20241224 - measurement model for mean/sum score

15-Factor-Analysis-PCA.Rmd

@@ -469,8 +469,8 @@ knitr::include_graphics("./Images/factorSolutionSimpleStructure.png")
 ```
 
 However, pure simple structure only occurs in simulations, not in real-life data.\index{factor analysis!decisions}\index{factor analysis}\index{principal component analysis}\index{simple structure}
-In reality, our measurement model in an unrotated [factor analysis](#factorAnalysis) model might look like the model in Figure \@ref(fig:factorSolutionUnrotatedExample).\index{factor analysis!decisions}\index{factor analysis}\index{principal component analysis}
-In this example, the measurement model does not show simple structure because the items have cross-loadings—that is, the items load onto more than one factor.\index{factor analysis!decisions}\index{factor analysis}\index{principal component analysis}\index{cross-loading}\index{simple structure}\index{structural equation modeling!measurement model}
+In reality, our [measurement model](#measurementModel-sem) in an unrotated [factor analysis](#factorAnalysis) model might look like the model in Figure \@ref(fig:factorSolutionUnrotatedExample).\index{factor analysis!decisions}\index{factor analysis}\index{principal component analysis}
+In this example, the [measurement model](#measurementModel-sem) does not show simple structure because the items have cross-loadings—that is, the items load onto more than one factor.\index{factor analysis!decisions}\index{factor analysis}\index{principal component analysis}\index{cross-loading}\index{simple structure}\index{structural equation modeling!measurement model}
 The cross-loadings make it difficult to interpret the factors, because all of the items load onto all of the factors, so the factors are not very distinct from each other, which makes it difficult to interpret what the factors mean.\index{cross-loading}\index{factor analysis!decisions}\index{factor analysis}\index{principal component analysis}\index{cross-loading}\index{simple structure}\index{structural equation modeling!measurement model}
 
 ```{r factorSolutionUnrotatedExample, out.width = "100%", fig.align = "center", fig.cap = "Example of a Measurement Model That Does Not Follow Simple Structure. 'INT' = internalizing problems; 'EXT' = externalizing problems; 'TD' = thought-disordered problems.", fig.scap = "Example of a Measurement Model That Does Not Follow Simple Structure.", echo = FALSE}
@@ -584,6 +584,7 @@ You can use them as predictors, mediators, moderators, or outcomes.\index{struct
 And, using latent factors in [SEM](#sem) helps disattenuate associations for measurement error, as described in Section \@ref(disattenuation).
 People often want to use factors outside of [SEM](#sem), but there is confusion here: When researchers find that three variables load onto Factor A, the researchers often combine those three using a sum or average—but this is not accurate.\index{structural equation modeling}\index{latent variable}\index{factor analysis!decisions}\index{factor analysis}\index{principal component analysis}
 If you just add or average them, this ignores the factor loadings and the error.\index{structural equation modeling!factor loading}\index{measurement error}\index{factor analysis!decisions}\index{factor analysis}\index{principal component analysis}
+A mean or sum score is a [measurement model](#measurementModel-sem) that assumes that all items have the same factor loading (i.e., a loading of 1) and no error (residual of 0), which is unrealistic.\index{factor analysis!decisions}\index{factor analysis}\index{principal component analysis}\index{structural equation modeling!measurement model}
 Another solution is to form a linear composite by adding and weighting the variables by the factor loadings, which retains the differences in correlations (i.e., a weighted sum), but this still ignores the estimated error, so it still may not be generalizable and meaningful.\index{factor analysis!decisions}\index{factor analysis}\index{principal component analysis}\index{linear composite}\index{measurement error}
 At the same time, weighted sums may be less generalizable than unit-weighted composites where each variable is given equal weight because some variability in factor loadings likely reflects sampling error.\index{factor analysis!decisions}\index{factor analysis}\index{principal component analysis}\index{linear composite}\index{structural equation modeling!factor loading}