Eigenvalues and the Number-of-Factors Problem
Now that we have a measure of how much variance each successive factor extracts, we can return to the question of how many factors to retain. As mentioned earlier, by its nature this is an arbitrary decision. However, there are some guidelines that are commonly used, and that, in practice, seem to yield the best results.

The Kaiser criterion. First, we can retain only factors with eigenvalues greater than 1. In essence this is like saying that, unless a factor extracts at least as much as the equivalent of one original variable, we drop it. This criterion was proposed by Kaiser (1960), and is probably the one most widely used. In our example above, using this criterion, we would retain 2 factors (principal components).

The scree test. A graphical method is the scree test first proposed by Cattell (1966). We can plot the eigenvalues shown above in a simple line plot.

Cattell suggests to find the place where the smooth decrease of eigenvalues appears to level off to the right of the plot. To the right of this point, presumably, one finds only "factorial scree" -- "scree" is the geological term referring to the debris which collects on the lower part of a rocky slope. According to this criterion, we would probably retain 2 or 3 factors in our example.

Which criterion to use. Both criteria have been studied in detail (Browne, 1968; Cattell & Jaspers, 1967; Hakstian, Rogers, & Cattell, 1982; Linn, 1968; Tucker, Koopman & Linn, 1969). Theoretically, one can evaluate those criteria by generating random data based on a particular number of factors. One can then see whether the number of factors is accurately detected by those criteria. Using this general technique, the first method (Kaiser criterion) sometimes retains too many factors, while the second technique (scree test) sometimes retains too few; however, both do quite well under normal conditions, that is, when there are relatively few factors and many cases. In practice, an additional important aspect is the extent to which a solution is interpretable. Therefore, one usually examines several solutions with more or fewer factors, and chooses the one that makes the best "sense." We will discuss this issue in the context of factor rotations below.

• Comprehensibility. Though not a strictly mathematical criterion, there is much to be said for limiting the number of factors to those whose dimension of meaning is readily comprehensible. Often this is the first two or three. Using one or more of the methods below, the researcher determines an appropriate range of solutions to investigate. For instance, the Kaiser criterion may suggest three factors and the scree test may suggest 5, so the researcher may request 3-, 4-, and 5-factor solutions and select the solution which generates the most comprehensible factor structure.

• Kaiser criterion: A common rule of thumb for dropping the least important factors from the analysis is the K1 rule. Though originated earlier by Guttman in 1954, the criterion is usually referenced in relation to Kaiser's 1960 work which relied upon it. The Kaiser rule is to drop all components with eigenvalues under 1.0. It may overestimate or underestimate the true number of factors; the preponderance of simulation study evidence suggests it usually overestimates the true number of factors, sometimes severely so (Lance, Butts, and Michels, 2006). The Kaiser criterion is the default in SPSS and most computer programs but is not recommended when used as the sole cut-off criterion for estimated the number of factors.

• Scree plot: The Cattell scree test plots the components as the X axis and the corresponding eigenvalues as the Y axis. As one moves to the right, toward later components, the eigenvalues drop. When the drop ceases and the curve makes an elbow toward less steep decline, Cattell's scree test says to drop all further components after the one starting the elbow. This rule is sometimes criticised for being amenable to researcher-controlled "fudging." That is, as picking the "elbow" can be subjective because the curve has multiple elbows or is a smooth curve, the researcher may be tempted to set the cut-off at the number of factors desired by his or her research agenda.Researcher bias may be introduced due to the subjectivity involved in selecting the elbow. The scree criterion may result in fewer or more factors than the Kaiser criterion. Scree plot example

• Parallel analysis (PA), also known as Humphrey-Ilgen parallel analysis. PA is now often recommended as the best method to assess the true number of factors (Velicer, Eaton, and Fava, 2000: 67; Lance, Butts, and Michels, 2006). PA selects the factors which are greater than random. The actual data are factor analyzed, and separately one does a factor analysis of a matrix of random numbers representing the same number of cases and variables. For both actual and random solutions, the number of factors on the x axis and cumulative eigenvalues on the y axis is plotted. Where the two lines intersect determines the number of factors to be extracted. Though not available directly in SPSS or SAS, O'Connor (2000) presents programs to implement PA in SPSS, SAS, and MATLAB. These programs are located at http://flash.lakeheadu.ca/~boconno2/nfactors.html.

• Minimum average partial (MAP) criterion. Developed by Velicer, this criterion is similar to PA in good resuls, but more complex to implement. O'Connor (2000), linked above, also presents programs for MAP.

• Variance explained criteria: Some researchers simply use the rule of keeping enough factors to account for 90% (sometimes 80%) of the variation. Where the researcher's goal emphasizes parsimony (explaining variance with as few factors as possible), the criterion could be as low as 50%.

• Joliffe criterion: A less used, more liberal rule of thumb which may result in twice as many factors as the Kaiser criterion. The Joliffe rule is to crop all components with eigenvalues under .7.

• Mean eigenvalue. This rule uses only the factors whose eigenvalues are at or above the mean eigenvalue. This strict rule may result in too few factors.
  Before dropping a factor below one's cut-off, however, the researcher should check its correlation with the dependent variable. A very small factor can have a large correlation with the dependent variable, in which case it should not be dropped. Also, as a rule of thumb, factors should have at least three high, interpretable loadings -- fewer may suggest that the reasearcher has asked for too many factors.