# Chapter 8 Factor

## 8.1 Principal Component Analysis

An example from Field (2018 pp. 795-796):

“I have noticed that a lot of students become very stressed about SPSS Statistics. Imagine that I wanted to design a questionnaire to measure a trait that I termed ‘SPSS anxiety’. I devised a questionnaire to measure various aspects of students’ anxiety towards learning SPSS, the SAQ. I generated questions based on interviews with anxious and non-anxious students and came up with 23 possible questions to include. Each question was a statement followed by a five-point Likert scale: ‘strongly disagree’, ‘disagree’, ‘neither agree nor disagree’, ‘agree’ and ‘strongly agree’ (SD, D, N, A and SA, respectively). What’s more, I wanted to know whether anxiety about SPSS could be broken down into specific forms of anxiety. In other words, what latent variables contribute to anxiety about SPSS? With a little help from a few lecturer friends I collected 2571 completed questionnaires.”

### 8.1.1 Results Overview

Table 8.1: Result Overview Exploratory Factor Analysis
JASP SPSS SAS Minitab R

### 8.1.3 SPSS

DATASET ACTIVATE DataSet1.
FACTOR
/VARIABLES Question_01 Question_02 Question_03 Question_04 Question_05 Question_06 Question_07
Question_08 Question_09 Question_10 Question_11 Question_12 Question_13 Question_14 Question_15
Question_16 Question_17 Question_18 Question_19 Question_20 Question_21 Question_22 Question_23
/MISSING LISTWISE
/ANALYSIS Question_01 Question_02 Question_03 Question_04 Question_05 Question_06 Question_07
Question_08 Question_09 Question_10 Question_11 Question_12 Question_13 Question_14 Question_15
Question_16 Question_17 Question_18 Question_19 Question_20 Question_21 Question_22 Question_23
/PRINT INITIAL ROTATION
/PLOT EIGEN
/CRITERIA MINEIGEN(1) ITERATE(25)
/EXTRACTION PC
/CRITERIA ITERATE(25)
/ROTATION VARIMAX
/METHOD=CORRELATION.

### 8.1.4 SAS

PROC FACTOR Data=work.PCA scree
Nfactors= 4
Method= p
Rotate=varimax;
Var Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23
;
Run;

### 8.1.6 R

## install.packages("psych")
## install.packages("factoextra")
## Principal Component Analysis
library("psych")
##
## Attaching package: 'psych'
## The following object is masked from 'package:car':
##
##     logit
library("factoextra")
## Loading required package: ggplot2
##
## Attaching package: 'ggplot2'
## The following objects are masked from 'package:psych':
##
##     %+%, alpha
## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa
fit1 <- prcomp(PCA.data, scale = TRUE) #eigenvalues
eig.val <- get_eigenvalue(fit1)
eig.val ## print results
##        eigenvalue variance.percent cumulative.variance.percent
## Dim.1   7.2900471        31.695857                    31.69586
## Dim.2   1.7388287         7.560125                    39.25598
## Dim.3   1.3167515         5.725007                    44.98099
## Dim.4   1.2271982         5.335644                    50.31663
## Dim.5   0.9878779         4.295121                    54.61175
## Dim.6   0.8953304         3.892741                    58.50449
## Dim.7   0.8055604         3.502436                    62.00693
## Dim.8   0.7828199         3.403565                    65.41050
## Dim.9   0.7509712         3.265092                    68.67559
## Dim.10  0.7169577         3.117207                    71.79280
## Dim.11  0.6835877         2.972121                    74.76492
## Dim.12  0.6695016         2.910876                    77.67579
## Dim.13  0.6119976         2.660859                    80.33665
## Dim.14  0.5777377         2.511903                    82.84855
## Dim.15  0.5491875         2.387772                    85.23633
## Dim.16  0.5231504         2.274567                    87.51089
## Dim.17  0.5083962         2.210418                    89.72131
## Dim.18  0.4559399         1.982347                    91.70366
## Dim.19  0.4238036         1.842624                    93.54628
## Dim.20  0.4077909         1.773004                    95.31929
## Dim.21  0.3794799         1.649912                    96.96920
## Dim.22  0.3640223         1.582705                    98.55191
## Dim.23  0.3330618         1.448095                   100.00000
fviz_eig(fit1) #scree plot

