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Factor analysis

Factor analysis is a method with which you can examine structures in your data by reducing a large number of variables to a smaller number of factors, which are, however, more central. This reduction can be done in two ways.

On the one hand, you can proceed exploratively and mathematically search for the best solution without orienting yourself in terms of content to a specific theory (= exploratory factor analysis).

On the other hand, with the help of the confirmatory factor analysis, you can test hypotheses whether a model you have set up is confirmed based on your data or not. In this case, a model refers to preliminary theoretical considerations that deal with the extent to which you can combine different variables into one factor.

Difference between exploratory and confirmatory factor analysis

The following figure shows the differences between exploratory and confirmatory factor analysis. In the exploratory factor analysis, you examine how well each individual variable fits all factors. For example, you could then find that your variables 1, 2 and 4 can best be summarized mathematically to form factor 1. Your variables 3, 5 and 6 match e.g. B. better to factor 2. The gray arrows in the figure show that z. For example, it was also calculated how well variable 1 fits factor 2.

In the confirmatory factor analysis, you make your calculations based on Hypotheses and could, for example, test whether the data pattern corresponds to your preliminary considerations and how well the variables 1, 2 and 4 reflect a common factor and the variables 3, 5 and 6 a second factor. To what extent variable 3 matches factor 1, you would not check with this method (indicated by the missing gray arrows).

Background of the factor analysis

The factor analysis is based on multivariate methods, i. H. it relates several AVs to one another. The aim is to summarize these AVs in such a way that you form factors that interact with one another as little as possible. In other words, those AVs that correlate strongly with one another are combined into one factor. A factor is therefore a purely theoretical construct that contains information from several related variables.

You can see which variable is assigned to which factor and how well it fits this factor from the so-called factor load. The factor load can be between +1 and -1 and the closer the factor load of a variable is to one of these two limits (i.e. the further it is numerically from zero), the more it loads on the factor.

Another term that you should be familiar with in the context of factor analysis is commonality. The commonality tells you how much each factor contributes to explaining the variance of each individual variable. The maximum commonality is 1 - if the commonality of a factor in relation to a single variable were 1, then almost 100% of the variance of the variable would be explained by the factor.

However, be careful not to confuse commonality with intrinsic value. The eigenvalue relates to the factors and indicates how much total variance (across all variables) is explained by a factor.

In order for you to be able to calculate a factor analysis that delivers valid results, your data set must meet the following requirements:

Requirements for a factor analysis

  • Your variables must be normally distributed
  • Your variables should be interval-scaled (you can only use dichotomous variables with reservations).
  • You should have collected at least 10 cases per variable
  • Sufficient variables must be available (four or more variables per factor)

If all requirements are met, there are various options with which you can extract factors from your data set. In principle, factor extraction works on the principle that each factor should explain as much variance in the data as possible. Two very common methods of performing factor extraction are Principal component analysis and the principal axis analysis.

How many factors you extract can either be derived from the theory in terms of content. Or you can determine how much a factor has to contribute to the explanation of variance in order for it to be included in your model. Alternatively, you can also proceed according to the eigenvalue criterion (= Kaiser-Guttman rule), according to which all factors are included that have an eigenvalue greater than 1, since the factor then definitely explains more variance than a single variable.

Interpretation of the results

It is relatively easy to carry out the factor analysis, but it is difficult to interpret the extracted factors in terms of their content. The factor analysis gives you a purely mathematical model, but does not offer any help on how to interpret it. In exploratory terms, the method is therefore primarily used to generate hypotheses. The confirmatory factor analysis is used, for example, in the development of measuring instruments. Because these are based on meticulous theoretical deductions. There are also precise hypotheses as to which variables can be combined into which factor.