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Factor Analysis: A Short Introduction, Part 1

by guest 95 Comments

by Maike Rahn, PhD

Why use factor analysis?

Factor analysis is a useful tool for investigating variable relationships for complex concepts such as socioeconomic status, dietary patterns, or psychological scales.

It allows researchers to investigate concepts that are not easily measured directly by collapsing a large number of variables into a few interpretable underlying factors.

What is a factor?

The key concept of factor analysis is that multiple observed variables have similar patterns of responses because they are all associated with a latent (i.e. not directly measured) variable.their association with an underlying latent variable, the factor, which cannot easily be measured.

For example, people may respond similarly to questions about income, education, and occupation, which are all associated with the latent variable socioeconomic status.

In every factor analysis, there are the same number of factors as there are variables.  Each factor captures a certain amount of the overall variance in the observed variables, and the factors are always listed in order of how much variation they explain.

The eigenvalue is a measure of how much of the variance of the observed variables a factor explains.  Any factor with an eigenvalue ≥1 explains more variance than a single observed variable.

So if the factor for socioeconomic status had an eigenvalue of 2.3 it would explain as much variance as 2.3 of the three variables.  This factor, which captures most of the variance in those three variables, could then be used in other analyses.

The factors that explain the least amount of variance are generally discarded.  Deciding how many factors are useful to retain will be the subject of another post.

What are factor loadings?

The relationship of each variable to the underlying factor is expressed by the so-called factor loading. Here is an example of the output of a simple factor analysis looking at indicators of wealth, with just six variables and two resulting factors.

Variables Factor 1 Factor 2
Income 0.65 0.11
Education 0.59 0.25
Occupation 0.48 0.19
House value 0.38 0.60
Number of public parks in neighborhood 0.13 0.57
Number of violent crimes per year in neighborhood 0.23 0.55

 

The variable with the strongest association to the underlying latent variable. Factor 1, is income, with a factor loading of 0.65.

Since factor loadings can be interpreted like standardized regression coefficients, one could also say that the variable income has a correlation of 0.65 with Factor 1. This would be considered a strong association for a factor analysis in most research fields.

Two other variables, education and occupation, are also associated with Factor 1. Based on the variables loading highly onto Factor 1, we could call it “Individual socioeconomic status.”

House value, number of public parks, and number of violent crimes per year, however, have high factor loadings on the other factor, Factor 2. They seem to indicate the overall wealth within the neighborhood, so we may want to call Factor 2 “Neighborhood socioeconomic status.”

Notice that the variable house value also is marginally important in Factor 1 (loading = 0.38). This makes sense, since the value of a person’s house should be associated with his or her income.

About the Author: Maike Rahn is a health scientist with a strong background in data analysis.   Maike has a Ph.D. in Nutrition from Cornell University.


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Principal Component Analysis
Summarize common variation in many variables... into just a few. Learn the 5 steps to conduct a Principal Component Analysis and the ways it differs from Factor Analysis.

Tagged With: Factor Analysis, factor loadings

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  • Factor Analysis: A Short Introduction, Part 5–Dropping unimportant variables from your analysis
  • How Big of a Sample Size do you need for Factor Analysis?
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  • How to Reduce the Number of Variables to Analyze

Reader Interactions

Comments

  1. Obioma says

    November 16, 2020 at 9:34 am

    Nice explanation thanks for the good work

    Reply
  2. Muhammad Karim says

    July 30, 2020 at 3:37 am

    Explained nicely. Now the meaning of factor loading is clear. But, there is still a confusion. What is eigen value. If eigen value is greater than 1, so what does it mean???

    Reply
  3. BIBHU BHUSAN NAYAK says

    May 29, 2020 at 1:51 am

    Thank you so much for my first understanding on FA

    Reply
  4. tiffany field says

    May 27, 2020 at 2:36 pm

    Very nice presentation. I have two questions: 1)on the SPSS output which of the analyses do you prefer-component, pattern or structure? and 2)how do you interpret negative sign loadings? Thanks so much. Tiffany

    Reply
  5. Barbara says

    April 20, 2020 at 7:27 am

    Hi,
    I am still confused about the factor analysis. If have 6 factors in my analysis table, is it necessary to reduce it to say only 2 factors only?
    Thanks

    Reply
  6. Idris shamsuddeen Yaradua says

    August 22, 2019 at 7:42 pm

    Thank you sir for this explanation.my question here can I add principal component analysis and factor analysis to make an analysis?

