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1 R 2012/10/05 Kosugi,E.Koji (Yamadai.R) Categorical Factor Analysis by using R 2012/10/05 1 / 9

2 Why we use Kosugi,E.Koji (Yamadai.R) Categorical Factor Analysis by using R 2012/10/05 2 / 9

3 FA vs categorical FA, Kosugi,E.Koji (Yamadai.R) Categorical Factor Analysis by using R 2012/10/05 3 / 9

4 One of the reasons 3 7 Kosugi,E.Koji (Yamadai.R) Categorical Factor Analysis by using R 2012/10/05 4 / 9

5 (2002) 1 2 polychoric correlation coefficient (polyserial correlation coefficient) 3 Kosugi,E.Koji (Yamadai.R) Categorical Factor Analysis by using R 2012/10/05 5 / 9

6 Polychoric Correlation Polyserial Correlation Tetrachoric Correlation Kosugi,E.Koji (Yamadai.R) Categorical Factor Analysis by using R 2012/10/05 6 / 9

7 images of latent continuity Figure : image of latent continuity and expression x ξ x ξ x = 1 ξ < a 1 x = 2 a 1 ξ < a 2 x = 3 a 2 ξ < a 3. x = s. a s 1 ξ (1) Kosugi,E.Koji (Yamadai.R) Categorical Factor Analysis by using R 2012/10/05 7 / 9

8 ρ X,Y (2step-ML) Kosugi,E.Koji (Yamadai.R) Categorical Factor Analysis by using R 2012/10/05 8 / 9

9 Follow me with R code... Kosugi,E.Koji (Yamadai.R) Categorical Factor Analysis by using R 2012/10/05 9 / 9

10 > library(psych) > library(polycor) > # sample statistics > sample <- read.csv("cefasample.csv",head=f,na.strings="*") > head(sample) V1 V2 V3 V4 V5 V6 V7 V > summary(sample) V1 V2 V3 V4 Min. :1.000 Min. :1.000 Min. :1.000 Min. : st Qu.: st Qu.: st Qu.: st Qu.:3.000 Median :4.000 Median :4.000 Median :4.000 Median :4.000 Mean :3.913 Mean :4.127 Mean :3.901 Mean : rd Qu.: rd Qu.: rd Qu.: rd Qu.:4.000 Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000 NA's :2 NA's :1 V5 V6 V7 V8 Min. :1.000 Min. :1.000 Min. :1.00 Min. : st Qu.: st Qu.: st Qu.:2.00 1st Qu.:3.000 Median :4.000 Median :3.000 Median :3.00 Median :4.000 Mean :3.955 Mean :3.138 Mean :2.78 Mean : rd Qu.: rd Qu.: rd Qu.:4.00 3rd Qu.:4.000 Max. :5.000 Max. :5.000 Max. :5.00 Max. :5.000 NA's :1 > table(sample$v1) > describe(sample) var n mean sd median trimmed mad min max range skew kurtosis se V V V V V V V V

11 > # peason cor > peason.cor <- cor(sample,use="complete.obs") > print(peason.cor,digit=2) V1 V2 V3 V4 V5 V6 V7 V8 V V V V V V V V > # polychoric cor > polychoric.cor <- polychoric(sample) > print(polychoric.cor$rho) V1 V2 V3 V4 V5 V6 V7 V V V V V V V V V8 V V V V V V V V > # > # compare, peason vs polycor > # > > # FA > fa.parallel(peason.cor,n.obs=355) Parallel analysis suggests that the number of factors = 3 and the number of components = 2

12 > fa.parallel(polychoric.cor$rho,n.obs=355) Parallel analysis suggests that the number of factors = 3 and the number of components = > fa.result.peason <- fa(peason.cor,n.obs=355,fm="gls",nfactors=3,rotate="promax") > fa.result.polych <- fa(polychoric.cor$rho,n.obs=355,fm="gls",nfactors=3,rotate="promax") > print(fa.result.peason,digit=3,sort=t) Factor Analysis using method = gls Call: fa(r = peason.cor, nfactors = 3, n.obs = 355, rotate = "promax", fm = "gls") Standardized loadings (pattern matrix) based upon correlation matrix item GLS2 GLS1 GLS3 h2 u2 V V V V V V V V GLS2 GLS1 GLS3 SS loadings Proportion Var Cumulative Var Proportion Explained Cumulative Proportion With factor correlations of GLS2 GLS1 GLS3 GLS GLS GLS Test of the hypothesis that 3 factors are sufficient. The degrees of freedom for the null model are 28 and the objective function was wit The degrees of freedom for the model are 7 and the objective function was 0.03 The root mean square of the residuals (RMSR) is The df corrected root mean square of the residuals is 0.04 The number of observations was 355 with Chi Square = with prob < Tucker Lewis Index of factoring reliability = RMSEA index = and the 90 % confidence intervals are NA BIC =

