9 Cognitive variables discussed by Tucker and Lewis (1973)

Tucker and Lewis (1973) introduced a reliability coefficient for ML factor analysis. Their example data set was previously reported by Tucker (1958) and taken from Thurstone and Thurstone (1941). The correlation matrix is a 9 x 9 for 710 subjects and has two correlated factors of ability: Word Fluency and Verbal.

data(Tucker)

Format

A data frame with 9 observations on the following 9 variables.

t42: Prefixes
t54: Suffixes
t45: Chicago Reading Test: Vocabulary
t46: Chicago Reading Test: Sentences
t23: First and last letters
t24: First letters
t27: Four letter words
t10: Completion
t51: Same or Opposite

Details

The correlation matrix from Tucker (1958) was used in Tucker and Lewis (1973) for the Tucker-Lewis Index of factoring reliability.

Source

Tucker, Ledyard (1958) An inter-battery method of factor analysis, Psychometrika, 23, 111-136.

References

L.~Tucker and C.~Lewis. (1973) A reliability coefficient for maximum likelihood factor analysis. Psychometrika, 38(1):1--10.

F.~J. Floyd and K.~F. Widaman. (1995) Factor analysis in the development and refinement of clinical assessment instruments., Psychological Assessment, 7(3):286 -- 299.

Examples

data(Tucker)
fa(Tucker,2,n.obs=710)
#> Factor Analysis using method =  minres
#> Call: fa(r = Tucker, nfactors = 2, n.obs = 710)
#> Standardized loadings (pattern matrix) based upon correlation matrix
#>       MR1   MR2   h2   u2 com
#> t42 -0.04  0.71 0.48 0.52 1.0
#> t54  0.02  0.71 0.51 0.49 1.0
#> t45  0.92 -0.04 0.82 0.18 1.0
#> t46  0.87 -0.06 0.71 0.29 1.0
#> t23 -0.02  0.71 0.50 0.50 1.0
#> t24  0.00  0.74 0.55 0.45 1.0
#> t27  0.11  0.59 0.42 0.58 1.1
#> t10  0.76  0.13 0.68 0.32 1.1
#> t51  0.78  0.05 0.65 0.35 1.0
#> 
#>                        MR1  MR2
#> SS loadings           2.84 2.47
#> Proportion Var        0.32 0.27
#> Cumulative Var        0.32 0.59
#> Proportion Explained  0.54 0.46
#> Cumulative Proportion 0.54 1.00
#> 
#>  With factor correlations of 
#>      MR1  MR2
#> MR1 1.00 0.43
#> MR2 0.43 1.00
#> 
#> Mean item complexity =  1
#> Test of the hypothesis that 2 factors are sufficient.
#> 
#> The degrees of freedom for the null model are  36  and the objective function was  4.49 with Chi Square of  3165.93
#> The degrees of freedom for the model are 19  and the objective function was  0.07 
#> 
#> The root mean square of the residuals (RMSR) is  0.02 
#> The df corrected root mean square of the residuals is  0.03 
#> 
#> The harmonic number of observations is  710 with the empirical chi square  22.23  with prob <  0.27 
#> The total number of observations was  710  with Likelihood Chi Square =  50.45  with prob <  0.00011 
#> 
#> Tucker Lewis Index of factoring reliability =  0.981
#> RMSEA index =  0.049  and the 90 % confidence intervals are  0.032 0.065
#> BIC =  -74.29
#> Fit based upon off diagonal values = 1
#> Measures of factor score adequacy             
#>                                                 MR1  MR2
#> Correlation of scores with factors             0.96 0.91
#> Multiple R square of scores with factors       0.92 0.84
#> Minimum correlation of possible factor scores  0.83 0.67
omega(Tucker,2)
#> 
#> Three factors are required for identification -- general factor loadings set to be equal. 
#> Proceed with caution. 
#> Think about redoing the analysis with alternative values of the 'option' setting.
#> Omega 
#> Call: omega(m = Tucker, nfactors = 2)
#> Alpha:                 0.86 
#> G.6:                   0.88 
#> Omega Hierarchical:    0.54 
#> Omega H asymptotic:    0.6 
#> Omega Total            0.9 
#> 
#> Schmid Leiman Factor loadings greater than  0.2 
#>        g   F1*   F2*   h2   u2   p2
#> t42 0.44        0.53 0.48 0.52 0.40
#> t54 0.48        0.54 0.51 0.49 0.44
#> t45 0.58  0.69       0.82 0.18 0.41
#> t46 0.53  0.65       0.71 0.29 0.39
#> t23 0.46        0.54 0.50 0.50 0.42
#> t24 0.48        0.56 0.55 0.45 0.43
#> t27 0.46        0.45 0.42 0.58 0.51
#> t10 0.58  0.57       0.68 0.32 0.50
#> t51 0.55  0.59       0.65 0.35 0.46
#> 
#> With eigenvalues of:
#>   g F1* F2* 
#> 2.3 1.6 1.4 
#> 
#> general/max  1.46   max/min =   1.15
#> mean percent general =  0.44    with sd =  0.04 and cv of  0.1 
#> Explained Common Variance of the general factor =  0.44 
#> 
#> The degrees of freedom are 19  and the fit is  0.07 
#> 
#> The root mean square of the residuals is  0.02 
#> The df corrected root mean square of the residuals is  0.03
#> 
#> Compare this with the adequacy of just a general factor and no group factors
#> The degrees of freedom for just the general factor are 27  and the fit is  1.94 
#> 
#> The root mean square of the residuals is  0.22 
#> The df corrected root mean square of the residuals is  0.25 
#> 
#> Measures of factor score adequacy             
#>                                                  g  F1*  F2*
#> Correlation of scores with factors            0.74 0.78 0.74
#> Multiple R square of scores with factors      0.55 0.60 0.55
#> Minimum correlation of factor score estimates 0.10 0.21 0.10
#> 
#>  Total, General and Subset omega for each subset
#>                                                  g  F1*  F2*
#> Omega total for total scores and subscales    0.90 0.91 0.83
#> Omega general for total scores and subscales  0.54 0.40 0.37
#> Omega group for total scores and subscales    0.34 0.51 0.46