Análise Fatorial - HBAT

---

library(readxl)
hbat <- read_excel("hbat.xls")
# x6 a x18 variáveis
dados <- hbat[,7:19]
head(dados)

# Estatísticas Descritivas
describe(dados)

dados 

 13  Variables      100  Observations
------------------------------------------------------------------------------------------
X6 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      43       1    7.81   5.595   5.790   6.575   8.000   9.100   9.410 
    .95 
  9.900 

lowest :  5.0  5.1  5.2  5.5  5.6, highest:  9.4  9.5  9.6  9.9 10.0 
------------------------------------------------------------------------------------------
X7 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      27       1   3.672   2.595   2.800   3.275   3.600   3.925   4.530 
    .95 
  5.100 

lowest : 2.2 2.4 2.5 2.6 2.7, highest: 4.9 5.1 5.5 5.6 5.7 
------------------------------------------------------------------------------------------
X8 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      50       1   5.365   2.700   3.280   4.250   5.400   6.625   7.210 
    .95 
  7.605 

lowest : 1.3 2.5 2.6 2.7 3.0, highest: 7.7 7.9 8.0 8.4 8.5 
------------------------------------------------------------------------------------------
X9 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      45       1   5.442   3.595   3.900   4.600   5.450   6.325   7.010 
    .95 
  7.305 

lowest : 2.6 3.0 3.2 3.5 3.6, highest: 7.4 7.5 7.6 7.7 7.8 
------------------------------------------------------------------------------------------
X10 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      41       1    4.01   2.200   2.400   3.175   4.000   4.800   5.510 
    .95 
  5.800 

lowest : 1.9 2.1 2.2 2.3 2.4, highest: 5.7 5.8 5.9 6.3 6.5 
------------------------------------------------------------------------------------------
X11 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      42       1   5.805   3.900   4.190   4.700   5.750   6.800   7.600 
    .95 
  7.805 

lowest : 2.3 2.9 3.3 3.6 3.9, highest: 7.7 7.8 7.9 8.3 8.4 
------------------------------------------------------------------------------------------
X12 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      35       1   5.123   3.385   3.790   4.500   4.900   5.800   6.610 
    .95 
  7.100 

lowest : 2.9 3.0 3.1 3.4 3.5, highest: 6.8 6.9 7.1 7.8 8.2 
------------------------------------------------------------------------------------------
X13 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      45       1   6.974   4.500   4.800   5.875   7.100   8.400   8.810 
    .95 
  9.105 

lowest : 3.7 3.8 4.4 4.5 4.6, highest: 9.2 9.3 9.6 9.7 9.9 
------------------------------------------------------------------------------------------
X14 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      34       1   6.043   4.795   5.000   5.400   6.100   6.600   7.200 
    .95 
  7.305 

lowest : 4.1 4.3 4.5 4.7 4.8, highest: 7.3 7.4 7.5 7.7 8.1 
------------------------------------------------------------------------------------------
X15 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      50       1    5.15   2.785   3.500   4.100   5.000   6.300   7.110 
    .95 
  7.505 

lowest : 1.7 2.4 2.5 2.8 3.0, highest: 7.5 7.6 7.7 9.2 9.5 
------------------------------------------------------------------------------------------
X16 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      37       1   4.278   2.595   3.000   3.700   4.400   4.800   5.400 
    .95 
  5.605 

lowest : 2.0 2.1 2.4 2.5 2.6, highest: 5.5 5.6 5.7 6.5 6.7 
------------------------------------------------------------------------------------------
X17 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      40       1    4.61    2.89    3.09    3.70    4.35    5.60    6.30 
    .95 
   6.60 

lowest : 2.6 2.7 2.9 3.0 3.1, highest: 6.5 6.6 6.7 6.9 7.3 
------------------------------------------------------------------------------------------
X18 
      n missing  unique    Info    Mean     .05     .10     .25     .50     .75     .90 
    100       0      30       1   3.886   2.595   2.990   3.400   3.900   4.425   4.710 
    .95 
  4.900 

lowest : 1.6 2.0 2.4 2.5 2.6, highest: 4.7 4.8 4.9 5.2 5.5 
------------------------------------------------------------------------------------------

boxplot(dados)

dados.pad <- scale(dados)
boxplot(dados.pad)

