wk.4.fa
William Revelle
10/05/2020
Always do this first!
library(psych)
library(psychTools)Factor analysis as a data description tool
Notes to accompany https://personality-project.org/courses/350/350.factoranalysis.pdf
Iterative fit
• Specifying a model and loss function
• Ideally supplying a first derivative to the loss function
• Can be done empirically
• Some way to evaluate the quality of the fit
• Minimization of function, but then how good is this minimum
Multivariate models
• Many procedures use this concept of iterative fitting
• Some, such as principal components are“closed form”and can just be solved directly
• Others, such as“factor analysis”need to find solutions by successive approximations
• The issue of factor or component rotation is an iterative procedure.
• Principal Components and Factor analysis are all the result of some basic matrix equations to approximate a matrix
• Conceptually, we are just taking the square root of a correlation matrix:
• R≈CC′ or R≈FF′+U2
• For any given correlation matrix, R, can we find a C or an F matrix which approximates it?
##simulate a correlation matrix
R <- sim.congeneric() #use the default values
R## V1 V2 V3 V4
## V1 1.00 0.56 0.48 0.40
## V2 0.56 1.00 0.42 0.35
## V3 0.48 0.42 1.00 0.30
## V4 0.40 0.35 0.30 1.00This looks like a simple multiplication table. See if we can factor it
f1 <- fa(R) #defaults to 1 factor
f1 #show the output## Factor Analysis using method = minres
## Call: fa(r = R)
## Standardized loadings (pattern matrix) based upon correlation matrix
## MR1 h2 u2 com
## V1 0.8 0.64 0.36 1
## V2 0.7 0.49 0.51 1
## V3 0.6 0.36 0.64 1
## V4 0.5 0.25 0.75 1
##
## MR1
## SS loadings 1.74
## Proportion Var 0.43
##
## Mean item complexity = 1
## Test of the hypothesis that 1 factor is sufficient.
##
## The degrees of freedom for the null model are 6 and the objective function was 0.9
## The degrees of freedom for the model are 2 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 0
##
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## MR1
## Correlation of (regression) scores with factors 0.89
## Multiple R square of scores with factors 0.78
## Minimum correlation of possible factor scores 0.57We can find the residuals by caculation
F <- f1$loadings
F##
## Loadings:
## MR1
## V1 0.8
## V2 0.7
## V3 0.6
## V4 0.5
##
## MR1
## SS loadings 1.740
## Proportion Var 0.435model <- F %*% t(F) #The factor model is that R = FF'
model #too many decimals## V1 V2 V3 V4
## V1 0.6399993 0.5599976 0.4800018 0.4000001
## V2 0.5599976 0.4899963 0.4200002 0.3499989
## V3 0.4800018 0.4200002 0.3600031 0.3000015
## V4 0.4000001 0.3499989 0.3000015 0.2500003round(model,2) #this looks better## V1 V2 V3 V4
## V1 0.64 0.56 0.48 0.40
## V2 0.56 0.49 0.42 0.35
## V3 0.48 0.42 0.36 0.30
## V4 0.40 0.35 0.30 0.25round(R - model,2) #show the resduals## V1 V2 V3 V4
## V1 0.36 0.00 0.00 0.00
## V2 0.00 0.51 0.00 0.00
## V3 0.00 0.00 0.64 0.00
## V4 0.00 0.00 0.00 0.75Or, we can ask for the residuals
resid(f1)## V1 V2 V3 V4
## V1 0.36
## V2 0.00 0.51
## V3 0.00 0.00 0.64
## V4 0.00 0.00 0.00 0.75Some example data sets
We have a number of nice example sets. Use the help function to find them. ?ability
?bfi
?sai
?msqR
?spi
The number of factor problems
No right answer
- Statistical
• Extracting factors until the χ2 of the residual matrix is not significant.
• Extracting factors until the change in χ2 from factor n to factor n+1 is not significant.