    Reply
  7. Jyotirmoy Pandit says

    June 23, 2019 at 12:16 pm

    Dear, In my study,l have selected some municipalities with their different indicators viz. Demographic, education, amenities, health. Here,my quarries is -by which analysis I am going to confirm that the situation of this or that municipality are good or bad. Pls reply.

    Reply
  8. Vithalani Bhargav says

    May 30, 2019 at 6:50 am

    Helpful thank you for help

    Reply
  9. maryam says

    April 13, 2019 at 2:44 am

    please help me

    how many variables minimum we need to run factor analysis? I saw some researchers use at least 15. Is it the rule of thumb?

    I have 3 varible and for evry vaible 150 observation
    can I use factor analysis?

    Reply
  10. Rahmatullah says

    April 1, 2019 at 11:11 am

    Well Explained, I found it very helpful and useful as described in the easiest way to understand it.
    Thank u.

    Reply
  11. Tareq says

    March 29, 2019 at 7:01 am

    Very clear example and useful coverage to the FA concept

    Reply
  12. Mariya Zheleva says

    November 20, 2018 at 4:45 am

    Dear Mr Rahn,

    I would like to ask for your piece of advice on the following questions in relation to factor analysis:
    1) How do you decide how many factors should be extracted? For instance, I have 44 variables in my survey and data is mainly categorical.
    2) Do you conduct the factor analysis for all of variables at once or it is best to first prepare a bunch of variables and conduct the analysis. In my case, should I make like for instance 4 bunches of 11 variables and on a separate case run the factor analysis for each of the bunches. Does this mean that I should in advance make a descriptive statistic for each variable?
    3) Once conducting a principle factor analysis for all variables, I see that the highest correlations have value 0,252 or 0,314 (in the correlation matrix). Does this mean that the model is insignificant?

    Thank you in advance for your kind guidance.

    Kind regards,
    Mariya Zheleva
    PhD student at Sofia University “St. Kliment Ohridski”, Bulgaria and at UVSQ in Paris, France

    Reply
    • Alphoncina says

      October 11, 2019 at 6:57 am

      can someone respond to this question please.

      I am facing the same problem

      Reply
  13. Hassan Golshani says

    October 30, 2018 at 6:16 am

    Easy to understand. thank you.

    Reply
  14. YJC says

    August 31, 2018 at 5:31 pm

    Really nice summary!
    Precise and comprehensive!
    Much appreciated,

    Reply
  15. eg tan says

    July 11, 2018 at 11:18 pm

    easy to understand.thks

    Reply
  16. Ioanna Karaoulani says

    May 4, 2018 at 10:54 am

    Clear, precise, simple to understand!

    Thank you.

    Reply
  17. Isah says

    May 4, 2018 at 2:47 am

    Hi, how are the factors obtained?

    Reply
  18. Tausif says

    April 24, 2018 at 8:05 am

    How you get factor 1 and Factor 2 ??

    Reply
  19. atheer says

    February 5, 2018 at 3:00 pm

    You are happy evening
    I would like to ask you about your effective position on whether it is possible to use counting variables with factor analysis
    thanks
    Best wishes from IRAQ

    Reply
    • Karen says

      February 14, 2018 at 11:23 am

      Atheer,

      It’s possible. The assumption is that all variables are normally distributed. Count variables are often skewed, but not always. So check your distributions.

      Reply
  20. Lanh says

    January 19, 2018 at 6:17 am

    Dear Maike,

    thank you so much for your clear and useful explanation. I totally understand how to apply it well.

    Best wishes from Germany

    Reply
  21. upasana says

    November 16, 2017 at 1:52 am

    Thank you. It was easy to understand.