13 Fit based upon off diagonal values = Measures of factor score adequacy GLS2 GLS1 GLS3 Correlation of scores with factors Multiple R square of scores with factors Minimum correlation of possible factor scores > print(fa.result.polych,digit=3,sort=t) Factor Analysis using method = gls Call: fa(r = polychoric.cor$rho, nfactors = 3, n.obs = 355, rotate = "promax", fm = "gls") Standardized loadings (pattern matrix) based upon correlation matrix item GLS3 GLS1 GLS2 h2 u2 V V V V V V V V GLS3 GLS1 GLS2 SS loadings Proportion Var Cumulative Var Proportion Explained Cumulative Proportion With factor correlations of GLS3 GLS1 GLS2 GLS GLS GLS Test of the hypothesis that 3 factors are sufficient. The degrees of freedom for the null model are 28 and the objective function was wit The degrees of freedom for the model are 7 and the objective function was The root mean square of the residuals (RMSR) is The df corrected root mean square of the residuals is The number of observations was 355 with Chi Square = with prob < Tucker Lewis Index of factoring reliability =

14 RMSEA index = and the 90 % confidence intervals are BIC = Fit based upon off diagonal values = Measures of factor score adequacy GLS3 GLS1 GLS2 Correlation of scores with factors Multiple R square of scores with factors Minimum correlation of possible factor scores > # > # sample <- subset(sample,select=c("v11","v13","v20","v5","v4","v17","v12","v15")) > # write.table(sample,"cefasample.csv",sep=",",row.name=f,col.name=f,na="*") > > > # mixed pattern > sample.cat <- data.frame(lapply(sample[1:3],factor),sample[4:8]) > summary(sample.cat) V1 V2 V3 V4 V5 V6 1: 2 1: 3 1 : 2 Min. :1.000 Min. :1.000 Min. : : 26 2: 13 2 : 18 1st Qu.: st Qu.: st Qu.: : 61 3: 31 3 : 60 Median :4.000 Median :4.000 Median : :178 4:197 4 :206 Mean :3.853 Mean :3.955 Mean : : 88 5:111 5 : 67 3rd Qu.: rd Qu.: rd Qu.:4.000 NA's: 2 Max. :5.000 Max. :5.000 Max. :5.000 NA's :1 V7 V8 Min. :1.00 Min. : st Qu.:2.00 1st Qu.:3.000 Median :3.00 Median :4.000 Mean :2.78 Mean : rd Qu.:4.00 3rd Qu.:4.000 Max. :5.00 Max. :5.000 NA's :1 > hetcor.cor <- hetcor(sample.cat) > hetcor.cor$correlations V1 V2 V3 V4 V5 V6 V7 V V V V V V V V

15 V8 V V V V V V V V > hetcor.cor$type [,1] [,2] [,3] [,4] [,5] [1,] "" "Polychoric" "Polychoric" "Polyserial" "Polyserial" [2,] "Polychoric" "" "Polychoric" "Polyserial" "Polyserial" [3,] "Polychoric" "Polychoric" "" "Polyserial" "Polyserial" [4,] "Polyserial" "Polyserial" "Polyserial" "" "Pearson" [5,] "Polyserial" "Polyserial" "Polyserial" "Pearson" "" [6,] "Polyserial" "Polyserial" "Polyserial" "Pearson" "Pearson" [7,] "Polyserial" "Polyserial" "Polyserial" "Pearson" "Pearson" [8,] "Polyserial" "Polyserial" "Polyserial" "Pearson" "Pearson" [,6] [,7] [,8] [1,] "Polyserial" "Polyserial" "Polyserial" [2,] "Polyserial" "Polyserial" "Polyserial" [3,] "Polyserial" "Polyserial" "Polyserial" [4,] "Pearson" "Pearson" "Pearson" [5,] "Pearson" "Pearson" "Pearson" [6,] "" "Pearson" "Pearson" [7,] "Pearson" "" "Pearson" [8,] "Pearson" "Pearson" "" > fa.parallel(hetcor.cor$correlations,n.obs=355) Parallel analysis suggests that the number of factors = 3 and the number of components = > fa.result.hetcor <- fa(hetcor.cor$correlations,n.obs=355,fm="gls",nfactors=3,rotate="proma > print(fa.result.hetcor,digit=3,sort=t) Factor Analysis using method = gls Call: fa(r = hetcor.cor$correlations, nfactors = 3, n.obs = 355, rotate = "promax", fm = "gls") Standardized loadings (pattern matrix) based upon correlation matrix item GLS1 GLS2 GLS3 h2 u2 V V V V

16 V V V V GLS1 GLS2 GLS3 SS loadings Proportion Var Cumulative Var Proportion Explained Cumulative Proportion With factor correlations of GLS1 GLS2 GLS3 GLS GLS GLS Test of the hypothesis that 3 factors are sufficient. The degrees of freedom for the null model are 28 and the objective function was wit The degrees of freedom for the model are 7 and the objective function was The root mean square of the residuals (RMSR) is The df corrected root mean square of the residuals is The number of observations was 355 with Chi Square = with prob < Tucker Lewis Index of factoring reliability = RMSEA index = and the 90 % confidence intervals are BIC = Fit based upon off diagonal values = Measures of factor score adequacy GLS1 GLS2 GLS3 Correlation of scores with factors Multiple R square of scores with factors Minimum correlation of possible factor scores > 7

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