## 5.11 Unrotated Factor Solution

According to the eigenvalue, scree plot and parallel analysis, we are to retain 4, 3 and 6 factors respectively. We will explore all these factors and then look out for best factor structure with good model data fit.

First, let’s explore the unrotated factor solution. Since the data is larger than 350, we can set a minimum factor loading of 0.3. In the code, we set rotate = "none" because we want results for unrotated solution; nfactors = 3 indicate the number of factor to extract, cut = 0.30 to get results with minimum loading of 0.30. Finally, we set sort = T so that we get results from the highest factor to the least.

# three factor solution
unrotated_three_factors <- fa(data.us.only, nfactors = 3, rotate = "none")
print(unrotated_three_factors, sort = T, cut = .30)
## Factor Analysis using method =  minres
## Call: fa(r = data.us.only, nfactors = 3, rotate = "none")
## Standardized loadings (pattern matrix) based upon correlation matrix
##          item  MR1   MR2   MR3   h2   u2 com
## COMPICT2    8 0.68             0.47 0.53 1.1
## AUICT4     15 0.67       -0.46 0.66 0.34 1.8
## COMPICT5   11 0.67             0.48 0.52 1.2
## COMPICT4   10 0.65             0.47 0.53 1.3
## AUICT3     14 0.63             0.48 0.52 1.4
## INTICT4     4 0.63             0.49 0.51 1.5
## AUICT5     16 0.61             0.47 0.53 1.5
## AUICT2     13 0.61       -0.31 0.53 0.47 1.9
## SOIAICT4   20 0.60 -0.55       0.72 0.28 2.3
## INTICT6     6 0.60  0.39       0.57 0.43 2.1
## COMPICT3    9 0.59  0.37       0.49 0.51 1.7
## AUICT1     12 0.57       -0.37 0.50 0.50 1.9
## SOIAICT2   18 0.57 -0.48       0.60 0.40 2.3
## SOIAICT5   21 0.56 -0.46       0.59 0.41 2.4
## COMPICT1    7 0.52             0.30 0.70 1.2
## SOIAICT3   19 0.52 -0.37       0.43 0.57 2.0
## SOIAICT1   17 0.50 -0.34       0.43 0.57 2.3
## INTICT3     3 0.50  0.42       0.50 0.50 2.6
## INTICT2     2 0.42  0.40       0.36 0.64 2.3
## INTICT5     5 0.38             0.23 0.77 2.1
## INTICT1     1 0.36             0.25 0.75 2.6
##
##                        MR1  MR2  MR3
## Proportion Var        0.33 0.09 0.06
## Cumulative Var        0.33 0.42 0.48
## Proportion Explained  0.68 0.19 0.13
## Cumulative Proportion 0.68 0.87 1.00
##
## Mean item complexity =  1.9
## Test of the hypothesis that 3 factors are sufficient.
##
## df null model =  210  with the objective function =  9.5 with Chi Square =  51537.58
## df of  the model are 150  and the objective function was  1.04
##
## 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 n.obs is  5433 with the empirical chi square  3868.82  with prob <  0
## The total n.obs was  5433  with Likelihood Chi Square =  5633.66  with prob <  0
##
## Tucker Lewis Index of factoring reliability =  0.85
## RMSEA index =  0.082  and the 90 % confidence intervals are  0.08 0.084
## BIC =  4343.62
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
##                                                    MR1  MR2  MR3
## Correlation of (regression) scores with factors   0.97 0.90 0.86
## Multiple R square of scores with factors          0.93 0.81 0.74
## Minimum correlation of possible factor scores     0.87 0.62 0.47

For the unrotated three factor solution, it is evident that all the items load on a single factor. This doesn’t align with our expectation.

Next, we can explore the unrotated four factors and six factor solution.

