Factor Rotation & Factor Scores: Interpreting & Using Factors

Factor Rotation & Factor Scores: Interpreting & Using Factors Well- & Ill-defined Factors Simple Structure Simple Structure & Factor Rotation Major Kinds of Factor Rotation Factor Interpretation Proper & Improper Factor Scores Uses of Factor Scores (interpretation & representation) Designing the “Next Study”

How the process really works… These decisions aren’t made independently in this order! • Here’s the series of steps we talked about earlier. • # factors decision • interpreting the factors • factor scores Considering the interpretations of the factors can aid the # factors decision! Considering how the factor scores (representing the factors) relate to each other and to variables external to the factoring can aid both the # factors decision and the interpretation of the factors. Remember that this is Exploratory Factor Analysis! Exploring means trying alternatives (# factor rules, rotations, cutoffs). If those alternatives agree we’re pretty confident in the agreed upon solution. If they do not agree, we must select the “best” exploratory solution for these data and then replicate and converge to see if it continues to look like the “best solution”.

Reminder of Goals & Process of Factoring Research • Remember that multiple programmatic studies and convergent findings are essential in factoring research (like all other kinds) • Many studies generate more questions than answers • When factoring we often learn about … • The factors or composite variables that can be formed -- their interpretations, meaning and utility • The variables you started with -- that often are different or more complex than is implied by their names • multi-vocal items, especially unexpected ones, are an often an indication that you have something to learn about that variable – it may be more interesting or complex than you thought

Kinds of well-defined factors • There is a trade-off between “parsimony” and “specificity” whenever we are factoring • This trade-off influences both the #-of-factors and cutoff decisions, both of which, in turn, influence factor interpretation • general and “larger” group factors include more variables, account for more variance -- are more parsimonious • unique and “smaller” group factors include fewer variables & many be more focused -- are often more specific • Preferences really depends upon ... • what you are expecting • what you are trying to accomplish with the factoring

Kinds of ill-defined factors Unique factors • hard to know what construct is represented by a 1-variable factor • especially if that variable is multi-vocal • then the factor is defined by “part” of that single variable -- but maybe not the part defined by its name Group factors can be ill-defined • “odd combinations” can be hard to interpret -- especially later factors comprised of multi-vocal variables (knowledge of variables & population is very important!)

Reasons for ill-defined factors • Ill-defined factors are particularly common when factoring a “closed set” of variables • especially when that set was chosen to be “efficient” and so the variables have low intercorrelations • When there is a general or large group factor, be careful about subsequent smaller group factors • they may be “left-over” parts of multi-vocal variables • factors may not represent the “named” parts of the vars • Keeping & rotating “too many” factors will increase the chances of finding ill-defined factors

Simple Structure • The idea of simple structure is very appealing ... • Each factor of any solution should have an unambiguous interpretation, because the variable loadings for each factor should be simple and clear. • There have been several different characterizations of this idea, and varying degrees of success with translating those characterizations into mathematical operations and objective procedures, here are some of the most common

Components of Simple Structure Each factor should have several variables with strong loadings • admonition for well-defined factors • remember that “strong” loadings can be “+” or “-” Each variable should have a strong loading for only one factor • admonition against multi-vocal items • admonition of conceptually separable factors • admonition that each variable should “belong” to some factor Each variable should have a large communality • implying that its membership “accounts” for its variance

The benefit of “simple structure” ? • Remember that … • we’re usually factoring to find “groups of variables” • But, the extraction process is trying to “reproduce variance” • the factor plot often looks simpler than the structure matrix PC2 PC1 PC2 V1 .7 .5 V2 .6 .6 V3 .6 -.5 V4 .7 -.6 V2 V1 PC1 V3 V4 • True, this gets more complicated with more variables and factors, but “simple structure” is basically about “seeing” in the structure matrix what is apparent in the plot

