Hi,
I have 74 items (Importance rating for security & privacy threats). Using a qualitative method (N=42) these items were already divided into 14 areas. However, I wanted to see how many areas/domains surfaces for N=861.
I applied EFA, though I was sensing CFA would be more appropriate here.
I followed the guidelines provided by:
1. Ledesma, R. D., & Valero-Mora, P. (2007). Determining the number of factors to retain in EFA: An easy-to-use computer program for carrying out parallel analysis. Practical assessment, research & evaluation, 12(2), 1-11.
2. Osborne, J. W., & Costello, A. B. (2009). Best practices in exploratory factor analysis: Four recommendations for getting the most from your analysis. Pan-Pacific Management Review, 12(2), 131-146.
I used Principal Axis Factors using Direct Oblimin rotation, with item loading >0,3. However, I changed factor extractions every time.
1. I used notorious Eigenvalue >1 rule, resulted in 10 factors (just to check)
2. Observed Sree plot. The plot has a big elbow for factors=4 and small elbows at Factor=, 6, 9, 12 and 14. [Attached]
3. Conducted Parallel Analysis (PA) (Monte Carlo Simulation) using 95 percentile, principal axis/common factor analysis and permutations for raw data set. (This was my first time running it and graph was not generated due to some error, which I couldn't figure out). However, using table I found that data should have 15 factors.
It was all confusing for me as priori factor structure was 14, Scree plot showed 4,6,9,12 & 14 possible factors and PA suggested 15. Therefore, I decided to conduct multiple factor analysis as suggested by (Osborne, J. W., & Costello, A. B. 2009). I ran EFA with 11 to 16 no. of factors to be extracted, item loading >0,3, items per factor not less than 3, and found following:
For 11:
No. of factors (items): 11(65)
Items to be dropped: 9
(5 non-related items conjugated into one factors, 4 items not loaded)
Additional: 1 item be removed from one of the 11 factors being irrelevant.
No. of items cross loading: 7
End result: 11(64)
For 12:
No. of factors (items): 12(69)
Items to be dropped: 5
(5 non-related items conjugated into one factors)
Additional: 2 items to be removed form 1 of 12 factors, being irrelevant to theme of the factor)
No. of items cross loading: 7
End result: 12(67)
For 13:
No. of factors (items): 12(65)
Items to be dropped: 9
(5 non-related items conjugated into one factors, 4 items not loaded)
Additional: 1 item be removed from one of the 11 factors being irrelevant.
No. of items cross loading: 7
End result: 12(64)
For 14:
No. of factors (items): 12(64)
Items to be dropped: 10
(5 non-related items conjugated into one factors, 2 items not loaded, 3 items were either along or less than 3 items).
Additional: 1 item be removed from one of the 11 factors being irrelevant.
No. of items cross loading: 9
End result: 12(63)
For 15:
No. of factors (items): 12(64)
Items to be dropped: 10
(5 non-related items conjugated into one factors, 2 items not loaded, 3 items were either along or less than 3 items).
No. of items cross loading: 8
End result: 12(64)
For 16:
No. of factors (items): 12(62)
Items to be dropped: 12
(5 non-related items conjugated into one factors, 3 items not loaded, 4 items were either alone or less than 3 items).
No. of items cross loading: 4
End result: 12(62)
It is pertinent to mention that in all the cases, none of the cross loadings were such that could be used for decreasing number of items to be dropped.
Now the question is that, If I want to retain as many items as possible without violating any of best practices of EFA, which solution is best. Personally, Factors0 12 looks promising to me.
I am open to criticism and comments.
Thanks,
Ali
1.
If you already have an existing model, you should run the CFA. EFA is susceptible to sample fluctuations and may not replicate from one sample to the next very well. That being said, your data seem to suggest that you have fewer than 14 factors. Based on your screen plot I would say either 3 or 5 big factors. It is entirely possible that you have both 14 factors and either 3 or 5. The hierarchical nature of most things that researchers study often results in smaller parcels coming together to form larger constructs or factors.
We conduct factor analysis when multi co-linearity exists in data. Exploratory find are tentative and they subject to change with the method (rotation). with different rotation you tend to get different result/factors. So if you have existing model, you should follow the existing modelling.
Dear Ali,
What you have done seems to be along the right lines. I believe EFA is the right technique here as an EFA does not appear to have been run before and you are basing your understanding on a prior qualitative analysis.
