Exploratory Factor Analysis
Mariana O. S. Silva
Last Updated: maio 15, 2020
tl;dr
Exploratory Factor Analysis (EFA) is a statistical approach for determining the correlation among the variables in a dataset. EFA can statistically explain the variance among the observed variable and condense a set of the observed variable into the unobserved variable called factors (a grouping of variables based on strong correlations). Observed variables are modeled as a linear combination of factors and error terms. Each factor explains a particular amount of variance in the observed variables and can help in data interpretations by reducing the number of variables.
This dataset contains 9 regions with 27 music genre networks, in total, containing 6 different topological edge measures. The edge measures were the following:
- preferential_attatchment = probability of connecting in the future (the more neighbors in common, the more likely they are to connect in the future)
- common_neighbors = number of common neighbors on the network
- neighborhood_overlap = fraction of the number of common neighbors (edges with low neighborhood_overlap consist of nodes traveling in “social circles”, having almost nobody in common = local bridges)
- edge_betweenness = number of the shortest paths that go through an edge in a graph or network (edges with a high edge betweenness centrality score represent a bridge-like connector between two parts of a network and the removal of which may affect the communication between many pairs of nodes through the shortest paths between them)
- resource_allocation = fraction of a resource (for example, information or something else) that a node can send to another node through its common neighbors
- weight = number of collaborations between two music genres (which is statistically correlated with the possibility that these two genres will collaborate in the future)
Global
Choosing the Number of Factors
Before conducting the EFA, we determine the number of factors via the Parallel Analysis to find an acceptable number of factors. Parallel Analysis is an alternative technique that compares the scree plot (line plot of the eigenvalues of factors) of the observed data with that of a random data matrix of the same size as the original.
The console would show the maximum number of factors we can consider.
## Parallel analysis suggests that the number of factors = 2 and the number of components = NA## Parallel analysis suggests that the number of factors = 3 and the number of components = NA
## Parallel analysis suggests that the number of factors = 3 and the number of components = NAIn the scree plot generated, the blue line shows eigenvalues of actual data and the two red lines (placed on top of each other) show simulated and resampled data. Here we look at the large drops in the actual data and spot the point where it levels off to the right. Also, we locate the point of inflection – the point where the gap between simulated data and actual data tends to be minimum.
Factor Analysis
Now that we have the probable number of factors, we can carry out exploratory factor analysis. Here, we will use the fa() function of the psych package. The following are the arguments we will provide:
- r – Raw data or correlation or covariance matrix
- nfactors – Number of factors to extract
- rotate – Type of rotation (
VarimaxandObliminare the most popular) - fm – Factoring method (e.g.,
Minimum Residual (OLS),Maximum Liklihood,Principal Axis, etc)
In our case, we will select oblique rotation (rotate = "oblimin") as we believe that there is a correlation in the factors. Note that Varimax rotation is used under the assumption that the factors are completely uncorrelated. We will use Ordinary Least Squared/ols factoring (fm = "ols"), as it is known to provide results similar to Maximum Likelihood without assuming a multivariate normal distribution and derives solutions through iterative eigendecomposition like a principal axis.
Adequacy Test
First, we should check the root mean square of residuals (RMSR). An acceptable value should be closer to 0. Finally, we must check the Tucker-Lewis Index (TLI). An acceptable value must be greater over 0.9.
## Factor Analysis using method = ols
## Call: fa(r = data_2017, nfactors = 2, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 h2 u2 com
## Weight 0.76 -0.04 0.55 0.450 1.0
## common_neighbors 0.82 0.25 0.98 0.019 1.2
## neighborhood_overlap 0.20 0.84 0.94 0.063 1.1
## resource_allocation 0.92 0.08 0.93 0.071 1.0
## preferential_attatchment 1.01 -0.12 0.90 0.102 1.0
## edge_betweenness 0.20 -0.70 0.37 0.632 1.2
##
## [,1] [,2]
## SS loadings 3.30 1.37
## Proportion Var 0.55 0.23
## Cumulative Var 0.55 0.78
## Proportion Explained 0.71 0.29
## Cumulative Proportion 0.71 1.00
##
## With factor correlations of
## [,1] [,2]
## [1,] 1.00 0.57
## [2,] 0.57 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 15 and the objective function was 7.52 with Chi Square of 4213.41
## The degrees of freedom for the model are 4 and the objective function was 1.01
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.04
##
## The harmonic number of observations is 564 with the empirical chi square 6.9 with prob < 0.14
## The total number of observations was 564 with Likelihood Chi Square = 563.86 with prob < 1e-120
##
## Tucker Lewis Index of factoring reliability = 0.499
## RMSEA index = 0.498 and the 90 % confidence intervals are 0.464 0.534
## BIC = 538.52
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2]
