R 28 February 2014

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1 R 28 February Timothy Bates *1 *2 2 * :1 10:1 1.0 (1) *4 (2) *1 * KMO MSA Bartlett *3 *4

2 (Parallel Analysis) X 1, X 2,..., X F 1 F 2 X 1 = β 1 1 F 1 + β 2 1 F 2 + ϵ 1 X 2 = β 1 2 F 1 + β 2 2 F 2 + ϵ 2. X 10 = β 1 10 F 1 + β 2 10 F 2 + ϵ 10 β (Factor loadings) ϵ (uniqueness) *5 F 1 F 2 *6 F 1 F 2 n n [1, 300] i X i (n) F S 1 (n) F S 2 (n) β F S 1 (n) = β 1 i X i (n) i=1 10 F S 2 (n) = β 2 i X i (n) i= % *5 1 (communality) rela *6

3 (very poor) 100 (poor) 200 (fair) 300 (good) 500 (very good) 1,000 (excellent) (Comfrey and Lee, 1992, p.217) KMO MSA KMO Kaiser-Meyer-Olkin MSA Measures of Sampling Adequacy KMO *7 MSA 0.5 KMO MSA kmo <- function(x) { x <- subset(x, complete.cases(x)) # Remove the cases with any missing value r <- cor(x) # Correlation matrix r2 <- r^2 # Squared correlation coefficients i <- solve(r) # Inverse matrix of correlation matrix d <- diag(i) # Diagonal elements of inverse matrix p2 <- (-i/sqrt(outer(d, d)))^2 # Squared partial correlation coefficients diag(r2) <- diag(p2) <- 0 # Delete diagonal elements KMO <- sum(r2)/(sum(r2)+sum(p2)) MSA <- colsums(r2)/(colsums(r2)+colsums(p2)) return(list(kmo=kmo, MSA=MSA)) } p *7 Kaiser (1974) (miserable) (mediocre) (middling) (meritorious) 0.9 (marvelous)

4 Bartlett.sphericity.test <- function(x) { method <- "Bartlett s test of sphericity" data.name <- deparse(substitute(x)) x <- subset(x, complete.cases(x)) # Remove the cases with any missing value n <- nrow(x) p <- ncol(x) chisq <- (1-n+(2*p+5)/6)*log(det(cor(x))) df <- p*(p-1)/2 p.value <- pchisq(chisq, df, lower.tail=false) names(chisq) <- "X-squared" names(df) <- "df" return(structure(list(statistic=chisq, parameter=df, p.value=p.value, method=method, data.name=data.name), class="htest")) } 7 R factanal paf rela rela eigencrit= 1 KMO MSA fa psych fm= "minres" "ml" "pa" nfactors= rotate= "none" "varimax" "quartimax" "bentlert" "geomint" "oblimin" "simplimax" "bentlerq" "geominq" "cluster" alpha psych α cortest.bartlett psych fa.parallel psych $nfact sem (confirmatory factor analysis; CFA) sem sem CFA

5 8 1 Tomothy Bates SPSS *8 p1-p40 Bates pdf *9 res$factor.loadings library(foreign) y <- read.spss(" x <- as.data.frame(y) for (i in 1:length(x)) { x[,i] <- ifelse(x[,i]==999,na,x[,i]) } # // Comments // ========================================================== # The data \verb!x! consists of 538 cases with 102 variables. # it can be saved as "factorexdata05.txt" by the following line # write.table(x,"factorexdata05.txt",quote=false,sep="\t",row.names=false) # if so, the data can be read by: # x <- read.delim("factorexdata05.txt") # ========================================================================= Ps <- x[,4:43] # Extract variables p1-p40 Ps <- subset(ps, complete.cases(ps)) # Omit missings (511 cases remain) library(rela) res <- paf(as.matrix(ps)) summary(res) # Automatically calculate KMO with MSA, determine the number of factors, # calculate chi-square of Bartlett s sphericity test, communalities and # factor loadings. Communalities are 1 minus uniquenesses. barplot(res$eigenvalues[,1]) # First column of eigenvalues. resv <- varimax(res$factor.loadings) # Varimax rotation is possible later. print(resv) barplot(sort(colsums(loadings(resv)^2),decreasing=true)) # screeplot using rotated SS loadings. scores <- as.matrix(ps) %*% as.matrix(resv$loadings) # Get factor scores in a simple manner. library(psych) cortest.bartlett(ps) # Bartlett s sphericity test. res2 <- fa.parallel(ps) res3 <- fa(ps, fm="minres", nfactors=8, rotate="oblimin") print(res3) # Factor loadings as $loadings 9 2 (2014) factanal() * 10 *8 *9 *10