- Rules of Thumb
• Parallel Extracting factors until the eigenvalues of the real data are less than the corresponding eigenvalues of a random data set of the same size (parallel analysis)
• Plotting the magnitude of the successive eigenvalues and applying the scree test.
- Interpretability
• Extracting factors as long as they are interpretable.
• Using the Very Simple Structure Criterion (VSS)
• Using the Minimum Average Partial criterion (MAP).
- Eigen Value of 1 rule
Simulations as way of knowing `Truth’
With real data, we do not know ‘truth’
With simulated data, we can know the right answer
Lets examine the sim item function
• defaults to certain parameter values
• processes the data (make it up)
• returns values
- Simulations can default to certain values but allow you to specify other values
sim.item #to show a function, just type the name## function (nvar = 72, nsub = 500, circum = FALSE, xloading = 0.6,
## yloading = 0.6, gloading = 0, xbias = 0, ybias = 0, categorical = FALSE,
## low = -3, high = 3, truncate = FALSE, cutpoint = 0)
## {
## avloading <- (xloading + yloading)/2
## errorweight <- sqrt(1 - (avloading^2 + gloading^2))
## g <- rnorm(nsub)
## truex <- rnorm(nsub) * xloading + xbias
## truey <- rnorm(nsub) * yloading + ybias
## if (circum) {
## radia <- seq(0, 2 * pi, len = nvar + 1)
## rad <- radia[which(radia < 2 * pi)]
## }
## else rad <- c(rep(0, nvar/4), rep(pi/2, nvar/4), rep(pi,
## nvar/4), rep(3 * pi/2, nvar/4))
## error <- matrix(rnorm(nsub * (nvar)), nsub)
## trueitem <- outer(truex, cos(rad)) + outer(truey, sin(rad))
## item <- gloading * g + trueitem + errorweight * error
## if (categorical) {
## item = round(item)
## item[(item <= low)] <- low
## item[(item > high)] <- high
## }
## if (truncate) {
## item[item < cutpoint] <- 0
## }
## colnames(item) <- paste("V", 1:nvar, sep = "")
## return(item)
## }
## <bytecode: 0x7fd1ea8374e0>
## <environment: namespace:psych>This simulates with two basic options and a host of other options
simple structure versus circumplex structure
set.seed(42)
my.data <- sim.item(12)
describe(my.data) #to see what they look like## vars n mean sd median trimmed mad min max range skew kurtosis se
## V1 1 500 0.01 1.01 0.03 0.01 0.97 -3.26 3.44 6.69 0.05 0.13 0.04
## V2 2 500 0.00 1.00 0.00 0.00 1.05 -3.16 3.12 6.28 0.02 -0.14 0.04
## V3 3 500 -0.03 1.05 -0.04 -0.04 1.06 -2.95 2.80 5.76 0.10 -0.28 0.05
## V4 4 500 -0.05 0.99 -0.04 -0.04 0.96 -2.96 2.82 5.78 -0.06 0.10 0.04
## V5 5 500 -0.04 0.96 -0.05 -0.05 0.96 -2.76 2.71 5.47 0.07 -0.20 0.04
## V6 6 500 -0.05 1.04 -0.06 -0.06 1.03 -3.65 3.07 6.72 0.05 0.09 0.05
## V7 7 500 0.01 0.97 0.01 0.02 0.95 -3.45 3.53 6.98 -0.13 0.38 0.04
## V8 8 500 0.02 1.04 0.02 0.02 1.09 -2.94 3.95 6.89 