# How many factors/components should I retain in a factor/principal components analysis?

The importance of each factor is represented by its eigenvalue. This corresponds to the amount of variance amongst the variables that each factor accounts for. The larger the eigenvalue the more influential is the factor relative to other factors.

1. Kaiser (1960) recommends retaining any factors with eigenvalues greater than 1. This is the default in SPSS.

2. The scree test orders and then plots the eigenvalues (largest to smallest). The recommendation is to retain all factors with eigenvalues in the sharp descent before the first one on the line where they start to level off.

3. Retain as many factors as will account for a specified proportion of total variance. Generally one would want to account for at least 70% of total variance. Proportion of total variance accounted by k factors is the sum of the eigenvalues of the k factors divided by the number of variables in the analysis.

An alternative approach which you can use in SPSS for exploratory factor analyses uses a goodness of fit criterion instead of an eigenvalue.

4. For an exploratory factor analysis using the maximum likelihood method in SPSS produces an overall chi-square fit measure. Large chi-squares are associated with worse fits so one can fit increasing numbers of factors and choose the one which first yields a non-significant chi-square. The chi-square, however, is sensitive to sample size so this approach may overestimate the number of factors. An alternative would be to do a scree plot of the chi-squares plotted against the number of factors and retain all factors with chi-squares in the sharp descent before the first one on the line where they start to level off.

In practice using a combination of the above will give a more rounded view of the data.

## Reference

Stevens, J. (1996) Applied multivariate statistics for the social sciences third Edition. Lawrence Erlbaoun, NJ.