EFA_Project1
Dani Grant
2/26/2021
1. Is data suitable for factor analysis?
a. is sample large enough? - Yes, N = 2571
b. variable type: 1-5 Likert scale (interval, so ok)
c. 23-items (more than 1 per 20 participants)
d. is there any missing data? - No
## missing
## 0
## 2571e. check for outliers and drop outliers
## [1] 49.72823## [1] 23## Mode FALSE TRUE
## logical 97 2474f. describe items
## vars n mean sd median trimmed mad min max range skew kurtosis se
## Q01 1 2474 3.64 0.80 4 3.67 0.00 1 5 4 -0.63 0.63 0.02
## Q02 2 2474 4.37 0.84 5 4.53 0.00 1 5 4 -1.41 1.74 0.02
## Q03 3 2474 2.61 1.06 3 2.60 1.48 1 5 4 0.06 -0.75 0.02
## Q04 4 2474 3.24 0.92 3 3.28 1.48 1 5 4 -0.37 -0.25 0.02
## Q05 5 2474 3.29 0.94 3 3.34 1.48 1 5 4 -0.46 -0.42 0.02
## Q06 6 2474 3.79 1.09 4 3.93 1.48 1 5 4 -0.93 0.23 0.02
## Q07 7 2474 3.11 1.07 3 3.13 1.48 1 5 4 -0.20 -0.81 0.02
## Q08 8 2474 3.77 0.85 4 3.85 0.00 1 5 4 -1.05 1.58 0.02
## Q09 9 2474 3.15 1.25 3 3.16 1.48 1 5 4 0.07 -1.13 0.03
## Q10 10 2474 3.73 0.85 4 3.80 0.00 1 5 4 -0.82 0.61 0.02
## Q11 11 2474 3.76 0.85 4 3.83 0.00 1 5 4 -0.81 0.94 0.02
## Q12 12 2474 2.87 0.88 3 2.90 1.48 1 5 4 -0.19 -0.13 0.02
## Q13 13 2474 3.57 0.92 4 3.61 1.48 1 5 4 -0.60 0.06 0.02
## Q14 14 2474 3.16 0.97 3 3.19 1.48 1 5 4 -0.28 -0.31 0.02
## Q15 15 2474 3.26 0.98 3 3.30 1.48 1 5 4 -0.43 -0.39 0.02
## Q16 16 2474 3.15 0.89 3 3.20 1.48 1 5 4 -0.39 -0.07 0.02
## Q17 17 2474 3.56 0.85 4 3.62 0.00 1 5 4 -0.75 0.56 0.02
## Q18 18 2474 3.46 1.02 4 3.50 1.48 1 5 4 -0.47 -0.18 0.02
## Q19 19 2474 3.71 1.09 4 3.78 1.48 1 5 4 -0.46 -0.73 0.02
## Q20 20 2474 2.40 1.02 2 2.35 1.48 1 5 4 0.36 -0.65 0.02
## Q21 21 2474 2.86 0.96 3 2.90 1.48 1 5 4 -0.13 -0.75 0.02
## Q22 22 2474 3.11 1.01 3 3.07 1.48 1 5 4 0.08 -0.66 0.02
## Q23 23 2474 2.57 1.02 2 2.52 1.48 1 5 4 0.59 -0.12 0.02i. histograms of item 2
ii. histogram of item 8
2. Testing Assumptions
a. correlation matrix
symnum(corr2)## Q01 Q02 Q03 Q04 Q05 Q06 Q07 Q08 Q09 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19