How rotation relates to “Simple Structure” Factor Rotations -- changing the “viewing angle” of the factor space-- have been the major approach to providing simple structure • structure is “simplified” if the factor vectors “spear” the variable clusters PC1’ Unrotated PC1 PC2 V1 .7 .5 V2 .6 .6 V3 .6 -.5 V4 .7 -.6 PC2 Rotated PC1 PC2 V1.7 -.1 V2.7 .1 V3 .1 .5 V4.2 .6 V2 V1 PC1 V3 V4 PC2’

Major Types of Rotation Remember -- extracted factors are orthogonal (uncorrelated) • Orthogonal Rotation -- resulting factors are uncorrelated • more parsimonious & efficient, but less “natural” • Oblique Rotation -- resulting factors are correlated • more “natural” & better “spearing”, but more complicated Orthogonal Rotation Oblique Rotation PC1’ PC1’ PC2 PC2 Angle less than 90o Angle is 90o V2 V2 V1 V1 PC1 PC1 V3 V3 V4 V4 PC2’ PC2’

Major Types of Orthogonal Rotation & their “tendencies” Varimax -- most commonly used and common default • “simplifies factors” by maximizing variance of loadings of variables of a factor (minimized #vars with high loadings) • tends to produce group factors Quartimax • “simplifies variables” by maximizing variance of loadings of a variable across factors (minimizes #factors a var loads on) • tends to “move” vars from extraction less than varimax • tends to produce a general & small group factors Equimax • designed to “balance” varimax and quartimax tendencies • didn’t work very well -- can’t do simultaneously - whichever is done first dominates the final structure

Major Types of Oblique Rotation & their “tendencies” Promax • computes best orthogonal solution and then “relaxes” orthogonality constraints to better “spear” variable clusters with factor vectors (give simpler structure) Direct Oblimin • spearing variable clusters as well as possible to produce lowest occurrence of multi-vocality All oblique rotations have a parameter (, , Κ) that set maximum correlation allowed between rotated factors • changing this parameter can “importantly” change the resulting rotation and interpretation • try at least a couple of values & look for consistency

Some things that are different (or not) when you use a Oblique Rotation Different things: • There will be a  (phi) matrix that holds the factor intercorrelations • The -values and variances accounted for by the rotated factors will be different than those of the extracted factors • compute  for each factor by summing the squared structure loadings for that factor • compute the variance accounted for as the newly computed  / k Same things: • the communality of each variable will be the same -- but can’t be computed by summing squared structure loadings for each variable (since factors are correlated)

Interpretation & Cut-offs • Interpretation is the process of naming factors based on the variables that “load on” them • Which variables “load” is decided based on a “cutoff” • cutoffs usually range from .3 to .4 ( + or - ) • Higher cutoffs limit # loading variables • factors may be ill-defined, some variables may not load • Lower cutoffs increases # loading variables • variables more likely to be multi-vocal • Worry & make a careful decision when your interpretation depends upon the cutoff that is chosen !!

Combining #-factors & Rotation to Select “the best Factor Solution” To specify “the solution” you must pick the #-factors, type or rotation & cutoff ! • Apply the different rules to arrive at an initial “best guess” of the #-factors • Obtain orthogonal and oblique rotations for that many factors, for one fewer and for one more • Compare the solutions to find for “your favorite” – remember this is exploratory factoring, so explore! • parsimony vs. specificity • different cutoffs (.3 - .4) • rotational survival • simple structure • conceptual sense • interesting surprises (about factors and/or variables)

Component Scores • A principal component is a composite variable formed as a linear combination of measure variables • A component SCORE is a person’s score on that composite variable -- when their variable values are applied to the component score formula • usually computed from Z-scores of measured variables • the resulting PC scores are also Z-scores (M=0, S=1) PC1 = 11Z1 + 21Z2 + … + k1Zk PC2 = 12Z1 + 22Z2 + … + k2Zk (etc.) • Component scores have the same properties as the components they represent (e.g., orthogonal or oblique)