fit2 <- principal(PCA.data, nfactors=4, rotate = "varimax") #varimax rotiation
fit2 ## print results
## Principal Components Analysis
## Call: principal(r = PCA.data, nfactors = 4, rotate = "varimax")
##                  RC3   RC1   RC4   RC2   h2   u2 com
## ï..Question_01  0.24  0.50  0.36  0.06 0.43 0.57 2.4
## Question_02    -0.01 -0.34  0.07  0.54 0.41 0.59 1.7
## Question_03    -0.20 -0.57 -0.18  0.37 0.53 0.47 2.3
## Question_04     0.32  0.52  0.31  0.04 0.47 0.53 2.4
## Question_05     0.32  0.43  0.24  0.01 0.34 0.66 2.5
## Question_06     0.80 -0.01  0.10 -0.07 0.65 0.35 1.0
## Question_07     0.64  0.33  0.16 -0.08 0.55 0.45 1.7
## Question_08     0.13  0.17  0.83  0.01 0.74 0.26 1.1
## Question_09    -0.09 -0.20  0.12  0.65 0.48 0.52 1.3
## Question_10     0.55  0.00  0.13 -0.12 0.33 0.67 1.2
## Question_11     0.26  0.21  0.75 -0.14 0.69 0.31 1.5
## Question_12     0.47  0.52  0.09 -0.08 0.51 0.49 2.1
## Question_13     0.65  0.23  0.23 -0.10 0.54 0.46 1.6
## Question_14     0.58  0.36  0.14 -0.07 0.49 0.51 1.8
## Question_15     0.46  0.22  0.29 -0.19 0.38 0.62 2.6
## Question_16     0.33  0.51  0.31 -0.12 0.49 0.51 2.6
## Question_17     0.27  0.22  0.75 -0.04 0.68 0.32 1.5
## Question_18     0.68  0.33  0.13 -0.08 0.60 0.40 1.5
## Question_19    -0.15 -0.37 -0.03  0.43 0.34 0.66 2.2
## Question_20    -0.04  0.68  0.07 -0.14 0.48 0.52 1.1
## Question_21     0.29  0.66  0.16 -0.07 0.55 0.45 1.5
## Question_22    -0.19  0.03 -0.10  0.65 0.46 0.54 1.2
## Question_23    -0.02  0.17 -0.20  0.59 0.41 0.59 1.4
##
##                        RC3  RC1  RC4  RC2
## Proportion Var        0.16 0.15 0.11 0.08
## Cumulative Var        0.16 0.31 0.42 0.50
## Proportion Explained  0.32 0.29 0.22 0.17
## Cumulative Proportion 0.32 0.61 0.83 1.00
##
## Mean item complexity =  1.8
## Test of the hypothesis that 4 components are sufficient.
##
## The root mean square of the residuals (RMSR) is  0.06
##  with the empirical chi square  4006.15  with prob <  0
##
## Fit based upon off diagonal values = 0.96

### 8.1.7 Remarks

The rotation used was “Varimax”. All differences in results between the software are due to rounding.

### 8.1.8 References

Field, A. (2018). Discovering statistics using IBM SPSS statistics. Los Angeles, CA: SAGE.

## 8.2 Exploratory Factor Analysis

An example from Field (2018 pp. 795-796):

“I have noticed that a lot of students become very stressed about SPSS Statistics. Imagine that I wanted to design a questionnaire to measure a trait that I termed ‘SPSS anxiety’. I devised a questionnaire to measure various aspects of students’ anxiety towards learning SPSS, the SAQ. I generated questions based on interviews with anxious and non-anxious students and came up with 23 possible questions to include. Each question was a statement followed by a five-point Likert scale: ‘strongly disagree’, ‘disagree’, ‘neither agree nor disagree’, ‘agree’ and ‘strongly agree’ (SD, D, N, A and SA, respectively). What’s more, I wanted to know whether anxiety about SPSS could be broken down into specific forms of anxiety. In other words, what latent variables contribute to anxiety about SPSS? With a little help from a few lecturer friends I collected 2571 completed questionnaires.”