## Q01 1
## Q02 1
## Q03 . . 1
## Q04 . . 1
## Q05 . . . 1
## Q06 1
## Q07 . . . . . 1
## Q08 . . . 1
## Q09 . . 1
## Q10 . 1
## Q11 . . . . . . , 1
## Q12 . . . . . . . 1
## Q13 . . . . . . . . . . 1
## Q14 . . . . . . . . . 1
## Q15 . . . . . . . . . . 1
## Q16 . . . . . . . . . . . 1
## Q17 . . . . . , , . . . . . 1
## Q18 . . . . . . . . . . . . . . 1
## Q19 . 1
## Q20 . .
## Q21 . . . . . . . . . . . . . .
## Q22
## Q23
## Q20 Q21 Q22 Q23
## Q01
## Q02
## Q03
## Q04
## Q05
## Q06
## Q07
## Q08
## Q09
## Q10
## Q11
## Q12
## Q13
## Q14
## Q15
## Q16
## Q17
## Q18
## Q19
## Q20 1
## Q21 . 1
## Q22 1
## Q23 1
## attr(,"legend")
## [1] 0 ' ' 0.3 '.' 0.6 ',' 0.8 '+' 0.9 '*' 0.95 'B' 1b. distribution of the correlation matrix
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 253 0.17 0.26 0.26 0.18 0.23 -0.36 0.65 1 -0.41 -1.33 0.02
c. Linearity
d. Normality
e. Homogeneity & Homoscidasticity
f. Additivity - Bartlett’s test of Sphericity
## $chisq
## [1] 19534.27
##
## $p.value
## [1] 0
##
## $df
## [1] 253g. Sample size - KMO test for sampling adequacy
## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = corr2)
## Overall MSA = 0.93
## MSA for each item =
## Q01 Q02 Q03 Q04 Q05 Q06 Q07 Q08 Q09 Q10 Q11 Q12 Q13 Q14 Q15 Q16
## 0.94 0.88 0.95 0.96 0.96 0.90 0.94 0.88 0.85 0.96 0.91 0.96 0.95 0.97 0.95 0.94
## Q17 Q18 Q19 Q20 Q21 Q22 Q23
## 0.93 0.95 0.95 0.89 0.93 0.89 0.79h. matrix determinants
## [1] 0.00036113395. How many factors to be retained?
a. old Kaiser (eigenvalues > 1) vs. new Kaiser criterion (eigenvalues > .7)
## Parallel analysis suggests that the number of factors = 6 and the number of components = NA## [1] 1## [1] 2b. Scree Plot and Parallel Analysis
6. Factor rotation? Orthogonal or Oblique?
a. without rotation
i. minres
M1 <-fa(d2, nfactors = 4, fm = "minres", SMC = TRUE, rotate="none")
print.psych(M1, cut=.3)## Factor Analysis using method = minres
## Call: fa(r = d2, nfactors = 4, rotate = "none", SMC = TRUE, fm = "minres")
## Standardized loadings (pattern matrix) based upon correlation matrix
## MR1 MR2 MR3 MR4 h2 u2 com
## Q01 0.57 0.39 0.61 1.5
## Q02 0.40 0.30 0.70 2.6
## Q03 0.61 0.48 0.52 1.6
## Q04 0.61 0.42 0.58 1.3
## Q05 0.53 0.32 0.68 1.2
## Q06 0.56 0.51 0.61 0.39 2.2
## Q07 0.67 0.51 0.49 1.3
## Q08 0.56 0.48 0.67 0.33 2.7
## Q09 0.47 0.36 0.64 2.3
## Q10 0.41 0.20 0.80 1.4
## Q11 0.66 0.32 0.65 0.35 2.0
## Q12 0.65 0.46 0.54 1.2
## Q13 0.66 0.49 0.51 1.2
## Q14 0.63 0.42 0.58 1.1
## Q15 0.57 0.33 0.67 1.1
## Q16 0.65 0.46 0.54 1.1
## Q17 0.64 0.36 0.59 0.41 1.9
## Q18 0.70 0.57 0.43 1.3
## Q19 -0.42 0.26 0.74 1.8
## Q20 0.42 0.28 0.72 2.2
## Q21 0.64 0.48 0.52 1.4
## Q22 -0.30 0.26 0.74 3.1
## Q23 0.11 0.89 3.1
##
## MR1 MR2 MR3 MR4
## SS loadings 6.98 1.15 0.83 0.64
## Proportion Var 0.30 0.05 0.04 0.03
## Cumulative Var 0.30 0.35 0.39 0.42
## Proportion Explained 0.73 0.12 0.09 0.07
## Cumulative Proportion 0.73 0.85 0.93 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 7.93 with Chi Square of 19534.27
## The degrees of freedom for the model are 167 and the objective function was 0.43
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.03
