EFA: New scale
This report narrates the process of factor analizing the proposed scale
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This warning is displayed once per session.Introduction
The purpose of this research was to develop a new scale for measuring subjective wellbing of people condending with substance use disorder using a mobile device. Participants were 427 students of Indiana State University.
Overview
By comparing various rotations and considering interpretive qualities of the solutions, we have decided to use orthogonal bifactor rotation, as the one that offers the greatest interpretability.
Analysis Steps
1.Scree
Scree plot is plotted and top eigen values are displayed
2.MAP
psych::nfactors call is applied, producing Very Simple Structure, Velicer’s MAP, and other criteria to determine the appropriate number of factors. See documentation
3.Parallel Analysis
psych::fa.parallel call is applied, comparing the number of factors in the correlation matrix to random “parallel” matrices. For details, see documentation
4.Fit
psych::fa call is applied to conduct maximum likelihood factor analysis (fm="ml") in order to obtain the chi-square of the proposed models, which incrementally increase the number of retained factors. CFI and TLI indices are then computed, following the formulae:
CFI = ((chisq_null-df_null) - (chisq-df))/(chisq_null-df_null)
TLI = ((chisq_null/df_null) - (chisq/df))/((chisq_null/df_null)-1)For details on psych::fa see documentation
5.RMSEA
RMSEA diagnostic is conducted using Advanced Factor Function by James Steiger. The routine relies on the maxim likelihood factor analysis conducted by stats::factanal call. For details on the latter see here
6.Estimate
Using Advanced Factor Function by James Steiger, we conduct maximum likelihood factor analysis, by obtaining the unrotated solution from stats::factanal call and then rotating solution using gradient projection algorithms (Bernaards & Jennrich, 2005).
7.Confirm
Applying “Exploratory-Confirmatory” procedure described by Joreskog(1978), we find the largest loading for each column of the factor pattern, then constrain all the other loadings in that row to be zero, and fit the resulting model as a confirmatory factor model. Given that we chose the orthogonal bifactor solution, we permit the the cross-loadings between general factor and subfactors.
EFA
We create a correlation matrix using all items on the administered scale and conduct eigen diagnostics:
Scree
eigen value
1 1 8.2120510
2 2 1.6938274
3 3 1.5492364
4 4 0.8865132
5 5 0.8534162
6 6 0.8341441
7 7 0.7270440
8 8 0.6757051
9 9 0.6091630
10 10 0.5387882
11 11 0.4550178
12 12 0.4447041
13 13 0.3993914
14 14 0.3738538
15 15 0.3482113Scree plot is somewhat ambiguious, suggesting a solution involving up to 3 factors, following the Keiser rule (eigenvalue > 1).
MAP
psych::nfactors call is applied, producing Very Simple Structure, Velicer’s MAP, and other criteria to determine the appropriate number of factors. See documentation
The solution again suggests that no more than 3 meaningful factors can be extracted from the correlation structure.
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 0.86 with 1 factors
VSS complexity 2 achieves a maximimum of 0.9 with 2 factors
The Velicer MAP achieves a minimum of 0.02 with 3 factors
Empirical BIC achieves a minimum of -603.72 with 3 factors
Sample Size adjusted BIC achieves a minimum of -87.25 with 8 factors
Statistics by number of factors
vss1 vss2 map dof chisq prob sqresid fit RMSEA BIC SABIC complex eChisq SRMR eCRMS eBIC
1 0.86 0.00 0.021 170 1.1e+03 5.2e-135 10.7 0.86 0.113 65.8 605.3 1.0 1.1e+03 8.3e-02 0.088 88.2
2 0.67 0.90 0.020 151 6.6e+02 1.4e-65 8.0 0.90 0.089 -249.6 229.6 1.4 5.8e+02 6.0e-02 0.067 -337.9
