS???)
52Slide53
Interpretation of factorsHighest loading(s)
Loading patternWhat’s a high loading?> |.30| rule or > |.40| ruleComes from some significance simulations. Reminder that, in n=100, b = |.4| and higher will tend to be significant. In
n=175, b = |.3| will tend to be significant
53Slide54
More art than science
Gorsuch said: “Resolving the number of factors issue is often more art than science. Although we generally try to explain 75% to 85% of the variance, the core procedure is an informal analysis of information gained by adding a factor. Simply increasing # factors to increase variance explained has costs (in terms of complexity, computation time, and replicability).”
All the other rules (Kaiser’s, scree plot, 70-80% variance explained) are flawed, contradictory, and perform differently in large vs. small samples
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The Gorsuch manifestoThe ultimate test of a factor solution is its replicability. This puts a premium on cross-validation.
55Slide56
EFA example: ACTIVE
Key factors were “memory”, “reasoning” and “speed”. Each of these were assessed with multiple measures. When we looked at this in Principal Components, we got one strong general factor, and the rest was a mess. Is there any evidence that these three factors are “real” in the ACTIVE baseline data?
56Slide57
Principal Components: ACTIVE Example
Memory Reasoning Speed
HVLT Word Series UFOV1
AVLT Letter Series UFOV2
Rivermead
Letter
Sets
UFOV3
UFOV4
57Slide58
Let’s start by running this as an orthogonal, unrotated
factor solution
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This is a big difference. Now, instead of trying to
dimensionalize
ALL the variance in our measures, we are only trying to
dimensionalize
the “reliable common variance”. SPSS needs an estimate of how much of the variance in our measures is reliable and common. It uses “prior” or “initial” communality estimates. These are typically the “squared multiple correlations”, which are the variance explained in the variable when all other variables being factored are used as predictors.
62Slide63
This eigenvalue table tells us, based on the initial communalities, and also on the extraction communalities, that we seem to have a strong first factor. Kaiser’s rule would extract a second factor, based on the initial communalities. Looking at the extraction communalities, our hypothesized three factors, memory reasoning and speed, don’t look good. Factors 2 and 3 look weak, and the cumulative variance explained is only 57%. We’re still in the valley of darkness, but hope is coming.
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Each loading is
squared in the table
below.
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The sum of extraction eigenvalues is 5.733…about 57% of the variance
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Reproduced correlations
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Original correlations70
.75
.80.85Slide71
Now, you run a factor model71Slide72
Reproduced correlations72
.68
.81.87
These are the ESTIMATED correlations based on the factor model
reproducedSlide73
Original and reproduced are seldom identical73
.75
.80
.85
.68
.81
.87
original
reproducedSlide74
Original and reproduced are seldom identical74
-.07
+.01
+.02
residual
Residual correlations
Difference between original and reproducedSlide75
The bigger the ‘average’ residual, the worse the model fits
75
-.07+.01
+.02
residual
Can’t take the average, because positives and negatives cancel each other outSlide76
First, square all the residuals. That removes the sign.
76
-.07+.01
+.02
residualSlide77
Second, compute the MEAN squared residual
77
-.07+.01
+.02
residualSlide78
The bigger the ‘average’ residual, the worse the model fits
78
-.07+.01
+.02
residualSlide79
Root mean squared residual (RMR or SRMR)
79
Convention is for
RMR to be < .05 if model fit is goodSlide80
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We still get “g”…but not much else. Why aren’t our three hypothesized factors jumping out?
We were told rotation increases interpretability.Let’s try an ORTHOGONAL rotation, which keeps the factors uncorrelatedThat helps us a little bit interpretatively (as we’ll see in a moment)But…it rotates the solution to maximize the variability of loadings (some higher, some lower…not so many “ambivalent” loadings that don’t clearly show the factor pattern).
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This is identical to what we
had before
Because we did not request
the unrotated solution, SPSS
suppresses the extraction
communalities, because it bases
them on the unrotated solution.
But we can figure them out.
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Wow! This is a huge difference. The first three factors, again, just like before, explain 57.329% of the variance in the original measures. BUT…and this is cool…the dominance of the first factor is gone. Now, the eigenvalues of the three factors are much more balanced…suggesting we have three roughly equal factors. This also shows you why you can’t take Kaiser’s rule too seriously. If we had, we would have missed this important third factor!
