Factor analysis and causal modeling

Transcript

RyanImagine you've written a twenty-item questionnaire about how medical trainees experience their workplace. Before you trust it, you want to know how the items relate to each other.

SoheeAnd an open-access editorial by Tavakol and Wetzel points out the problem. A basic analysis of inter-item relationships for twenty items produces a twenty-by-twenty grid of correlations, four hundred cells. Nobody can hold that in their head.

RyanSo we need a way to see the forest instead of four hundred trees.

SoheeWelcome to EPPP Prep, the podcast. I'm Sohee.

RyanI'm Ryan. Today, two families of analysis named in the knowledge statement: factor analysis, and causal modeling. We'll skip the matrix algebra and keep the ideas.

SoheeTavakol and Wetzel describe factor analysis as a way to simplify a complex set of variables or items, to explore the underlying dimensions that explain the relationships among them.

RyanIn plainer words: when certain items rise and fall together, there may be one hidden thing behind them. That hidden thing is a factor, a latent variable you can't measure directly.

SoheeAnd the editorial says factor analysis is widely used to develop and refine assessment instruments, and to produce evidence for construct validity.

RyanThere are two main types. First, exploratory factor analysis, often shortened to EFA. It's used early in developing an instrument, when you don't yet know the structure.

SoheeThe key number in EFA is the factor loading. The editorial defines it as the correlation between an item and a factor. And a loading above 0.30 usually indicates a moderate correlation.

RyanTheir example: thirty questions yield two factors. The items loading on the first factor are all about professionalism. The items loading on the second are about leadership. So the researcher learns the questionnaire is measuring two distinct things.

SoheeNow two more terms that show up on exams. Communality and eigenvalue. Both come from squaring loadings.

RyanSquare an item's loading and you get the share of that item's variance explained by the factor. The editorial's example has loadings of 0.86, 0.75, 0.66 and 0.58, which square to about 74, 56, 43 and 33 percent.

SoheeNow sum the squared loadings two different ways. Across the factors for one item, and you get the communality: how much of that item's variance all the factors together explain.

RyanDown the items for one factor, and you get the eigenvalue. In the example, it's 2.1. Divide by the four items and that factor accounts for about 52 percent of the variance.

SoheeA memory hook: communality belongs to an item. The eigenvalue belongs to a factor.

RyanThe editorial adds a caution. Principal component analysis is often described, in error, as exploratory factor analysis. Principal components is appropriate for reducing a set of items to a smaller set. If you're trying to measure a latent construct, exploratory factor analysis is the appropriate method.

SoheeSecond type: confirmatory factor analysis, or CFA. Here you already have a model, maybe from earlier EFA studies. The editorial calls CFA a theory-driven approach that tests how well the data fit the proposed model.

RyanSo the difference is direction. EFA asks, what structure is in here? CFA asks, does the structure I predicted actually fit? And in CFA, the researchers must specify the factor model before analyzing the data.

SoheeOne more piece: rotation, a step applied to the factor solution. The editorial notes that an orthogonal rotation produces factors that are uncorrelated.

RyanAnd since most constructs in social science are assumed to be interrelated, it says an oblique rotation should usually be applied. Oblique allows the factors to correlate.

SoheeOkay, pause and think. A researcher believes anxiety and depression are separate but related factors in her scale. Which rotation fits that belief?

RyanOblique, because it allows the factors to correlate. Orthogonal would force them to be uncorrelated.

SoheeNow causal modeling. The roots go back to a geneticist named Wright. An open-access history by Tarka explains that Wright was dissatisfied with partial correlations, which remained far from a causal explanation.

RyanSo he developed path analysis, to impose a causal structure on observed correlations. With path diagrams, he could break correlations down into direct effects, indirect effects and common causes.

SoheeThat grew into structural equation modeling, or SEM. Gunzler and colleagues, in an open-access introduction, call it a very general, very powerful multivariate technique. It uses a conceptual model, a path diagram, and a system of linked regression-style equations.