# correlação 
mcor <- rcorr(as.matrix(dados))
mcor

       X6    X7    X8    X9   X10   X11   X12   X13   X14   X15   X16   X17   X18
X6   1.00 -0.14  0.10  0.11 -0.05  0.48 -0.15 -0.40  0.09  0.03  0.10 -0.49  0.03
X7  -0.14  1.00  0.00  0.14  0.43 -0.05  0.79  0.23  0.05 -0.03  0.16  0.27  0.19
X8   0.10  0.00  1.00  0.10 -0.06  0.19  0.02 -0.27  0.80 -0.07  0.08 -0.19  0.03
X9   0.11  0.14  0.10  1.00  0.20  0.56  0.23 -0.13  0.14  0.06  0.76  0.39  0.87
X10 -0.05  0.43 -0.06  0.20  1.00 -0.01  0.54  0.13  0.01  0.08  0.18  0.33  0.28
X11  0.48 -0.05  0.19  0.56 -0.01  1.00 -0.06 -0.49  0.27  0.05  0.42 -0.38  0.60
X12 -0.15  0.79  0.02  0.23  0.54 -0.06  1.00  0.26  0.11  0.03  0.20  0.35  0.27
X13 -0.40  0.23 -0.27 -0.13  0.13 -0.49  0.26  1.00 -0.24  0.02 -0.11  0.47 -0.07
X14  0.09  0.05  0.80  0.14  0.01  0.27  0.11 -0.24  1.00  0.04  0.20 -0.17  0.11
X15  0.03 -0.03 -0.07  0.06  0.08  0.05  0.03  0.02  0.04  1.00  0.07  0.09  0.11
X16  0.10  0.16  0.08  0.76  0.18  0.42  0.20 -0.11  0.20  0.07  1.00  0.41  0.75
X17 -0.49  0.27 -0.19  0.39  0.33 -0.38  0.35  0.47 -0.17  0.09  0.41  1.00  0.50
X18  0.03  0.19  0.03  0.87  0.28  0.60  0.27 -0.07  0.11  0.11  0.75  0.50  1.00

n= 100 


P
    X6     X7     X8     X9     X10    X11    X12    X13    X14    X15    X16    X17   
X6         0.1736 0.3441 0.2922 0.5972 0.0000 0.1316 0.0000 0.3823 0.7898 0.3017 0.0000
X7  0.1736        0.9932 0.1642 0.0000 0.6026 0.0000 0.0216 0.6081 0.7865 0.1208 0.0065
X8  0.3441 0.9932        0.3387 0.5343 0.0549 0.8668 0.0064 0.0000 0.4669 0.4282 0.0638
X9  0.2922 0.1642 0.3387        0.0496 0.0000 0.0215 0.2046 0.1635 0.5572 0.0000 0.0000
X10 0.5972 0.0000 0.5343 0.0496        0.9092 0.0000 0.1831 0.9151 0.4051 0.0665 0.0007
X11 0.0000 0.6026 0.0549 0.0000 0.9092        0.5445 0.0000 0.0060 0.6483 0.0000 0.0001
X12 0.1316 0.0000 0.8668 0.0215 0.0000 0.5445        0.0078 0.2873 0.7547 0.0517 0.0003
X13 0.0000 0.0216 0.0064 0.2046 0.1831 0.0000 0.0078        0.0140 0.8191 0.2564 0.0000
X14 0.3823 0.6081 0.0000 0.1635 0.9151 0.0060 0.2873 0.0140        0.7281 0.0494 0.0904
X15 0.7898 0.7865 0.4669 0.5572 0.4051 0.6483 0.7547 0.8191 0.7281        0.4980 0.3516
X16 0.3017 0.1208 0.4282 0.0000 0.0665 0.0000 0.0517 0.2564 0.0494 0.4980        0.0000
X17 0.0000 0.0065 0.0638 0.0000 0.0007 0.0001 0.0003 0.0000 0.0904 0.3516 0.0000       
X18 0.7843 0.0561 0.8016 0.0000 0.0055 0.0000 0.0063 0.4712 0.2786 0.2950 0.0000 0.0000
    X18   
X6  0.7843
X7  0.0561
X8  0.8016
X9  0.0000
X10 0.0055
X11 0.0000
X12 0.0063
X13 0.4712
X14 0.2786
X15 0.2950
X16 0.0000
X17 0.0000
X18       