### 8.2.1 Results Overview

Table 8.2: Result Overview Exploratory Factor Analysis
JASP SPSS SAS Minitab R

### 8.2.3 SPSS

FACTOR
/VARIABLES Question_01 Question_02 Question_03 Question_04 Question_05 Question_06 Question_07
Question_08 Question_09 Question_10 Question_11 Question_12 Question_13 Question_14 Question_15
Question_16 Question_17 Question_18 Question_19 Question_20 Question_21 Question_22 Question_23
/MISSING LISTWISE
/ANALYSIS Question_01 Question_02 Question_03 Question_04 Question_05 Question_06 Question_07
Question_08 Question_09 Question_10 Question_11 Question_12 Question_13 Question_14 Question_15
Question_16 Question_17 Question_18 Question_19 Question_20 Question_21 Question_22 Question_23
/PRINT INITIAL KMO EXTRACTION ROTATION
/CRITERIA MINEIGEN(1) ITERATE(25)
/EXTRACTION PAF
/CRITERIA ITERATE(25)
/ROTATION VARIMAX
/METHOD=CORRELATION.

### 8.2.4 SAS

PROC FACTOR Data=work.efa scree
Nfactors= 4
Method= prinit
Rotate=varimax;
Var Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23
;
Run;

### 8.2.5 Minitab

Exploratory Factor Analysis with Principal Axis Factoring is not available in Minitab.

