因子分析
library(psych)
library(MASS)
#3D表示
options(rgl.printRglwidget = TRUE)
#WebGLに変換してRStudio, VSCode, ウェブブラウザなど出力
library(rgl)1 因子分析の基礎用語
h2：共通性
u2：独自性
com 複雑性（item complexity）
１つの因子からのみ影響を受けている単純構造であれば 1 に近く、
「どっちつかず」だと値が大きくなる。(単純構造に近づけたい時に便利な指標)
（Mean item complexity はその平均で、１に近いほどよい）loadings（因子負荷量）：元の変量とその因子との相関係数
SS loadings（因子寄与）：固有値（Sum of Square loadings 因子負荷量の平方和）
因子軸の回転
因子を解釈するためには、因子が単純構造である方が便利。
単純構造とは、各項目の因子負荷が1つの因子のみ大きな値を持ち、
それ以外はゼロに近い構造。
そのような状態にするために利用されるのが因子軸の回転です。
RMSR
モデルによって説明できなかったデータの分散の大きさを指標化したもの
データの分散共分散モデルと推定値の分散共分散モデルの差の二乗の平均の平方根。
０に近いほど良い。
empirical chi square
因子分析が「うまくいった」と判断される因子負荷量は「1観測変数に対して，1共通因子だけ1もしくは-1に近く，他の共通因子では0に近い」である．逆に．1つの観測変数に対して因子負荷量の共通因子間の差が小さい場合は因子分析がうまくいかなかったことを示している． 大きいほどP値は小さく、モデルの当てはまりは良い
1.1 ３次元データ
mu <- c(0,0,0)
Sigma <- matrix(c(1, 0.7,0.6,0.7,1,0.5,0.6,0.5, 1), 3, 3)
data <- mvrnorm(100, mu, Sigma)
d <- data.frame(X=data[,1],Y=data[,2],Z=data[,3])
attach(d)1.2 none回転
fit <- fa(r=d, nfactors=2 ,rotate="", fm="ml", scores=T)## Specified rotation not found, rotate='none' usedprint(fit, digits=5,sort=T)# 因子得点を出すためにはscores=T## Factor Analysis using method = ml
## Call: fa(r = d, nfactors = 2, rotate = "", scores = T, fm = "ml")
## Standardized loadings (pattern matrix) based upon correlation matrix
## item ML1 ML2 h2 u2 com
## Y 2 0.94920 -0.29958 0.99072 0.0092762 1.1973
## X 1 0.90967 0.40073 0.98808 0.0119205 1.3740
## Z 3 0.64062 0.06796 0.41502 0.5849842 1.0225
##
## ML1 ML2
## SS loadings 2.13887 0.25495
## Proportion Var 0.71296 0.08498
## Cumulative Var 0.71296 0.79794
## Proportion Explained 0.89350 0.10650
## Cumulative Proportion 0.89350 1.00000
##
## Mean item complexity = 1.2
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 3 and the objective function was 1.33608 with Chi Square of 129.8224
## The degrees of freedom for the model are -2 and the objective function was 0
##
## The root mean square of the residuals (RMSR) is 0
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 100 with the empirical chi square 0 with prob < NA
## The total number of observations was 100 with Likelihood Chi Square = 0 with prob < NA
##
## Tucker Lewis Index of factoring reliability = 1.023992
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## ML1 ML2
## Correlation of (regression) scores with factors 0.99702 0.97908
## Multiple R square of scores with factors 0.99406 0.95860
## Minimum correlation of possible factor scores 0.98811 0.91720biplot(fit)
fa.diagram( fit )
plot3d(X, Y, Z, type = "s", col = "gray", size = 1)
v <- fit$ loadings
#v[1,1];v[2,1];v[3,1]
#v[1,2];v[2,2];v[3,2]
str(v)## 'loadings' num [1:3, 1:2] 0.91 0.949 0.641 0.401 -0.3 ...