R <- cor(dados)
corrplot(R, method="number",type="upper", tl.srt = 45)

corrplot(R, method="number",type="upper", order = "hclust", tl.srt = 45)

# corrplot(R, method="circle",type="full", order = "hclust", tl.srt = 45)

29 das 78 (37%) correlações são significativas a 1% de significância

################################################################
#Partial correlation matrix
################################################################
partial.cor <- function (x)
{
R <- cor(x)
RI <- solve(R)
D <- 1/sqrt(diag(RI))
Rp <- -RI * (D %o% D)
diag(Rp) <- 0
rownames(Rp) <- colnames(Rp) <- colnames(x)
Rp
}
mat_anti_imagem <- -partial.cor(dados)
corrplot(mat_anti_imagem, method="number",type="upper", order = "hclust", tl.srt = 45)

################################################################
# The Bartlett's test statistic indicates to what extent we deviate from the reference situation |R| = 1.
################################################################
Bartlett.sphericity.test <- function(x)
{
  method <- "Bartlett's test of sphericity"
  data.name <- deparse(substitute(x))
  x <- subset(x, complete.cases(x)) # Omit missing values
  n <- nrow(x)
  p <- ncol(x)
  chisq <- (1-n+(2*p+5)/6)*log(det(cor(x)))
  df <- p*(p-1)/2
  p.value <- pchisq(chisq, df, lower.tail=FALSE)
  names(chisq) <- "X-squared"
  names(df) <- "df"
  return(structure(list(statistic=chisq, parameter=df, p.value=p.value,
                        method=method, data.name=data.name), class="htest"))
}
Bartlett.sphericity.test(dados)

    Bartlett's test of sphericity

data:  dados
X-squared = 948.98, df = 78, p-value < 2.2e-16

################################################################
# KMO index
################################################################
kmo <- function(x)
{
  x <- subset(x, complete.cases(x)) # Omit missing values
  r <- cor(x) # Correlation matrix
  r2 <- r^2 # Squared correlation coefficients
  i <- solve(r) # Inverse matrix of correlation matrix
  d <- diag(i) # Diagonal elements of inverse matrix
  p2 <- (-i/sqrt(outer(d, d)))^2 # Squared partial correlation coefficients
  diag(r2) <- diag(p2) <- 0 # Delete diagonal elements
  KMO <- sum(r2)/(sum(r2)+sum(p2))
  MSA <- colSums(r2)/(colSums(r2)+colSums(p2))
  return(list(KMO=KMO, MSA=MSA))
}
kmo(dados)

$KMO
[1] 0.6087046

$MSA
       X6        X7        X8        X9       X10       X11       X12       X13       X14 
0.8733349 0.6201707 0.5270181 0.8895112 0.8066328 0.4478469 0.5862796 0.8791773 0.5293082 
      X15       X16       X17       X18 
0.3144405 0.8588912 0.4423739 0.5331944 

X11 X15 e X17 MSA < 0,50

estratégia: omitir X15 (menor MSA)

dados <- dados[,-10] #rodar apenas uma vez!
dados

kmo(dados)

$KMO
[1] 0.6121225

$MSA
       X6        X7        X8        X9       X10       X11       X12       X13       X14 
0.8843310 0.6244785 0.5310965 0.8898236 0.8009663 0.4492636 0.5873693 0.8799355 0.5328693 
      X16       X17       X18 
0.8584831 0.4440650 0.5328216 

estratégia: omitir X17 (menor MSA)

dados <- dados[,-11] #rodar apenas uma vez!
dados

kmo(dados)

$KMO
[1] 0.6531422

$MSA
       X6        X7        X8        X9       X10       X11       X12       X13       X14 
0.5087065 0.6255547 0.5190081 0.7865552 0.7793615 0.6223277 0.6218422 0.7528265 0.5107613 
      X16       X18 
0.7600148 0.6655768 

mat_anti_imagem <- -partial.cor(dados)
corrplot(mat_anti_imagem, method="number",type="upper", tl.srt = 45)

n.dados <- length(dados)
fit <- principal(dados, nfactors=n.dados, rotate="none")
print(fit, sort=T)