## Correlation of (regression) scores with factors 0.99 0.97
## Multiple R square of scores with factors 0.99 0.95
## Minimum correlation of possible factor scores 0.98 0.89## Factor Analysis using method = ols
## Call: fa(r = data_2018, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight 0.78 0.05 -0.07 0.63 0.3741 1.0
## common_neighbors 0.10 0.71 0.32 1.00 0.0047 1.4
## neighborhood_overlap 0.23 0.03 0.82 0.95 0.0534 1.2
## resource_allocation 0.91 0.05 0.08 1.00 0.0011 1.0
## preferential_attatchment 0.02 1.02 -0.08 0.99 0.0062 1.0
## edge_betweenness 0.24 -0.07 -0.68 0.38 0.6170 1.3
##
## [,1] [,2] [,3]
## SS loadings 1.74 1.80 1.40
## Proportion Var 0.29 0.30 0.23
## Cumulative Var 0.29 0.59 0.82
## Proportion Explained 0.35 0.36 0.28
## Cumulative Proportion 0.35 0.72 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.86 0.5
## [2,] 0.86 1.00 0.5
## [3,] 0.50 0.50 1.0
##
## Mean item complexity = 1.2
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.55 with Chi Square of 4374
## The degrees of freedom for the model are 0 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 NA
##
## The harmonic number of observations is 583 with the empirical chi square 0.06 with prob < NA
## The total number of observations was 583 with Likelihood Chi Square = 2.85 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2] [,3]
## Correlation of (regression) scores with factors 1 1.00 0.99
## Multiple R square of scores with factors 1 1.00 0.97
## Minimum correlation of possible factor scores 1 0.99 0.95## Factor Analysis using method = ols
## Call: fa(r = data_2019, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight -0.02 0.76 -0.04 0.53 0.47406 1.0
## common_neighbors 0.69 0.11 0.33 1.00 0.00441 1.5
## neighborhood_overlap 0.00 0.19 0.85 0.93 0.07432 1.1
## resource_allocation 0.12 0.86 0.08 1.00 0.00057 1.1
## preferential_attatchment 1.01 0.03 -0.08 1.00 0.00482 1.0
## edge_betweenness -0.09 0.27 -0.65 0.34 0.66067 1.4
##
## [,1] [,2] [,3]
## SS loadings 1.78 1.60 1.41
## Proportion Var 0.30 0.27 0.23
## Cumulative Var 0.30 0.56 0.80
## Proportion Explained 0.37 0.33 0.29
## Cumulative Proportion 0.37 0.71 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.85 0.51
## [2,] 0.85 1.00 0.51
## [3,] 0.51 0.51 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.17 with Chi Square of 5053.6
## The degrees of freedom for the model are 0 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 NA
##
## The harmonic number of observations is 709 with the empirical chi square 0.08 with prob < NA
## The total number of observations was 709 with Likelihood Chi Square = 1.67 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2] [,3]
## Correlation of (regression) scores with factors 1.00 1 0.99
## Multiple R square of scores with factors 1.00 1 0.97
## Minimum correlation of possible factor scores 0.99 1 0.94Australia
Choosing the Number of Factors
## 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
## Parallel analysis suggests that the number of factors = 2 and the number of components = NAFactor Analysis
Adequacy Test
## Factor Analysis using method = ols
## Call: fa(r = data_2017, nfactors = 2, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 h2 u2 com
## Weight 0.84 -0.15 0.59 0.412 1.1
## common_neighbors 0.78 0.30 0.98 0.021 1.3
## neighborhood_overlap 0.22 0.83 0.95 0.050 1.1
## resource_allocation 0.88 0.11 0.91 0.088 1.0
## preferential_attatchment 1.00 -0.06 0.93 0.073 1.0
## edge_betweenness 0.19 -0.70 0.37 0.632 1.2
##
## [,1] [,2]
## SS loadings 3.30 1.42
## Proportion Var 0.55 0.24
## Cumulative Var 0.55 0.79
## Proportion Explained 0.70 0.30
## Cumulative Proportion 0.70 1.00
##
## With factor correlations of
## [,1] [,2]
## [1,] 1.00 0.59
## [2,] 0.59 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 15 and the objective function was 7.78 with Chi Square of 3952.04
## The degrees of freedom for the model are 4 and the objective function was 0.85
##
## 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 512 with the empirical chi square 4.92 with prob < 0.3
## The total number of observations was 512 with Likelihood Chi Square = 428.95 with prob < 1.5e-91
##
## Tucker Lewis Index of factoring reliability = 0.594
## RMSEA index = 0.456 and the 90 % confidence intervals are 0.42 0.493
## BIC = 404
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2]
## Correlation of (regression) scores with factors 0.99 0.98
## Multiple R square of scores with factors 0.98 0.97
## Minimum correlation of possible factor scores 0.95 0.94## Factor Analysis using method = ols
## Call: fa(r = data_2018, nfactors = 2, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 h2 u2 com
## Weight 0.89 -0.17 0.66 0.339 1.1
## common_neighbors 0.81 0.29 0.98 0.020 1.2
## neighborhood_overlap 0.31 0.74 0.88 0.123 1.3
## resource_allocation 0.92 0.07 0.92 0.082 1.0
## preferential_attatchment 0.95 -0.03 0.87 0.130 1.0
## edge_betweenness 0.16 -0.73 0.44 0.556 1.1
##
## [,1] [,2]
## SS loadings 3.43 1.32
## Proportion Var 0.57 0.22
## Cumulative Var 0.57 0.79
## Proportion Explained 0.72 0.28
## Cumulative Proportion 0.72 1.00
##
## With factor correlations of
## [,1] [,2]
## [1,] 1.00 0.51
## [2,] 0.51 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 15 and the objective function was 7.77 with Chi Square of 3962.07
## The degrees of freedom for the model are 4 and the objective function was 1.42
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.05