0.06 0.05 0.05
## V9 9 500 0.04 1.04 -0.02 0.04 1.05 -3.07 2.78 5.85 0.04 -0.25 0.05
## V10 10 500 0.04 0.98 0.10 0.07 0.97 -2.75 2.47 5.22 -0.23 -0.15 0.04
## V11 11 500 -0.01 0.98 -0.03 -0.01 1.02 -3.31 2.88 6.19 -0.01 -0.02 0.04
## V12 12 500 -0.01 0.97 0.00 -0.01 0.91 -3.04 2.57 5.61 -0.08 0.15 0.04lowerCor(my.data)## V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12
## V1 1.00
## V2 0.36 1.00
## V3 0.38 0.37 1.00
## V4 -0.01 -0.04 -0.01 1.00
## V5 0.05 -0.02 0.01 0.34 1.00
## V6 0.03 0.01 0.01 0.37 0.35 1.00
## V7 -0.35 -0.37 -0.38 -0.09 -0.01 -0.05 1.00
## V8 -0.40 -0.34 -0.39 0.00 0.08 0.11 0.34 1.00
## V9 -0.41 -0.36 -0.32 0.00 0.02 -0.03 0.32 0.39 1.00
## V10 0.06 0.07 0.01 -0.33 -0.32 -0.39 -0.04 -0.11 -0.06 1.00
## V11 0.02 0.03 0.05 -0.37 -0.35 -0.32 0.02 -0.12 -0.01 0.41 1.00
## V12 0.01 0.01 -0.11 -0.31 -0.30 -0.33 0.08 -0.02 0.00 0.36 0.39 1.00corPlot(my.data) #to see it graphically
## Try parallel analysis
One way to see how many factors are in the data is to compare how well many alternative models fit.
parallel analysis simulates the data with no factors and compares factor solutions to simulated (0 factor) data to the observed solution. When the random data solution is better than the real solution, you have gone too far.
fa.parallel(my.data)
## Parallel analysis suggests that the number of factors = 2 and the number of components = 2This suggests 2 factors (which is correct, since that is how we made the data)
Try a 2 factor solution
f2 <- fa(my.data,nfactors=2)## Loading required namespace: GPArotationf2## Factor Analysis using method = minres
## Call: fa(r = my.data, nfactors = 2)
## Standardized loadings (pattern matrix) based upon correlation matrix
## MR1 MR2 h2 u2 com
## V1 0.64 -0.02 0.41 0.59 1.0
## V2 0.59 0.02 0.35 0.65 1.0
## V3 0.61 -0.04 0.37 0.63 1.0
## V4 0.03 -0.58 0.34 0.66 1.0
## V5 0.01 -0.55 0.30 0.70 1.0
## V6 0.03 -0.60 0.36 0.64 1.0
## V7 -0.58 0.08 0.34 0.66 1.0
## V8 -0.62 -0.10 0.39 0.61 1.1
## V9 -0.59 0.00 0.35 0.65 1.0
## V10 0.07 0.61 0.38 0.62 1.0
## V11 0.03 0.63 0.39 0.61 1.0
## V12 -0.06 0.57 0.33 0.67 1.0
##
## MR1 MR2
## SS loadings 2.21 2.12
## Proportion Var 0.18 0.18
## Cumulative Var 0.18 0.36
## Proportion Explained 0.51 0.49
## Cumulative Proportion 0.51 1.00
##
## With factor correlations of
## MR1 MR2
## MR1 1.00 0.04
## MR2 0.04 1.00
##
## Mean item complexity = 1
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 66 and the objective function was 2.52 with Chi Square of 1246.71