##
## The harmonic number of observations is 2474 with the empirical chi square 779.08 with prob < 1e-79
## The total number of observations was 2474 with Likelihood Chi Square = 1065.21 with prob < 1.1e-130
##
## Tucker Lewis Index of factoring reliability = 0.929
## RMSEA index = 0.047 and the 90 % confidence intervals are 0.044 0.049
## BIC = -239.66
## 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 0.83 0.80 0.74
## Multiple R square of scores with factors 0.93 0.69 0.64 0.54
## Minimum correlation of possible factor scores 0.86 0.38 0.29 0.08ii. maximum likelihood
M2 <- fa(d2, nfactors = 4, fm = "ml", SMC = TRUE, rotate = "none")
print.psych(M2, cut=.3)## Factor Analysis using method = ml
## Call: fa(r = d2, nfactors = 4, rotate = "none", SMC = TRUE, fm = "ml")
## Standardized loadings (pattern matrix) based upon correlation matrix
## ML1 ML2 ML3 ML4 h2 u2 com
## Q01 0.57 0.405 0.59 1.5
## Q02 0.32 0.311 0.69 3.8
## Q03 0.59 -0.31 0.477 0.52 1.7
## Q04 0.60 0.426 0.57 1.3
## Q05 0.52 0.323 0.68 1.4
## Q06 0.58 0.48 0.637 0.36 2.4
## Q07 0.67 0.514 0.49 1.3
## Q08 0.61 0.55 0.671 0.33 2.0
## Q09 0.32 0.34 0.365 0.64 3.8
## Q10 0.40 0.193 0.81 1.4
## Q11 0.69 0.38 0.663 0.34 1.7
## Q12 0.63 0.459 0.54 1.3
## Q13 0.67 0.488 0.51 1.2
## Q14 0.62 0.424 0.58 1.2
## Q15 0.57 0.330 0.67 1.0
## Q16 0.64 0.464 0.54 1.3
## Q17 0.67 0.37 0.582 0.42 1.6
## Q18 0.70 0.573 0.43 1.4
## Q19 -0.40 0.253 0.75 2.2
## Q20 0.40 -0.31 0.267 0.73 2.0
## Q21 0.62 0.458 0.54 1.4
## Q22 0.32 0.232 0.77 2.9
## Q23 0.095 0.90 2.3
##
## ML1 ML2 ML3 ML4
## SS loadings 6.93 1.09 0.90 0.69
## Proportion Var 0.30 0.05 0.04 0.03
## Cumulative Var 0.30 0.35 0.39 0.42
## Proportion Explained 0.72 0.11 0.09 0.07
## Cumulative Proportion 0.72 0.84 0.93 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 7.93 with Chi Square of 19534.27
## The degrees of freedom for the model are 167 and the objective function was 0.43
##
## 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 2474 with the empirical chi square 794.58 with prob < 2.2e-82
## The total number of observations was 2474 with Likelihood Chi Square = 1053.65 with prob < 1.5e-128
##
## Tucker Lewis Index of factoring reliability = 0.93
## RMSEA index = 0.046 and the 90 % confidence intervals are 0.044 0.049
## BIC = -251.22
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## ML1 ML2 ML3 ML4
## Correlation of (regression) scores with factors 0.97 0.85 0.80 0.73
## Multiple R square of scores with factors 0.93 0.72 0.63 0.53
## Minimum correlation of possible factor scores 0.87 0.44 0.27 0.06iii. principal axis
M3 <- fa(d2, nfactors = 4, fm = "pa", SMC = TRUE, rotate = "none")
print.psych(M3, cut=.3)## Factor Analysis using method = pa