## - attr(*, "dimnames")=List of 2
## ..$ : chr [1:3] "X" "Y" "Z"
## ..$ : chr [1:2] "ML1" "ML2"arrow3d(p0 = c(0, 0, 0), p1 = c(v[1,1],v[2,1],v[3,1]), col = "red")
arrow3d(p0 = c(0, 0, 0), p1 = c(v[1,2],v[2,2],v[3,2]), col = "blue")1.3 promax回転
fit.ml.promax <- fa(r=d, nfactors=2 ,rotate="promax", fm="ml", scores=T)# 因子得点を出すためにはscores=T## Loading required namespace: GPArotationprint(fit.ml.promax, digits=3)## Factor Analysis using method = ml
## Call: fa(r = d, nfactors = 2, rotate = "promax", scores = T, fm = "ml")
## Standardized loadings (pattern matrix) based upon correlation matrix
## ML2 ML1 h2 u2 com
## X 1.026 -0.042 0.988 0.01192 1.00
## Y -0.005 0.999 0.991 0.00928 1.00
## Z 0.401 0.282 0.415 0.58498 1.79
##
## ML2 ML1
## SS loadings 1.264 1.130
## Proportion Var 0.421 0.377
## Cumulative Var 0.421 0.798
## Proportion Explained 0.528 0.472
## Cumulative Proportion 0.528 1.000
##
## With factor correlations of
## ML2 ML1
## ML2 1.000 0.771
## ML1 0.771 1.000
##
## Mean item complexity = 1.3
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 3 and the objective function was 1.336 with Chi Square of 129.822
## The degrees of freedom for the model are -2 and the objective function was 0
##
## The root mean square of the residuals (RMSR) is 0
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 100 with the empirical chi square 0 with prob < NA
## The total number of observations was 100 with Likelihood Chi Square = 0 with prob < NA
##
## Tucker Lewis Index of factoring reliability = 1.024
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## ML2 ML1
## Correlation of (regression) scores with factors 0.994 0.995
## Multiple R square of scores with factors 0.989 0.991
## Minimum correlation of possible factor scores 0.978 0.982biplot(fit.ml.promax)
fa.diagram( fit.ml.promax)
plot3d(X, Y, Z, type = "s", col = "gray", size = 1)
v <- fit.ml.promax$ loadings
arrow3d(p0 = c(0, 0, 0), p1 = c(v[1,1],v[2,1],v[3,1]), col = "red")
arrow3d(p0 = c(0, 0, 0), p1 = c(v[1,2],v[2,2],v[3,2]), col = "blue")1.4 varimax回転
fit.ml.varimax <- fa(r=d, nfactors=2 ,rotate="varimax", fm="ml", scores=T)
print(fit.ml.varimax, digits=3)## Factor Analysis using method = ml
## Call: fa(r = d, nfactors = 2, rotate = "varimax", scores = T, fm = "ml")
## Standardized loadings (pattern matrix) based upon correlation matrix
## ML2 ML1 h2 u2 com
## X 0.911 0.397 0.988 0.01192 1.37
## Y 0.423 0.901 0.991 0.00928 1.42
## Z 0.484 0.425 0.415 0.58498 1.97
##
## ML2 ML1
## SS loadings 1.244 1.150
## Proportion Var 0.415 0.383
## Cumulative Var 0.415 0.798
## Proportion Explained 0.520 0.480
## Cumulative Proportion 0.520 1.000
##
## Mean item complexity = 1.6
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 3 and the objective function was 1.336 with Chi Square of 129.822
## The degrees of freedom for the model are -2 and the objective function was 0
##
## The root mean square of the residuals (RMSR) is 0
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 100 with the empirical chi square 0 with prob < NA
## The total number of observations was 100 with Likelihood Chi Square = 0 with prob < NA
##
## Tucker Lewis Index of factoring reliability = 1.024
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## ML2 ML1
## Correlation of (regression) scores with factors 0.987 0.989
## Multiple R square of scores with factors 0.975 0.978
## Minimum correlation of possible factor scores 0.950 0.956biplot(fit.ml.varimax)
fa.diagram( fit.ml.varimax)
plot3d(X, Y, Z, type = "s", col = "gray", size = 1)
v <- fit.ml.varimax$ loadings
arrow3d(p0 = c(0, 0, 0), p1 = c(v[1,1],v[2,1],v[3,1]), col = "red")
arrow3d(p0 = c(0, 0, 0), p1 = c(v[1,2],v[2,2],v[3,2]), col = "blue")c(v[1,1],v[2,1],v[3,1])## [1] 0.9112822 0.4232714 0.48426211.5 最小残差法minres
rotate = “oblimin”
result = fa( d, nfactors = 2, fm = "minres", rotate = "oblimin", use = "complete.obs" )
print( result$loadings, digits = 2, cutoff = 0.3 )##
## Loadings:
## MR1 MR2
## X 0.87
## Y 0.85
## Z 0.69
##
## MR1 MR2
## SS loadings 1.97 0.01
## Proportion Var 0.66 0.00
## Cumulative Var 0.66 0.66biplot(result)
fa.diagram( result)
plot3d(X, Y, Z, type = "s", col = "gray", size = 1)
v <- result$ loadings
arrow3d(p0 = c(0, 0, 0), p1 = c(v[1,1],v[2,1],v[3,1]), col = "red")
arrow3d(p0 = c(0, 0, 0), p1 = c(v[1,2],v[2,2],v[3,2]), col = "blue")