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My factors made sense!!! They are looking close to how I hypothesized them. (Reminder:
Varimax
is an orthogonal rotation, so they are still uncorrelated. Does this make sense?)
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Matches what we got before
from SPSS—unique eigenvalues
If we took the mean of these communlaties, we would know roughly how much variance the factors explain in the measured variables.
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Benefits of the varimax
It GREATLY improved the interpretability of the solutionIt got rid of that less-than-useful general factor, and told us a more differentiated story (that conformed with hypotheses)Because the factors are uncorrelated, we can still calculate the unique eigenvalues of each factor, and we can still calculate the communalities of each variable
But does the orthogonality make sense? No!
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Oblique rotation
Oblique rotations make sense. They allow the factors to be correlatedIf the factors are NOT, in fact, correlated, they will be estimated to have a zero correlation…and the solution won’t look much different from a varimaxBut, if the factors ARE correlated, we’ll now actually get an estimate of how much
As with canonical correlations, the factor correlations should exceed variable correlations, largely because they are “disattenuated for measurement error”
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Same as before
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Oblique rotation
Two choices:Oblimin: an “okay” approach, but Tucker’s tests (with a data set with a known, perfect solution) reveal that it does not reproduce factors as well as PromaxPromax is generally betterKappa controls the degree of correlation you allow…”0” means orthogonal, and is equivalent to
varimax; “4” means “allow the maximum possible correlation, whatever it may be”. This is what SPSS selects by default, and you should leave it that way
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No different, useless now,
because it is based on the
unrotated solution.
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Now, this is a bit different. Again, (and SPSS screwed up the print and I couldn’t fix it), the eigenvalues for the three factors seem balanced at extraction. No evidence of a dominant “g”. But look at how big the extraction eigenvalues are now. What happened?
This is the one cost of correlated factors. The eigenvalues are no longer
unique
, and no longer tells us how much variance each factor UNIQUELY explains. Now they tell us about the total bivariate relationships between each factor and the measures, NOT controlling for the other factors. That is why SPSS is no longer presenting “% variance explained” figures.
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Hmmmm….this is new.
But it draws on a distinction we have made repeatedly
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The pattern matrix is the matrix of standardized regression coefficients when each factor is allowed to predict each indicator.
Thus, like all regression coefficients, it tells us about the UNIQUE association between each factor and the measured indicators, controlling for all other factors.
This is the preferred matrix from which to report factor loadings in an oblique rotation. Let’s expand this matrix and look how beautifully it gives us
Thurstone’s
simple structure. This is such a beautifully clear matrix to interpret!
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Beautiful! This is the
definition
of simple structure. Most of the “off-factor” loadings are just about zero.
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This is new. But look at how HIGHLY correlated the factors are.
Given this, how could we have held the orthogonality assumption?
Reasoning Speed Memory
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PromaxClear advantages
Factor pattern offers a HIGHLY interpretable simple structureAllows for factor correlation—a state of affairs that is usually trueDisadvantage…with correlated factors, the variance of each factor is no longer independent of all other factors…so interpretability is hurt.
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Aiding interpetation
SAS offers two matrices: (A) “Variance explained by each factor ignoring other factors” and (B) “Variance explained by each factor controlling for other factors”We can calculate these ourselves, using the factor loadingsAgain, we’ll take each loading, square it, and sum it.
When we use the factor structure, we will get (A)When we use the factor pattern, we will get (B)
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It really makes no sense, though,
to try to calculate communalities,
because the variance explained in
a variable that is
shared
by these
correlated factors cannot be easily
deduced. So, for the variables,
there is no easy “r-squared”
analog.
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So, the promax rotation is pretty cool for producing interpretable simple structure
But…there is something vaguely dissatisfying to the inferential scientists hereI had started with a three factor hypothesis—and sure enough that is what I gotBut I’m not getting any STATISTICAL guidelines for whether the solution fits well. I’m not getting a test of whether my hypotheses are confirmed.For that test, I’ll need to consider CFA.
