FACTOR ANALYSIS WITH MISSING DATA *

Factor analysis has been used primarily in application to problems of multiple correlation. It has been most frequently applied to matrices of correlation coefficients. However, while the classical model for factor analysis is based on this data framework, the underlying assumptions on the nature of the data do not restrict it to this purpose and it may be used fruitfully on a much broader scale in the analysis of data which may be arranged in tabular form. Since even data which are inherently continuous must be sampled a t discrete intervals for digital computer analysis, continuous as well as discrete data may find factor methods useful in their analysis. In particular, we deal with measurements which may be placed in the form of a rectangular two-dimensional table or matrix of a function of two variables. All the entries in the matrix are measurements of the same dependent variable under different conditions. The row in which the measurement is entered is determined by one independent variable, and the column by another. An example of such a table is shown in TABLE 1, in which the entries are values of the ratio of sulfanilamide concentration in tissue and plasma; the column number is determined by the location in the body at which the concentration is measured, and the row number by the type of sulfanilamide used. Such tables are, of course, extremely common in biomedical work. Factor analysis may be of assistance in the analysis of such data when the dependent variable is equal to a function of one independent variable times a function of another, or to a sum of such functions. The terms “function” and “independent variable” must here be viewed in a very broad sense. They may be continuous or discrete, or a combination of these. For example, one independent variable might be the concentration of the drug, taking on many values from zero (control) to some maximum; this would be a sampled continuous variable. Or, an independent variable might be, as in TABLE 1, the type of drug administered; this is a nominal discrete variable. For two continuous independent variables, the data are fitted by: