Factor Rotation & Factor Scores: Interpreting & Using FactorsHow the process really works…Reminder of Goals & Process of Factoring ResearchKinds of well-defined factorsKinds of ill-defined factorsReasons for ill-defined factorsSimple StructureComponents of Simple StructureThe benefit of “simple structure” ?How rotation relates to “Simple Structure”Major Types of RotationMajor Types of Orthogonal Rotation & their “tendencies”Major Types of Oblique Rotation & their “tendencies”Some things that are different (or not) when you use a Oblique RotationInterpretation & Cut-offsCombining #-factors & Rotation to Select “the best Factor Solution”Component ScoresProper & Improper Component ScoresProper Component ScoresImproper Component ScoresWhy many folks like Improper Component ScoresUses of Component ScoresUses of Component Scores, cont.Using Component Scores to Help with Factor InterpretationDesigning the next studyDesigning the next study, cont.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…•Here’s the series of steps we talked about earlier. •# factors decision•interpreting the factors•factor scoresConsidering the interpretations of the factors can aid the # factors decision!These decisions aren’t made independently in this order!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 thoughtKinds 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 factoringKinds of ill-defined factorsUnique 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 factorsSimple 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 commonComponents of Simple StructureEach 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 factorEach variable should have a large communality•implying that its membership “accounts” for its varianceThe 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 PC1 PC2V1 .7 .5V2 .6 .6V3 .6 -.5V4 .7 -.6PC1PC2V1V2V3V4•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 plotHow 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 Unrotated PC1 PC2V1 .7 .5V2 .6 .6V3 .6 -.5V4 .7 -.6PC2V1V2V3V4PC1PC1’PC2’ Rotated PC1 PC2V1 .7 -.1V2 .7 .1V3 .1 .5V4 .2 .6Major Types of RotationRemember --