====== EFA datasets: Car buying factors ======
{{:r:EFA.csv}}
# read the dataset into R variable using the read.csv(file) function
data <- read.csv("http://commres.net/wiki/_media/r/efa.csv")
head(data)
# install the package
# install.packages("psych")
# install.packages("GPArotation")
# load the package
library(psych)
library(GPArotation)
# calculate the correlation matrix
corMat <- cor(data)
# display the correlation matrix
round(corMat,3)
# use fa() to conduct an oblique principal-axis exploratory factor analysis
# save the solution to an R variable
fa.ini <- fa(r = corMat)
# display the solution output
fa.ini
names(fa.ini)
fa.ini$e.values
num <- 5
fa.5 <- fa(r=corMat, nfactors=num, rotate="varimax", fm="pa")
fa.5.oblimin.minres <- fa(r=corMat, nfactors=num, rotate="oblimin", fm="minres")
fa.5.vm.minres <- fa(r=corMat, nfactors=num, rotate="varimax", fm="minres")
fa.sort(fa.5.oblimin.minres)
fa.sort(fa.5.vm.minres)
> fa.sort(fa.5.oblimin.minres)
Factor Analysis using method = minres
Call: fa(r = corMat, nfactors = num, rotate = "oblimin", fm = "minres")
Standardized loadings (pattern matrix) based upon correlation matrix
MR1 MR2 MR4 MR5 MR3 h2 u2 com
Resale_Value 0.73 -0.20 0.08 -0.05 -0.10 0.59 0.413 1.2
Price 0.58 0.23 -0.11 -0.01 -0.07 0.35 0.645 1.4
Maintenance 0.55 0.11 -0.01 0.26 0.01 0.45 0.549 1.5
Fuel_Efficiency 0.38 0.01 0.33 0.23 0.22 0.49 0.512 3.3
Safety -0.31 0.23 0.17 0.07 -0.24 0.21 0.790 3.6
Space_comfort -0.02 0.88 0.04 -0.06 0.03 0.83 0.172 1.0
Fuel_Type 0.03 0.52 0.11 0.02 -0.13 0.30 0.699 1.2
Technology 0.00 0.28 0.14 0.06 0.01 0.13 0.873 1.6
After_Sales_Service -0.01 0.04 0.95 -0.01 -0.01 0.92 0.085 1.0
Color 0.03 -0.05 0.02 0.78 0.09 0.67 0.329 1.0
Exterior_Looks 0.09 0.05 0.03 -0.54 0.31 0.33 0.669 1.7
Testimonials -0.23 0.04 -0.04 0.08 0.63 0.47 0.535 1.3
Test_drive 0.06 -0.02 0.23 -0.06 0.44 0.26 0.745 1.6
Product_reviews 0.33 0.25 -0.08 0.12 0.35 0.36 0.642 3.2
MR1 MR2 MR4 MR5 MR3
SS loadings 1.64 1.41 1.21 1.12 0.97
Proportion Var 0.12 0.10 0.09 0.08 0.07
Cumulative Var 0.12 0.22 0.30 0.38 0.45
Proportion Explained 0.26 0.22 0.19 0.18 0.15
Cumulative Proportion 0.26 0.48 0.67 0.85 1.00
With factor correlations of
MR1 MR2 MR4 MR5 MR3
MR1 1.00 -0.03 0.16 0.25 0.03
MR2 -0.03 1.00 0.32 -0.07 0.21
MR4 0.16 0.32 1.00 0.13 0.07
MR5 0.25 -0.07 0.13 1.00 0.17
MR3 0.03 0.21 0.07 0.17 1.00
Mean item complexity = 1.8
Test of the hypothesis that 5 factors are sufficient.
The degrees of freedom for the null model are 91 and the objective function was 2.97
The degrees of freedom for the model are 31 and the objective function was 0.34
The root mean square of the residuals (RMSR) is 0.04
The df corrected root mean square of the residuals is 0.06
Fit based upon off diagonal values = 0.97
Measures of factor score adequacy
MR1 MR2 MR4 MR5 MR3
Correlation of (regression) scores with factors 0.87 0.92 0.96 0.86 0.79
Multiple R square of scores with factors 0.76 0.85 0.92 0.73 0.63
Minimum correlation of possible factor scores 0.52 0.70 0.85 0.46 0.26
> fa.sort(fa.5.vm.minres)
Factor Analysis using method = minres
Call: fa(r = corMat, nfactors = num, rotate = "varimax", fm = "minres")
Standardized loadings (pattern matrix) based upon correlation matrix
MR1 MR2 MR3 MR5 MR4 h2 u2 com
Resale_Value 0.69 -0.22 -0.17 0.16 0.01 0.59 0.413 1.5
Maintenance 0.61 0.07 0.07 0.07 0.25 0.45 0.549 1.4
Price 0.57 0.15 -0.05 -0.04 -0.02 0.35 0.645 1.2
Fuel_Efficiency 0.46 0.09 0.28 0.37 0.23 0.49 0.512 3.2
Space_comfort 0.02 0.87 0.21 0.00 -0.17 0.83 0.172 1.2
Fuel_Type 0.06 0.54 -0.02 0.08 -0.04 0.30 0.699 1.1
Technology 0.04 0.32 0.09 0.13 0.03 0.13 0.873 1.5
Safety -0.28 0.30 -0.16 0.11 0.05 0.21 0.790 2.9
Testimonials -0.18 0.04 0.66 -0.05 0.02 0.47 0.535 1.2
Test_drive 0.09 0.06 0.43 0.23 -0.07 0.26 0.745 1.7
Product_reviews 0.38 0.18 0.41 -0.03 0.07 0.36 0.642 2.4
After_Sales_Service 0.06 0.37 0.06 0.88 0.04 0.92 0.085 1.4
Color 0.21 -0.05 0.26 0.07 0.74 0.67 0.329 1.4
Exterior_Looks -0.01 0.05 0.20 0.01 -0.54 0.33 0.669 1.3
MR1 MR2 MR3 MR5 MR4
SS loadings 1.72 1.50 1.09 1.03 1.00
Proportion Var 0.12 0.11 0.08 0.07 0.07
Cumulative Var 0.12 0.23 0.31 0.38 0.45
Proportion Explained 0.27 0.24 0.17 0.16 0.16
Cumulative Proportion 0.27 0.51 0.68 0.84 1.00
Mean item complexity = 1.7
Test of the hypothesis that 5 factors are sufficient.
