library("psych")
library("readxl")
library("REdaS")
## Loading required package: grid
places <- read.table("2nd Sem/MULTIVARIATE DATA ANALYSIS/Factor-Analysis/places.txt", quote="\"", comment.char="")
head(places)
##    V1   V2   V3   V4   V5   V6   V7   V8   V9 V10
## 1 521 6200  237  923 4031 2757  996 1405 7633   1
## 2 575 8138 1656  886 4883 2438 5564 2632 4350   2
## 3 468 7339  618  970 2531 2560  237  859 5250   3
## 4 476 7908 1431  610 6883 3399 4655 1617 5864   4
## 5 659 8393 1853 1483 6558 3026 4496 2612 5727   5
## 6 520 5819  640  727 2444 2972  334 1018 5254   6
data <- log10(places[,1:9])
head(data)
##         V1       V2       V3       V4       V5       V6       V7       V8
## 1 2.716838 3.792392 2.374748 2.965202 3.605413 3.440437 2.998259 3.147676
## 2 2.759668 3.910518 3.219060 2.947434 3.688687 3.387034 3.745387 3.420286
## 3 2.670246 3.865637 2.790988 2.986772 3.403292 3.408240 2.374748 2.933993
## 4 2.677607 3.898067 3.155640 2.785330 3.837778 3.531351 3.667920 3.208710
## 5 2.818885 3.923917 3.267875 3.171141 3.816771 3.480869 3.652826 3.416973
## 6 2.716003 3.764848 2.806180 2.861534 3.388101 3.473049 2.523746 3.007748
##         V9
## 1 3.882695
## 2 3.638489
## 3 3.720159
## 4 3.768194
## 5 3.757927
## 6 3.720490

Bartletts test of spherecity

bart_spher(places)
##  Bartlett's Test of Sphericity
## 
## Call: bart_spher(x = places)
## 
##      X2 = 1057.162
##      df = 45
## p-value < 2.22e-16
KMO(places) 
## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = places)
## Overall MSA =  0.7
## MSA for each item = 
##   V1   V2   V3   V4   V5   V6   V7   V8   V9  V10 
## 0.56 0.69 0.69 0.64 0.86 0.73 0.71 0.81 0.38 0.65
fa(r = data, nfactors = 9, rotate = "varimax")
## Factor Analysis using method =  minres
## Call: fa(r = data, nfactors = 9, rotate = "varimax")
## Standardized loadings (pattern matrix) based upon correlation matrix
##     MR1  MR7   MR2  MR5   MR4   MR3   MR8   MR6 MR9   h2   u2 com
## V1 0.06 0.06 -0.08 0.04  0.13  0.74  0.02 -0.01   0 0.58 0.42 1.1
## V2 0.54 0.32  0.36 0.05 -0.15  0.35  0.41  0.04   0 0.84 0.16 4.5
## V3 0.21 0.77 -0.01 0.40  0.08  0.08  0.09 -0.01   0 0.81 0.19 1.8
## V4 0.20 0.09  0.18 0.01  0.77  0.17 -0.02  0.01   0 0.70 0.30 1.4
## V5 0.52 0.18 -0.08 0.38  0.24 -0.10  0.12  0.24   0 0.60 0.40 3.5
## V6 0.06 0.24  0.10 0.66 -0.01  0.06  0.00  0.00   0 0.51 0.49 1.4
## V7 0.57 0.58  0.05 0.25  0.20  0.10 -0.06  0.10   0 0.78 0.22 2.8
## V8 0.69 0.11  0.12 0.01  0.15  0.08  0.01 -0.05   0 0.53 0.47 1.3
## V9 0.09 0.00  0.80 0.08  0.16 -0.09  0.03 -0.01   0 0.69 0.31 1.2
## 
##                        MR1  MR7  MR2  MR5  MR4  MR3  MR8  MR6  MR9
## SS loadings           1.46 1.15 0.84 0.81 0.78 0.74 0.20 0.07 0.00
## Proportion Var        0.16 0.13 0.09 0.09 0.09 0.08 0.02 0.01 0.00
## Cumulative Var        0.16 0.29 0.38 0.47 0.56 0.64 0.66 0.67 0.67
## Proportion Explained  0.24 0.19 0.14 0.13 0.13 0.12 0.03 0.01 0.00
## Cumulative Proportion 0.24 0.43 0.57 0.70 0.83 0.95 0.99 1.00 1.00
## 
## Mean item complexity =  2.1
## Test of the hypothesis that 9 factors are sufficient.
## 
## The degrees of freedom for the null model are  36  and the objective function was  2.59 with Chi Square of  839.43
## The degrees of freedom for the model are -9  and the objective function was  0 
## 
## The root mean square of the residuals (RMSR) is  0 
## The df corrected root mean square of the residuals is  NA 
## 
## The harmonic number of observations is  329 with the empirical chi square  0  with prob <  NA 
## The total number of observations was  329  with Likelihood Chi Square =  0  with prob <  NA 
## 
## Tucker Lewis Index of factoring reliability =  1.046
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy             
##                                                    MR1  MR7  MR2  MR5  MR4  MR3
## Correlation of (regression) scores with factors   0.82 0.82 0.84 0.72 0.82 0.79
## Multiple R square of scores with factors          0.68 0.67 0.70 0.52 0.68 0.62
## Minimum correlation of possible factor scores     0.36 0.34 0.40 0.04 0.35 0.24
##                                                     MR8   MR6 MR9
## Correlation of (regression) scores with factors    0.58  0.33   0
## Multiple R square of scores with factors           0.34  0.11   0
## Minimum correlation of possible factor scores     -0.33 -0.78  -1
pa <- fa(r = data, nfactors = 3, rotate = "varimax", fm = "pa", residuals = TRUE)
## maximum iteration exceeded
## Warning in fa.stats(r = r, f = f, phi = phi, n.obs = n.obs, np.obs = np.obs, :
## The estimated weights for the factor scores are probably incorrect. Try a
## different factor score estimation method.
## Warning in fac(r = r, nfactors = nfactors, n.obs = n.obs, rotate = rotate, : An
## ultra-Heywood case was detected. Examine the results carefully
pa
## Factor Analysis using method =  pa
## Call: fa(r = data, nfactors = 3, rotate = "varimax", residuals = TRUE, 
##     fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##     PA1  PA3   PA2   h2    u2 com
## V1 0.07 0.07  1.05 1.11 -0.11 1.0
## V2 0.36 0.50  0.18 0.41  0.59 2.1
## V3 0.87 0.13  0.09 0.78  0.22 1.1
## V4 0.11 0.44  0.15 0.22  0.78 1.4
## V5 0.47 0.39 -0.03 0.38  0.62 1.9
## V6 0.51 0.05  0.02 0.27  0.73 1.0
## V7 0.68 0.52  0.09 0.74  0.26 1.9
## V8 0.20 0.66  0.07 0.48  0.52 1.2
## V9 0.02 0.37 -0.09 0.15  0.85 1.1
## 
##                        PA1  PA3  PA2
## SS loadings           1.90 1.46 1.19
## Proportion Var        0.21 0.16 0.13
## Cumulative Var        0.21 0.37 0.51
## Proportion Explained  0.42 0.32 0.26
## Cumulative Proportion 0.42 0.74 1.00
## 
## Mean item complexity =  1.4
## Test of the hypothesis that 3 factors are sufficient.
## 
## The degrees of freedom for the null model are  36  and the objective function was  2.59 with Chi Square of  839.43
## The degrees of freedom for the model are 12  and the objective function was  0.3 
## 
## The root mean square of the residuals (RMSR) is  0.05 
## The df corrected root mean square of the residuals is  0.09 
## 
## The harmonic number of observations is  329 with the empirical chi square  68.23  with prob <  6.9e-10 
## The total number of observations was  329  with Likelihood Chi Square =  96.38  with prob <  2.8e-15 
## 
## Tucker Lewis Index of factoring reliability =  0.683
## RMSEA index =  0.146  and the 90 % confidence intervals are  0.12 0.174
## BIC =  26.83
## Fit based upon off diagonal values = 0.97
ml <- fa(r = data, nfactors = 3, rotate = "varimax", fm = "ml", residuals = TRUE)