##
## The harmonic number of observations is 514 with the empirical chi square 9.21 with prob < 0.056
## The total number of observations was 514 with Likelihood Chi Square = 723.36 with prob < 3e-155
##
## Tucker Lewis Index of factoring reliability = 0.315
## RMSEA index = 0.592 and the 90 % confidence intervals are 0.556 0.629
## BIC = 698.39
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2]
## Correlation of (regression) scores with factors 0.99 0.97
## Multiple R square of scores with factors 0.99 0.94
## Minimum correlation of possible factor scores 0.98 0.87## Factor Analysis using method = ols
## Call: fa(r = data_2019, nfactors = 2, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 h2 u2 com
## Weight 0.83 -0.13 0.58 0.418 1.1
## common_neighbors 0.77 0.32 0.98 0.019 1.3
## neighborhood_overlap 0.15 0.83 0.86 0.138 1.1
## resource_allocation 0.96 0.05 0.97 0.030 1.0
## preferential_attatchment 0.95 -0.05 0.85 0.146 1.0
## edge_betweenness 0.20 -0.62 0.28 0.716 1.2
##
## [,1] [,2]
## SS loadings 3.26 1.28
## Proportion Var 0.54 0.21
## Cumulative Var 0.54 0.76
## Proportion Explained 0.72 0.28
## Cumulative Proportion 0.72 1.00
##
## With factor correlations of
## [,1] [,2]
## [1,] 1.00 0.58
## [2,] 0.58 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 15 and the objective function was 7.39 with Chi Square of 4234.66
## The degrees of freedom for the model are 4 and the objective function was 1.71
##
## The root mean square of the residuals (RMSR) is 0.03
## The df corrected root mean square of the residuals is 0.05
##
## The harmonic number of observations is 577 with the empirical chi square 11.74 with prob < 0.019
## The total number of observations was 577 with Likelihood Chi Square = 978.3 with prob < 1.8e-210
##
## Tucker Lewis Index of factoring reliability = 0.132
## RMSEA index = 0.65 and the 90 % confidence intervals are 0.616 0.685
## BIC = 952.86
## Fit based upon off diagonal values = 1Brazil
Choosing the Number of Factors
## Parallel analysis suggests that the number of factors = 2 and the number of components = NA## Parallel analysis suggests that the number of factors = 3 and the number of components = NA
## Parallel analysis suggests that the number of factors = 3 and the number of components = NAFactor Analysis
Adequacy Test
## Factor Analysis using method = ols
## Call: fa(r = data_2017, nfactors = 2, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 h2 u2 com
## Weight 0.85 -0.11 0.64 0.360 1.0
## common_neighbors 0.80 0.30 0.97 0.032 1.3
## neighborhood_overlap 0.17 0.89 0.98 0.015 1.1
## resource_allocation 0.89 0.13 0.92 0.079 1.0
## preferential_attatchment 1.01 -0.10 0.93 0.074 1.0
## edge_betweenness 0.18 -0.75 0.45 0.548 1.1
##
## [,1] [,2]
## SS loadings 3.32 1.57
## Proportion Var 0.55 0.26
## Cumulative Var 0.55 0.82
## Proportion Explained 0.68 0.32
## Cumulative Proportion 0.68 1.00
##
## With factor correlations of
## [,1] [,2]
## [1,] 1.00 0.51
## [2,] 0.51 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 15 and the objective function was 7.78 with Chi Square of 3496.47
## The degrees of freedom for the model are 4 and the objective function was 0.82
##
## 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 453 with the empirical chi square 4.42 with prob < 0.35
## The total number of observations was 453 with Likelihood Chi Square = 366.16 with prob < 5.7e-78
##
## Tucker Lewis Index of factoring reliability = 0.609
## RMSEA index = 0.447 and the 90 % confidence intervals are 0.409 0.487
## BIC = 341.69
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2]
## Correlation of (regression) scores with factors 0.99 1.00
## Multiple R square of scores with factors 0.98 0.99
## Minimum correlation of possible factor scores 0.95 0.99## Factor Analysis using method = ols
## Call: fa(r = data_2018, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight 0.54 -0.02 0.29 0.50 0.4983 1.5
## common_neighbors 0.85 0.28 -0.03 1.00 0.0041 1.2
## neighborhood_overlap 0.07 0.89 0.13 0.93 0.0650 1.1
## resource_allocation 0.68 0.20 0.29 0.91 0.0920 1.6
## preferential_attatchment 1.07 -0.13 -0.05 0.98 0.0237 1.0
## edge_betweenness 0.04 -0.73 0.22 0.49 0.5050 1.2
##
## [,1] [,2] [,3]
## SS loadings 2.86 1.59 0.36
## Proportion Var 0.48 0.26 0.06
## Cumulative Var 0.48 0.74 0.80
## Proportion Explained 0.59 0.33 0.08
## Cumulative Proportion 0.59 0.92 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.48 0.42
## [2,] 0.48 1.00 0.22
## [3,] 0.42 0.22 1.00
##
## Mean item complexity = 1.3
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 6.71 with Chi Square of 3490.18
## The degrees of freedom for the model are 0 and the objective function was 0.01
##
## The root mean square of the residuals (RMSR) is 0
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 524 with the empirical chi square 0.03 with prob < NA
## The total number of observations was 524 with Likelihood Chi Square = 4.91 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2] [,3]
## Correlation of (regression) scores with factors 1.00 0.98 0.80
## Multiple R square of scores with factors 0.99 0.95 0.64
## Minimum correlation of possible factor scores 0.98 0.91 0.28## Factor Analysis using method = ols
## Call: fa(r = data_2019, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight 0.27 -0.13 0.61 0.63 0.3744 1.5
## common_neighbors 0.67 0.35 0.16 0.99 0.0130 1.6
## neighborhood_overlap -0.03 0.84 0.23 0.92 0.0846 1.2
## resource_allocation 0.05 0.10 0.89 0.98 0.0241 1.0
## preferential_attatchment 1.00 -0.07 0.03 1.00 0.0037 1.0
## edge_betweenness -0.10 -0.73 0.22 0.46 0.5415 1.2
##
## [,1] [,2] [,3]
## SS loadings 1.85 1.50 1.60
## Proportion Var 0.31 0.25 0.27
## Cumulative Var 0.31 0.56 0.83
## Proportion Explained 0.37 0.30 0.32
## Cumulative Proportion 0.37 0.68 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.35 0.81
## [2,] 0.35 1.00 0.48
## [3,] 0.81 0.48 1.00
##
## Mean item complexity = 1.3
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 6.78 with Chi Square of 2631.74
