Psychometric theory: Latent variable is “true score”
Observed score (\(Y\)) is a function of true score and error
\(Y_i = T_i + e_{ij}\)
FA partitions variance in each item into
Big idea in factor analysis: Any correlations between items are due to what they have in common (i.e. a common latent factor)
x1 x2 x3 x4 x5 x6
x1 1.871 0.912 0.944 0.312 0.344 0.226
x2 0.912 1.830 0.994 0.367 0.385 0.315
x3 0.944 0.994 2.059 0.287 0.362 0.267
x4 0.312 0.367 0.287 2.150 1.112 1.091
x5 0.344 0.385 0.362 1.112 2.117 1.041
x6 0.226 0.315 0.267 1.091 1.041 2.016
Loadings:
PA1 PA2
x1 0.535 0.422
x2 0.590 0.420
x3 0.549 0.446
x4 0.605 -0.417
x5 0.607 -0.368
x6 0.573 -0.424
PA1 PA2
SS loadings 1.999 1.043
Proportion Var 0.333 0.174
Cumulative Var 0.333 0.507
x1 x2 x3 x4 x5 x6
0.4635790 0.5250197 0.5010631 0.5416307 0.5028439 0.5079958
Item | F1 | F2 | F3 |
---|---|---|---|
1 | 0.618 | 0.094 | -0.049 |
2 | 0.440 | -0.075 | 0.065 |
3 | 0.671 | 0.037 | 0.041 |
4 | 0.031 | 0.731 | -0.079 |
5 | 0.126 | 0.705 | 0.053 |
6 | 0.265 | 0.296 | 0.603 |
\(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) |
Item | F1 | F2 | F3 |
---|---|---|---|
1 | 0.620 | 0 | 0 |
2 | 0.450 | 0 | 0 |
3 | 0.665 | 0 | 0 |
4 | 0 | 0.725 | 0 |
5 | 0 | 0.689 | 0 |
6 | 0 | 0 | 0.613 |
\(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) |
\[Y_i = T_i + e_{ij}\]
\(\textbf{A} \; \textbf{R}_F \; \textbf{A}'=\)
\(\begin{bmatrix}a_{1,1} & a_{1,2}\\ a_{2,1} & a_{2,2}\\ a_{3,1} & a_{3,2}\\ a_{4,1} & a_{4,2}\\ a_{5,1} & a_{5,2}\\ a_{6,1} & a_{6,2}\\\end{bmatrix}\; \begin{bmatrix}\sigma^2_{F_1} & \sigma_{F_1F_2} \\ \sigma_{F_2F_1} & \sigma^2_{F_2}\\ \end{bmatrix} \; \begin{bmatrix}a_{1,1} & a_{2,1} & a_{3,1} & a_{4,1} & a_{5,1} & a_{6,1} \\ a_{1,2} & a_{2,2} & a_{3,2} & a_{4,2} & a_{5,2} & a_{6,2}\\ \end{bmatrix}\)
\(\textbf{D} = \begin{bmatrix}d_1 & 0 & 0 & 0 & 0 & 0\\0 & d_2 & 0 & 0 & 0 & 0\\ 0 & 0 & d_3 & 0 & 0 & 0\\ 0 & 0 & 0 & d_4 & 0 & 0\\ 0 & 0 & 0 & 0 & d_5 & 0\\ 0 & 0 & 0 & 0 & 0 & d_6\\ \end{bmatrix}\)
\(\textbf{R}_{YY} = \begin{bmatrix} 1 & r_{12} & r_{13} & r_{14} & r_{15} & r_{16}\\ r_{21} & 1 & r_{23} & r_{24} & r_{25} & r_{26}\\ r_{31} & r_{32} & 1 & r_{34} & r_{35} & r_{36}\\ r_{41} & r_{42} & r_{43} & 1 & r_{45} & r_{46}\\ r_{51} & r_{52} & r_{53} & r_{54} & 1 & r_{56}\\ r_{61} & r_{62} & r_{63} & r_{64} & r_{65} & 1\\ \end{bmatrix}\)
\(\textbf{S}_{YY} = \begin{bmatrix} s_1^2 & s_{12} & s_{13} & s_{14} & s_{15} & s_{16}\\ s_{21} & s_2^2 & s_{23} & s_{24} & s_{25} & s_{26}\\ s_{31} & s_{32} & s_3^2 & s_{34} & s_{35} & s_{36}\\ s_{41} & s_{42} & s_{43} & s_4^2 & s_{45} & s_{46}\\ s_{51} & s_{52} & s_{53} & s_{54} & s_5^2 & s_{56}\\ s_{61} & s_{62} & s_{63} & s_{64} & s_{65} & s_6^2\\ \end{bmatrix}\)
\(\textbf{R}_{reduced} = \begin{bmatrix} \color{OrangeRed}{h_1^2} & r_{12} & r_{13} & r_{14} & r_{15} & r_{16}\\ r_{21} & \color{OrangeRed}{h_2^2} & r_{23} & r_{24} & r_{25} & r_{26}\\ r_{31} & r_{32} & \color{OrangeRed}{h_3^2} & r_{34} & r_{35} & r_{36}\\ r_{41} & r_{42} & r_{43} & \color{OrangeRed}{h_4^2} & r_{45} & r_{46}\\ r_{51} & r_{52} & r_{53} & r_{54} & \color{OrangeRed}{h_5^2} & r_{56}\\ r_{61} & r_{62} & r_{63} & r_{64} & r_{65} & \color{OrangeRed}{h_6^2}\\ \end{bmatrix}\)