## The degrees of freedom for the model are 43 and the objective function was 0.11
##
## 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 500 with the empirical chi square 43.95 with prob < 0.43
## The total number of observations was 500 with Likelihood Chi Square = 54.57 with prob < 0.11
##
## Tucker Lewis Index of factoring reliability = 0.985
## RMSEA index = 0.023 and the 90 % confidence intervals are 0 0.04
## BIC = -212.66
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## MR1 MR2
## Correlation of (regression) scores with factors 0.88 0.88
## Multiple R square of scores with factors 0.78 0.77
## Minimum correlation of possible factor scores 0.56 0.53fa.sort(f2) #a cleaner output## Factor Analysis using method = minres
## Call: fa(r = my.data, nfactors = 2)
## Standardized loadings (pattern matrix) based upon correlation matrix
## MR1 MR2 h2 u2 com
## V1 0.64 -0.02 0.41 0.59 1.0
## V8 -0.62 -0.10 0.39 0.61 1.1
## V3 0.61 -0.04 0.37 0.63 1.0
## V2 0.59 0.02 0.35 0.65 1.0
## V9 -0.59 0.00 0.35 0.65 1.0
## V7 -0.58 0.08 0.34 0.66 1.0
## V11 0.03 0.63 0.39 0.61 1.0
## V10 0.07 0.61 0.38 0.62 1.0
## V6 0.03 -0.60 0.36 0.64 1.0
## V4 0.03 -0.58 0.34 0.66 1.0
## V12 -0.06 0.57 0.33 0.67 1.0
## V5 0.01 -0.55 0.30 0.70 1.0
##
## MR1 MR2
## SS loadings 2.21 2.12
## Proportion Var 0.18 0.18
## Cumulative Var 0.18 0.36
## Proportion Explained 0.51 0.49
## Cumulative Proportion 0.51 1.00
##
## With factor correlations of
## MR1 MR2
## MR1 1.00 0.04
## MR2 0.04 1.00
##
## Mean item complexity = 1
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 66 and the objective function was 2.52 with Chi Square of 1246.71
## The degrees of freedom for the model are 43 and the objective function was 0.11
##
## 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 500 with the empirical chi square 43.95 with prob < 0.43
## The total number of observations was 500 with Likelihood Chi Square = 54.57 with prob < 0.11
##
## Tucker Lewis Index of factoring reliability = 0.985
## RMSEA index = 0.023 and the 90 % confidence intervals are 0 0.04
## BIC = -212.66
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## MR1 MR2
## Correlation of (regression) scores with factors 0.88 0.88
## Multiple R square of scores with factors 0.78 0.77
## Minimum correlation of possible factor scores 0.56 0.53fa.diagram(f2) #show the structure
corPlot(fa.sort(f2))
fa.plot(f2)
Lets simulate another structure that does not have simple structure
set.seed(47) #reset the seed for reproducibility
my.circ <- sim.item(12,circum=TRUE)
describe(my.circ)## vars n mean sd median trimmed mad min max range skew kurtosis se