## Call: fa(r = d2, nfactors = 4, rotate = "none", SMC = TRUE, fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA2 PA3 PA4 h2 u2 com
## Q01 0.57 0.39 0.61 1.5
## Q02 0.40 0.30 0.70 2.6
## Q03 0.61 0.48 0.52 1.6
## Q04 0.61 0.42 0.58 1.3
## Q05 0.53 0.32 0.68 1.2
## Q06 0.56 0.51 0.61 0.39 2.2
## Q07 0.67 0.51 0.49 1.3
## Q08 0.56 0.48 0.67 0.33 2.7
## Q09 0.47 0.36 0.64 2.2
## Q10 0.41 0.20 0.80 1.4
## Q11 0.66 0.32 0.65 0.35 2.0
## Q12 0.65 0.46 0.54 1.2
## Q13 0.66 0.49 0.51 1.2
## Q14 0.63 0.42 0.58 1.1
## Q15 0.57 0.33 0.67 1.1
## Q16 0.65 0.46 0.54 1.1
## Q17 0.64 0.36 0.59 0.41 1.9
## Q18 0.70 0.57 0.43 1.3
## Q19 -0.42 0.26 0.74 1.8
## Q20 0.42 0.28 0.72 2.2
## Q21 0.64 0.48 0.52 1.4
## Q22 -0.30 0.26 0.74 3.1
## Q23 0.11 0.89 3.1
##
## PA1 PA2 PA3 PA4
## SS loadings 6.98 1.15 0.83 0.64
## Proportion Var 0.30 0.05 0.04 0.03
## Cumulative Var 0.30 0.35 0.39 0.42
## Proportion Explained 0.73 0.12 0.09 0.07
## Cumulative Proportion 0.73 0.85 0.93 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 7.93 with Chi Square of 19534.27
## The degrees of freedom for the model are 167 and the objective function was 0.43
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.03
##
## The harmonic number of observations is 2474 with the empirical chi square 779.09 with prob < 1e-79
## The total number of observations was 2474 with Likelihood Chi Square = 1065.48 with prob < 1e-130
##
## Tucker Lewis Index of factoring reliability = 0.929
## RMSEA index = 0.047 and the 90 % confidence intervals are 0.044 0.049
## BIC = -239.39
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## PA1 PA2 PA3 PA4
## Correlation of (regression) scores with factors 0.97 0.83 0.80 0.74
## Multiple R square of scores with factors 0.93 0.69 0.64 0.54
## Minimum correlation of possible factor scores 0.86 0.37 0.28 0.08b. with rotations
i. with orthogonal (varimax) rotation
M4 <- fa(d2, nfactors = 4, fm = "pa", SMC=TRUE, rotate = "varimax")
print.psych(M4, cut=.3)## Factor Analysis using method = pa
## Call: fa(r = d2, nfactors = 4, rotate = "varimax", SMC = TRUE, fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA3 PA4 PA2 h2 u2 com
## Q01 0.52 0.39 0.61 1.9
## Q02 -0.51 0.30 0.70 1.3
## Q03 0.50 0.42 0.48 0.52 2.5
## Q04 0.54 0.42 0.58 1.9
## Q05 0.46 0.32 0.68 2.0
## Q06 0.76 0.61 0.39 1.1
## Q07 0.36 0.58 0.51 0.49 2.0
## Q08 0.76 0.67 0.33 1.3
## Q09 -0.58 0.36 0.64 1.2
## Q10 0.38 0.20 0.80 1.7
## Q11 0.69 0.65 0.35 1.7
## Q12 0.50 0.40 0.46 0.54 2.3
## Q13 0.57 0.49 0.51 2.1
## Q14 0.39 0.48 0.42 0.58 2.4
## Q15 0.39 0.33 0.67 3.3
## Q16 0.54 0.46 0.54 2.2
## Q17 0.30 0.65 0.59 0.41 1.8
## Q18 0.37 0.63 0.57 0.43 1.9
## Q19 -0.38 0.26 0.74 2.3
## Q20 0.48 0.28 0.72 1.5
## Q21 0.61 0.48 0.52 1.7
## Q22 -0.47 0.26 0.74 1.3
## Q23 -0.31 0.11 0.89 1.2
##
## PA1 PA3 PA4 PA2
## SS loadings 3.09 2.94 2.02 1.55
## Proportion Var 0.13 0.13 0.09 0.07
## Cumulative Var 0.13 0.26 0.35 0.42
## Proportion Explained 0.32 0.31 0.21 0.16
## Cumulative Proportion 0.32 0.63 0.84 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 7.93 with Chi Square of 19534.27
## The degrees of freedom for the model are 167 and the objective function was 0.43
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.03