## The degrees of freedom for the model are 0 and the objective function was 0.01
##
## The root mean square of the residuals (RMSR) is 0
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 392 with the empirical chi square 0.02 with prob < NA
## The total number of observations was 392 with Likelihood Chi Square = 4.06 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2] [,3]
## Correlation of (regression) scores with factors 1.00 0.98 0.99
## Multiple R square of scores with factors 1.00 0.95 0.97
## Minimum correlation of possible factor scores 0.99 0.91 0.95Canada
Choosing the Number of Factors
## Parallel analysis suggests that the number of factors = 3 and the number of components = NA## Parallel analysis suggests that the number of factors = 3 and the number of components = NA
## Parallel analysis suggests that the number of factors = 3 and the number of components = NAFactor Analysis
Adequacy Test
## Factor Analysis using method = ols
## Call: fa(r = data_2017, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight 0.09 0.62 -0.09 0.44 0.5638 1.1
## common_neighbors 0.77 0.08 0.27 1.00 0.0023 1.3
## neighborhood_overlap 0.02 0.33 0.76 0.95 0.0451 1.4
## resource_allocation 0.14 0.85 0.08 1.00 -0.0047 1.1
## preferential_attatchment 1.01 0.03 -0.10 0.98 0.0227 1.0
## edge_betweenness -0.11 0.22 -0.72 0.47 0.5307 1.2
##
## [,1] [,2] [,3]
## SS loadings 1.95 1.53 1.36
## Proportion Var 0.32 0.26 0.23
## Cumulative Var 0.32 0.58 0.81
## Proportion Explained 0.40 0.32 0.28
## Cumulative Proportion 0.40 0.72 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.83 0.48
## [2,] 0.83 1.00 0.47
## [3,] 0.48 0.47 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.27 with Chi Square of 3900.18
## The degrees of freedom for the model are 0 and the objective function was 0.02
##
## The root mean square of the residuals (RMSR) is 0
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 540 with the empirical chi square 0.33 with prob < NA
## The total number of observations was 540 with Likelihood Chi Square = 12.21 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1## Factor Analysis using method = ols
## Call: fa(r = data_2018, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight -0.02 -0.07 0.79 0.55 0.4530 1.0
## common_neighbors 0.76 0.28 0.06 1.00 0.0014 1.3
## neighborhood_overlap 0.02 0.87 0.18 0.97 0.0251 1.1
## resource_allocation 0.16 0.15 0.78 1.01 -0.0089 1.2
## preferential_attatchment 1.02 -0.10 0.03 0.98 0.0192 1.0
## edge_betweenness -0.07 -0.68 0.23 0.39 0.6094 1.2
##
## [,1] [,2] [,3]
## SS loadings 1.91 1.49 1.50
## Proportion Var 0.32 0.25 0.25
## Cumulative Var 0.32 0.57 0.82
## Proportion Explained 0.39 0.30 0.31
## Cumulative Proportion 0.39 0.69 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.58 0.83
## [2,] 0.58 1.00 0.49
## [3,] 0.83 0.49 1.00
##
## Mean item complexity = 1.1
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.84 with Chi Square of 4343.58
## The degrees of freedom for the model are 0 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 NA
##
## The harmonic number of observations is 558 with the empirical chi square 0.59 with prob < NA
## The total number of observations was 558 with Likelihood Chi Square = 44.1 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1## Factor Analysis using method = ols
## Call: fa(r = data_2019, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight -0.02 0.76 -0.05 0.52 0.4782 1.0
## common_neighbors 0.76 0.08 0.29 1.00 0.0016 1.3
## neighborhood_overlap -0.01 0.15 0.90 0.95 0.0542 1.1
## resource_allocation 0.15 0.82 0.11 1.01 -0.0086 1.1
## preferential_attatchment 1.00 0.04 -0.10 0.98 0.0204 1.0
## edge_betweenness -0.12 0.26 -0.63 0.35 0.6541 1.4
##
## [,1] [,2] [,3]
## SS loadings 1.86 1.52 1.43
## Proportion Var 0.31 0.25 0.24
## Cumulative Var 0.31 0.56 0.80
## Proportion Explained 0.39 0.32 0.30
## Cumulative Proportion 0.39 0.70 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.85 0.50
## [2,] 0.85 1.00 0.47
## [3,] 0.50 0.47 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.28 with Chi Square of 4924
## The degrees of freedom for the model are 0 and the objective function was 0.03
##
## The root mean square of the residuals (RMSR) is 0.01
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 680 with the empirical chi square 0.71 with prob < NA
## The total number of observations was 680 with Likelihood Chi Square = 20.46 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1France
Choosing the Number of Factors
## Parallel analysis suggests that the number of factors = 2 and the number of components = NA## Parallel analysis suggests that the number of factors = 3 and the number of components = NA
## Parallel analysis suggests that the number of factors = 2 and the number of components = NAFactor Analysis
Adequacy Test