# four factor solution
unrotated_four_factors <- fa(data.us.only, nfactors = 4, rotate = "none")
print(unrotated_four_factors, sort = T, cut = .30)
## Factor Analysis using method =  minres
## Call: fa(r = data.us.only, nfactors = 4, rotate = "none")
## Standardized loadings (pattern matrix) based upon correlation matrix
##          item  MR1   MR2   MR3   MR4   h2   u2 com
## COMPICT2    8 0.69             -0.31 0.59 0.41 1.5
## COMPICT5   11 0.68             -0.39 0.67 0.33 1.9
## AUICT4     15 0.67       -0.45       0.68 0.32 1.9
## COMPICT4   10 0.66             -0.35 0.62 0.38 1.8
## AUICT3     14 0.64                   0.56 0.44 1.8
## INTICT4     4 0.62                   0.49 0.51 1.6
## AUICT5     16 0.62                   0.54 0.46 1.8
## AUICT2     13 0.61       -0.30       0.54 0.46 2.0
## INTICT6     6 0.60  0.38             0.57 0.43 2.1
## SOIAICT4   20 0.60 -0.55             0.72 0.28 2.3
## COMPICT3    9 0.59  0.37             0.50 0.50 1.8
## AUICT1     12 0.57       -0.36       0.51 0.49 2.1
## SOIAICT2   18 0.56 -0.48             0.59 0.41 2.3
## SOIAICT5   21 0.56 -0.46             0.59 0.41 2.4
## COMPICT1    7 0.52                   0.34 0.66 1.5
## SOIAICT3   19 0.51 -0.37             0.43 0.57 2.0
## SOIAICT1   17 0.50 -0.34             0.43 0.57 2.3
## INTICT3     3 0.50  0.42             0.54 0.46 2.9
## INTICT2     2 0.42  0.40             0.40 0.60 2.6
## INTICT5     5 0.38                   0.23 0.77 2.1
## INTICT1     1 0.36                   0.26 0.74 2.7
##
##                        MR1  MR2  MR3  MR4
## SS loadings           6.89 1.92 1.31 0.70
## Proportion Var        0.33 0.09 0.06 0.03
## Cumulative Var        0.33 0.42 0.48 0.51
## Proportion Explained  0.64 0.18 0.12 0.06
## Cumulative Proportion 0.64 0.81 0.94 1.00
##
## Mean item complexity =  2.1
## Test of the hypothesis that 4 factors are sufficient.
##
## df null model =  210  with the objective function =  9.5 with Chi Square =  51537.58
## df of  the model are 132  and the objective function was  0.52
##
## 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 n.obs is  5433 with the empirical chi square  1598.07  with prob <  5.9e-250
## The total n.obs was  5433  with Likelihood Chi Square =  2829.87  with prob <  0
##
## Tucker Lewis Index of factoring reliability =  0.916
## RMSEA index =  0.061  and the 90 % confidence intervals are  0.059 0.063
## BIC =  1694.64
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
##                                                    MR1  MR2  MR3  MR4
## Correlation of (regression) scores with factors   0.97 0.90 0.87 0.79
## Multiple R square of scores with factors          0.94 0.82 0.75 0.63
## Minimum correlation of possible factor scores     0.88 0.63 0.50 0.26
# six factor solution
unrotated_six_factors <- fa(data.us.only, nfactors = 6, rotate = "none")
print(unrotated_six_factors, sort = T, cut = .30)
## Factor Analysis using method =  minres
## Call: fa(r = data.us.only, nfactors = 6, rotate = "none")
## Standardized loadings (pattern matrix) based upon correlation matrix
##          item  MR1   MR2   MR3   MR4   MR5   MR6   h2   u2 com
## COMPICT2    8 0.68             -0.32             0.58 0.42 1.5
## COMPICT5   11 0.68             -0.40             0.67 0.33 1.9
## AUICT4     15 0.67       -0.43                   0.65 0.35 1.8
## COMPICT4   10 0.66             -0.36             0.63 0.37 1.9
## AUICT3     14 0.65              0.30             0.69 0.31 2.4
## INTICT4     4 0.62                               0.52 0.48 1.7
## AUICT5     16 0.62                               0.60 0.40 2.2
## AUICT2     13 0.61       -0.31                   0.64 0.36 2.6
## INTICT6     6 0.60  0.38                         0.57 0.43 2.1
## SOIAICT4   20 0.60 -0.54                         0.72 0.28 2.4
## COMPICT3    9 0.59  0.37                         0.54 0.46 2.0
## AUICT1     12 0.58       -0.38                   0.63 0.37 2.8
## SOIAICT2   18 0.56 -0.47                         0.59 0.41 2.3
## SOIAICT5   21 0.55 -0.45                         0.60 0.40 2.5
## COMPICT1    7 0.52                               0.35 0.65 1.6
## SOIAICT3   19 0.51 -0.36                         0.43 0.57 2.1
## SOIAICT1   17 0.50 -0.34                         0.46 0.54 2.6
## INTICT3     3 0.50  0.43                         0.56 0.44 3.1
## INTICT5     5 0.40                    0.38       0.46 0.54 3.8
## INTICT1     1 0.36                               0.27 0.73 3.1
## INTICT2     2 0.44  0.45                    0.30 0.59 0.41 4.0
##
##                        MR1  MR2  MR3  MR4  MR5  MR6
## SS loadings           6.92 1.96 1.35 0.74 0.42 0.37
## Proportion Var        0.33 0.09 0.06 0.04 0.02 0.02
## Cumulative Var        0.33 0.42 0.49 0.52 0.54 0.56
## Proportion Explained  0.59 0.17 0.12 0.06 0.04 0.03
## Cumulative Proportion 0.59 0.75 0.87 0.93 0.97 1.00
##
## Mean item complexity =  2.4
## Test of the hypothesis that 6 factors are sufficient.
##
## df null model =  210  with the objective function =  9.5 with Chi Square =  51537.58
## df of  the model are 99  and the objective function was  0.21
##
## The root mean square of the residuals (RMSR) is  0.01
## The df corrected root mean square of the residuals is  0.02
##
## The harmonic n.obs is  5433 with the empirical chi square  478.16  with prob <  4.9e-51
## The total n.obs was  5433  with Likelihood Chi Square =  1121.08  with prob <  9.3e-173
##
## Tucker Lewis Index of factoring reliability =  0.958
## RMSEA index =  0.044  and the 90 % confidence intervals are  0.041 0.046
## BIC =  269.65
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
##                                                    MR1  MR2  MR3  MR4   MR5   MR6
## Correlation of (regression) scores with factors   0.97 0.91 0.88 0.81  0.70  0.69
## Multiple R square of scores with factors          0.94 0.83 0.77 0.66  0.48  0.48
## Minimum correlation of possible factor scores     0.89 0.65 0.54 0.32 -0.03 -0.05

Results from the unrotated four and six factor solution also show that almost all the items load on one factor only.

As you can see, it is quite challenging to identify a clear pattern for the unrotated factor solution. To overcome this challenge, we need to rotate the factors to get a finer solution.