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Factor Analysis: A Short Introduction, Part 1

by guest contributer 97 Comments

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 they cannot measure directly. It does this by using a large number of variables to esimate 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 variable (i.e. not directly measured).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 one fewer factors than 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 common 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 factor loadings express the relationship of each variable to the underlying factor. 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. Most research fields consider this a strong association for a factor analysis.

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

Related Posts

  • 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?
  • One of the Many Advantages to Running Confirmatory Factor Analysis with a Structural Equation Model
  • How to Reduce the Number of Variables to Analyze

Reader Interactions

Comments

  1. Samir K Mahajan says

    August 29, 2021 at 12:11 am

    this is what i was searching. the interpretation.

    Reply
  2. lavenda says

    March 16, 2021 at 2:44 am

    Thanks for posting the best information and the blog is very informative seku.

    Reply
  3. Obioma says

    November 16, 2020 at 9:34 am

    Nice explanation thanks for the good work

    Reply
  4. 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
  5. BIBHU BHUSAN NAYAK says

    May 29, 2020 at 1:51 am

    Thank you so much for my first understanding on FA

    Reply
  6. 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
  7. 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
  8. 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
  9. 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
  10. Vithalani Bhargav says

    May 30, 2019 at 6:50 am

    Helpful thank you for help

    Reply
  11. 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
  12. 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
  13. Tareq says

    March 29, 2019 at 7:01 am

    Very clear example and useful coverage to the FA concept

    Reply
  14. 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
  15. Hassan Golshani says

    October 30, 2018 at 6:16 am

    Easy to understand. thank you.

    Reply
  16. YJC says

    August 31, 2018 at 5:31 pm

    Really nice summary!
    Precise and comprehensive!
    Much appreciated,

    Reply
  17. eg tan says

    July 11, 2018 at 11:18 pm

    easy to understand.thks

    Reply
  18. Ioanna Karaoulani says

    May 4, 2018 at 10:54 am

    Clear, precise, simple to understand!

    Thank you.

    Reply
  19. Isah says

    May 4, 2018 at 2:47 am

    Hi, how are the factors obtained?

    Reply
  20. Tausif says

    April 24, 2018 at 8:05 am

    How you get factor 1 and Factor 2 ??

    Reply
  21. 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
  22. 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
  23. upasana says

    November 16, 2017 at 1:52 am

    Thank you. It was easy to understand.

    Reply
  24. Morobi Mothulatshipi says

    November 13, 2017 at 10:10 am

    thanks a lot for the information

    Reply
  25. 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
  26. Alex Hamed says

    September 30, 2017 at 1:44 pm

    This is a clear and straight forward explanation.

    Reply
  27. Alex says

    September 30, 2017 at 1:42 pm

    This clear and straight forward explanation.
    Thank you

    Reply
  28. Daniel Lim says

    September 14, 2017 at 5:25 am

    Thank you for the clear explanation!

    Reply
  29. Fairouz says

    September 4, 2017 at 7:50 am

    Thanks for the simplicity and clear info 🙂

    Reply
  30. Sarah Andalib says

    August 17, 2017 at 7:22 pm

    Thanks. It was explained very well.

    Reply
  31. Ashenafi says

    June 29, 2017 at 2:04 am

    Thank you

    Reply
  32. 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
  33. Mark says

    May 30, 2017 at 9:59 am

    commendable . best explanation so far

    Reply
  34. 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
  35. 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
  36. 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
  37. 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
  38. 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
  39. 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
  40. 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
  41. 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
  42. Sakhila Thapa says

    July 17, 2016 at 12:34 pm

    Thank you for easier explanation. It definitely will helpful for my next step of data analysis.

    Reply
  43. Annonymous says

    July 5, 2016 at 7:10 am

    Excellent description, very helpful to build understanding of the topic.

    Reply
  44. Jayashree Ramanan says

    June 29, 2016 at 5:10 am

    Explained in the simplest way even a lay man can understand. Thanks a bunch.