The iterative process sometimes causes problems
Causes: too few cases, bad start values, too many factors, too few factors, non-linear relationships between factors
Some solutions:
Parallel analysis suggests that the number of factors = 2 and the number of components = NA
Parallel analysis suggests that the number of factors = 2 and the number of components = NA
Number of factors
Call: vss(x = x, n = n, rotate = rotate, diagonal = diagonal, fm = fm,
n.obs = n.obs, plot = FALSE, title = title, use = use, cor = cor)
VSS complexity 1 achieves a maximimum of Although the vss.max shows 5 factors, it is probably more reasonable to think about 2 factors
VSS complexity 2 achieves a maximimum of 0.85 with 3 factors
The Velicer MAP achieves a minimum of 0.1 with 2 factors
Empirical BIC achieves a minimum of -26.76 with 2 factors
Sample Size adjusted BIC achieves a minimum of -13.16 with 2 factors
Statistics by number of factors
vss1 vss2 map dof chisq prob sqresid fit RMSEA BIC SABIC complex
1 0.56 0.00 0.12 9 5.8e+02 3.1e-118 4.2 0.56 0.25 514 542 1.0
2 0.79 0.85 0.10 4 1.8e+00 7.8e-01 1.5 0.85 0.00 -26 -13 1.1
3 0.69 0.85 0.22 0 2.8e-02 NA 1.1 0.88 NA NA NA 1.3
4 0.79 0.85 0.42 -3 1.6e-09 NA 1.4 0.86 NA NA NA 1.1
5 0.79 0.84 1.00 -5 0.0e+00 NA 1.4 0.86 NA NA NA 1.1
6 0.75 0.81 NA -6 2.6e+01 NA 1.8 0.81 NA NA NA 1.1
eChisq SRMR eCRMS eBIC
1 1.0e+03 1.8e-01 0.24 940
2 8.7e-01 5.4e-03 0.01 -27
3 1.7e-02 7.5e-04 NA NA
4 8.5e-10 1.7e-07 NA NA
5 2.7e-16 9.5e-11 NA NA
6 3.6e+01 3.4e-02 NA NA
“The total number of observations was 1000 with Likelihood Chi Square = 1.77 with prob < 0.78”
Factor Analysis using method = ml
Call: fa(r = FA_data, nfactors = 2, rotate = "none", SMC = TRUE, warnings = TRUE,
fm = "ml")
Standardized loadings (pattern matrix) based upon correlation matrix
ML1 ML2 h2 u2 com
x1 0.52 0.44 0.46 0.54 1.9
x2 0.58 0.44 0.53 0.47 1.9
x3 0.53 0.46 0.50 0.50 2.0
x4 0.62 -0.40 0.54 0.46 1.7
x5 0.62 -0.35 0.50 0.50 1.6
x6 0.59 -0.41 0.51 0.49 1.8
ML1 ML2
SS loadings 2.00 1.04
Proportion Var 0.33 0.17
Cumulative Var 0.33 0.51
Proportion Explained 0.66 0.34
Cumulative Proportion 0.66 1.00
Mean item complexity = 1.8
Test of the hypothesis that 2 factors are sufficient.
The degrees of freedom for the null model are 15 and the objective function was 1.49 with Chi Square of 1483.93
The degrees of freedom for the model are 4 and the objective function was 0
The root mean square of the residuals (RMSR) is 0.01
The df corrected root mean square of the residuals is 0.01
The harmonic number of observations is 1000 with the empirical chi square 0.87 with prob < 0.93
The total number of observations was 1000 with Likelihood Chi Square = 1.77 with prob < 0.78
Tucker Lewis Index of factoring reliability = 1.006
RMSEA index = 0 and the 90 % confidence intervals are 0 0.032
BIC = -25.86
Fit based upon off diagonal values = 1
Measures of factor score adequacy
ML1 ML2
Correlation of (regression) scores with factors 0.90 0.82
Multiple R square of scores with factors 0.80 0.68
Minimum correlation of possible factor scores 0.61 0.36