RyanThe twist that makes SEM different from ordinary regression: a variable that's dependent in one equation can be an independent variable in another. Variables that are only ever predictors are called exogenous. Variables that are predicted somewhere in the model are called endogenous.

SoheeAnd the path diagrams have a grammar. Latent variables, like depression, are drawn as circles or ovals. Observed variables, like a score on a rating scale, are rectangles. A single straight arrow means a causal path. A curved two-headed arrow means the two may be associated.

RyanThe Tarka history also notes that SEM is general enough to include the familiar methods as special cases: the t-test, ANOVA, analysis of covariance, multivariate analysis of variance, and multiple regression.

SoheeNow here's the part that keeps us honest. Does drawing arrows prove cause?

RyanNo. Tarka describes a long debate. One side argues that using SEM with nonexperimental data creates the illusion of causal process modeling. The current consensus leans toward careful application of SEM for investigating causal relationships.

SoheeAnd the history quotes a point Simon made in 1954: deciding whether a correlation is spurious requires assuming, in advance, that certain other causal relations don't hold. So a good fit means the data are consistent with your model and its assumptions. It doesn't turn correlational data into an experiment.

RyanLast topic, and an exam favorite. Mediators and moderators.

SoheeGunzler and colleagues define a mediator as an intermediate variable that helps explain how or why an independent variable influences an outcome.

RyanTheir example is a tobacco prevention program that teaches people to stop taking smoking breaks at work. That changes their social norms about tobacco use, the mediator, which then leads to less smoking.

SoheeSo the mediator sits in the middle of the chain. Program, then norms, then smoking.

RyanWhich gives us three effects. The direct effect is the path from the program to smoking while controlling for the mediator. The indirect effect is the path through the mediator. And the total effect is the direct plus the indirect.

SoheeIf the mediator carries all of the effect, that's full mediation, and the direct path is completely broken. If some direct effect remains, that's partial mediation. The paper says partial mediation is more common.

RyanHistorically, Baron and Kenny tested mediation with a series of regression equations. Gunzler and colleagues argue that SEM is a more appropriate framework, partly because it gives information about how well the whole model fits the data.

SoheeAnd moderation? Baron and Kenny distinguished it: a moderator is a third variable that affects the strength or direction of the relationship between an independent variable and an outcome.

RyanThe paper's example: if men and women react differently to a cholesterol-lowering intervention, there's a gender by treatment interaction, and gender is a moderator.

SoheeSo, a moderator answers for whom or when. A mediator answers how or why.

RyanPause and think, second one. A study finds exercise reduces depressive symptoms because it improves sleep quality. Another finds exercise reduces depressive symptoms more for older adults than for younger ones. Name the role of sleep, and of age.

SoheeSleep is a mediator: it's the mechanism, the how. Age is a moderator: it changes the strength of the effect, the for whom.

RyanSo how does the exam test this? Expect definitions in disguise: a study that confirms a predicted structure, a question about which rotation allows correlated factors, or a scenario with a third variable you have to label.

SoheeThe tempting wrong answers: swapping exploratory and confirmatory, swapping communality and eigenvalue, swapping mediator and moderator, and claiming that a well-fitting causal model proves causation.

RyanRecap. Factor analysis finds latent dimensions behind many correlated items. A loading is the item-factor correlation.

SoheeCommunality is an item's variance explained by all the factors. An eigenvalue is a factor's total explained variance across the items.

RyanEFA explores. CFA tests a structure specified in advance. Orthogonal rotation means uncorrelated factors, oblique allows them to correlate.

SoheePath analysis and SEM model networks of causal paths, with exogenous and endogenous variables, and a good fit is consistent with, not proof of, cause.

RyanAnd mediators explain how, moderators change how much or for whom.

SoheeThese are some of the most abstract ideas on the exam, and you stuck with them. That's real persistence, and it pays off when the words show up in a stem.

RyanTry a few items on factor analysis and mediators next, and draw a tiny arrow diagram for each one. Thanks for listening to EPPP Prep.

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