## Factor Analysis using method = ols
## Call: fa(r = data_2017, nfactors = 2, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 h2 u2 com
## Weight 0.86 -0.21 0.59 0.413 1.1
## common_neighbors 0.70 0.41 0.97 0.026 1.6
## neighborhood_overlap 0.18 0.87 0.97 0.030 1.1
## resource_allocation 0.95 0.06 0.96 0.040 1.0
## preferential_attatchment 0.90 0.04 0.85 0.147 1.0
## edge_betweenness 0.17 -0.75 0.45 0.553 1.1
##
## [,1] [,2]
## SS loadings 3.13 1.67
## Proportion Var 0.52 0.28
## Cumulative Var 0.52 0.80
## Proportion Explained 0.65 0.35
## Cumulative Proportion 0.65 1.00
##
## With factor correlations of
## [,1] [,2]
## [1,] 1.00 0.57
## [2,] 0.57 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 8.22 with Chi Square of 3792.84
## The degrees of freedom for the model are 4 and the objective function was 2.04
##
## The root mean square of the residuals (RMSR) is 0.03
## The df corrected root mean square of the residuals is 0.05
##
## The harmonic number of observations is 465 with the empirical chi square 10 with prob < 0.04
## The total number of observations was 465 with Likelihood Chi Square = 939.23 with prob < 5.3e-202
##
## Tucker Lewis Index of factoring reliability = 0.069
## RMSEA index = 0.709 and the 90 % confidence intervals are 0.672 0.748
## BIC = 914.66
## Fit based upon off diagonal values = 1## Factor Analysis using method = ols
## Call: fa(r = data_2018, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight 0.00 0.79 -0.14 0.53 0.4737 1.1
## common_neighbors 0.70 0.09 0.34 1.00 0.0045 1.5
## neighborhood_overlap 0.05 0.16 0.85 0.96 0.0411 1.1
## resource_allocation 0.06 0.89 0.12 1.00 -0.0022 1.0
## preferential_attatchment 1.03 0.01 -0.08 0.99 0.0063 1.0
## edge_betweenness 0.01 0.16 -0.72 0.43 0.5668 1.1
##
## [,1] [,2] [,3]
## SS loadings 1.76 1.60 1.54
## Proportion Var 0.29 0.27 0.26
## Cumulative Var 0.29 0.56 0.82
## Proportion Explained 0.36 0.33 0.31
## Cumulative Proportion 0.36 0.69 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.84 0.53
## [2,] 0.84 1.00 0.50
## [3,] 0.53 0.50 1.00
##
## Mean item complexity = 1.1
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.43 with Chi Square of 3420.05
## The degrees of freedom for the model are 0 and the objective function was 0.13
##
## The root mean square of the residuals (RMSR) is 0
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 464 with the empirical chi square 0.35 with prob < NA
## The total number of observations was 464 with Likelihood Chi Square = 57.81 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1## Factor Analysis using method = ols
## Call: fa(r = data_2019, nfactors = 2, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 h2 u2 com
## Weight 0.87 -0.20 0.62 0.376 1.1
## common_neighbors 0.66 0.47 0.96 0.043 1.8
## neighborhood_overlap 0.10 0.91 0.93 0.069 1.0
## resource_allocation 0.93 0.06 0.93 0.074 1.0
## preferential_attatchment 0.87 0.04 0.80 0.200 1.0
## edge_betweenness 0.17 -0.72 0.43 0.566 1.1
##
## [,1] [,2]
## SS loadings 2.95 1.72
## Proportion Var 0.49 0.29
## Cumulative Var 0.49 0.78
## Proportion Explained 0.63 0.37
## Cumulative Proportion 0.63 1.00
##
## With factor correlations of
## [,1] [,2]
## [1,] 1.00 0.48
## [2,] 0.48 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.06 with Chi Square of 3036.14
## The degrees of freedom for the model are 4 and the objective function was 1.77
##
## The root mean square of the residuals (RMSR) is 0.04
## The df corrected root mean square of the residuals is 0.07
##
## The harmonic number of observations is 434 with the empirical chi square 17.56 with prob < 0.0015
## The total number of observations was 434 with Likelihood Chi Square = 758.95 with prob < 6e-163
##
## Tucker Lewis Index of factoring reliability = 0.06
## RMSEA index = 0.659 and the 90 % confidence intervals are 0.621 0.7
## BIC = 734.66
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2]
## Correlation of (regression) scores with factors 1.00 0.98
## Multiple R square of scores with factors 0.99 0.96
## Minimum correlation of possible factor scores 0.99 0.92Germany
Choosing the Number of Factors
## Parallel analysis suggests that the number of factors = 2 and the number of components = NA## Parallel analysis suggests that the number of factors = 3 and the number of components = NA
## Parallel analysis suggests that the number of factors = 3 and the number of components = NAFactor Analysis
Adequacy Test
## Factor Analysis using method = ols
## Call: fa(r = data_2017, nfactors = 2, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 h2 u2 com
## Weight 0.74 -0.13 0.45 0.5494 1.1
## common_neighbors 0.80 0.28 0.99 0.0098 1.2
## neighborhood_overlap 0.25 0.79 0.93 0.0744 1.2
## resource_allocation 0.90 0.08 0.89 0.1078 1.0
## preferential_attatchment 0.99 -0.08 0.89 0.1076 1.0
## edge_betweenness 0.16 -0.74 0.43 0.5676 1.1
##
## [,1] [,2]
## SS loadings 3.18 1.40
## Proportion Var 0.53 0.23
## Cumulative Var 0.53 0.76
## Proportion Explained 0.69 0.31
## Cumulative Proportion 0.69 1.00
##
## With factor correlations of
## [,1] [,2]
## [1,] 1.0 0.6
## [2,] 0.6 1.0
##
## Mean item complexity = 1.1
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.27 with Chi Square of 4009.33
## The degrees of freedom for the model are 4 and the objective function was 0.86
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.04
##
## The harmonic number of observations is 555 with the empirical chi square 6.14 with prob < 0.19
## The total number of observations was 555 with Likelihood Chi Square = 475.17 with prob < 1.6e-101
##
## Tucker Lewis Index of factoring reliability = 0.557
## RMSEA index = 0.461 and the 90 % confidence intervals are 0.427 0.497
## BIC = 449.89
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2]
