# Factor Analysis: A Short Introduction, Part 1

### 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.

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.

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.

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???

2. tiffany field says

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

3. Barbara says

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

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

5. Jyotirmoy Pandit says

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.

6. maryam says

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?

7. Rahmatullah says

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

8. Mariya Zheleva says

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?

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

9. atheer says

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

• Karen says

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.

10. Lanh says

Dear Maike,

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

Best wishes from Germany

11. Mark Norman says

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.

• Karen says

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.

12. Dr. Ramnath Takiar says

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?

13. samuel says

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

14. Arslan Saleem says

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

15. David Akiiki Kalenzi says

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

16. Tamanna says

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.

17. James Tan says

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

18. Mike says

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

• Karen says

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.

19. IceSwan says

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

20. Roy says

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.

21. Monica says

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 !

22. Prof Sreekumar Pillai says

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!

23. Rishi says

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)

24. Sat says

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?
Sat

25. magda says

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

26. Lucia Sauti says

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.”

27. Ali says

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

28. Abel says

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.

29. Jason lee says

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.

30. Hassan says

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?

31. DR..H.K.LAKSHMANRAO says

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.

• Rizwan says

Dear sir,

32. mohammed ibrahim, fut minna, nger state. nigeria says

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.

33. Bibi says

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.

34. john says

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

35. Amaa says

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.

36. sangeetha says

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

37. Emily says

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!

38. hari says

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

39. Jakob says

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:

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:

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

or maybe this formular:

((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 🙂

40. Wilbert says

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

41. Clint says

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

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