3 0.54 0.85 0.017 133 3.9e+02 2.1e-26 5.9 0.92 0.067 -420.1 2.0 1.6 2.0e+02 3.5e-02 0.042 -603.7
4 0.54 0.84 0.021 116 2.9e+02 1.1e-16 5.3 0.93 0.059 -414.1 -46.0 1.7 1.4e+02 3.0e-02 0.038 -558.1
5 0.54 0.79 0.027 100 2.3e+02 7.0e-12 4.7 0.94 0.055 -378.6 -61.2 1.9 1.0e+02 2.5e-02 0.034 -505.1
6 0.52 0.73 0.033 85 1.7e+02 3.4e-07 4.3 0.95 0.047 -348.6 -78.9 2.1 6.6e+01 2.0e-02 0.030 -449.1
7 0.48 0.75 0.041 71 1.2e+02 8.4e-05 3.8 0.95 0.042 -305.2 -79.9 2.1 4.6e+01 1.7e-02 0.027 -384.5
8 0.46 0.71 0.049 58 8.0e+01 2.9e-02 3.3 0.96 0.030 -271.3 -87.3 2.3 2.6e+01 1.3e-02 0.023 -325.7
9 0.40 0.75 0.059 46 5.0e+01 3.1e-01 2.9 0.96 0.014 -228.5 -82.5 2.1 1.7e+01 1.0e-02 0.021 -261.8
10 0.41 0.75 0.066 35 3.3e+01 5.7e-01 2.5 0.97 0.000 -179.1 -68.1 2.0 8.9e+00 7.4e-03 0.017 -203.1
11 0.42 0.64 0.081 25 2.3e+01 6.0e-01 2.8 0.96 0.000 -128.7 -49.4 2.5 6.0e+00 6.1e-03 0.017 -145.5
12 0.46 0.65 0.094 16 1.3e+01 6.8e-01 2.1 0.97 0.000 -84.0 -33.2 2.2 2.8e+00 4.2e-03 0.014 -94.1
13 0.37 0.62 0.116 8 7.1e+00 5.3e-01 2.3 0.97 0.000 -41.4 -16.0 2.5 1.5e+00 3.0e-03 0.015 -47.0
14 0.23 0.42 0.136 1 1.2e+00 2.8e-01 2.3 0.97 0.019 -4.9 -1.7 3.0 2.1e-01 1.1e-03 0.016 -5.8
15 0.23 0.38 0.177 -5 2.0e-04 NA 2.1 0.97 NA NA NA 3.3 3.2e-05 1.4e-05 NA NA
16 0.34 0.56 0.245 -10 1.1e-05 NA 2.0 0.97 NA NA NA 2.6 1.7e-06 3.2e-06 NA NA
17 0.33 0.55 0.369 -14 4.3e-08 NA 2.1 0.97 NA NA NA 2.7 7.7e-09 2.2e-07 NA NA
18 0.33 0.56 0.553 -17 1.9e-08 NA 2.1 0.97 NA NA NA 2.7 3.3e-09 1.4e-07 NA NA
19 0.33 0.56 1.000 -19 4.4e-10 NA 2.1 0.97 NA NA NA 2.7 8.2e-11 2.2e-08 NA NA
20 0.33 0.56 NA -20 4.4e-10 NA 2.1 0.97 NA NA NA 2.7 8.2e-11 2.2e-08 NA NAParallel
psych::fa.parallel call is applied, comparing the number of factors in the correlation matrix to random “parallel” matrices. For details, see documentation
There are only two non-general factors that appear to be more distinguishable than their simulated counterparts. While the analysis technically suggests 4 factors, the last one barely makes the cut ( 0.2696 vs 0.2557) and is not supported by the scree test and MAP analysis.
Parallel analysis suggests that the number of factors = 4 and the number of components = NA observed_eigens simulated_eigens
1 7.655685295 0.61044984
2 0.994542517 0.34685612
3 0.844239726 0.29598141
4 0.269632660 0.23974758
5 0.199970175 0.20371859
6 0.129517289 0.16847548
7 0.039944804 0.12398564
8 -0.006467331 0.09037350
9 -0.026975661 0.06092811
10 -0.052193362 0.01959554
11 -0.060142500 -0.00938798
12 -0.069033568 -0.03984756
13 -0.119911700 -0.06892370
14 -0.176364116 -0.09745562
15 -0.222915845 -0.13561839Fit
psych::fa call is applied to conduct maximum likelihood factor analysls (fm="ml") in order to obtain the chi-square of the proposed models, which incrementally increase the number of retained factors. CFI and TLI indices are then computed from the produced criteria. For details on psych::fa see documentation
n_factors chisq_null df_null chisq df CFI TLI
1 1 4263.511 190 1117.88350 170 0.7673055 0.7399297
2 2 4263.511 190 576.66302 151 0.8955046 0.8685158
3 3 4263.511 190 201.83722 133 0.9831013 0.9758589
4 4 4263.511 190 144.50271 116 0.9930029 0.9885393
5 5 4263.511 190 100.53314 100 0.9998691 0.9997513
6 6 4263.511 190 65.77164 85 1.0047203 1.0105513RMSEA
RMSEA diagnostic is conducted using Advanced Factor Function by James Steiger. The routine relies on the maxim likelihood factor analysis conducted by stats::factanal call. For details on the latter see here
The confidence intervale of RMSEA point estimate does not include values below the threshhold (<.08) for the models with two factors. When a more liberal threshhold is adopted (<.05), the model with 3 factors appear to be preferable. 