ml
## Factor Analysis using method =  ml
## Call: fa(r = data, nfactors = 3, rotate = "varimax", residuals = TRUE, 
##     fm = "ml")
## Standardized loadings (pattern matrix) based upon correlation matrix
##      ML2  ML1  ML3    h2    u2 com
## V1  0.11 0.25 0.06 0.077 0.923 1.5
## V2  0.28 0.95 0.11 0.995 0.005 1.2
## V3  0.87 0.18 0.17 0.815 0.185 1.2
## V4  0.09 0.07 0.50 0.258 0.742 1.1
## V5  0.37 0.17 0.48 0.398 0.602 2.1
## V6  0.51 0.06 0.08 0.264 0.736 1.1
## V7  0.61 0.29 0.56 0.771 0.229 2.4
## V8  0.10 0.39 0.56 0.479 0.521 1.9
## V9 -0.03 0.30 0.16 0.117 0.883 1.5
## 
##                        ML2  ML1  ML3
## SS loadings           1.61 1.37 1.19
## Proportion Var        0.18 0.15 0.13
## Cumulative Var        0.18 0.33 0.46
## Proportion Explained  0.39 0.33 0.29
## Cumulative Proportion 0.39 0.71 1.00
## 
## Mean item complexity =  1.6
## Test of the hypothesis that 3 factors are sufficient.
## 
## The degrees of freedom for the null model are  36  and the objective function was  2.59 with Chi Square of  839.43
## The degrees of freedom for the model are 12  and the objective function was  0.26 
## 
## The root mean square of the residuals (RMSR) is  0.06 
## The df corrected root mean square of the residuals is  0.1 
## 
## The harmonic number of observations is  329 with the empirical chi square  86.12  with prob <  2.8e-13 
## The total number of observations was  329  with Likelihood Chi Square =  82.31  with prob <  1.5e-12 
## 
## Tucker Lewis Index of factoring reliability =  0.736
## RMSEA index =  0.133  and the 90 % confidence intervals are  0.107 0.162
## BIC =  12.75
## Fit based upon off diagonal values = 0.96
## Measures of factor score adequacy             
##                                                    ML2  ML1  ML3
## Correlation of (regression) scores with factors   0.89 0.99 0.80
## Multiple R square of scores with factors          0.80 0.98 0.64
## Minimum correlation of possible factor scores     0.60 0.96 0.29
pa <- fa(r = data, nfactors = 3, rotate = "varimax", fm = "pa", residuals = TRUE)
## maximum iteration exceeded
## Warning in fa.stats(r = r, f = f, phi = phi, n.obs = n.obs, np.obs = np.obs, :
## The estimated weights for the factor scores are probably incorrect. Try a
## different factor score estimation method.
## Warning in fac(r = r, nfactors = nfactors, n.obs = n.obs, rotate = rotate, : An
## ultra-Heywood case was detected. Examine the results carefully
fa.diagram(pa,main="data")