Factor Analysis

Detailed data attached at the end
Detailed code see github.com

Factor analysis using psych package: A brief example

For more details of Parallel Analysis, see Factor Retention Decisions in Exploratory Factor Analysis: A Tutorial on Parallel Analysis

判断公共因子个数的方法：

• 先验经验和理论知识
  
• 事先确定的累计贡献率阈值
  
• 根据Kaiser-Harris准则: 保留特征值大于0的因子
  
• 进行平行分析: 基于多次模拟数据矩阵的特征值均值来选取因子

当摇摆不定时, 高估因子比低估好, 因为高估因子较少曲解“真实”情况

library(psych)
fa.parallel(mydata, fa = "fa", n.iter = 100, 
            show.legend = FALSE, main = "Scree plot with parallel analysis")

## Parallel analysis suggests that the number of factors =  4  and the number of components =  NA

按照parallel analysis的结果, 选取四个因子

fa0=factanal(~., factors=4, data=mydata, rotation="none")   ##不旋转因子难以解释
print(fa0)

## 
## Call:
## factanal(x = ~., factors = 4, data = mydata, rotation = "none")
## 
## Uniquenesses:
##    FL   APP    AA    LA    SC    LC   HON   SMS   EXP   DRV   AMB   GSP 
## 0.443 0.685 0.521 0.185 0.119 0.198 0.339 0.138 0.357 0.226 0.137 0.153 
##   POT    KJ  SUIT 
## 0.090 0.005 0.252 
## 
## Loadings:
##      Factor1 Factor2 Factor3 Factor4
## FL    0.473           0.549  -0.177 
## APP   0.334   0.426           0.140 
## AA   -0.272   0.475   0.336   0.258 
## LA    0.700           0.124   0.555 
## SC    0.552   0.641  -0.396         
## LC    0.600   0.649  -0.134         
## HON   0.465          -0.273   0.603 
## SMS   0.634   0.643          -0.205 
## EXP   0.237   0.189   0.717  -0.192 
## DRV   0.673   0.538          -0.155 
## AMB   0.618   0.652  -0.137  -0.191 
## GSP   0.625   0.664           0.115 
## POT   0.617   0.671   0.178   0.220 
## KJ    0.992  -0.105                 
## SUIT  0.440   0.372   0.621  -0.175 
## 
##                Factor1 Factor2 Factor3 Factor4
## SS loadings      5.023   3.452   1.647   1.033
## Proportion Var   0.335   0.230   0.110   0.069
## Cumulative Var   0.335   0.565   0.675   0.744
## 
## Test of the hypothesis that 4 factors are sufficient.
## The chi square statistic is 84 on 51 degrees of freedom.
## The p-value is 0.00247

fa1=factanal(~., factors=4, data=mydata, scores="Bartlett")  ##默认旋转
print(fa1)

## 
## Call:
## factanal(x = ~., factors = 4, data = mydata, scores = "Bartlett")
## 
## Uniquenesses:
##    FL   APP    AA    LA    SC    LC   HON   SMS   EXP   DRV   AMB   GSP 
## 0.443 0.685 0.521 0.185 0.119 0.198 0.339 0.138 0.357 0.226 0.137 0.153 
##   POT    KJ  SUIT 
## 0.090 0.005 0.252 
## 
## Loadings:
##      Factor1 Factor2 Factor3 Factor4
## FL    0.129   0.717   0.113  -0.117 
## APP   0.458   0.142   0.243   0.164 
## AA            0.126           0.677 
## LA    0.231   0.239   0.838         
## SC    0.918           0.142         
## LC    0.838   0.111   0.291         
## HON   0.252  -0.216   0.742         
## SMS   0.885   0.258                 
## EXP           0.778           0.165 
## DRV   0.767   0.389   0.172         
## AMB   0.904   0.181                 
## GSP   0.792   0.275   0.351   0.148 
## POT   0.735   0.349   0.432   0.247 
## KJ    0.424   0.389   0.554  -0.598 
## SUIT  0.364   0.770           0.142 
## 
##                Factor1 Factor2 Factor3 Factor4
## SS loadings      5.570   2.473   2.099   1.013
## Proportion Var   0.371   0.165   0.140   0.068
## Cumulative Var   0.371   0.536   0.676   0.744
## 
## Test of the hypothesis that 4 factors are sufficient.
## The chi square statistic is 84 on 51 degrees of freedom.
## The p-value is 0.00247

结果显示因子变得更好解释了:

• 第一公共因子中, 系数绝对值大的变量主要与求职者外露能力有关
  
• 第二公共因子中, 系数绝对值大的变量主要反映求职者的经验
  
• 第二公共因子主要反映了求职者是否讨人喜欢
  
• 第四公共因子系数绝对值较小, 说明其相对次要一些, 主要反映求职者的专业能力

apply(fa1$scores,2,which.max)  ## 各因子得分最高者

## Factor1 Factor2 Factor3 Factor4 
##      10      42      46      46

apply(fa1$scores,2,which.min)  ## 各因子得分最低者

## Factor1 Factor2 Factor3 Factor4 
##      42      47      42      29

plot(fa1$scores[, 1:2], type="n", main="前两个公共因子得分图")
text(fa1$scores[,1], fa1$scores[,2])

Appendix: datasets

applicant

X	FL	APP	AA	LA	SC	LC	HON	SMS	EXP	DRV	AMB	GSP	POT	KJ	SUIT
1	6	7	2	5	8	7	8	8	3	8	9	7	5	7	10
2	9	10	5	8	10	9	9	10	5	9	9	8	8	8	10
3	7	8	3	6	9	8	9	7	4	9	9	8	6	8	10
4	5	6	8	5	6	5	9	2	8	4	5	8	7	6	5
5	6	8	8	8	4	4	9	5	8	5	5	8	8	7	7
6	7	7	7	6	8	7	10	5	9	6	5	8	6	6	6
7	9	9	8	8	8	8	8	8	10	8	10	8	9	8	10
8	9	9	9	8	9	9	8	8	10	9	10	9	9	9	10
9	9	9	7	8	8	8	8	5	9	8	9	8	8	8	10
10	4	7	10	2	10	10	7	10	3	10	10	10	9	3	10
11	4	7	10	0	10	8	3	9	5	9	10	8	10	2	5
12	4	7	10	4	10	10	7	8	2	8	8	10	10	3	7
13	6	9	8	10	5	4	9	4	4	4	5	4	7	6	8
14	8	9	8	9	6	3	8	2	5	2	6	6	7	5	6
15	4	8	8	7	5	4	10	2	7	5	3	6	6	4	6
16	6	9	6	7	8	9	8	9	8	8	7	6	8	6	10
17	8	7	7	7	9	5	8	6	6	7	8	6	6	7	8
18	6	8	8	4	8	8	6	4	3	3	6	7	2	6	4
19	6	7	8	4	7	8	5	4	4	2	6	8	3	5	4
20	4	8	7	8	8	9	10	5	2	6	7	9	8	8	9
21	3	8	6	8	8	8	10	5	3	6	7	8	8	5	8
22	9	8	7	8	9	10	10	10	3	10	8	10	8	10	8
23	7	10	7	9	9	9	10	10	3	9	9	10	9	10	8
24	9	8	7	10	8	10	10	10	2	9	7	9	9	10	8
25	6	9	7	7	4	5	9	3	2	4	4	4	4	5	4
26	7	8	7	8	5	4	8	2	3	4	5	6	5	5	6
27	2	10	7	9	8	9	10	5	3	5	6	7	6	4	5
28	6	3	5	3	5	3	5	0	0	3	3	0	0	5	0
29	4	3	4	3	3	0	0	0	0	4	4	0	0	5	0
30	4	6	5	6	9	4	10	3	1	3	3	2	2	7	3
31	5	5	4	7	8	4	10	3	2	5	5	3	4	8	3
32	3	3	5	7	7	9	10	3	2	5	3	7	5	5	2
33	2	3	5	7	7	9	10	3	2	2	3	6	4	5	2
34	3	4	6	4	3	3	8	1	1	3	3	3	2	5	2
35	6	7	4	3	3	0	9	0	1	0	2	3	1	5	3
36	9	8	5	5	6	6	8	2	2	2	4	5	6	6	3
37	4	9	6	4	10	8	8	9	1	3	9	7	5	3	2
38	4	9	6	6	9	9	7	9	1	2	10	8	5	5	2
39	10	6	9	10	9	10	10	10	10	10	8	10	10	10	10
40	10	6	9	10	9	10	10	10	10	10	10	10	10	10	10
41	10	7	8	0	2	1	2	0	10	2	0	3	0	0	10
42	10	3	8	0	1	1	0	0	10	0	0	0	0	0	10
43	3	4	9	8	2	4	5	3	6	2	1	3	3	3	8
44	7	7	7	6	9	8	8	6	8	8	10	8	8	6	5
45	9	6	10	9	7	7	10	2	1	5	5	7	8	4	5
46	9	8	10	10	7	9	10	3	1	5	7	9	9	4	4
47	0	7	10	3	5	0	10	0	0	2	2	0	0	0	0
48	0	6	10	1	5	0	10	0	0	2	2	0	0	0	0