## Correlation of (regression) scores with factors 0.99 0.97
## Multiple R square of scores with factors 0.99 0.95
## Minimum correlation of possible factor scores 0.98 0.89## Factor Analysis using method = ols
## Call: fa(r = data_2018, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight 0.06 0.74 -0.11 0.54 0.4583 1.1
## common_neighbors 0.70 0.13 0.29 1.00 0.0032 1.4
## neighborhood_overlap 0.04 0.26 0.79 0.94 0.0573 1.2
## resource_allocation 0.09 0.85 0.13 1.00 -0.0029 1.1
## preferential_attatchment 1.03 0.01 -0.07 0.99 0.0053 1.0
## edge_betweenness -0.08 0.23 -0.68 0.38 0.6169 1.3
##
## [,1] [,2] [,3]
## SS loadings 1.86 1.63 1.37
## Proportion Var 0.31 0.27 0.23
## Cumulative Var 0.31 0.58 0.81
## Proportion Explained 0.38 0.34 0.28
## Cumulative Proportion 0.38 0.72 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.84 0.54
## [2,] 0.84 1.00 0.53
## [3,] 0.54 0.53 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.81 with Chi Square of 4577.09
## The degrees of freedom for the model are 0 and the objective function was 0.05
##
## The root mean square of the residuals (RMSR) is 0
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 590 with the empirical chi square 0.22 with prob < NA
## The total number of observations was 590 with Likelihood Chi Square = 27.5 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1## Factor Analysis using method = ols
## Call: fa(r = data_2019, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight -0.06 0.74 -0.09 0.42 0.5772 1.0
## common_neighbors 0.70 0.09 0.34 1.00 0.0046 1.5
## neighborhood_overlap 0.05 0.18 0.83 0.92 0.0801 1.1
## resource_allocation 0.08 0.89 0.07 1.00 0.0034 1.0
## preferential_attatchment 1.02 0.02 -0.08 1.00 0.0047 1.0
## edge_betweenness 0.01 0.17 -0.71 0.41 0.5925 1.1
##
## [,1] [,2] [,3]
## SS loadings 1.74 1.54 1.45
## Proportion Var 0.29 0.26 0.24
## Cumulative Var 0.29 0.55 0.79
## Proportion Explained 0.37 0.32 0.31
## Cumulative Proportion 0.37 0.69 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.83 0.53
## [2,] 0.83 1.00 0.52
## [3,] 0.53 0.52 1.00
##
## Mean item complexity = 1.1
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 6.8 with Chi Square of 3529.51
## The degrees of freedom for the model are 0 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 NA
##
## The harmonic number of observations is 523 with the empirical chi square 0.02 with prob < NA
## The total number of observations was 523 with Likelihood Chi Square = 0.84 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2] [,3]
## Correlation of (regression) scores with factors 1.00 1.00 0.99
## Multiple R square of scores with factors 1.00 1.00 0.97
## Minimum correlation of possible factor scores 0.99 0.99 0.94Japan
Choosing the Number of Factors
## 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
## Parallel analysis suggests that the number of factors = 3 and the number of components = NAFactor Analysis
Adequacy Test
## Factor Analysis using method = ols
## Call: fa(r = data_2017, nfactors = 2, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 h2 u2 com
## Weight 0.80 -0.08 0.57 0.425 1.0
## common_neighbors 0.79 0.31 0.98 0.022 1.3
## neighborhood_overlap 0.18 0.87 0.96 0.039 1.1
## resource_allocation 0.80 0.21 0.87 0.129 1.1
## preferential_attatchment 1.03 -0.13 0.92 0.076 1.0
## edge_betweenness 0.17 -0.73 0.43 0.568 1.1
##
## [,1] [,2]
## SS loadings 3.14 1.60
## Proportion Var 0.52 0.27
## Cumulative Var 0.52 0.79
## Proportion Explained 0.66 0.34
## Cumulative Proportion 0.66 1.00
##
## With factor correlations of
## [,1] [,2]
## [1,] 1.00 0.54
## [2,] 0.54 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 15 and the objective function was 7.12 with Chi Square of 2465.63
## The degrees of freedom for the model are 4 and the objective function was 0.57
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.04
##
## The harmonic number of observations is 350 with the empirical chi square 4.26 with prob < 0.37
## The total number of observations was 350 with Likelihood Chi Square = 197.78 with prob < 1.1e-41
##
## Tucker Lewis Index of factoring reliability = 0.702
## RMSEA index = 0.372 and the 90 % confidence intervals are 0.329 0.418
## BIC = 174.35
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2]
## Correlation of (regression) scores with factors 0.99 0.99
## Multiple R square of scores with factors 0.97 0.98
## Minimum correlation of possible factor scores 0.94 0.96## Factor Analysis using method = ols
## Call: fa(r = data_2018, nfactors = 2, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 h2 u2 com
## Weight 0.86 -0.17 0.62 0.377 1.1
## common_neighbors 0.79 0.30 0.97 0.032 1.3
## neighborhood_overlap 0.27 0.80 0.93 0.066 1.2
## resource_allocation 0.87 0.12 0.88 0.118 1.0
## preferential_attatchment 0.96 -0.06 0.87 0.134 1.0
## edge_betweenness 0.17 -0.76 0.48 0.521 1.1
##
## [,1] [,2]
## SS loadings 3.28 1.47
## Proportion Var 0.55 0.25
## Cumulative Var 0.55 0.79
## Proportion Explained 0.69 0.31
## Cumulative Proportion 0.69 1.00
##
## With factor correlations of
## [,1] [,2]
## [1,] 1.00 0.52
## [2,] 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 15 and the objective function was 7.4 with Chi Square of 3605.06
## The degrees of freedom for the model are 4 and the objective function was 1.32
##
## The root mean square of the residuals (RMSR) is 0.03
## The df corrected root mean square of the residuals is 0.06
##
## The harmonic number of observations is 491 with the empirical chi square 12.38 with prob < 0.015
## The total number of observations was 491 with Likelihood Chi Square = 642.15 with prob < 1.2e-137
##
## Tucker Lewis Index of factoring reliability = 0.332
## RMSEA index = 0.57 and the 90 % confidence intervals are 0.534 0.608
## BIC = 617.37
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2]
