When one designates factor analysis (actually factor analyses) a collection of frequently together applied statistic procedures, with which several variables can be seized to some few factors together. One speaks also of variable bundling. The factor analysis is arranged to the datareducing (also dimension-reducing) statistic procedures and used due to numerous advantages very frequently.

For the computation of the factors a multiplicity of extraction methods is available. A frequently used extraction method is the main component analysis PCA. Likewise to the factor analysis count procedures, which measure the quality of the factors, by setting them in purchase to the output variables. Also for this a multiplicity of alternative computations is available. Further analysis steps add, which facilitate the contentwise interpretation of the factors, as bsw. the rotation procedures.

A frequent conceptual mistake results from the acceptance, the factor analysis is identical to the main component analysis. Actually the main component analysis is today only one Extraktionsverfahren, which besides regarding the model criteria is adapted.

Philosophy of the factor analysis

Compression of information

Each variable contains observed information, which can give information on the looked for regularity to the scientist, interested in those it actually. Here often the sample parameter variance near pulled, so also with the factor analysis.

These information (variance portions) may during statistic computations not due to mathematical operations be lost, but must in as high a degree as possible remain preserved. They must be consolidated however, in particular with extensive data records, also so far that on the one hand further computations remain practically feasible, on the other hand a contentwise interpretation are still permissible. Similarly as with the compression of music and video files, which lose with rising compression at quality, a compromise is to be closed also with the factor analysis. As many and which factors in a specific application be used may, is criterion-led fixed.

Differently expressed the philosophy consists of extracting from many variables few factors which contain the same information. If the factors explain the variance of the variables well "„"“can, then the factors can be used with further computations in place of the variables. One speaks also of the fact that the factors may do the variables "„represented "“.

Hypothesis change

With the computation of the factors ideally the information, which is often redundantly present in the numerous original variables, remains large. A characteristic while handling factors consists however of the fact that the bundled information may not be interpreted contentwise no more in the same way as the raw data. As soon as the factors developed, own hypotheses must be set up for their further use and/or the original hypotheses be adapted. The factor analysis is therefore always also hypothesis-generating.

Conditions of the factor analysis

1. All variables must be at least interval-scaled.
2. of the sample
3. Contentwise good conception over the meaning of the variables, since uncertainties affect the interpretation of the developing factors and increase the interpretation clearance.
4. The developing factors must be able to be distinguished regarding their eigenvalues and charges sufficiently well from each other.

Historical to the factor analysis

The factor analysis was developed in the scientific psychology (Spearman) and is originally used today frequently in the sociological and psychological research. It is isolated in physical to find biological or chemical disciplines. The goal consisted of it, numerous very similar variables, as they occurred in questionnaires frequently, summarizes to be able, in order to arrange the further use of the data expenditure-poor.

Grundproblem: In questionnaires a personality characteristic can be never seized with an individual question. Always numerous Items must be given. Items, which place similar questions, can be interconnected:

1. Do you feel often
2. Are you during the day often
3. Do you have the need to put on the day to
4. Do you feel yourselves often
5. Do you drink during the day much

Historically one searched a procedure, with which it is possible to bundle out these 5 Items developing 5 variables i.e. connections between these variables (measured variables) to represent, by classifying these on the basis its correlations into as few, not overlapping a factors as possible. In this example a factor would be desirable, which could be interpreted contentwise as "„tired reducingness "“. Bundling should take place in such a way the fact that "„tired reducingness explains "“as large a part of the common variance of the variable as possible 1-5 so that with the further computation on this a factor must be only counted. From this practical necessity developed a collection of procedures, which rank today all together among the factor analysis.

Originally by Spearman a explorative factor analysis was developed, those the main component analysis (PCA) is very similar, however from it differs. Both together is first their model acceptance:

y = Fx+e.

where:

• y the vector of the variables which can be explained
• F the charge matrix
• x a vector of Faktorenwerten
• e a vector with residues

A fundamental difference between explorativer factor analysis and Hauptkomponentenanlyse (PCA) exists in an acceptance concerning the correlation between the residues (measuring errors). In the explorativen factor analysis one assumes, which residues would not correlate with one another, while into the PCA the residues with one another are quite correlated/can be. The whole looks then in such a way that the correlation matrix of the residues is with the factor analysis a diagonal matrix (i.e. the elements outside of the diagonals are directly 0) and the same matrix with the PCA also values not equal 0 outside of the diagonals to have can.

This "„small "“however extremely purifies difference led to a controversy over the validity of the explorativen factor analysis (not the PCA), which until today (thus scarcely 100 years) continues (see riser, J.H. (1979). Factor indeterminacy into the 1930 ' s and into the 1970 ' s"… some interesting parallels. Psychometrika, 44, 157-167.).

Were historically continued to develop bsw. the procedures:

• Main component analysis
• Main factor analysis (descriptively, explorativ)
• konfirmatorische factor analysis (structure-searchingly)

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