    Reply
  45. Rajendran says

    June 26, 2016 at 1:57 pm

    Simple and very clear explanation. It’s very clear for me now. Thank you.

    Reply
  46. Dr Altaf says

    June 18, 2016 at 6:20 pm

    Thaks sir,

    Very nice explained, as simple as lay mans language

    Reply
  47. Jeremy says

    June 6, 2016 at 3:08 am

    I wish everything had such an easy to understand definition! Thank you

    Reply
  48. surag_1 says

    June 6, 2016 at 12:31 am

    Very crisp, clear and concise explanation. Thanks a ton.

    Reply
  49. IceSwan says

    May 30, 2016 at 7:37 am

    have been through many documents about factor analysis, yours is the most clear explanation. Thanks big time

    Reply
  50. Baloyi says

    May 19, 2016 at 4:02 pm

    this is the best explanation that i have understand, keep on the standard Dr,,

    Reply
  51. J. O. Kwapong says

    May 19, 2016 at 9:56 am

    I like it. kudos!

    Reply
  52. Roy says

    May 19, 2016 at 3:11 am

    Very nice explanation of factor analysis. Keep up the nice work. A small request to you sir – please start small regular tutorials on statistics & data analysis.

    Reply
  53. CMB says

    April 14, 2016 at 5:42 pm

    Just adding my thanks to the list so you keep the posts coming!

    Reply
  54. Monica says

    April 10, 2016 at 10:21 am

    OMG !
    As I have searched many of websites for factor analysis. This was the best and easiest explanation i found yet.
    Really helpful ! Great attempt ! Keep on doing social service !

    Reply
  55. A3 (assalafiy) says

    March 11, 2016 at 6:44 pm

    that is very nice explanation.
    you are so wonderful

    Reply
  56. R says

    March 9, 2016 at 6:56 am

    Very lucid introduction on factors which would be useful to any novice to FA.

    Reply
  57. Eric Francis Eshun says

    March 5, 2016 at 9:37 am

    Thank you

    Reply
  58. Godwin Kodituwakku says

    February 20, 2016 at 1:26 am

    Simple but valuable explanation. Thanks.

    Reply
  59. Rebecca mcmullen says

    December 30, 2015 at 2:08 pm

    Thank you for your clear explanation of factor loading!

    Reply
  60. baba iddi says

    December 8, 2015 at 5:56 am

    thanks for the introduction on factor analysis

    Reply
  61. Prof Sreekumar Pillai says

    November 27, 2015 at 11:28 pm

    Excellent explanation of the basics,
    in my language there is a saying ( around 2000 years old) “Good teachings should have the quality of mothers milk,being good ,simple,digestable and sustaining) and I feel I have found it for Factor analysis.
    Keep up the good work!

    Reply
  62. Rishi says

    September 30, 2015 at 7:40 am

    Explained in one of the best ways possible!!! Helps you understand by just reading it once (quite the contrary for the definitions on the other websites)

    Reply
  63. Sat says

    September 29, 2015 at 4:57 am

    Hi Maike,
    I have a survey with 15 q, 3 measure reading ability, 3 writing, 3 understanding, 3 measure monetary values and 3 measure literacy unrelated aspects.
    I am confused
    do I pick the read, write and understanding on the SPSS for factor analysis? how about the literacy unrelated q which are controls?
    Thanks for your help.
    Sat

    Reply
  64. Sanelise says

    September 19, 2015 at 6:27 am

    Very simple and straight forward…Thanx

    Reply
  65. magda says

    September 16, 2015 at 7:18 pm

    Very clear explanation and useful examples. Thanks. I woudl liek to aks you somehting. I have a questionnaire of 52 items (I used it for Pilot Sutdy)and I have done FA obtaining 1O factors after reduction. I need to reduce the number of questions since 52 is too much and leave the most ‘powerful’ can I use the FA analysis to reduce the number of questions? Thank you

    Reply
  66. Lucia Sauti says

    July 30, 2015 at 7:15 am

    I would like to design a questionnaire using Likert scale that I can use for factor analysis. my challenge is should I mix positive statements and negative statements in my compilation of the questionnaire? e.g. Let us say I need to find out the view of a student if they have a negative attitude towards learning a subject. Should I say in my questionnaire, “I have a negative attitude towards Mathematics.” or I do not have a negative attitude towards Mathematics.”