Principal Components Analysis
Call: principal(r = dados, nfactors = n.dados, rotate = "none")
Standardized loadings (pattern matrix) based upon correlation matrix
    item   PC1   PC2   PC3   PC4   PC5   PC6   PC7   PC8   PC9  PC10  PC11 h2       u2
X18   11  0.88  0.12 -0.30 -0.21 -0.03 -0.02  0.15 -0.03  0.01  0.04 -0.24  1  7.8e-16
X9     4  0.87  0.03 -0.27 -0.22  0.02  0.02  0.01 -0.26 -0.04 -0.19  0.11  1  1.3e-15
X16   10  0.81  0.04 -0.22 -0.25  0.02  0.09 -0.42  0.18  0.04  0.11  0.06  1  1.6e-15
X11    6  0.72 -0.45 -0.15  0.21  0.09 -0.07  0.39  0.17  0.04  0.09  0.12  1  7.8e-16
X12    7  0.38  0.75  0.31  0.23  0.16 -0.08  0.00 -0.08 -0.29  0.13  0.03  1 -4.4e-16
X7     2  0.31  0.71  0.31  0.28  0.33 -0.20 -0.04  0.06  0.24 -0.10 -0.02  1 -1.8e-15
X13    8 -0.28  0.66 -0.07 -0.35  0.19  0.53  0.20  0.05  0.04  0.02  0.03  1  1.2e-15
X8     3  0.29 -0.37  0.79 -0.20 -0.02  0.08  0.01 -0.23  0.16  0.14  0.02  1  1.9e-15
X14    9  0.39 -0.31  0.78 -0.19 -0.03  0.10  0.02  0.22 -0.14 -0.16 -0.04  1  1.9e-15
X6     1  0.25 -0.50 -0.08  0.67  0.18  0.42 -0.12 -0.07 -0.01 -0.02 -0.06  1 -2.2e-16
X10    5  0.34  0.58  0.11  0.33 -0.63  0.15  0.04  0.03  0.07 -0.01  0.03  1 -4.4e-16
    com
X18 1.7
X9  1.7
X16 2.1
X11 3.0
X12 2.8
X7  3.3
X13 3.4
X8  2.3
X14 2.5
X6  3.3
X10 3.4

                       PC1  PC2  PC3  PC4  PC5  PC6  PC7  PC8  PC9 PC10 PC11
SS loadings           3.43 2.55 1.69 1.09 0.61 0.55 0.40 0.25 0.20 0.13 0.10
Proportion Var        0.31 0.23 0.15 0.10 0.06 0.05 0.04 0.02 0.02 0.01 0.01
Cumulative Var        0.31 0.54 0.70 0.80 0.85 0.90 0.94 0.96 0.98 0.99 1.00
Proportion Explained  0.31 0.23 0.15 0.10 0.06 0.05 0.04 0.02 0.02 0.01 0.01
Cumulative Proportion 0.31 0.54 0.70 0.80 0.85 0.90 0.94 0.96 0.98 0.99 1.00

Mean item complexity =  2.7
Test of the hypothesis that 11 components are sufficient.

The root mean square of the residuals (RMSR) is  0 
 with the empirical chi square  0  with prob <  NA 

Fit based upon off diagonal values = 1

#fit$values
#fit$scores
#fit$weights
#fit$loadings
#fit$communality
# Escolha a quantidade de fatores
fit1 <- principal(dados, nfactors=4, rotate="none")
print(fit1, sort=T)

Principal Components Analysis
Call: principal(r = dados, nfactors = 4, rotate = "none")
Standardized loadings (pattern matrix) based upon correlation matrix
    item   PC1   PC2   PC3   PC4   h2    u2 com
X18   11  0.88  0.12 -0.30 -0.21 0.91 0.086 1.4
X9     4  0.87  0.03 -0.27 -0.22 0.88 0.119 1.3
X16   10  0.81  0.04 -0.22 -0.25 0.77 0.234 1.3
X11    6  0.72 -0.45 -0.15  0.21 0.79 0.213 2.0
X12    7  0.38  0.75  0.31  0.23 0.86 0.141 2.1
X7     2  0.31  0.71  0.31  0.28 0.78 0.223 2.1
X13    8 -0.28  0.66 -0.07 -0.35 0.64 0.359 1.9
X10    5  0.34  0.58  0.11  0.33 0.58 0.424 2.4
X8     3  0.29 -0.37  0.79 -0.20 0.89 0.107 1.9
X14    9  0.39 -0.31  0.78 -0.19 0.89 0.108 2.0
X6     1  0.25 -0.50 -0.08  0.67 0.77 0.232 2.2