## Correlation of (regression) scores with factors 0.99 0.98
## Multiple R square of scores with factors 0.97 0.96
## Minimum correlation of possible factor scores 0.95 0.92## Factor Analysis using method = ols
## Call: fa(r = data_2019, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight 0.16 0.71 -0.16 0.63 0.3746 1.2
## common_neighbors 0.76 0.05 0.32 0.99 0.0092 1.4
## neighborhood_overlap 0.04 0.22 0.84 0.96 0.0390 1.1
## resource_allocation 0.04 0.90 0.14 1.00 -0.0028 1.0
## preferential_attatchment 0.99 0.05 -0.09 1.00 0.0030 1.0
## edge_betweenness -0.11 0.22 -0.66 0.39 0.6077 1.3
##
## [,1] [,2] [,3]
## SS loadings 1.90 1.64 1.43
## Proportion Var 0.32 0.27 0.24
## Cumulative Var 0.32 0.59 0.83
## Proportion Explained 0.38 0.33 0.29
## Cumulative Proportion 0.38 0.71 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.84 0.46
## [2,] 0.84 1.00 0.45
## [3,] 0.46 0.45 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.46 with Chi Square of 3088.4
## The degrees of freedom for the model are 0 and the objective function was 0.26
##
## The root mean square of the residuals (RMSR) is 0.01
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 418 with the empirical chi square 0.56 with prob < NA
## The total number of observations was 418 with Likelihood Chi Square = 108.52 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1UK
Choosing the Number of Factors
## Parallel analysis suggests that the number of factors = 2 and the number of components = NA## Parallel analysis suggests that the number of factors = 3 and the number of components = NA
## Parallel analysis suggests that the number of factors = 3 and the number of components = NAFactor Analysis
Adequacy Test
## Factor Analysis using method = ols
## Call: fa(r = data_2017, nfactors = 2, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 h2 u2 com
## Weight 0.85 -0.13 0.62 0.383 1.0
## common_neighbors 0.77 0.32 0.97 0.033 1.3
## neighborhood_overlap 0.15 0.89 0.95 0.048 1.1
## resource_allocation 0.89 0.11 0.90 0.099 1.0
## preferential_attatchment 1.00 -0.07 0.93 0.074 1.0
## edge_betweenness 0.18 -0.70 0.38 0.622 1.1
##
## [,1] [,2]
## SS loadings 3.25 1.50
## Proportion Var 0.54 0.25
## Cumulative Var 0.54 0.79
## Proportion Explained 0.68 0.32
## Cumulative Proportion 0.68 1.00
##
## With factor correlations of
## [,1] [,2]
## [1,] 1.00 0.54
## [2,] 0.54 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 15 and the objective function was 7.25 with Chi Square of 4397.54
## The degrees of freedom for the model are 4 and the objective function was 0.73
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.04
##
## The harmonic number of observations is 610 with the empirical chi square 6.13 with prob < 0.19
## The total number of observations was 610 with Likelihood Chi Square = 443.54 with prob < 1.1e-94
##
## Tucker Lewis Index of factoring reliability = 0.623
## RMSEA index = 0.424 and the 90 % confidence intervals are 0.392 0.459
## BIC = 417.88
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2]
## Correlation of (regression) scores with factors 0.99 0.98
## Multiple R square of scores with factors 0.97 0.96
## Minimum correlation of possible factor scores 0.94 0.93## Factor Analysis using method = ols
## Call: fa(r = data_2018, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight 0.80 0.09 -0.12 0.69 0.31053 1.1
## common_neighbors 0.09 0.72 0.32 1.00 0.00418 1.4
## neighborhood_overlap 0.21 0.03 0.84 0.94 0.05927 1.1
## resource_allocation 0.91 0.03 0.11 1.00 0.00032 1.0
## preferential_attatchment 0.03 1.01 -0.08 1.00 0.00448 1.0
## edge_betweenness 0.22 -0.08 -0.70 0.43 0.56954 1.2
##
## [,1] [,2] [,3]
## SS loadings 1.76 1.83 1.47
## Proportion Var 0.29 0.30 0.24
## Cumulative Var 0.29 0.60 0.84
## Proportion Explained 0.35 0.36 0.29
## Cumulative Proportion 0.35 0.71 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.85 0.45
## [2,] 0.85 1.00 0.50
## [3,] 0.45 0.50 1.00
##
## Mean item complexity = 1.1
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.83 with Chi Square of 4708.92
## The degrees of freedom for the model are 0 and the objective function was 0.04
##
## The root mean square of the residuals (RMSR) is 0
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 605 with the empirical chi square 0.14 with prob < NA
## The total number of observations was 605 with Likelihood Chi Square = 24.88 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1## Factor Analysis using method = ols
## Call: fa(r = data_2019, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight 0.13 0.67 -0.06 0.58 0.4242 1.1
## common_neighbors 0.76 0.08 0.28 1.00 0.0028 1.3
## neighborhood_overlap -0.01 0.21 0.86 0.93 0.0706 1.1
## resource_allocation 0.12 0.85 0.11 1.00 -0.0031 1.1
## preferential_attatchment 0.98 0.06 -0.11 0.98 0.0187 1.0
## edge_betweenness -0.16 0.29 -0.65 0.38 0.6187 1.5
##
## [,1] [,2] [,3]
## SS loadings 1.92 1.56 1.39
## Proportion Var 0.32 0.26 0.23
## Cumulative Var 0.32 0.58 0.81
## Proportion Explained 0.40 0.32 0.29
## Cumulative Proportion 0.40 0.71 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.85 0.46
## [2,] 0.85 1.00 0.43
## [3,] 0.46 0.43 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.21 with Chi Square of 4492.7
## The degrees of freedom for the model are 0 and the objective function was 0.03
##
## The root mean square of the residuals (RMSR) is 0
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 627 with the empirical chi square 0.28 with prob < NA