    Reply
  67. Ahmed Muhammed says

    June 2, 2015 at 8:36 pm

    A very good work, thank you sir.

    Reply
  68. Ali says

    May 9, 2015 at 10:11 pm

    It seems to me you have mixed up the difference between factor analysis and PCA (Principal Component Analysis).
    Where you talked about the amount of variance a factor captures and eigenvalue that measures that. it is principal components in PCA that tells you that because each principal component is orthogonal to the others and associated with an eigen-vector with a corresponding eigenvalue.

    If not please let me know how eigenvalues of factors are calculated in factor anlysis

    Reply
  69. Dr. Akhter says

    April 23, 2015 at 2:24 pm

    Very simple and nice explainations

    Reply
  70. issa stambuli says

    April 17, 2015 at 6:48 am

    Well done

    Reply
  71. Abel says

    March 30, 2015 at 10:47 am

    Thanks Doc
    This has been the most understandable explanation I have so far had. You mentioned something about your next post? about determination of number of factors. May you please also talk about factor analysis using R.

    Reply
  72. Jason lee says

    March 29, 2015 at 6:32 am

    Dear Dr.

    Good day to you. I have a question on factor analysis. I have a pool of 30 items for my construct, then I conducted the PCs, with nine items. After conducted the CFA, it only has three items. Does this acceptable ? Thank you.

    Reply
  73. Al-Amin says

    March 26, 2015 at 10:31 am

    Fantastic explanation!! Thank you

    Reply
  74. Hassan says

    January 12, 2015 at 3:38 am

    I have two kinds of questions: one with a 5-option response and another with a 7-option one. Can I run exploratory FA on both at the same time? When I run them with SPSS it lead to 8 factors that can explain 61% of the variance. But, mathematically, is it right?

    Reply
  75. S.S. says

    December 31, 2014 at 11:28 am

    Hi Rahn,
    Great Job.!!!
    How am I suppose to put citations to your web site?

    Reply
  76. DR..H.K.LAKSHMANRAO says

    December 11, 2014 at 10:16 pm

    FACTOR ANALYSIS IS VERY USEFUL METHOD FOR ANALYSING SCIENTIFIC DATA PARTICULARLY FOR DATA RELATING TO BIOTECH AND FOOD TECNOLOGY AND ANIMAL BEHAVIOUR
    ALSO;Principal component analysis and exploratory factor analysis are both data reduction techniques — techniques to combine a group of correlated variables into fewer variables. You can then use those combination variables — indices or subscales — in other analyses.

    Reply
    • Rizwan says

      September 8, 2015 at 1:33 am

      Dear sir,
      I am a new research student please help me about ”Comparatively study on data reduction method between factor analysis and principal component analysis”. Kindly guide me about this I will waiting for your answer.

      Reply
  77. mohammed ibrahim, fut minna, nger state. nigeria says

    October 25, 2014 at 5:57 pm

    I am grateful to have little idea on how to apply factor analysis. But stil sir! How would I enter data on exel spreat sheet and how will I start running the analysis? I am ph.D student and one of my objective of the study has to do with factor analysis. I have identify four factors with twenty three variable in question. Pls explain step by step for me. Thanks and best regard. Looking forward to hear from you sir.

    Reply
  78. Zimula says

    October 24, 2014 at 3:15 pm

    Good stuff

    Reply
  79. Bibi says

    October 7, 2014 at 6:25 am

    Thank you very much Dr. Rahn. I have struggled 13 months to understand Factor Analysis, and this has been the simple and very helpful. Thank you again.