##
## The harmonic number of observations is 2474 with the empirical chi square 779.09 with prob < 1e-79
## The total number of observations was 2474 with Likelihood Chi Square = 1065.48 with prob < 1e-130
##
## Tucker Lewis Index of factoring reliability = 0.929
## RMSEA index = 0.047 and the 90 % confidence intervals are 0.044 0.049
## BIC = -239.39
## 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 0.87 0.78
## Multiple R square of scores with factors 0.69 0.74 0.75 0.61
## Minimum correlation of possible factor scores 0.39 0.49 0.51 0.22ii. with oblique (promax) rotation
M5 <- fa(d2, nfactors = 4, fm = "pa", SMC=TRUE, rotate = "promax")
print.psych(M5, cut=.3)## Factor Analysis using method = pa
## Call: fa(r = d2, nfactors = 4, rotate = "promax", SMC = TRUE, fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA3 PA4 PA2 h2 u2 com
## Q01 0.58 0.39 0.61 1.3
## Q02 0.50 0.30 0.70 1.4
## Q03 0.51 -0.31 0.48 0.52 1.7
## Q04 0.59 0.42 0.58 1.2
## Q05 0.49 0.32 0.68 1.2
## Q06 0.98 0.61 0.39 1.2
## Q07 0.58 0.51 0.49 1.3
## Q08 0.84 0.67 0.33 1.1
## Q09 0.59 0.36 0.64 1.1
## Q10 0.41 0.20 0.80 1.0
## Q11 0.72 0.65 0.35 1.1
## Q12 0.50 0.46 0.54 1.7
## Q13 0.57 0.49 0.51 1.1
## Q14 0.44 0.42 0.58 1.7
## Q15 0.31 0.33 0.67 2.1
## Q16 0.56 0.46 0.54 1.1
## Q17 0.66 0.59 0.41 1.1
## Q18 0.65 0.57 0.43 1.2
## Q19 0.33 0.26 0.74 1.9
## Q20 0.60 0.28 0.72 1.3
## Q21 0.69 0.48 0.52 1.0
## Q22 0.49 0.26 0.74 1.3
## Q23 0.35 0.11 0.89 1.4
##
## PA1 PA3 PA4 PA2
## SS loadings 3.47 2.93 1.89 1.32
## Proportion Var 0.15 0.13 0.08 0.06
## Cumulative Var 0.15 0.28 0.36 0.42
## Proportion Explained 0.36 0.30 0.20 0.14
## Cumulative Proportion 0.36 0.67 0.86 1.00
##
## With factor correlations of
## PA1 PA3 PA4 PA2
## PA1 1.00 0.69 0.57 -0.46
## PA3 0.69 1.00 0.58 -0.41
## PA4 0.57 0.58 1.00 -0.23
## PA2 -0.46 -0.41 -0.23 1.00
##
## Mean item complexity = 1.3
## Test of the hypothesis that 4 factors are sufficient.
##
## The degrees of freedom for the null model are 253 and the objective function was 7.93 with Chi Square of 19534.27
## The degrees of freedom for the model are 167 and the objective function was 0.43
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.03
##
## The harmonic number of observations is 2474 with the empirical chi square 779.09 with prob < 1e-79
## The total number of observations was 2474 with Likelihood Chi Square = 1065.48 with prob < 1e-130
##
## Tucker Lewis Index of factoring reliability = 0.929
## RMSEA index = 0.047 and the 90 % confidence intervals are 0.044 0.049
## BIC = -239.39
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## PA1 PA3 PA4 PA2
