class: center, middle, inverse, title-slide # Training Workshop on the basics of<br> SEM using R ## Session 1: Exploratory Factor Analysis (EFA) --- layout: true <style>.shareagain-bar { --shareagain-foreground: rgb(255, 255, 255); --shareagain-background: rgba(0, 0, 0, 0.5); --shareagain-pinterest: none; --shareagain-pocket: none; --shareagain-reddit: none; }</style> --- ## Factor analysis process **Stage 1**: Objectives of factor analysis **Stage 2**: Designing an Exploratory factor analysis **Stage 3**: Assumptions in Exploratory factor analysis **Stage 4**: Deriving factors and assessing overall fit **Stage 5**: Interpreting the factors --- class: middle center # Stage 1 : Objectives of factor analysis ---- --- ## Types of factor analysis .leftcol[ #### Exploratory factor analysis + Use when you do not have a well-developed theory + Estimate all possible variable/ factor relationships + Looking for patterns in the data ] .rightcol[ #### Confirmatory factor analysis + Testing a theory that you know in advance + Only specified variables/factor relationships ] --- ## Types of factor analysis .left-column-50[ #### Exploratory factor analysis - Difficult to interpret without a theory. - factor loadings: meanings can sometimes be inferred from patterns. ] <br> .right-column-50[ <img src="image/efa_interpret.png" width="50%" style="display: block; margin: auto;" /> ] --- ## Types of factor analysis .left-column-50[ #### Confirmatory factor analysis - Model fit: how well the hypothesized model fits the data. - Factor loadings: how well items measure their corresponding constructs. ] <br> .right-column-50[ <img src="image/cfa_interpret.png" width="50%" style="display: block; margin: auto;" /> ] --- class: middle center # Stage 2: Designing an EFA ---- --- ## Variable selection and measurement issues -- What types of variables can be used in factor analysis? -- + *Primary requirement: a correlation value can be calculated among all variables.* + *e.g., metric variables, scale items, dummy variables to represent nonmetric variables.* <br> How many variables should be included? + *Five or more per factor for scale development.* + *Three or more per factor for factor measurement (based on how degrees of freedom is computed).* --- ## Sample size Some recommended guidelines: Absolute size of the dataset + *should not fewer than 50 observation* + *preferably 100 and larger* + *200 and larger as the number of variables and expected factors incerases* Ratio of cases to variables + *observation is 5x as the number of variables* + *sample size is 10:1 ratio* + *some proposes 20 cases per variables* --- class: middle center # Stage 3: Assumptions in EFA ---- --- ## Sample Dataset .leftcol40[ + HBAT Industries, manufacturer of paper products. + Perceptions on a set of business functions. + Rating scale: + `0 "poor"` to `10 "excellent"` ] .rightcol60[ <img src="image/hbat_data.png" width="80%" style="display: block; margin: auto;" /> ] --- ## Sample Dataset .leftcol40[ + `\(X_6\)` product quality + `\(X_7\)` e-commerce + `\(X_8\)` technical support + `\(X_9\)` complaint resolution + `\(X_{10}\)` advertising + `\(X_{11}\)` product line + `\(X_{12}\)` salesforce image + `\(X_{13}\)` competitive pricing + `\(X_{14}\)` warranty claims + `\(X_{15}\)` packaging + `\(X_{16}\)` order & billing + `\(X_{17}\)` price flexibility + `\(X_{18}\)` delivery speed ] .rightcol60[ .code80[ <div data-pagedtable="false"> <script data-pagedtable-source type="application/json"> 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</script> </div> ] .font80[Source: J.F. Hair (2019): Multivariate data analysis.] ] --- ## Conceptual assumptions .leftcol70[ + Some uderlying structure does exist in the set of selected variables. + correlated variables and subsequent definition of factors do not guarantee relevance + *even if they meet the statistical requirement!