    Reply
  80. john says

    September 24, 2014 at 12:00 pm

    Dear Dr Thanks very much for you explanation on factor analysis, even those who beginners in statistics like me can follow your elaborations. its so illuminating. have gone through several text on factor analysis but could hardly capture the concept,
    Thanks

    Reply
  81. Amaa says

    September 23, 2014 at 3:55 pm

    As i am using Factor analysis by SPSS in my master research, i got five factors related to my research. At the end of the results by spss there is a 5*5 matrix ( 5 are the factors ). What does this matrix endicated for? in the beginning i thought it is a correlation matrix of the factors, but then i’ve been told no it isn’t ( without giving me what it is exactly). Can you help please?
    p.s ; welcome to everybodys’ answer.

    Thank you.

    Reply
  82. ashish says

    August 3, 2014 at 2:42 am

    This was simple and clear with commonsense.

    Reply
  83. sangeetha says

    July 21, 2014 at 7:40 am

    very usefull an understandable explanation.saved lit if time bcoz if this easy explationation..thank you…sir mikhe…

    Reply
  84. rania says

    July 18, 2014 at 7:14 am

    Thanks a lot this made my life a lot easier in the PHD
    Thanks again!!

    Reply
  85. Emily says

    July 13, 2014 at 8:33 pm

    Dr. Rahn- I’ve been trying all afternoon to understand a research article that used this method and this was the first explanation that has helped me. Thank you very much for posting it!

    Reply
  86. jessica says

    June 30, 2014 at 11:01 am

    Thanks, this was great. simple and to the point. many thanks.

    Reply
  87. seatlathebe ephraim lepomane says

    March 11, 2014 at 4:54 am

    Dear Dr.

    very simple and informative.

    thanks

    Reply
  88. hari says

    November 10, 2013 at 10:53 am

    the first one is correct. the Factor is a linear combination of the original variable. Hence, your first formula, represents the required info.

    Reply
  89. Jakob says

    September 17, 2013 at 10:14 pm

    Dear Dr. Rahn,

    I would like to hear your opinion if this method is valid:

    I have used a PLS model and created an ‘factor’ (lets called it “Loyalty”). To make that factor I’ve used four variables and the factor loadings are the following:

    s1 factorloading: 0,934
    s2 factorloading: 0,886
    s3 factorloading: 0,913
    s4 factorloading: 0,937

    Next I would like to estimate the loyalty of a respondent, who has the following values:

    s1 = 3
    s2 = 4
    s3 = 4
    s4 = 2

    How can I emerge these values to one value and group each respondent into e.g. two groups (e.g. high loyalty, low loyalty)

    I have an idea:
    I use this formular:

    Sum of (factorloading (si) * values(si))

    (0.934 * 3) + (0.886 * 4) + (0.913 * 4) * (0.937 * 2) = 11.872

    or maybe this formular:

    Sum of (factorloadings(si) / (sum of factorloadings(s1,s2,s3,s4)) * values(si)

    ((0.934/(0.934+0.886+0.913+0.937)) * 3) + ((0.886/ (0.934+0.886+0.913+0.937)) * 4 + ((0.913 * (0.934+0.886+0.913+0.937)) * 4 + ((0.937 * (0.934+0.886+0.913+0.937)) * 2) = 3.23
    Using this formular in this example would give the respondent a value of:

    which formular is the right one (if any), and if either of them are the right one, what is?

    thanks

    p.s. Anyone is welcome to answer this question 🙂

    Reply
  90. Wilbert says

    September 12, 2013 at 9:04 am

    Very clear and useful description, also understandable for non-mathematicians, e.g. linguists. Many thanks for posting this!

    Reply
  91. Clint says

    August 17, 2013 at 7:33 pm

    Hello Dr. Rahn

    This was the best and and easiest to understand explanation of Factor Analysis I have found. I will book mark your page as a future reference. Thanks

    Clint

    Reply

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