1 CLP 6529, Applied Multivariate - PowerPoint Presentation

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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

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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.

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Preview to EFAh

2 = communality = proportion of variance that can be accounted for by common factors

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Preview to EFAh

2 = communality = proportion of variance that can be accounted for by common factorsu2 = uniqueness = 1 - h2

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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

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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

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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

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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)

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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

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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?

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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?

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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.

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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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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!

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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.

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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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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.

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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

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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”

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FA: Oblique rotationPromax is short for “Procrustean maximization”.

Procrustes ????????????????????????

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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.

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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.

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Procrustes

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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

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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”

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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)

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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.

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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.

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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

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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.

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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.

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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???)

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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

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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.

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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?

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Principal Components: ACTIVE Example

Memory Reasoning Speed

HVLT Word Series UFOV1

AVLT Letter Series UFOV2

Rivermead

Letter

Sets

UFOV3

UFOV4

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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.

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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

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+.02

residual

Can’t take the average, because positives and negatives cancel each other outSlide76

First, square all the residuals. That removes the sign.

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+.02

residualSlide77

Second, compute the MEAN squared residual

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+.02

residualSlide78

The bigger the ‘average’ residual, the worse the model fits

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+.02

residualSlide79

Root mean squared residual (RMR or SRMR)

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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

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Where is the elbow?Slide113

ADEPT:

Promax

EFA, retain 7 factors

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ADEPT:

Promax

EFA, retain 7 factors

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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

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Factor patternSlide118

ADEPT: Promax EFA, retain 7 factors

Some factors are QUITE correlated…suggests redundancy….maybe the smaller number of factors makes more sense

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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

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From 4 factorSlide122

ADEPT:

Promax

EFA, retain 4 factors

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From 4 factor

From 7 factorSlide123

ADEPT:

Promax

EFA, retain 4 factors

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ADEPT:

Promax

EFA, retain 4 factors

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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

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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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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

131Slide132

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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