## V1 1 500 -0.03 1.00 -0.05 -0.03 1.06 -2.51 2.59 5.09 0.06 -0.42 0.04
## V2 2 500 -0.06 0.98 0.02 -0.03 0.95 -3.57 3.33 6.90 -0.28 0.24 0.04
## V3 3 500 -0.13 1.01 -0.12 -0.15 1.10 -3.01 3.22 6.23 0.17 -0.07 0.05
## V4 4 500 -0.12 0.99 -0.16 -0.16 1.07 -2.43 3.50 5.93 0.36 0.08 0.04
## V5 5 500 -0.04 0.99 -0.05 -0.04 0.99 -2.93 3.93 6.86 0.16 0.38 0.04
## V6 6 500 -0.06 1.06 -0.06 -0.05 0.99 -3.09 3.46 6.56 -0.05 0.15 0.05
## V7 7 500 0.02 1.06 0.02 0.03 1.04 -3.41 3.51 6.92 -0.06 0.24 0.05
## V8 8 500 0.07 1.01 0.06 0.06 0.95 -2.73 3.44 6.17 0.19 0.25 0.05
## V9 9 500 0.06 1.04 0.09 0.05 1.01 -3.65 3.19 6.83 0.07 0.20 0.05
## V10 10 500 0.05 1.02 0.01 0.06 1.07 -4.29 2.76 7.05 -0.19 0.27 0.05
## V11 11 500 0.05 0.98 0.01 0.05 0.94 -3.08 3.26 6.34 0.04 0.01 0.04
## V12 12 500 0.01 1.03 0.06 0.01 1.02 -3.17 2.81 5.99 -0.03 -0.02 0.05corPlot(my.circ)
How many factors? (we put in two, do we get that back)
fa.parallel(my.circ)
## Parallel analysis suggests that the number of factors = 2 and the number of components = 2extract 2 factors
f2.circ <- fa(my.circ,2) #note that we know that the second parameter is nfactors
fa.sort(f2.circ)## Factor Analysis using method = minres
## Call: fa(r = my.circ, nfactors = 2)
## Standardized loadings (pattern matrix) based upon correlation matrix
## MR1 MR2 h2 u2 com
## V1 0.60 -0.22 0.42 0.58 1.3
## V2 0.59 0.15 0.37 0.63 1.1
## V8 -0.58 -0.09 0.34 0.66 1.0
## V7 -0.58 0.10 0.35 0.65 1.1
## V3 0.49 0.40 0.38 0.62 1.9
## V12 0.48 -0.40 0.40 0.60 1.9
## V9 -0.46 -0.37 0.33 0.67 1.9
## V5 -0.09 0.66 0.45 0.55 1.0
## V4 0.11 0.63 0.40 0.60 1.1
## V11 0.11 -0.58 0.35 0.65 1.1
## V10 -0.21 -0.54 0.33 0.67 1.3
## V6 -0.42 0.45 0.39 0.61 2.0
##
## MR1 MR2
## SS loadings 2.32 2.20
## Proportion Var 0.19 0.18
## Cumulative Var 0.19 0.38
## Proportion Explained 0.51 0.49
## Cumulative Proportion 0.51 1.00
##
## With factor correlations of
## MR1 MR2
## MR1 1.00 -0.04
## MR2 -0.04 1.00
##
## Mean item complexity = 1.4
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 66 and the objective function was 2.73 with Chi Square of 1348.83
## The degrees of freedom for the model are 43 and the objective function was 0.11
##
## 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 500 with the empirical chi square 42.45 with prob < 0.5
## The total number of observations was 500 with Likelihood Chi Square = 54.8 with prob < 0.11
##
## Tucker Lewis Index of factoring reliability = 0.986
## RMSEA index = 0.023 and the 90 % confidence intervals are 0 0.04
## BIC = -212.43
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## MR1 MR2
## Correlation of (regression) scores with factors 0.89 0.88
## Multiple R square of scores with factors 0.79 0.78
## Minimum correlation of possible factor scores 0.58 0.57fa.diagram(f2.circ)
corPlot(fa.sort(f2.circ))
fa.plot(f2.circ)
Compare the two plots
par(mfrow=c(1,2)) #make a two column plot
fa.plot(f2,title="simple structure")
fa.plot(f2.circ,title = "circumplex structure")
par(mfrow=c(1,1)) #return to 1 column plotsDo real data show circumplex structure?
Yes, mood.
