The relation of sample size to the number of variables in using factor analysis techniques

The relationship of sample size to number of variables in the use of factor analysis has been treated by many investigators. In attempting to explore what the minimum sample size should be, none of these investigators pointed out the constraints imposed on the dimension-ality of the variables by using a sample size smaller than the number of variables. A review of studies in this area is made as well as suggestions for resolution of the problem. ONE of the basic axioms of factor analysis is that the number of variables ( V) in the correlation matrix should not exceed the number of observations (N). In fact, this axiom is so taken for granted by Thurstone (1947), Guilford (1954), and Harman (1960) that they state it without any supporting reasons. Of course, the most obvious reason for having N greater than V is that otherwise one restricts the maximum number of linearly inde-pendent factors that can be extracted from the correlation matrix. This can readily be shown by the following: The rank of a matrix is the maximum number of linearly indepen-dent row (or column) vectors of that matrix (Murdoch, 1957). The rank of the product of two matrices is less than or equal to the rank of either matrix. Whenever the number of rows is not equal to the number of columns of a matrix the maximum number of linearly independent vectors is equal to the smaller number of rows or columns of that matrix. Thus, the maximum number of linearly independent factors that can be extracted from a correlation matrix R is equal to the rank of either matrix that is used to generate R