## Correlation of (regression) scores with factors 0.93 0.93 0.92 0.83
## Multiple R square of scores with factors 0.87 0.87 0.84 0.68
## Minimum correlation of possible factor scores 0.73 0.75 0.69 0.37- Drop item 3 for promax rotation (load onto multiple factors)
d3 <- d2[,c(1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23)]
M6 <- fa(d3, nfactors = 4, fm = "pa", SMC = TRUE, rotate = "promax")
print.psych(M6, cut=.3)## Factor Analysis using method = pa
## Call: fa(r = d3, nfactors = 4, rotate = "promax", SMC = TRUE, fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA3 PA1 PA4 PA2 h2 u2 com
## Q01 0.57 0.39 0.61 1.3
## Q02 0.49 0.28 0.72 1.4
## Q04 0.57 0.42 0.58 1.2
## Q05 0.49 0.32 0.68 1.1
## Q06 0.99 -0.31 0.61 0.39 1.2
## Q07 0.60 0.51 0.49 1.2
## Q08 0.85 0.67 0.33 1.1
## Q09 0.59 0.36 0.64 1.1
## Q10 0.41 0.20 0.80 1.0
## Q11 0.72 0.65 0.35 1.1
## Q12 0.49 0.46 0.54 1.8
## Q13 0.57 0.49 0.51 1.1
## Q14 0.45 0.42 0.58 1.7
## Q15 0.31 0.34 0.66 2.2
## Q16 0.56 0.46 0.54 1.1
## Q17 0.66 0.59 0.41 1.1
## Q18 0.67 0.57 0.43 1.2
## Q19 0.33 0.25 0.75 1.9
## Q20 0.61 0.29 0.71 1.4
## Q21 0.69 0.48 0.52 1.0
## Q22 0.50 0.27 0.73 1.2
## Q23 0.34 0.10 0.90 1.4
##
## PA3 PA1 PA4 PA2
## SS loadings 2.95 3.10 1.90 1.18
## Proportion Var 0.13 0.14 0.09 0.05
## Cumulative Var 0.13 0.28 0.36 0.41
## Proportion Explained 0.32 0.34 0.21 0.13
## Cumulative Proportion 0.32 0.66 0.87 1.00
##
## With factor correlations of
## PA3 PA1 PA4 PA2
## PA3 1.00 0.70 0.58 -0.42
## PA1 0.70 1.00 0.57 -0.44
## PA4 0.58 0.57 1.00 -0.23
## PA2 -0.42 -0.44 -0.23 1.00
##
## Mean item complexity = 1.3
## Test of the hypothesis that 4 factors are sufficient.
##
## The degrees of freedom for the null model are 231 and the objective function was 7.4 with Chi Square of 18233.31
## The degrees of freedom for the model are 149 and the objective function was 0.41
##
## 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 2474 with the empirical chi square 743.33 with prob < 1e-79
## The total number of observations was 2474 with Likelihood Chi Square = 1012.53 with prob < 2.4e-128
##
## Tucker Lewis Index of factoring reliability = 0.926
## RMSEA index = 0.048 and the 90 % confidence intervals are 0.046 0.051
## BIC = -151.7
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## PA3 PA1 PA4 PA2
## Correlation of (regression) scores with factors 0.94 0.92 0.92 0.81
## Multiple R square of scores with factors 0.87 0.85 0.84 0.66
## Minimum correlation of possible factor scores 0.75 0.71 0.69 0.31- Drop item 6 for promax rotation (loads onto multiple factors)
d4 <- d2[,c(1, 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23)]
M7 <- fa(d4, nfactors = 4, fm = "pa", SMC = TRUE, rotate = "promax")
print.psych(M7, cut=.3)## Factor Analysis using method = pa
## Call: fa(r = d4, nfactors = 4, rotate = "promax", SMC = TRUE, fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA3 PA2 PA4 h2 u2 com