                       PC1  PC2  PC3  PC4
SS loadings           3.43 2.55 1.69 1.09
Proportion Var        0.31 0.23 0.15 0.10
Cumulative Var        0.31 0.54 0.70 0.80
Proportion Explained  0.39 0.29 0.19 0.12
Cumulative Proportion 0.39 0.68 0.88 1.00

Mean item complexity =  1.9
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  39.02  with prob <  0.0018 

Fit based upon off diagonal values = 0.97

#fit1$values
#fit1$scores
#fit1$weights
#fit1$loadings
#fit1$communality

# varimax rotation
fit2 <- principal(dados, nfactors=4, rotate="varimax")
print(fit2, sort=T)

Principal Components Analysis
Call: principal(r = dados, nfactors = 4, rotate = "varimax")
Standardized loadings (pattern matrix) based upon correlation matrix
    item   RC1   RC2   RC3   RC4   h2    u2 com
X18   11  0.94  0.18  0.00  0.05 0.91 0.086 1.1
X9     4  0.93  0.12  0.05  0.09 0.88 0.119 1.1
X16   10  0.86  0.11  0.08  0.04 0.77 0.234 1.1
X12    7  0.13  0.90  0.08 -0.16 0.86 0.141 1.1
X7     2  0.06  0.87  0.05 -0.12 0.78 0.223 1.1
X10    5  0.14  0.74 -0.08  0.01 0.58 0.424 1.1
X8     3  0.02 -0.02  0.94  0.10 0.89 0.107 1.0
X14    9  0.11  0.05  0.93  0.10 0.89 0.108 1.1
X6     1  0.00 -0.01 -0.03  0.88 0.77 0.232 1.0
X13    8 -0.09  0.23 -0.25 -0.72 0.64 0.359 1.5
X11    6  0.59 -0.06  0.15  0.64 0.79 0.213 2.1

                       RC1  RC2  RC3  RC4
SS loadings           2.89 2.23 1.86 1.77
Proportion Var        0.26 0.20 0.17 0.16
Cumulative Var        0.26 0.47 0.63 0.80
Proportion Explained  0.33 0.26 0.21 0.20
Cumulative Proportion 0.33 0.59 0.80 1.00

Mean item complexity =  1.2
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  39.02  with prob <  0.0018 

Fit based upon off diagonal values = 0.97

#fit2$values
#fit2$scores
#fit2$weights
#fit2$loadings
#fit2$communality

eliminar X11 pois ainda há carga cruzada

dados <- dados[,-6]
# varimax rotation
fit3 <- principal(dados, nfactors=4, rotate="varimax")
print(fit3, sort=T)

Principal Components Analysis
Call: principal(r = dados, nfactors = 4, rotate = "varimax")
Standardized loadings (pattern matrix) based upon correlation matrix
    item   RC1   RC2   RC3   RC4   h2   u2 com
X9     4  0.93  0.10  0.06  0.08 0.89 0.11 1.0
X18   10  0.93  0.17  0.00  0.01 0.89 0.11 1.1
X16    9  0.89  0.10  0.09  0.07 0.81 0.19 1.1
X12    6  0.14  0.90  0.08 -0.17 0.86 0.14 1.1
X7     2  0.06  0.87  0.05 -0.14 0.78 0.22 1.1
X10    5  0.16  0.74 -0.08  0.04 0.58 0.42 1.1
X8     3  0.02 -0.02  0.94  0.10 0.89 0.11 1.0
X14    8  0.10  0.05  0.93  0.08 0.89 0.11 1.0
X6     1  0.03 -0.01 -0.02  0.89 0.80 0.20 1.0
X13    7 -0.10  0.23 -0.26 -0.73 0.66 0.34 1.5

                       RC1  RC2  RC3  RC4
SS loadings           2.59 2.22 1.85 1.41
Proportion Var        0.26 0.22 0.18 0.14
Cumulative Var        0.26 0.48 0.67 0.81
Proportion Explained  0.32 0.28 0.23 0.17
Cumulative Proportion 0.32 0.60 0.83 1.00

Mean item complexity =  1.1
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  35.71  with prob <  0.00019 

Fit based upon off diagonal values = 0.96

#fit2$values
#fit2$scores
#fit2$weights
#fit2$loadings # cargas fatoriais
#fit2$communality