Factors Cum.Eigen Chi-Square Df p.value RMSEA.Pt RMSEA.Lo RMSEA.Hi
[1,] 1 8.212051 1090.4368 170 0.000000e+00 0.11273733 0.10639932 0.11917319
[2,] 2 9.905878 655.9242 151 0.000000e+00 0.08859716 0.08170447 0.09560747
[3,] 3 11.455115 376.4958 133 0.000000e+00 0.06555641 0.05781148 0.07340431
[4,] 4 12.341628 282.2465 116 6.661338e-16 0.05800193 0.04943997 0.06663338
[5,] 5 13.195044 221.9364 100 3.058820e-11 0.05350100 0.04403827 0.06298407
[6,] 6 14.029188 161.2841 85 1.170769e-06 0.04589894 0.03496809 0.05662521
Factors Cum.Eigen Chi-Square Df p.value RMSEA.Pt RMSEA.Lo RMSEA.Hi
[1,] 1 8.212051 1090.4368 170 0.000000e+00 0.11273733 0.10639932 0.11917319
[2,] 2 9.905878 655.9242 151 0.000000e+00 0.08859716 0.08170447 0.09560747
[3,] 3 11.455115 376.4958 133 0.000000e+00 0.06555641 0.05781148 0.07340431
[4,] 4 12.341628 282.2465 116 6.661338e-16 0.05800193 0.04943997 0.06663338
[5,] 5 13.195044 221.9364 100 3.058820e-11 0.05350100 0.04403827 0.06298407
[6,] 6 14.029188 161.2841 85 1.170769e-06 0.04589894 0.03496809 0.05662521Estimate
Using Advanced Factor Function by James Steiger, we conduct maximum likelihood factor analysis, by obtaining the unrotated solution from stats::factanal call and then rotating solution using gradient projection algorithms (Bernaards & Jennrich, 2005).
This will take a moment..........exiting
Loadings above threashold (.3) are masked to see the simpler structure
Confirm
Applying “Exploratory-Confirmatory” procedure described by Joreskog(1978), we find the largest loading for each column of the factor pattern, then constrain all the other loadings in that row to be zero, and fit the resulting model as a confirmatory factor model. Given that we chose the orthogonal bifactor solution, we permit the the cross-loadings between general factor and subfactors.
Model Chiquare = 521.5535 | df model = 162 | df null = 190
Goodness-of-fit index = 0.8866976
Adjusted Goodness-of-fit index = 0.8531266
RMSEA index = .0722 90% CI: (.065,.079)
Comparitive Fit Index (CFI = 0.9133589
Tucker Lewis Index (TLI/NNFI) = 0.8983839
Akaike Information Criterion (AIC) = 617.5535
Bayesian Information Criterion (BIC) = -459.6456
Rotations
Varimax
geominT
bifactorT
quartimax
oblimin
geominQ
bifactorQ
Correlations
Decisions
What items should be included into the daily questionnaire?
Guidelines for preference:
- High item-total correlation to the new scale
- High item-total correlation to the CHU and Warwick scales
- Items should cover all factors that seem to be present (3)
- Items with more pronounced weights onto a single factore should be preferred
- Even though the new scale shows good unidimentionality, diagnostic tests revealed that there appear to be two subfactors:
- “Self-Care”
Q4_13(Today I feel energetic)
Q4_5(Today I feel rested)
Q6_1(Today I woke up feeling well-rested)
Q6_2(Today I ate well)
Q6_3(Today I took good care of myself)
- “Distress”
Q4_11(Today I feel sad)
Q4_15(Today I feel angry)
Q4_4(Today I feel annoyed/irritable)
Q4_7(Today I feel in physical pain)
Q4_9(Today I feel worried/anxious)
Item
Q4_4(Today I feel annoyed/irritable) was selected because it consistened loaded the strongest onto the “Distress” factor.Items
Q4_5(Today I feel rested) andQ6_1(Today I woke up feeling well-rested) had highest loadings on the “Self-Care” factors, consistently across rotations. Deciding between them, we opted forQ4_5because it had slightly higher item-total correlations with both the new and existing scales. These two items had a high correlation (R = .75).Two other items had the next highest loadings on the “Self-Care” factor:
Q4_13(Today I feel energetic) andQ6_3(Today I took good care of myself). We felt thatQ4_13was similar toQ4_5, whicleQ6_3provided a different facet of well-being. Therefore, choosing between the two we settled on the latter.We inclued
Q4_6(Today I feel happy) because it had the highest loading on the general factor and highest item-total correlation on all scales.With 4 items included into the new questionnaire, we had 2-3 items to assist
Q4_6in capturing the genral factor. The following group stood out:
Q4_8(Today I feel confident)Q4_16(Today I feel in charge of my life)Q4_10(Today I feel that my life has a purpose)Q4_2(Today I feel optimistic about the future)
All of these appear to have similar loadings, with minor variations. We chose Q4_10 because this items appears to cover the spiritual/self-actualization domain. We also chose Q4_8 it had consistently higher loading on the general factor that other two (Q4_16 and Q4_2) while being similar to them in term of the content.