## Q01 0.44 0.36 0.64 1.7
## Q02 0.46 0.28 0.72 1.4
## Q04 0.47 0.40 0.60 1.6
## Q05 0.45 0.30 0.70 1.3
## Q07 0.70 0.46 0.54 1.0
## Q08 0.87 0.68 0.32 1.0
## Q09 0.58 0.36 0.64 1.1
## Q10 0.52 0.20 0.80 1.2
## Q11 0.73 0.64 0.36 1.1
## Q12 0.67 0.47 0.53 1.1
## Q13 0.71 0.48 0.52 1.1
## Q14 0.69 0.43 0.57 1.0
## Q15 0.44 0.33 0.67 1.4
## Q16 0.49 0.44 0.56 1.2
## Q17 0.66 0.59 0.41 1.1
## Q18 0.85 0.55 0.45 1.0
## Q19 0.31 0.25 0.75 2.1
## Q20 0.69 0.43 0.57 1.1
## Q21 0.57 0.57 0.43 1.5
## Q22 0.51 0.28 0.72 1.1
## Q23 0.34 0.11 0.89 1.2
##
## PA1 PA3 PA2 PA4
## SS loadings 4.31 1.92 1.15 1.21
## Proportion Var 0.21 0.09 0.05 0.06
## Cumulative Var 0.21 0.30 0.35 0.41
## Proportion Explained 0.50 0.22 0.13 0.14
## Cumulative Proportion 0.50 0.73 0.86 1.00
##
## With factor correlations of
## PA1 PA3 PA2 PA4
## PA1 1.00 0.64 -0.43 0.63
## PA3 0.64 1.00 -0.21 0.42
## PA2 -0.43 -0.21 1.00 -0.32
## PA4 0.63 0.42 -0.32 1.00
##
## Mean item complexity = 1.3
## Test of the hypothesis that 4 factors are sufficient.
##
## The degrees of freedom for the null model are 210 and the objective function was 6.81 with Chi Square of 16788.88
## The degrees of freedom for the model are 132 and the objective function was 0.36
##
## 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 2474 with the empirical chi square 652.57 with prob < 7.2e-70
## The total number of observations was 2474 with Likelihood Chi Square = 875.54 with prob < 5.2e-110
##
## Tucker Lewis Index of factoring reliability = 0.929
## RMSEA index = 0.048 and the 90 % confidence intervals are 0.045 0.051
## BIC = -155.86
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## PA1 PA3 PA2 PA4
## Correlation of (regression) scores with factors 0.95 0.92 0.80 0.85
## Multiple R square of scores with factors 0.90 0.85 0.64 0.72
## Minimum correlation of possible factor scores 0.79 0.70 0.29 0.44iii. with oblique (oblimin) rotation
M8 <- fa(d2, nfactors = 4, fm = "pa", rotate = "oblimin")
print.psych(M8, cut=.3)## Factor Analysis using method = pa
## Call: fa(r = d2, nfactors = 4, rotate = "oblimin", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA3 PA4 PA2 h2 u2 com