The degrees of freedom for the null model are 91 and the objective function was 2.97
The degrees of freedom for the model are 31 and the objective function was 0.34
The root mean square of the residuals (RMSR) is 0.04
The df corrected root mean square of the residuals is 0.06
Fit based upon off diagonal values = 0.97
Measures of factor score adequacy
MR1 MR2 MR3 MR5 MR4
Correlation of (regression) scores with factors 0.86 0.91 0.80 0.93 0.82
Multiple R square of scores with factors 0.75 0.82 0.64 0.87 0.67
Minimum correlation of possible factor scores 0.50 0.64 0.27 0.75 0.34
>
>
Factor Analysis using method = minres
Call: fa(r = corMat, nfactors = num, rotate = "varimax", fm = "minres")
Standardized loadings (pattern matrix) based upon correlation matrix
MR1 MR2 MR3 MR5 MR4 h2 u2 com
----------------------------------------------------------------
Resale_Value 0.69 -0.22 -0.17 0.16 0.01 0.59 0.413 1.5
Maintenance 0.61 0.07 0.07 0.07 0.25 0.45 0.549 1.4
Price 0.57 0.15 -0.05 -0.04 -0.02 0.35 0.645 1.2
Fuel_Efficiency 0.46 0.09 0.28 0.37 0.23 0.49 0.512 3.2
----------------------------------------------------------------
Space_comfort 0.02 0.87 0.21 0.00 -0.17 0.83 0.172 1.2
Fuel_Type 0.06 0.54 -0.02 0.08 -0.04 0.30 0.699 1.1
Technology 0.04 0.32 0.09 0.13 0.03 0.13 0.873 1.5
Safety -0.28 0.30 -0.16 0.11 0.05 0.21 0.790 2.9
----------------------------------------------------------------
Testimonials -0.18 0.04 0.66 -0.05 0.02 0.47 0.535 1.2
Test_drive 0.09 0.06 0.43 0.23 -0.07 0.26 0.745 1.7
Product_reviews 0.38 0.18 0.41 -0.03 0.07 0.36 0.642 2.4
----------------------------------------------------------------
After_Sales_Service 0.06 0.37 0.06 0.88 0.04 0.92 0.085 1.4
----------------------------------------------------------------
Color 0.21 -0.05 0.26 0.07 0.74 0.67 0.329 1.4
Exterior_Looks -0.01 0.05 0.20 0.01 -0.54 0.33 0.669 1.3
MR1 MR2 MR3 MR5 MR4
SS loadings 1.72 1.50 1.09 1.03 1.00
Proportion Var 0.12 0.11 0.08 0.07 0.07
Cumulative Var 0.12 0.23 0.31 0.38 0.45
Proportion Explained 0.27 0.24 0.17 0.16 0.16
Cumulative Proportion 0.27 0.51 0.68 0.84 1.00
====== 2 ======
food <- read.csv("http://commres.net/wiki/_media/r/food-texture.csv",
row.names = "X")
str(food)
food.ini <- fa(food)
food.ini$e.values # we decide to get two factors based on eigenvalues
food.fa <- factanal(food, factors = 2)
food.fa
food.fa$uniquenesses
apply(food.fa$loadings^2,1,sum) # communality
1 - apply(food.fa$loadings^2,1,sum) # 1 - communality = uniq
Lambda <- food.fa$loadings
Psi <- diag(food.fa$uniquenesses)
S <- food.fa$correlation
Sigma <- Lambda %*% t(Lambda) + Psi
round(S - Sigma, 6)
food.fa.none <- factanal(food, factors = 2, rotation = "none")
food.fa.varimax <- factanal(food, factors = 2, rotation = "varimax")
food.fa.promax <- factanal(food, factors = 2, rotation = "promax")
par(mfrow = c(1,3))
plot(food.fa.none$loadings[,1],
food.fa.none$loadings[,2],
xlab = "Factor 1",
ylab = "Factor 2",
ylim = c(-1,1),
xlim = c(-1,1),
main = "No rotation")
abline(h = 0, v = 0)
plot(food.fa.varimax$loadings[,1],
food.fa.varimax$loadings[,2],
xlab = "Factor 1",
ylab = "Factor 2",
ylim = c(-1,1),
xlim = c(-1,1),
main = "Varimax rotation")
text(food.fa.varimax$loadings[,1]-0.08,
food.fa.varimax$loadings[,2]+0.08,
colnames(food),
col="blue")
abline(h = 0, v = 0)
plot(food.fa.promax$loadings[,1],
food.fa.promax$loadings[,2],
xlab = "Factor 1",
ylab = "Factor 2",
ylim = c(-1,1),
xlim = c(-1,1),
main = "Promax rotation")
abline(h = 0, v = 0)
par(mfrow = c(1,1))