Escores Fatoriais

fit2$weights

            RC1          RC2          RC3         RC4
X6  -0.13906658  0.147235978 -0.131782402  0.60690554
X7  -0.07975823  0.421013424  0.021541189  0.05589414
X8  -0.04000256 -0.017857872  0.527173116 -0.05913410
X9   0.34848625 -0.053278604 -0.020396958 -0.07376774
X10 -0.04492535  0.368809456 -0.069941866  0.12791365
X11  0.14231359 -0.005697205 -0.019329512  0.31728383
X12 -0.04695514  0.418071580  0.039751067  0.01627490
X13  0.06046002  0.007312344 -0.049627429 -0.41425781
X14 -0.01328373  0.010973416  0.517162446 -0.05794101
X16  0.33135641 -0.059245362  0.009101153 -0.10569317
X18  0.35339032 -0.028889175 -0.047766436 -0.08508483

# 10 primeiros escores
fit2$scores[1:10,]

             RC1        RC2          RC3        RC4
 [1,]  0.1274910  0.7698686 -1.878446273  0.3664848
 [2,]  1.2216666 -1.6458617 -0.614030010  0.8130648
 [3,]  0.6158214  0.5800037  0.003689252  1.5699769
 [4,] -0.8446267 -0.2719218  1.267493254 -1.2541645
 [5,] -0.3197943 -0.8340650 -0.008096627  0.4475377
 [6,] -0.6470292 -1.0672683 -1.303198892 -1.0527792
 [7,] -2.6267985 -0.2458827 -0.555423494 -1.2260147
 [8,] -0.2793639 -0.1573204 -0.749311481 -1.0146418
 [9,]  1.0515134 -0.1722883 -0.092252815 -1.6580963
[10,]  0.4287538  0.7635327 -0.450377116 -0.8911659

Análise de fatores comuns

a matriz de correlação é reduzida com comunalidades iniciais estimadas na diagonal em vez de unidades

pa <- fa(dados, nfactors=4, fm="pa", rotate="varimax")
print(pa, sort=T)

Factor Analysis using method =  pa
Call: fa(r = dados, nfactors = 4, rotate = "varimax", fm = "pa")
Standardized loadings (pattern matrix) based upon correlation matrix
    item   PA1   PA2   PA3   PA4   h2    u2 com
X18   10  0.92  0.17  0.01  0.01 0.88 0.115 1.1
X9     4  0.91  0.12  0.06  0.10 0.86 0.140 1.1
X16    9  0.79  0.12  0.09  0.10 0.66 0.340 1.1
X12    6  0.12  0.97  0.06 -0.15 0.99 0.011 1.1
X7     2  0.07  0.78  0.03 -0.14 0.63 0.367 1.1
X10    5  0.17  0.53 -0.05 -0.05 0.32 0.683 1.2
X8     3  0.02 -0.02  0.89  0.12 0.81 0.191 1.0
X14    8  0.10  0.05  0.88  0.10 0.80 0.201 1.1
X6     1  0.05 -0.06  0.02  0.65 0.43 0.575 1.0
X13    7 -0.10  0.21 -0.21 -0.58 0.44 0.559 1.6

                       PA1  PA2  PA3  PA4
SS loadings           2.39 1.95 1.63 0.85
Proportion Var        0.24 0.19 0.16 0.08
Cumulative Var        0.24 0.43 0.60 0.68
Proportion Explained  0.35 0.29 0.24 0.12
Cumulative Proportion 0.35 0.64 0.88 1.00

Mean item complexity =  1.1
Test of the hypothesis that 4 factors are sufficient.

The degrees of freedom for the null model are  45  and the objective function was  5.3 with Chi Square of  502.97
The degrees of freedom for the model are 11  and the objective function was  0.08 

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 number of observations is  100 with the empirical chi square  0.88  with prob <  1 
The total number of observations was  100  with MLE Chi Square =  7.45  with prob <  0.76 

Tucker Lewis Index of factoring reliability =  1.033
RMSEA index =  0  and the 90 % confidence intervals are  NA 0.074
BIC =  -43.21
Fit based upon off diagonal values = 1
Measures of factor score adequacy             
                                                PA1  PA2  PA3  PA4
Correlation of scores with factors             0.97 0.99 0.94 0.74
Multiple R square of scores with factors       0.93 0.98 0.88 0.55
Minimum correlation of possible factor scores  0.87 0.95 0.76 0.11