## Q01 0.52 0.39 0.61 1.3
## Q02 0.53 0.30 0.70 1.1
## Q03 0.40 -0.38 0.48 0.52 2.1
## Q04 0.53 0.42 0.58 1.2
## Q05 0.45 0.32 0.68 1.2
## Q06 0.82 0.61 0.39 1.0
## Q07 0.50 0.51 0.49 1.6
## Q08 0.86 0.67 0.33 1.0
## Q09 0.61 0.36 0.64 1.0
## Q10 0.35 0.20 0.80 1.2
## Q11 0.76 0.65 0.35 1.1
## Q12 0.48 0.46 0.54 1.6
## Q13 0.49 0.49 0.51 1.4
## Q14 0.32 0.38 0.42 0.58 2.0
## Q15 0.33 0.67 2.9
## Q16 0.49 0.46 0.54 1.3
## Q17 0.69 0.59 0.41 1.0
## Q18 0.56 0.57 0.43 1.5
## Q19 0.37 0.26 0.74 1.5
## Q20 0.48 0.28 0.72 1.6
## Q21 0.61 0.48 0.52 1.1
## Q22 0.48 0.26 0.74 1.4
## Q23 0.34 0.11 0.89 1.6
##
## PA1 PA3 PA4 PA2
## SS loadings 3.10 2.53 2.37 1.61
## Proportion Var 0.13 0.11 0.10 0.07
## Cumulative Var 0.13 0.24 0.35 0.42
## Proportion Explained 0.32 0.26 0.25 0.17
## Cumulative Proportion 0.32 0.59 0.83 1.00
##
## With factor correlations of
## PA1 PA3 PA4 PA2
## PA1 1.00 0.50 0.55 -0.40
## PA3 0.50 1.00 0.47 -0.33
## PA4 0.55 0.47 1.00 -0.22
## PA2 -0.40 -0.33 -0.22 1.00
##
## Mean item complexity = 1.4
## Test of the hypothesis that 4 factors are sufficient.
##
## The degrees of freedom for the null model are 253 and the objective function was 7.93 with Chi Square of 19534.27
## The degrees of freedom for the model are 167 and the objective function was 0.43
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.03
##
## The harmonic number of observations is 2474 with the empirical chi square 779.09 with prob < 1e-79
## The total number of observations was 2474 with Likelihood Chi Square = 1065.48 with prob < 1e-130
##
## Tucker Lewis Index of factoring reliability = 0.929
## RMSEA index = 0.047 and the 90 % confidence intervals are 0.044 0.049
## BIC = -239.39
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## PA1 PA3 PA4 PA2
## Correlation of (regression) scores with factors 0.91 0.91 0.92 0.83
## Multiple R square of scores with factors 0.83 0.82 0.85 0.70
## Minimum correlation of possible factor scores 0.66 0.64 0.71 0.39- Drop items 3, 14 (loads onto multiple factors) and 15 (no loadings) for oblimin rotation
# after Q03 and Q14 are dropped
d5 <- d2[,c(1, 2, 4, 5,6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23)]
M9 <- fa(d5, nfactors = 4, fm = "pa", SMC = TRUE, rotate = "oblimin")
print.psych(M9, cut=.3)## Factor Analysis using method = pa
## Call: fa(r = d5, nfactors = 4, rotate = "oblimin", SMC = TRUE, fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA4 PA3 PA2 h2 u2 com
## Q01 0.53 0.40 0.60 1.3
## Q02 0.51 0.28 0.72 1.1
## Q04 0.55 0.42 0.58 1.1
## Q05 0.48 0.32 0.68 1.1
## Q06 0.81 0.61 0.39 1.0
## Q07 0.49 0.51 0.49 1.6
## Q08 0.86 0.67 0.33 1.0
## Q09 0.59 0.35 0.65 1.0
## Q10 0.33 0.19 0.81 1.3
## Q11 0.75 0.65 0.35 1.1
## Q12 0.50 0.46 0.54 1.5
## Q13 0.48 0.49 0.51 1.5
## Q16 0.52 0.45 0.55 1.2
## Q17 0.68 0.58 0.42 1.1
## Q18 0.55 0.58 0.42 1.5
## Q19 0.36 0.25 0.75 1.6
## Q20 0.51 0.29 0.71 1.5
## Q21 0.64 0.49 0.51 1.0
## Q22 0.50 0.28 0.72 1.3
## Q23 0.33 0.11 0.89 1.6
##
## PA1 PA4 PA3 PA2
## SS loadings 2.83 2.18 2.06 1.31
## Proportion Var 0.14 0.11 0.10 0.07
## Cumulative Var 0.14 0.25 0.35 0.42
## Proportion Explained 0.34 0.26 0.25 0.16
## Cumulative Proportion 0.34 0.60 0.84 1.00
##
## With factor correlations of
## PA1 PA4 PA3 PA2
## PA1 1.00 0.56 0.51 -0.37
## PA4 0.56 1.00 0.45 -0.18
## PA3 0.51 0.45 1.00 -0.31
## PA2 -0.37 -0.18 -0.31 1.00
##
## Mean item complexity = 1.3
## Test of the hypothesis that 4 factors are sufficient.
##
## The degrees of freedom for the null model are 190 and the objective function was 6.48 with Chi Square of 15970.51
## The degrees of freedom for the model are 116 and the objective function was 0.33
##
## 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 2474 with the empirical chi square 605.58 with prob < 2.6e-67
## The total number of observations was 2474 with Likelihood Chi Square = 812.47 with prob < 5.4e-105
##
## Tucker Lewis Index of factoring reliability = 0.928
## RMSEA index = 0.049 and the 90 % confidence intervals are 0.046 0.052
## BIC = -93.91
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## PA1 PA4 PA3 PA2
## Correlation of (regression) scores with factors 0.91 0.92 0.90 0.81
## Multiple R square of scores with factors 0.83 0.85 0.81 0.65
## Minimum correlation of possible factor scores 0.65 0.70 0.61 0.308. Results
a. factor loadings
fa.diagram(M8, main = "R Anxiety Questionnaire")
b. Are factors reliable?
c. Do factors make sense?
in sum i chose to go with oblimin because the factor groups make more sense than with promax.