Proper & Improper Component Scores A proper component score is a linear combination of all the variables in the analysis • the appropriate s applied to variable Z-scores An improper component score is a linear combination of the variables which “define” that component • usually an additive combination of the Z-scores of the variables with structure weights beyond the chosen cut-off value (Note: improper doesn’t mean “wrong” -- it means “not derived from optimal OLS weightings”)

Proper Component Scores • Proper component scores are the “instantiation” of the components as they were mathematically derived from R(a linear combination of all the variables) • Proper component scores have the same properties as components • they are correlated with each other the same as are the PCs • PC scores from orthogonal components are orthogonal • PC scores from oblique components have r =  • they can be used to produce the structure matrix (corr of component scores and variables scores), communalities, variance accounted for, etc.

Improper Component Scores • Improper component scores are the “instantiation” of the components as they were interpreted by the researcher(a linear combination of the variables which define that component) • Improper component scores usually don’t have exactly the same properties as components • they are usually correlated with each other whether based on orthogonal or oblique solutions • they can not be used to produce the structure matrix (corr of component scores and variables scores), communalities, variance accounted for, etc.

Why many folks like ImproperComponent Scores • When we talk about a component we seldom conceptualize it as a linear combination of all the variables -- shouldn’t we generate a score for the PC the same way be define it ?? • The weights of the “noncontributing” variables for a given PC are primarily chosen to produce a desired  -- if orthogonality is unlikely, can’t we just treat these as “0” ?? • Aren’t we fooling ourselves using 5-decimal s when the variables themselves probably have a single significant digit ? • Much simpler application -- if you only want a score for a given PC, you need only measure the variables used to define it • Improper PC scores replicate and generalize across populations better than proper PC scores (fewer parameters to “drift”)

Uses of Component Scores • PC scores can be used as predictors or criteria in subsequent dependent model analyses • truncated components analysis -- using as predictors the PCs selected & derived from a large set of collinear predictors (common reason for factoring “closed sets” of variables) • watch the “parsimony/stability vs. specificity” trade-off • answers using proper and improper PC scores often differ (more collinearity & specificity w/ improper) • similar approaches can be taken in ldf, ancova, etc.

Uses of Component Scores, cont. • PC scores can be used as “variables” in additional factor analyses • Higher order factoring • factoring the factors • looking for “more basic” or “more aggregated” variables • watch the parsimony/stability vs. specificity trade-off • PC scores can be used as the basis for cluster analysis & MDScaling interpretation (more later) • parsimony/stability vs. specificity again)

Using Component Scores to Help with Factor Interpretation We can use component scores (especially improper scores) two ways to compare different factor solutions • Compare the inter-correlations of component scores from different “solutions” (#-factors, type of rotations & cutoff) • Important differences may help identify the best solution • Compare the correlations of component scores from different “solutions” (#-factors, type of rotations & cutoff) and additional variables (that were not part of the factor analysis) The information in these correlations can be especially helpful in determining what to do about multi-vocal variables and naming factors – what are the differences in the patterns of correlations for different versions of the factor solution?

Designing the next study • Usually the next study involves “additional variables” • Additional variables might be selected to “join” a factor from this study OR to “create” a new factor • One way to convincing folks you know how to interpret this factor solution is to be able to anticipate the result of including selected additional variables in the next study • One way to solve “difficulties” of multi-vocality and ill-defined factors it to include selected additional variables in the next study • “Additional Variables” might be more indices of the same construct(s), or indices of new “constructs “

Designing the next study, cont. • Consider different ways of measuring constructs within or across studies • different “tests” measuring the variable • especially helpful if you know about the similarities and differences among the measures (e.g., “anxiety”) • self-report, other-report, observation, “ lab tasks”, etc. • Consider how the factor structure might differ across populations (e.g., intelligence) • Consider how the factor structure might differ across “social time” (e.g., racism)

Factor Rotation & Factor Scores: Interpreting & Using Factors