You have a large sample size which should make your results relatively more reliable. The choice of cut off value for component loadings depends upon sample size (the larger the sample, the lower the cut-off). Cross factor loadings should be measured according to the square of the ratio of their loadings (e.g. 0.4 to 0.3 gives a ratio of 16 to 9).
You have not stated which EFA method you have used. There are big differences between Principal Component analysis, orthogonal Factor Analysis and oblique factor analysis and you need to justify your choice. There are certain other checks that you would be advised to do (e.g KMO, determinant and communalities). I would not focus entirely on the scree plot as a means for deciding how many factors to include.
Have a look at my (draft) guide:
https://www.researchgate.net/publication/304490328_Advice_on_Exploratory_Factor_Analysis
Working Paper Advice on Exploratory Factor Analysis
Thanks, Rafael, Pratiksha, Peter and Juan.
Peter, I used Principal Axis Factors using Direct Oblimin rotation. I also checked for KMO was 0.966 and Bartlett's test of sphericity was significant for all the cases suggesting that data was a good candidate for factor analysis.
Juan, O'Connor's code for SPSS to conduct PA, here is the source:
O'Connor, B. P. (2000). SPSS and SAS programs for determining the number of components using parallel analysis and Velicer's MAP test. Behavior Research Methods, Instrumentation, and Computers, 32, 396-402.
and here is the code:
https://people.ok.ubc.ca/brioconn/nfactors/rawpar.sps
Here is output of the table from which I found there should be 15 factors:
Run MATRIX procedure:
PARALLEL ANALYSIS:
PAF/Common Factor Analysis & Raw Data Permutation
Specifications for this Run:
Ncases 493
Nvars 74
Ndatsets 1000
Percent 95
Raw Data Eigenvalues, & Mean & Percentile Random Data Eigenvalues
Root Raw Data Means Prcntyle
1,000000 27,230775 1,019795 1,090128
2,000000 3,240269 ,950995 1,004136
3,000000 2,403540 ,899506 ,944288
4,000000 1,851533 ,856555 ,901782
5,000000 1,713000 ,818160 ,857280
6,000000 1,434135 ,781901 ,818517
7,000000 1,303594 ,747956 ,785206
8,000000 1,168184 ,716043 ,751682
9,000000 1,089739 ,685930 ,717879
10,000000 ,984410 ,657838 ,687420
11,000000 ,903921 ,629845 ,659777
12,000000 ,786649 ,603328 ,632840
13,000000 ,715080 ,577642 ,605873
14,000000 ,613253 ,552722 ,580596
15,000000 ,555564 ,528314 ,553778
16,000000 ,511523 ,505470 ,532534
17,000000 ,485715 ,482959 ,508028
18,000000 ,426304 ,460397 ,486575
19,000000 ,369343 ,438330 ,463644
20,000000 ,340054 ,416921 ,439745
As we can see eigenvalue at 16 (0,511523) is lower than 95 percentile value (0,532534) thus there are 15 factors. [This is what understood, if I am correct.].
Thanks,
Ali
Try my VARIMIN rotation which excludes all simple structure errors.
http://www.varimin.com/about.htm
Don`t be afraid of dropping seeming evidences and take granted what is right, not what the expert majority consider to be right.
Suitbert
Thank you everyone. I have found the solution to my problem and I would like to share it here so that anyone reading this thread in the future can know the final solution.
I executed multiple factor analysis (several iterations) until the best solution is found. The initial criteria for final solution was:
1. Factor containing 3-5 items with loading more than 0.5
2. Items with cross loading greater than 0.32 were removed'
3. Items with loading less than 0.3 were removed
4. Extreme cases (Heywood and reverse loadings) were removed
5. Total Variance Explained (TVE) should be greater than 50
Once the above anomalies removed I checked for face validity.
6. Face validity: items related to same concept loading under same factor, otherwise removed the item.
However, I found that in order to preserve the data points, I had to relax the first criterion, Factors must contain 3-5 items.
Below is the table with final results:
Initial Factors 9 10 11 12 13 14 15
Factors NA 9 12 13 13 14 15
Items 41 47 49 54 46 62
TVE 65,29 63,81 64,64 65,26 67,29 65,374
Items dropped 33 27 25 20 28 12
The solution given by Parallel Analysis was found to be the most feasible as it gives move TVE, less items to be dropped (keep data integrity) and provides factors with best explanation.
N.B. In case of 9 initial factors, the result was not appropriate for factorization- many items were loading haphazardly.
Hope it helps those who need guidance in the future.
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