*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.

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.*

{ 26 comments… read them below or add one }

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

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

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

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

Dear Dr.

very simple and informative.

thanks

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

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!

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

Thanks again!!

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

This was simple and clear with commonsense.

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.

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

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.

Good stuff

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.

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.

Hi Rahn,

Great Job.!!!

How am I suppose to put citations to your web site?

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?

Fantastic explanation!! Thank you

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.

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.

Well done

Very simple and nice explainations

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

A very good work, thank you sir.

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.”