## Factor analysis

**Factor analysis is an arithmetic tool to see whether the items have enough variance in common so they can be treated as a single construct.**

**What is factor analysis used for**

Suppose you want to know if people are satisfied with the house they live in. For this a single question can be asked: ‘Are you satisfied with the house you live in? Yes / No.’ This question leads to a dichotomy and this does not give a nicely and finetuned vision. It may even be a reason why research questions about pleasant living are difficult to answer.

A question like ‘How satisfied are you with the house you live in? Very satisfied, satisfied, neutral unsatisfied, very unsatisfied’ is an improvement. Such a question is measured at an ordinal sale, but is still is not finetuned. It is still too general.

A further improvement is creating a list of questions. Now a nicely finetuned variable can be realised. This list might look like:

- Are you satisfied with your living room?

- Are you satisfied with your kitchen?

- Are you satisfied with your bedroom?

- Are you satisfied with your doorway?

- Are you satisfied with your garden?

- Are you satisfied with your shelter?

These question can be presented in a matrix-form with the answers very satisfied, satisfied, neutral unsatisfied, very unsatisfied. In this questionnaire, the answers can be given numbers. Very satisfied will score a 5, satisfied a 4 and so on and very unsatisfied will score 1. You can sum these numbers or compute a mean to get a score for the total. A high outcome reflects a high degree of satisfaction and a low outcome low satisfaction. Besides you have created a variable at interval level with which a lot more statistical analysis can be done.

Do not be too happy yet. Any statement in science should be tested. Are you sure the created variable reflects the construct? What supports your statement? First of all, support can be found from specialists in the field. If they all say you are measuring the satisfaction with the house that people live in, it will get some support in validity. More support can be obtained by performing a factor analysis on the respondents' answers.

**The principle of factor analysis**

The use of factor analyses is described in more detail in my paper Factor analysis and Cronbach´s alpha. In short, the goal of factor analysis is to reduce the number of dimensions. The six questions above represent 6 dimensions. With factor analysis this can be reduced. Each question has loading on these factors, for instance like the table below:

Oops. It looks like something went wrong. Instead of one factor ‘satisfaction with the house you live in’ two factors are obtained. What might be the reasons for this outcome?

In dimension I the questions 1, 2, 3 and 4 have a rather high score and the questions 5 and 6 score rather low. In dimension II it is the opposite. And wait, the questions 1 to 4 are about the house itself, and the questions 5 and 6 are more about the extra’s extras or the surrounding of the house. So the conclusion is not true that the six items reflect one dimension. These six questions reflect two dimensions. One dimension can be called satisfaction with the building. The other one can be called satisfaction with the space outside the house.

**Unrotated and rotated factor loadings**

The above outcome is the pure result of a factor analysis. Of each item the score on the dimension is shown indicating how much the item contributes to the dimension. This outcome of the factor analysis is called the unrotated solution.

As you can guess, when there is an unrotated solution, there must be a rotated solution. A handful rotated solutions can be distinguished. The most widely used is varimax rotation. The name already explains what is opted with this technique. It tries to maximise the explained variance. Compared to the unrotated solution, applying varimax rotation leads to higher factor loadings. However, the scores of an item might change drastically and the item may change from one factor to another. This might help to interpret a factor, but it is not always helpful. It could even make rubbish.

Much more can be told about factor analysis. If you want to know more, I recommend reading my paper about Factor analysis and Cronbach’s alpha.