library(psych)
library(moments)
library(lavaan)
## This is lavaan 0.6-11
## lavaan is FREE software! Please report any bugs.
## 
## 載入套件：'lavaan'
## 下列物件被遮斷自 'package:psych':
## 
##     cor2cov
library(reshape2)
library(tidyverse)
## -- Attaching packages --------------------------------------- tidyverse 1.3.1 --
## v ggplot2 3.3.6     v purrr   0.3.4
## v tibble  3.1.6     v dplyr   1.0.9
## v tidyr   1.2.0     v stringr 1.4.0
## v readr   2.1.2     v forcats 0.5.1
## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x ggplot2::%+%()   masks psych::%+%()
## x ggplot2::alpha() masks psych::alpha()
## x dplyr::filter()  masks stats::filter()
## x dplyr::lag()     masks stats::lag()
data(PoliticalDemocracy)
dta <- PoliticalDemocracy
str(dta)
## 'data.frame':    75 obs. of  11 variables:
##  $ y1: num  2.5 1.25 7.5 8.9 10 7.5 7.5 7.5 2.5 10 ...
##  $ y2: num  0 0 8.8 8.8 3.33 ...
##  $ y3: num  3.33 3.33 10 10 10 ...
##  $ y4: num  0 0 9.2 9.2 6.67 ...
##  $ y5: num  1.25 6.25 8.75 8.91 7.5 ...
##  $ y6: num  0 1.1 8.09 8.13 3.33 ...
##  $ y7: num  3.73 6.67 10 10 10 ...
##  $ y8: num  3.333 0.737 8.212 4.615 6.667 ...
##  $ x1: num  4.44 5.38 5.96 6.29 5.86 ...
##  $ x2: num  3.64 5.06 6.26 7.57 6.82 ...
##  $ x3: num  2.56 3.57 5.22 6.27 4.57 ...

#感覺這11個items可以被分配成兩個factors

探索性因素分析

#factor loading 因素負載量越高越好，越能反映子構念

print.psych(fa(dta[, 1:11],  nfactor = 2,  fm = "pa",  rotate = "promax"),  cut = .3) 
## Factor Analysis using method =  pa
## Call: fa(r = dta[, 1:11], nfactors = 2, rotate = "promax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##      PA1   PA2   h2    u2 com
## y1  0.88       0.72 0.277 1.0
## y2  0.83       0.59 0.411 1.1
## y3  0.70       0.48 0.523 1.0
## y4  0.83       0.74 0.257 1.0
## y5  0.68       0.66 0.344 1.2
## y6  0.79       0.62 0.385 1.0
## y7  0.80       0.67 0.327 1.0
## y8  0.79       0.70 0.295 1.0
## x1        0.92 0.87 0.131 1.0
## x2        0.98 0.94 0.064 1.0
## x3        0.89 0.75 0.247 1.0
## 
##                        PA1  PA2
## SS loadings           5.04 2.70
## Proportion Var        0.46 0.25
## Cumulative Var        0.46 0.70
## Proportion Explained  0.65 0.35
## Cumulative Proportion 0.65 1.00
## 
##  With factor correlations of 
##      PA1  PA2
## PA1 1.00 0.49
## PA2 0.49 1.00
## 
## Mean item complexity =  1
## Test of the hypothesis that 2 factors are sufficient.
## 
## The degrees of freedom for the null model are  55  and the objective function was  9.74 with Chi Square of  677.07
## The degrees of freedom for the model are 34  and the objective function was  0.84 
## 
## The root mean square of the residuals (RMSR) is  0.04 
## The df corrected root mean square of the residuals is  0.05 
## 
## The harmonic number of observations is  75 with the empirical chi square  14.95  with prob <  1 
## The total number of observations was  75  with Likelihood Chi Square =  57.06  with prob <  0.0079 
## 
## Tucker Lewis Index of factoring reliability =  0.939
## RMSEA index =  0.094  and the 90 % confidence intervals are  0.049 0.138
## BIC =  -89.74
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy             
##                                                    PA1  PA2
## Correlation of (regression) scores with factors   0.97 0.98
## Multiple R square of scores with factors          0.94 0.96
## Minimum correlation of possible factor scores     0.88 0.93

平行分析

fa.parallel(dta[, 1:11], fa = "pc", show.legend = FALSE)

## Parallel analysis suggests that the number of factors =  NA  and the number of components =  2

#建議分成兩個因素。若分成2個因素，特徵值皆會大於1

接著以factor loading來篩選題目

#準則:高fl，高h2，低u2，接近1的com

#共有PA1, PA2兩個因素

#x1/x2/x3這三個items分配到PA2這個factor，且都是高fl高h2低u2，接近1的com，三題都保留

#y1~y8這八個items分配到PA1這個factor，其中去掉fl較低的y5、低h2高u2的y3

#最後保留的items有9個，x1、x2、x3分配到PA2；y1、y2、y4、y6、y7、y8分配到PA1