    Reply
  22. Morobi Mothulatshipi says

    November 13, 2017 at 10:10 am

    thanks a lot for the information

    Reply
  23. Mark Norman says

    October 21, 2017 at 3:02 pm

    The article states “In every factor analysis, there are the same number of factors as there are variables”. However the table used in the example shows 6 variables and 2 factors. Why are the two numbers not equal? Does “variable” have different meanings in the statement and the table?

    Thanks in advance for any clarification.

    Reply
    • Karen says

      January 29, 2018 at 12:18 pm

      Mark,
      Because although there are as many factors as variables, they aren’t all useful. So part of the job of the data analyst is to decide how many factors are useful and therefore retained.

      Reply
  24. Alex Hamed says

    September 30, 2017 at 1:44 pm

    This is a clear and straight forward explanation.

    Reply
  25. Alex says

    September 30, 2017 at 1:42 pm

    This clear and straight forward explanation.
    Thank you

    Reply
  26. Daniel Lim says

    September 14, 2017 at 5:25 am

    Thank you for the clear explanation!

    Reply
  27. Fairouz says

    September 4, 2017 at 7:50 am

    Thanks for the simplicity and clear info 🙂

    Reply
  28. Sarah Andalib says

    August 17, 2017 at 7:22 pm

    Thanks. It was explained very well.

    Reply
  29. Ashenafi says

    June 29, 2017 at 2:04 am

    Thank you

    Reply
  30. Dr. Ramnath Takiar says

    June 21, 2017 at 10:13 am

    It is a well written article. If I understood correctly, we may use many questionnaire to assess some construct like Motivation. For this, I may include questions related to Work environment, Supervisor relationship, pay and other benefits, job satisfaction, training facilities etc., So there are five subcategories under which I have framed the questions. A factor analysis, if done properly should result at least in five factors. So, a factor analysis tries to stratify the questions included in the survey to homogeneous sub groups. Whether my understanding is correct?

    Reply
  31. Mark says

    May 30, 2017 at 9:59 am

    commendable . best explanation so far

    Reply
  32. samuel says

    April 5, 2017 at 6:59 pm

    so if i understood it well, the FA can be used to analyse a data on “barroriers” to effective communication. That is when i have about 20 factors of the barriers to analyse. Thank you

    Reply
  33. Arslan Saleem says

    March 29, 2017 at 1:46 am

    God Bless you. it was an interesting, simple and understandable. it was well written and to the point. helped me a lot

    Reply
  34. Jimoh says

    January 15, 2017 at 3:22 am

    Thanks for your contribution of FA. It’s is helping but need a hypothesis to support it

    Reply
  35. David Akiiki Kalenzi says

    October 16, 2016 at 3:58 am

    Dr Maike Rahn, Thanks so much for the short explanation of what factor analysis is all about. I fully understand how to apply. I wish one day you read my piece of work.
    Kindest regards from Queenstown in Eastern Cape-South Africa

    Reply
  36. Tamanna says

    October 14, 2016 at 2:42 pm

    Hey, could you please name 4 psychological tests based on factor analysis, such as 16 PF and NEO, any other tests that you have come across?
    Thanks.

    Reply
  37. James Tan says

    September 29, 2016 at 6:27 pm

    I have read several articles trying to explain factor analysis. This one is the easiest to understand because it is clear and concise.

    Reply
  38. Mike says

    July 26, 2016 at 3:07 am

    Hi,

    Is it safe to say that factor analysis is the the analysis done in seeking the relationship of demographic and the variables (dependent, mediator, moderator) in the study? or Or is it the analysis done on every items under a construct? to see the loading among the items that represent the construct.
    Do help me as I still cant figure out what factor analysis is. Kindly assist. Many thanks.

    Mike

    Reply
    • Karen says

      October 14, 2016 at 11:47 am

      Hi Mike,
      No, FA isn’t done to seek relationship between different variables in a relationship model.

      Factor Analysis is a measurement model for an unmeasured variable (a construct). So it’s closer to your latter definition.

      Reply
  39. Pablo Ramos says

    July 18, 2016 at 4:24 am

    Thank you very much!
    The clearest explanation I ever read.
    Regards from Spain.

    Reply
    • Morobi Mothulatshipi says

      November 13, 2017 at 10:08 am

      Thank you very much. I fully understand how to apply it.

      Reply
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