6 install.packages("misaki", repos=" library(misaki) demo(part2) # factanal() rotation="none" > # demo(part2) data(tests) > result <- factanal(, factors=2, scores="regression") > result$loadings Loadings: Factor1 Factor Factor1 Factor2 SS loadings Proportion Var Cumulative Var > biplot(result$scores, result$loading, cex = 2) biplot()

7 rela paf 2 40 KMO 0.7 MSA 0.5 > library(rela) > summary(paf(as.matrix( ))) $KMO [1] $MSA MSA $Bartlett [1] $Communalities Initial Communalities Final Extraction $Factor.Loadings [,1] [,2] $RMS [1] psych > cortest.bartlett(cor( ), n=40)$p.value # [1] e-21 > # cortest.bartlett( ) > # > res2 <- fa.parallel( ) Loading required package: parallel Loading required package: MASS Parallel analysis suggests that the number of factors = 2 and the number of components = 2

8 p 2 2 fa() > print(res3 <- fa(, nfactors=2, rotate="varimax", fm = "ml")) Factor Analysis using method = ml Call: fa(r =, nfactors = 2, rotate = "varimax", fm = "ml") Standardized loadings (pattern matrix) based upon correlation matrix ML1 ML2 h2 u2 com ML1 ML2 SS loadings Proportion Var Cumulative Var Proportion Explained Cumulative Proportion Mean item complexity = 1.2 Test of the hypothesis that 2 factors are sufficient. The degrees of freedom for the null model are 15 and the objective function was 3.71 with Chi Square of The degrees of freedom for the model are 4 and the objective function was 0.06 The root mean square of the residuals (RMSR) is 0.02 The df corrected root mean square of the residuals is 0.05 The harmonic number of observations is 40 with the empirical chi square 0.44 with prob < 0.98 The total number of observations was 40 with MLE Chi Square = 2.16 with prob < 0.71 Tucker Lewis Index of factoring reliability = RMSEA index = 0 and the 90 % confidence intervals are NA BIC = Fit based upon off diagonal values = 1 Measures of factor score adequacy ML1 ML2 Correlation of scores with factors Multiple R square of scores with factors Minimum correlation of possible factor scores > biplot(res3$scores, res3$structure) factanal() 40

9 sem * 11 * 12 library(misaki); data(tests) # library(sem) # sem cor1 <- cor( ) # model1 <- specifymodel() # <-, a1 <-, a2 <-, a3 <-, b1 <-, b2 <-, b3 <-, b4 <->, e1, NA <->, e2, NA <->, e3, NA <->, e4, NA <->, e5, NA <->, e6, NA <->, NA, 1 <->, NA, 1 sem1 <- sem(model1, cor1, N=40) # sem summary(sem1, fit.indices=c("gfi", "AGFI", "RMSEA", "CFI", "AIC", "BIC")) *11 install.packages(sem, dep=true) sem *12

10 CFA * 13 AGFI 0.9 CFI RMSEA Model Chisquare = Df = 8 Pr(>Chisq) = Goodness-of-fit index = Adjusted goodness-of-fit index = RMSEA index = % CI: (NA, ) Bentler CFI = AIC = BIC = Normalized Residuals Min. 1st Qu. Median Mean 3rd Qu. Max R-square for Endogenous Variables Parameter Estimates Estimate Std Error z value Pr(> z ) a e-04 <--- a e-08 <--- a e-08 <--- b e-04 <--- b e-15 <--- b e-09 <--- b e-10 <--- e e-04 <--> e e-02 <--> e e-01 <--> e e-01 <--> e e-04 <--> e e-04 <--> Iterations = 22 *13 model1 <- cfa(covs=null, reference.indicators=false) :,, :,,,

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