### 8.2.6 R

## install.packages("psych")
## Principal Axis Factor Analysis
library("psych")
fit <- factor.pa(EFA.data, nfactors=4, rotate = "varimax")
## Warning: factor.pa is deprecated. Please use the fa function with fm=pa
fit ## print results
## Factor Analysis using method =  pa
## Call: factor.pa(r = EFA.data, nfactors = 4, rotate = "varimax")
##                  PA1   PA3   PA4   PA2   h2   u2   H2   U2
## ï..Question_01  0.50  0.22 -0.27  0.00 0.37 0.63 0.37 0.63
## Question_02    -0.21 -0.03 -0.01  0.46 0.26 0.74 0.26 0.74
## Question_03    -0.50 -0.18  0.16  0.40 0.47 0.53 0.47 0.53
## Question_04     0.53  0.28 -0.25 -0.03 0.42 0.58 0.42 0.58
## Question_05     0.44  0.27 -0.19 -0.05 0.30 0.70 0.30 0.70
## Question_06     0.05  0.75 -0.12 -0.10 0.59 0.41 0.59 0.41
## Question_07     0.36  0.56 -0.16 -0.13 0.49 0.51 0.49 0.51
## Question_08     0.22  0.15 -0.76  0.00 0.65 0.35 0.65 0.35
## Question_09    -0.13 -0.07 -0.06  0.56 0.34 0.66 0.34 0.66
## Question_10     0.14  0.38 -0.14 -0.12 0.20 0.80 0.20 0.80
## Question_11     0.24  0.27 -0.69 -0.17 0.63 0.37 0.63 0.37
## Question_12     0.51  0.40 -0.11 -0.15 0.45 0.55 0.45 0.55
## Question_13     0.29  0.56 -0.23 -0.14 0.47 0.53 0.47 0.53
## Question_14     0.39  0.49 -0.15 -0.13 0.42 0.58 0.42 0.58
## Question_15     0.28  0.38 -0.25 -0.20 0.32 0.68 0.32 0.68
## Question_16     0.54  0.28 -0.25 -0.16 0.46 0.54 0.46 0.54
## Question_17     0.29  0.27 -0.64 -0.05 0.58 0.42 0.58 0.42
## Question_18     0.37  0.61 -0.14 -0.13 0.54 0.46 0.54 0.46
## Question_19    -0.28 -0.15  0.06  0.38 0.24 0.76 0.24 0.76
## Question_20     0.46  0.04 -0.09 -0.20 0.27 0.73 0.27 0.73
## Question_21     0.59  0.26 -0.15 -0.15 0.47 0.53 0.47 0.53
## Question_22    -0.03 -0.16  0.07  0.47 0.25 0.75 0.25 0.75
## Question_23     0.03 -0.04  0.07  0.33 0.12 0.88 0.12 0.88
##
##                        PA1  PA3  PA4  PA2
## Proportion Var        0.13 0.12 0.09 0.06
## Cumulative Var        0.13 0.26 0.34 0.40
## Proportion Explained  0.33 0.31 0.21 0.15
## Cumulative Proportion 0.33 0.63 0.85 1.00
##
##                item   PA1   PA3   PA4   PA2   h2   u2
## ï..Question_01    1  0.50  0.22 -0.27  0.00 0.37 0.63
## Question_02       2 -0.21 -0.03 -0.01  0.46 0.26 0.74
## Question_03       3 -0.50 -0.18  0.16  0.40 0.47 0.53
## Question_04       4  0.53  0.28 -0.25 -0.03 0.42 0.58
## Question_05       5  0.44  0.27 -0.19 -0.05 0.30 0.70
## Question_06       6  0.05  0.75 -0.12 -0.10 0.59 0.41
## Question_07       7  0.36  0.56 -0.16 -0.13 0.49 0.51
## Question_08       8  0.22  0.15 -0.76  0.00 0.65 0.35
## Question_09       9 -0.13 -0.07 -0.06  0.56 0.34 0.66
## Question_10      10  0.14  0.38 -0.14 -0.12 0.20 0.80
## Question_11      11  0.24  0.27 -0.69 -0.17 0.63 0.37
## Question_12      12  0.51  0.40 -0.11 -0.15 0.45 0.55
## Question_13      13  0.29  0.56 -0.23 -0.14 0.47 0.53
## Question_14      14  0.39  0.48 -0.15 -0.13 0.42 0.58
## Question_15      15  0.28  0.38 -0.25 -0.20 0.32 0.68
## Question_16      16  0.54  0.28 -0.25 -0.16 0.46 0.54
## Question_17      17  0.30  0.27 -0.64 -0.05 0.58 0.42
## Question_18      18  0.37  0.61 -0.14 -0.13 0.54 0.46
## Question_19      19 -0.28 -0.15  0.06  0.37 0.24 0.76
## Question_20      20  0.47  0.04 -0.09 -0.20 0.27 0.73
## Question_21      21  0.60  0.26 -0.15 -0.15 0.47 0.53
## Question_22      22 -0.03 -0.16  0.07  0.47 0.25 0.75
## Question_23      23  0.03 -0.04  0.07  0.33 0.12 0.88
##
##                  PA1  PA3  PA4  PA2
## Proportion Var  0.13 0.12 0.09 0.06
## Cumulative Var  0.13 0.26 0.34 0.40
## Cum. factor Var 0.33 0.63 0.85 1.00
##
## Mean item complexity =  1.8
## Test of the hypothesis that 4 factors are sufficient.
##
## The degrees of freedom for the null model are  253  and the objective function was  7.55 with Chi Square of  19334.49
## The degrees of freedom for the model are 167  and the objective function was  0.46
##
## The root mean square of the residuals (RMSR) is  0.03
## The df corrected root mean square of the residuals is  0.03
##
## The harmonic number of observations is  2571 with the empirical chi square  880.48  with prob <  2.3e-97
## The total number of observations was  2571  with Likelihood Chi Square =  1166.49  with prob <  2.1e-149
##
## Tucker Lewis Index of factoring reliability =  0.921
## RMSEA index =  0.048  and the 90 % confidence intervals are  0.046 0.051
## BIC =  -144.8
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
##                                                    PA1  PA3  PA4  PA2
## Correlation of (regression) scores with factors   0.83 0.86 0.86 0.77
## Multiple R square of scores with factors          0.69 0.73 0.74 0.59
## Minimum correlation of possible factor scores     0.37 0.46 0.49 0.19

### 8.2.7 Remarks

The method used was “Principal Axis Factoring” and the rotation used was “Varimax”. All differences in results between the software are due to rounding.

### 8.2.8 References

Field, A. (2018). Discovering statistics using IBM SPSS statistics. Los Angeles, CA: SAGE.