f2 <- fa(msqR[1:72],2)
fa.plot(f2,labels=colnames(msqR)[1:72],title="2 dimensions of affect at the item level")
Do that graph again, but omit the points, just show the labels
fa.plot(f2,labels=colnames(msqR)[1:72],title="2 dimensions of affect at the item level",show.points=FALSE)
### Real data (continued)
The ability data set has 1525 participants taking 16 ability items
describe(ability)## vars n mean sd median trimmed mad min max range skew kurtosis se
## reason.4 1 1442 0.68 0.47 1 0.72 0 0 1 1 -0.75 -1.44 0.01
## reason.16 2 1463 0.73 0.45 1 0.78 0 0 1 1 -1.02 -0.96 0.01
## reason.17 3 1440 0.74 0.44 1 0.80 0 0 1 1 -1.08 -0.84 0.01
## reason.19 4 1456 0.64 0.48 1 0.68 0 0 1 1 -0.60 -1.64 0.01
## letter.7 5 1441 0.63 0.48 1 0.67 0 0 1 1 -0.56 -1.69 0.01
## letter.33 6 1438 0.61 0.49 1 0.63 0 0 1 1 -0.43 -1.82 0.01
## letter.34 7 1455 0.64 0.48 1 0.68 0 0 1 1 -0.59 -1.65 0.01
## letter.58 8 1438 0.47 0.50 0 0.46 0 0 1 1 0.12 -1.99 0.01
## matrix.45 9 1458 0.55 0.50 1 0.56 0 0 1 1 -0.20 -1.96 0.01
## matrix.46 10 1470 0.57 0.50 1 0.59 0 0 1 1 -0.28 -1.92 0.01
## matrix.47 11 1465 0.64 0.48 1 0.67 0 0 1 1 -0.57 -1.67 0.01
## matrix.55 12 1459 0.39 0.49 0 0.36 0 0 1 1 0.45 -1.80 0.01
## rotate.3 13 1456 0.20 0.40 0 0.13 0 0 1 1 1.48 0.19 0.01
## rotate.4 14 1460 0.22 0.42 0 0.15 0 0 1 1 1.34 -0.21 0.01
## rotate.6 15 1456 0.31 0.46 0 0.27 0 0 1 1 0.80 -1.35 0.01
## rotate.8 16 1460 0.19 0.39 0 0.12 0 0 1 1 1.55 0.41 0.01lowerCor(ability)## rsn.4 rs.16 rs.17 rs.19 ltt.7 lt.33 lt.34 lt.58 mt.45 mt.46 mt.47 mt.55 rtt.3 rtt.4
## reason.4 1.00
## reason.16 0.28 1.00
## reason.17 0.40 0.32 1.00
## reason.19 0.30 0.25 0.34 1.00
## letter.7 0.28 0.27 0.29 0.25 1.00
## letter.33 0.23 0.20 0.26 0.25 0.34 1.00
## letter.34 0.29 0.26 0.29 0.27 0.40 0.37 1.00
## letter.58 0.29 0.21 0.29 0.25 0.33 0.28 0.32 1.00
## matrix.45 0.25 0.18 0.20 0.22 0.20 0.20 0.21 0.19 1.00
## matrix.46 0.25 0.18 0.24 0.18 0.24 0.23 0.27 0.21 0.33 1.00
## matrix.47 0.24 0.24 0.27 0.23 0.27 0.23 0.30 0.23 0.24 0.23 1.00
## matrix.55 0.16 0.15 0.16 0.15 0.14 0.17 0.14 0.23 0.21 0.14 0.21 1.00
## rotate.3 0.23 0.16 0.17 0.18 0.18 0.17 0.19 0.24 0.16 0.15 0.20 0.18 1.00
## rotate.4 0.25 0.20 0.20 0.21 0.23 0.21 0.21 0.27 0.17 0.17 0.20 0.18 0.53 1.00
## rotate.6 0.25 0.20 0.27 0.19 0.20 0.21 0.19 0.26 0.15 0.20 0.18 0.17 0.43 0.45
## rotate.8 0.21 0.16 0.18 0.16 0.13 0.14 0.15 0.22 0.16 0.15 0.17 0.19 0.43 0.44
## rtt.6 rtt.8
## rotate.6 1.00
## rotate.8 0.42 1.00corPlot(ability)
fa.parallel(ability)