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Next stepsWe’ll look at a few more EFAs this week
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Another EFA exampleThe ADEPT Study
The first cognitive training study. Older adult participants received a large battery of measuresI’m starting with this, despite data similarity, just to help extend what we already did…then we’ll go to other types
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ADEPT DataModel One: Seven primary factors
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ADEPT: Promax EFA, retain 7 factors
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ADEPT: Promax EFA, retain 7 factors
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ADEPT: Promax EFA, retain 7 factors
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ADEPT:
Promax
EFA, retain 7 factors
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ADEPT:
Promax
EFA, retain 7 factors
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ADEPT:
Promax
EFA, retain 7 factors
112
Where is the elbow?Slide113
ADEPT:
Promax
EFA, retain 7 factors
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ADEPT:
Promax
EFA, retain 7 factors
114
There were 21 variables here. That means there are (21*20)/2 unique correlation elements, i.e., 210.
Of these 210 elements, only 1 (!) exceeds |.05|. Slide115
To compute SRMR115
You would dump into excel, and restrict yourself to the lower triangular elementsSlide116
To compute SRMR116
Then
Square each elementTake the averageTake the square root of thatResult: 0.015047635 Slide117
ADEPT:
Promax
EFA, retain 7 factors
117
Factor patternSlide118
ADEPT: Promax EFA, retain 7 factors
Some factors are QUITE correlated…suggests redundancy….maybe the smaller number of factors makes more sense
118
1 Inductive Reasoning +
2. Memory span
3. Verbal + Social
4. Perceptual Speed
5. Figural + Speed
6. Figural + Social + Verbal
7 Inductive againSlide119
ADEPT Data: Model 2: Seven is too many
Gf = fluid;
Gc
= crystallized; Ms = memory span; Ps = perceptual speed
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ADEPT:
Promax
EFA, retain 4 factors
121
From 4 factorSlide122
ADEPT:
Promax
EFA, retain 4 factors
122
From 4 factor
From 7 factorSlide123
ADEPT:
Promax
EFA, retain 4 factors
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ADEPT:
Promax
EFA, retain 4 factors
124
19 out of 210 correlations are now
mis
-estimated by more than |.05
When I move the matrix into Excel and compute SRMR, I get:
0.031058357|
(still under threshold, but 2 times higher than the more complicated solution)Slide125
ADEPT:
Promax
EFA, retain 4 factors
125
Factor patternSlide126
ADEPT:
Promax
EFA, retain 4 factors
1=Gf fluid
2=Gc crystal.
3=Ms memory
4=Ps speed
Factors 1 & 2 are surprisingly correlated.
That’s the finding that launched a thousand papers, to this day….
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127
The Big PictureSlide128
The Big Picture
This week, we considered the exploratory factor analysis.We considered key concepts in factor analysis, like communality, uniqueness, reliable but specific variance versus errorWe considered factors as “pockets of local covariation” among measures
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The Big Picture
We considered the “number of factors problem”, and considered heuristics like Kaiser’s, scree plots, and the 70-80% ruleUltimately, though, factors are more art than science, and theory and replication are most important
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The Big Picture
We considered rotation, which renders a solution mathematically equivalent to the unrotated solution, but more interpretableOrthogonal rotations, specifically varimax
, retail orthogonal factors, but maximize the variance among loadingsOblique rotations, specifically promax, relax the orthogonality rule and aim for loadings that are 0 or 1 (simple structure)130Slide131
The Big Picture
We talked about initial or primary communality estimates giving the program starting values about how much variance in each indicator is reliable and sharedSquared multiple correlationEach measure is the DV predicted by all other measures; resulting R2 is SMC
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The Big Picture
We looked at how loading drive factor interpretationSum of squared loadings in a column = each factor’s eigenvalueOblique: squared columns of pattern matrix = “variance controlling for other factors”Oblique:
squared columns of structure matrix = “variance ignoring other factors”Sum of squared loadings in a row = each variable’s communality (but only in orthogonal rotations)132Slide133
The Big Picture
We considered the reproduced and residual correlation matricesResiduals are an index of misfitSPSS tells us the % of residuals that exceed |.05|Root mean square residual gives us an overall index of fit, and we want it to be less than |.05|
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The Big Picture
We worked through a theory guided example where 7 or 4 factors might fit7 explained more variance, but was greatly over-fit per Kaiser and screeMore important, 7 factors didn’t make much sense4 factors (with a few explainable split loadings) fit. Theory trumped variance
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