- To represent social facet, we chose
Q4_14(Today I feel supported) overQ4_3(Today I feel loved), because the former had higher loading, and higher item-total correlations.
Chosen items
Q4_6(Today I feel happy)Q4_8(Today I feel confident)Q4_10(Today I feel that my life has a purpose)Q4_14(Today I feel supported)Q6_3(Today I took good care of myself) - SELFCARE subfactorQ4_5(Today I feel rested) - SELFCARE subfactorQ4_4(Today I feel annoyed/irritable) - DISTRESS subfactor
Reproducibility
R version 3.6.2 (2019-12-12)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 18363)
Matrix products: default
locale:
[1] LC_COLLATE=English_United States.1252 LC_CTYPE=English_United States.1252 LC_MONETARY=English_United States.1252
[4] LC_NUMERIC=C LC_TIME=English_United States.1252
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] corrgram_1.13 corrplot_0.84 GPArotation_2014.11-1 sem_3.1-9 plotrix_3.7-7
[6] psych_1.9.12.31 dplyr_0.8.4 ggplot2_3.2.1 magrittr_1.5 knitr_1.28
loaded via a namespace (and not attached):
[1] nlme_3.1-142 bitops_1.0-6 RColorBrewer_1.1-2 mi_1.0 tools_3.6.2 utf8_1.1.4
[7] R6_2.4.1 KernSmooth_2.23-16 lazyeval_0.2.2 colorspace_1.4-1 withr_2.1.2 tidyselect_1.0.0
[13] gridExtra_2.3 mnormt_1.5-6 compiler_3.6.2 extrafontdb_1.0 cli_2.0.1 TSP_1.1-9
[19] labeling_0.3 caTools_1.18.0 scales_1.1.0 readr_1.3.1 stringr_1.4.0 digest_0.6.24
[25] minqa_1.2.4 rmarkdown_2.1 dichromat_2.0-0 pkgconfig_2.0.3 htmltools_0.4.0 extrafont_0.17
[31] lme4_1.1-21 labelled_2.2.2 rlang_0.4.4 farver_2.0.3 testit_0.11 gtools_3.8.2
[37] dendextend_1.13.4 Matrix_1.2-18 Rcpp_1.0.3 munsell_0.5.0 fansi_0.4.1 abind_1.4-5
[43] viridis_0.5.1 lifecycle_0.1.0 stringi_1.4.5 yaml_2.2.1 MASS_7.3-51.4 gplots_3.0.3
[49] plyr_1.8.6 matrixcalc_1.0-3 grid_3.6.2 parallel_3.6.2 gdata_2.18.0 forcats_0.4.0
[55] crayon_1.3.4 lattice_0.20-38 haven_2.2.0 splines_3.6.2 hms_0.5.3 pillar_1.4.3
[61] boot_1.3-23 reshape2_1.4.3 codetools_0.2-16 stats4_3.6.2 glue_1.3.1 gclus_1.3.2
[67] evaluate_0.14 vctrs_0.2.2 nloptr_1.2.2 foreach_1.5.0 Rttf2pt1_1.3.8 gtable_0.3.0
[73] purrr_0.3.3 tidyr_1.0.2 assertthat_0.2.1 xfun_0.12 coda_0.19-3 viridisLite_0.3.0
[79] seriation_1.2-8 tibble_2.1.3 arm_1.10-1 iterators_1.0.12 registry_0.5-1 cluster_2.1.0