## Parallel analysis suggests that the number of factors = 4 and the number of components = 2f1.ab <- fa(ability)
f1.ab## Factor Analysis using method = minres
## Call: fa(r = ability)
## Standardized loadings (pattern matrix) based upon correlation matrix
## MR1 h2 u2 com
## reason.4 0.55 0.30 0.70 1
## reason.16 0.45 0.20 0.80 1
## reason.17 0.54 0.29 0.71 1
## reason.19 0.47 0.22 0.78 1
## letter.7 0.52 0.27 0.73 1
## letter.33 0.48 0.23 0.77 1
## letter.34 0.54 0.29 0.71 1
## letter.58 0.53 0.28 0.72 1
## matrix.45 0.41 0.17 0.83 1
## matrix.46 0.43 0.18 0.82 1
## matrix.47 0.47 0.22 0.78 1
## matrix.55 0.35 0.12 0.88 1
## rotate.3 0.50 0.25 0.75 1
## rotate.4 0.55 0.30 0.70 1
## rotate.6 0.53 0.28 0.72 1
## rotate.8 0.46 0.21 0.79 1
##
## MR1
## SS loadings 3.81
## Proportion Var 0.24
##
## Mean item complexity = 1
## Test of the hypothesis that 1 factor is sufficient.
##
## The degrees of freedom for the null model are 120 and the objective function was 3.28 with Chi Square of 4973.83
## The degrees of freedom for the model are 104 and the objective function was 0.7
##
## The root mean square of the residuals (RMSR) is 0.07
## The df corrected root mean square of the residuals is 0.07
##
## The harmonic number of observations is 1426 with the empirical chi square 1476.62 with prob < 3e-241
## The total number of observations was 1525 with Likelihood Chi Square = 1063.25 with prob < 9.5e-159
##
## Tucker Lewis Index of factoring reliability = 0.772
## RMSEA index = 0.078 and the 90 % confidence intervals are 0.074 0.082
## BIC = 300.96
## Fit based upon off diagonal values = 0.93
## Measures of factor score adequacy
## MR1
## Correlation of (regression) scores with factors 0.91
## Multiple R square of scores with factors 0.84
## Minimum correlation of possible factor scores 0.67f2.ab <- fa(ability,2)
f2.ab## Factor Analysis using method = minres
## Call: fa(r = ability, nfactors = 2)
## Standardized loadings (pattern matrix) based upon correlation matrix
## MR1 MR2 h2 u2 com
## reason.4 0.51 0.08 0.31 0.69 1.0
## reason.16 0.45 0.03 0.21 0.79 1.0
## reason.17 0.57 0.00 0.32 0.68 1.0
## reason.19 0.48 0.02 0.24 0.76 1.0
## letter.7 0.60 -0.05 0.33 0.67 1.0
## letter.33 0.52 -0.01 0.26 0.74 1.0
## letter.34 0.63 -0.07 0.35 0.65 1.0
## letter.58 0.45 0.12 0.28 0.72 1.2
## matrix.45 0.41 0.02 0.18 0.82 1.0
## matrix.46 0.45 0.01 0.20 0.80 1.0
## matrix.47 0.46 0.04 0.23 0.77 1.0
## matrix.55 0.24 0.15 0.12 0.88 1.7
## rotate.3 -0.03 0.72 0.49 0.51 1.0
## rotate.4 0.02 0.71 0.52 0.48 1.0
## rotate.6 0.09 0.59 0.40 0.60 1.0
## rotate.8 -0.03 0.65 0.40 0.60 1.0
##
## MR1 MR2
## SS loadings 2.96 1.91
## Proportion Var 0.19 0.12
## Cumulative Var 0.19 0.30
## Proportion Explained 0.61 0.39
## Cumulative Proportion 0.61 1.00
##
## With factor correlations of
## MR1 MR2
## MR1 1.00 0.52
## MR2 0.52 1.00
##
## Mean item complexity = 1.1
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 120 and the objective function was 3.28 with Chi Square of 4973.83
## The degrees of freedom for the model are 89 and the objective function was 0.18
##
## 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 1426 with the empirical chi square 288.87 with prob < 5.9e-23
## The total number of observations was 1525 with Likelihood Chi Square = 277.64 with prob < 2.9e-21
##
## Tucker Lewis Index of factoring reliability = 0.948
## RMSEA index = 0.037 and the 90 % confidence intervals are 0.032 0.042
## BIC = -374.71
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## MR1 MR2
## Correlation of (regression) scores with factors 0.90 0.89
## Multiple R square of scores with factors 0.81 0.79
## Minimum correlation of possible factor scores 0.63 0.58fa.diagram(f2.ab)