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Hence a set of factors and factor loadings is unique only up to an orthogonal transformation.
Example[ edit ] Suppose a psychologist has the hypothesis that there are two kinds of intelligence"verbal intelligence" and "mathematical intelligence", neither of which is directly observed.
Evidence for the hypothesis is sought in the examination scores from each of 10 different academic fields of students.
If each student is chosen randomly from a large populationthen each student's 10 scores are random variables. The psychologist's hypothesis may say that for each of the 10 academic fields, the score averaged over the group of all students who share some common pair of values for verbal and mathematical "intelligences" is some constant times their level of verbal intelligence plus another constant times their level of mathematical intelligence, i.
The numbers for a particular subject, by which the two kinds of intelligence are multiplied to obtain the expected score, are posited by the hypothesis to be the same for all intelligence level pairs, and are called "factor loading" for this subject.
The numbers 10 and 6 are the factor loadings associated with astronomy. Other academic subjects may have different factor loadings.
Two students having identical degrees of verbal intelligence and identical degrees of mathematical intelligence may have different aptitudes in astronomy because individual aptitudes differ from average aptitudes.
That difference is called the "error" — a statistical term that means the amount by which an individual differs from what is average for his or her levels of intelligence see errors and residuals in statistics. The observable data that go into factor analysis would be 10 scores of each of the students, a total of 10, numbers.
The factor loadings and levels of the two kinds of intelligence of each student must be inferred from the data.
Mathematical model of the same example[ edit ] In the following, matrices will be indicated by indexed variables.disadvantages. Debt financing has its limitations and drawbacks.
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