## The total number of observations was 627 with Likelihood Chi Square = 16.24 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1US
Choosing the Number of Factors
## Parallel analysis suggests that the number of factors = 3 and the number of components = NA## Parallel analysis suggests that the number of factors = 3 and the number of components = NA
## Parallel analysis suggests that the number of factors = 3 and the number of components = NAFactor Analysis
Adequacy Test
## Factor Analysis using method = ols
## Call: fa(r = data_2017, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight 0.14 0.60 -0.08 0.46 0.5380 1.1
## common_neighbors 0.76 0.10 0.28 1.00 0.0017 1.3
## neighborhood_overlap 0.01 0.31 0.76 0.90 0.1040 1.3
## resource_allocation 0.11 0.87 0.07 1.00 0.0012 1.0
## preferential_attatchment 1.00 0.03 -0.09 0.98 0.0180 1.0
## edge_betweenness -0.13 0.25 -0.67 0.39 0.6072 1.4
##
## [,1] [,2] [,3]
## SS loadings 1.93 1.54 1.26
## Proportion Var 0.32 0.26 0.21
## Cumulative Var 0.32 0.58 0.79
## Proportion Explained 0.41 0.33 0.27
## Cumulative Proportion 0.41 0.73 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.83 0.45
## [2,] 0.83 1.00 0.48
## [3,] 0.45 0.48 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 6.75 with Chi Square of 3635.07
## The degrees of freedom for the model are 0 and the objective function was 0.02
##
## The root mean square of the residuals (RMSR) is 0
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 542 with the empirical chi square 0.12 with prob < NA
## The total number of observations was 542 with Likelihood Chi Square = 8.79 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## [,1] [,2] [,3]
## Correlation of (regression) scores with factors 1.00 1 0.97
## Multiple R square of scores with factors 0.99 1 0.95
## Minimum correlation of possible factor scores 0.98 1 0.90## Factor Analysis using method = ols
## Call: fa(r = data_2018, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight -0.02 0.81 -0.10 0.57 0.4278 1.0
## common_neighbors 0.74 0.08 0.31 1.00 0.0038 1.4
## neighborhood_overlap 0.03 0.19 0.86 0.96 0.0401 1.1
## resource_allocation 0.09 0.87 0.11 1.01 -0.0074 1.1
## preferential_attatchment 1.02 0.02 -0.09 0.99 0.0117 1.0
## edge_betweenness -0.03 0.18 -0.74 0.46 0.5418 1.1
##
## [,1] [,2] [,3]
## SS loadings 1.82 1.64 1.53
## Proportion Var 0.30 0.27 0.25
## Cumulative Var 0.30 0.58 0.83
## Proportion Explained 0.36 0.33 0.31
## Cumulative Proportion 0.36 0.69 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.83 0.50
## [2,] 0.83 1.00 0.48
## [3,] 0.50 0.48 1.00
##
## Mean item complexity = 1.1
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.47 with Chi Square of 3870.36
## The degrees of freedom for the model are 0 and the objective function was 0.09
##
## The root mean square of the residuals (RMSR) is 0.01
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 522 with the empirical chi square 0.66 with prob < NA
## The total number of observations was 522 with Likelihood Chi Square = 47.86 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1## Factor Analysis using method = ols
## Call: fa(r = data_2019, nfactors = 3, rotate = "oblimin", fm = "ols")
## Standardized loadings (pattern matrix) based upon correlation matrix
## 1 2 3 h2 u2 com
## Weight -0.01 -0.06 0.69 0.43 0.5721 1.0
## common_neighbors 0.72 0.31 0.09 1.00 0.0024 1.4
## neighborhood_overlap -0.01 0.88 0.17 0.96 0.0430 1.1
## resource_allocation 0.16 0.12 0.80 1.01 -0.0070 1.1
## preferential_attatchment 1.00 -0.09 0.05 0.98 0.0186 1.0
## edge_betweenness -0.13 -0.63 0.25 0.34 0.6611 1.4
##
## [,1] [,2] [,3]
## SS loadings 1.85 1.45 1.41
## Proportion Var 0.31 0.24 0.24
## Cumulative Var 0.31 0.55 0.78
## Proportion Explained 0.39 0.31 0.30
## Cumulative Proportion 0.39 0.70 1.00
##
## With factor correlations of
## [,1] [,2] [,3]
## [1,] 1.00 0.53 0.85
## [2,] 0.53 1.00 0.53
## [3,] 0.85 0.53 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 15 and the objective function was 7.3 with Chi Square of 4866.33
## The degrees of freedom for the model are 0 and the objective function was 0.03
##
## The root mean square of the residuals (RMSR) is 0.01
## The df corrected root mean square of the residuals is NA
##
## The harmonic number of observations is 670 with the empirical chi square 0.62 with prob < NA
## The total number of observations was 670 with Likelihood Chi Square = 21.8 with prob < NA
##
## Tucker Lewis Index of factoring reliability = -Inf
## Fit based upon off diagonal values = 1Naming the Factors
After establishing the adequacy of the factors, it is time for us to name the factors. This is the theoretical side of the analysis where we form the factors depending on the variable loadings. In this case, here is how the factors can be created:
| Attractiveness | Influence | Affinity |
|---|---|---|
| preferential_attatchment | neighborhood_overlap | resource_allocation |
| common_neighbors | edge_betweenness | weight |
- Attractiveness = corresponds to the predisposition of two nodes to connect in the future
- Influence = corresponds to the importance level of an edge in relation to access to different regions in the graph
- Affinity = corresponds to the frequency of collaboration between two nodes
A work by Mariana O. S. Silva
marianaossilva@gmail.com