* + It is the responsibility of the researcher to ensure that observed patterns are conceptually valid and appropriate. ] --- ## Determining appropriateness of EFA 1. Bartlett Test 2. Measure of Sampling Adequacy --- ## Determining the appropriateness of EFA .leftcol[ #### 1. Bartlett Test + Examines the entire correlation matrix + Test the hypothesis that correlation matrix is an identity matrix. + A significant result signifies data are appropriate for factor analysis. ] .rightcol[ .details[ ```r library(EFAtools) BARTLETT(data, N= nrow(data)) ``` ``` v The Bartlett's test of sphericity was significant at an alpha level of .05. These data are probably suitable for factor analysis. <U+0001D712>²(55) = 619.27, p < .001 ``` ] ] --- ## Determining the appropriateness of EFA #### 2. Kaiser-Meyen-Olkin (KMO Test) + Measure of sampling adequacy + Indicate the proportion of variance explained by the underlying factor. + Guidelines: - `\(\ge 0.90\)` - marvelous - `\(\ge 0.80\)` - meritorious - `\(\ge 0.70\)` - middling - `\(\ge 0.60\)` - mediocre - `\(\ge 0.50\)` - miserable - `\(< 0.50\)` - unacceptable --- ## Determining the appropriateness of EFA #### 2. Kaiser-Meyen-Olkin (KMO Test) .code70[ .details[ ``` -- Kaiser-Meyer-Olkin criterion (KMO) ------------------------------------------ ! The overall KMO value for your data is mediocre. These data are probably suitable for factor analysis. Overall: 0.653 For each variable: x6 x7 x8 x9 x10 x11 x12 x13 x14 x16 x18 0.509 0.626 0.519 0.787 0.779 0.622 0.622 0.753 0.511 0.760 0.666 ``` ] ] --- ## Determining the appropriateness of EFA #### 2. Kaiser-Meyen-Olkin (KMO Test) + When overall MSA is less than 0.50 + Identify variables with lowest MSA subject for deletion. + Recalculate MSA + Repeat unitl overall MSA is 0.50 and above + Deletion of variables with MSA under 0.50 means variable's correlation with <br>other variables are poorly representing the extracted factor. --- class: middle center # Let's practice! --- class: middle center # Stage 4: Deriving factors and <br>assessing overall fit ---- --- ## Partitioning the variance of a variable .leftcol[ #### Unique variance + Variance associated with only a specific variable. + Not represented in the correlations among variables. + *Specific variance* + associated uniquely with a single variable. + *Error variance* + May be due to unreliability of data gathering process, measurement error, or a random component in the measured phenomenom. ] .rightcol[ #### Common variance + Shared variance with all other variables. + High common variance are more amenable for factor analysis. + Derived factors represents the shared or common variance among the variables. ] --- ## Partitioning the variance of a variable .center[ <img src="image/deriving_factors.png" width="55%" /> .font70[_Source: JF Hair et al. (2019) Multivariate data analysis._<br>] ] --- ## PCA vs Common factor analysis .leftcol[ #### Principal component analysis (PCA) + Considers the total variance + data reduction is a primary concern #### Common factor analysis + Considers only the common variance or shared variance + Primary objective is to identify the latent dimensions or constructs ] .rightcol[ <br> <br> <br> .center[ <img src="image/pca_cfa.png" width="85%" /> .font70[_Source: JF Hair et al. (2019) Multivariate data analysis._]] ] --- ## Exploring possible factors .leftcol40[ #### 1. Kaiser-Guttman Criterion + Only consider factors whose eigenvalues is greater than 1. + Rationale is that factor should account for the variance of at least a single variable if it is to be retained for interpretation. ] .rightcol60[ ```r library(EFAtools) KGC(Data, eigen_type = "EFA") ``` <img src="image/kgc_plot.png" width="75%" style="display: block; margin: auto;" /> ] --- ## Exploring possible factors .leftcol40[ #### 2. Scree test + Identify the optimum number of factors that can be extracted before the amount of unique variance begins to dominate the common variance. + Inflection point or the "elbow" ] .rightcol60[ ```r library(psych) scree(data) ``` <img src="image/scree_plot.png" width="75%" style="display: block; margin: auto;" /> ] --- ## Exploring possible factors .leftcol[ #### 3. Parallel Test + Generates a large number of simulated dataset. + Each simulated dataset is factor analyzed. + Results is the average eigenvalues across simulation. + Values are then compared to the eigenvalues extracted from the original dataset. + All factors with eigenvalues above those average eigenvalues are retained. ] .rightcol[ ```r library(psych) fa.parallel(data, fa = "fa") ``` <img src="image/parallel_plot.png" width="90%" style="display: block; margin: auto;" /> ] --- class: middle center # Let's practice! --- class: middle center # Stage 5: Interpreting the factors ---- --- ## Three process of factor intepretation #### 1. Factor extraction #### 2. Factor rotation #### 3. Factor interpretation and re-specification --- ## Factor extraction .rightcol[ .code40[ ```r fa_unrotated <- fa(r = data, nfactors = 4,rotate = "none") print(fa_unrotated$loadings) ``` ``` Loadings: MR1 MR2 MR3 MR4 x6 0.201 -0.408 0.463 x7 0.290 0.656 0.267 0.210 x8 0.278 -0.382 0.744 -0.169 x9 0.862 -0.255 -0.184 x10 0.287 0.456 0.127 x11 0.689 -0.454 -0.141 0.316 x12 0.398 0.807 0.348 0.255 x13 -0.231 0.553 -0.287 x14 0.378 -0.322 0.730 -0.151 x16 0.747 -0.176 -0.181 x18 0.895 -0.304 -0.198 MR1 MR2 MR3 MR4 SS loadings 3.215 2.226 1.500 0.679 Proportion Var 0.292 0.202 0.136 0.062 Cumulative Var 0.292 0.495 0.631 0.693 ``` ] ] .leftcol[ #### Loadings + Correlation of each variable and the factor. + Indicate the degree of correspondence between variable and factor. + Higher loadings making the variable representative of the factor. ] --- ## Factor extraction .rightcol[ .code30[ ```r fa_unrotated <- fa(r = data, nfactors = 4,rotate = "none") print(fa_unrotated$loadings) ``` ``` Loadings: MR1 MR2 MR3 MR4 x6 0.201 -0.408 0.463 x7 0.290 0.656 0.267 0.210 x8 0.278 -0.382 0.744 -0.169 x9 0.862 -0.255 -0.184 x10 0.287 0.456 0.127 x11 0.689 -0.454 -0.141 0.316 x12 0.398 0.807 0.348 0.255 x13 -0.231 0.553 -0.287 x14 0.378 -0.322 0.730 -0.151 x16 0.747 -0.176 -0.181 x18 0.895 -0.304 -0.198 MR1 MR2 MR3 MR4 SS loadings 3.215 2.226 1.500 0.679 Proportion Var 0.292 0.202 0.136 0.062 Cumulative Var 0.292 0.495 0.631 0.693 ``` ] ] .leftcol[ #### Loadings + `\(\le \pm 0.10 \approx\)` zero + `\(\pm 0.10\)` to `\(\pm 0.40\)` meet the minimal level + `\(\ge \pm 0.50\)` practically significant + `\(\ge \pm 0.70 \approx\)` well-defined structure #### SS loadings + Eigenvalues - column sum of squared factor loadings. + Relative importance of each factor in accounting for the variance associated with the set of variables. ] --- ## Factor rotation .leftcol[ #### Why do factor rotation? + To simplify the complexity of factor loadings. + Distribute the loadings more clearly into the factors. + Facilitate interpretation. ] .rightcol[ <img src="image/factor_rotation.png" width="80%" /> ] --- ## Factor rotation .leftcol35[ .code30[ ```{} par(mfrow = c(1, 2)) plot(fa_unrotated$loadings[,1], fa_unrotated$loadings[,2], xlab = "Factor 1", ylab = "Factor 2", ylim = c(-1, 1), xlim = c(-1, 1), main = "No rotation", pch = 19, col = "#6c757d") abline(h=0, v=0) text(fa_unrotated$loadings[,1], fa_unrotated$loadings[,2], labels = rownames(fa_unrotated$loadings), pos = 4, cex = 0.5) plot(fa_rotated$loadings[,1], fa_rotated$loadings[,2], xlab = "Factor 1", ylab = "Factor 2", ylim = c(-1, 1), xlim = c(-1, 1), main = "With rotation", pch = 19, col = "#6c757d") abline(h=0, v=0) text(fa_rotated$loadings[,1], fa_rotated$loadings[,2], labels = rownames(fa_unrotated$loadings), pos = 4, cex = 0.5) ``` ] ] .rightcol65[ <img src="efa_presentation_files/figure-html/unnamed-chunk-18-1.png" width="100%" /> ] --- ## Factor rotation .leftcol[ #### Orthogonal rotation + axes are maintained at 90 degrees + orthogonal rotation methods + Varimax - *most commonly used* + Quartimax + Equimax + Check-out some of these references + [IBM](https://www.ibm.com/docs/de/spss-statistics/24.0.0?topic=analysis-factor-rotation) + [Factor analysis](http://statweb.stanford.edu/~susan/courses/stats305c/examplesFA.html) ] .rightcol[ <img src="image/orthogonal.png" width="90%" /> ] --- ## Factor rotation #### Orthogonal rotation <img src="efa_presentation_files/figure-html/unnamed-chunk-20-1.png" width="100%" /> --- ## Factor rotation .leftcol[ #### Oblique rotation rotation + allow correlated factors + suited to the goal of theoretically meaningful constructs + oblique rotation methods + Promax + Oblimin ] .rightcol[ <img src="image/oblique.png" width="85%" /> ] --- ## Factor rotation #### Oblique rotation <img src="efa_presentation_files/figure-html/unnamed-chunk-22-1.png" width="100%" /> --- class: middle center # Let's practice! --- ## Factor interpretation and respecification .leftcol40[ + each variable has a high loadings on one factor only + each factor has a high loadings for only a subset of the items. ] .rightcol60[ .code40[ ```r fa_varimax <- fa(r = data, nfactors = 4, rotate = "varimax") print(fa_varimax$loadings, sort = TRUE) ``` ``` Loadings: MR1 MR2 MR3 MR4 x9 0.897 0.130 0.132 x16 0.768 0.127 x18 0.949 0.185 x7 0.781 -0.115 x10 0.166 0.529 x12 0.114 0.980 -0.133 x8 0.890 0.115 x14 0.103 0.879 0.129 x6 0.647 x11 0.525 0.127 0.712 x13 0.213 -0.209 -0.590 MR1 MR2 MR3 MR4 SS loadings 2.635 1.973 1.641 1.371 Proportion Var 0.240 0.179 0.149 0.125 Cumulative Var 0.240 0.419 0.568 0.693 ``` ] ] --- ## Factor interpretation and respecification .leftcol40[ + each variable has a high loadings on one factor only + each factor has a high loadings for only a subset of the items. ] .rightcol60[ .code40[ ```r fa_varimax <- fa(r = data, nfactors = 4, rotate = "varimax") print(fa_varimax$loadings, sort = TRUE, cutoff = 0.4) ``` ``` Loadings: MR1 MR2 MR3 MR4 x9 0.897 x16 0.768 x18 0.949 x7 0.781 x10 0.529 x12 0.980 x8 0.890 x14 0.879 x6 0.647 x11 0.525 0.712 x13 -0.590 MR1 MR2 MR3 MR4 SS loadings 2.635 1.973 1.641 1.371 Proportion Var 0.240 0.179 0.149 0.125 Cumulative Var 0.240 0.419 0.568 0.693 ``` ] ] --- ## Factor interpretation and respecification .leftcol40[ What to do with cross-loadings? Ratio of variance (*JF Hair et al. 2019*) + 1 to 1.5 - problematic + 1.5 to 2.0 - potential cross-loading + 2.0 and higher - ignorable Example: + `\(X_{11}\)` + `MR1`: 0.525 + `MR2`: 0.712 + `\(0.712^2 \div 0.525^2 = 1.8\)` ] .rightcol60[ .code40[ ```r fa_varimax <- fa(r = data, nfactors = 4, rotate = "varimax") print(fa_varimax$loadings, sort = TRUE, cutoff = 0.4) ``` ``` Loadings: MR1 MR2 MR3 MR4 x9 0.897 x16 0.768 x18 0.949 x7 0.781 x10 0.529 x12 0.980 x8 0.890 x14 0.879 x6 0.647 x11 0.525 0.712 x13 -0.590 MR1 MR2 MR3 MR4 SS loadings 2.635 1.973 1.641 1.371 Proportion Var 0.240 0.179 0.149 0.125 Cumulative Var 0.240 0.419 0.568 0.693 ``` ] ] --- class: middle center # Let's practice! --- class: middle center # Thank you! #### Slides created via the R packages: .leftcol[ <img src="image/xaringan.png" style="display:inline-block; margin: 0" width=20%/> ### xaringan by Yihui ] .rightcol[ <img src="image/xaringanthemer.png" style="display:inline-block; margin: 0" width=25%/> ### xaringanthemer and xaringanExtra by Garrick ]