Teaching Factor Analysis in Terms of Variable Space and Subject Space Using Multimedia Visualization

Figures & data

Table 1. Disciplines of Respondents to the Survey

Graduate programs	Frequency	Percentage
Education	9	36%
Social sciences	6	24%
Unknown	5	20%
Physical sciences & engineering	3	12%
Business	1	4%
Nursing	1	4%
	25	100%

Figure 1a. Regression Represented in Subject Space

Figure 1b. Factor Analysis Represented in Subject Space

Table 2. Mapping Between Variable Space and Subject Space

	Variable space	Subject space
Graphical representation	The axes are variables whereas the data points are people.	The axes are people whereas the data points are variables.
Reduction	The purpose of regression analysis is to reduce a large number of people's responses into a small manageable number of trends called regression lines.	The purpose of factor analysis is to reduce a large number of variables into a small manageable number of factors which are represented by eigenvectors.
Fit	This reduction of people's responses is essentially to make the scattered data form a meaningful pattern. To find the pattern in variable space we “fit” the regression line to the people's responses. In statistical terminology we call it the best fit.	In subject space we look for the fit between the variables and the factors. We want each variable to “load” into the factor most related to it. In statistical terminology we call this factor loading.
Criterion	In regression we sum the squares of residuals and make the best fit based on the theory of least squares. These are the criteria used to make the reduction and the fit.	In factor analysis we sum the squares of factor loadings to get the eigenvalue. The sizes of the eigenvalues determine how many factors are “extracted” from the variables.
Structure	In regression we want the regression line to represent the trend for as many points as possible.	In factor analysis the eigenvalue is geometrically expressed with the eigenvector. We want the eigenvector to represent variability for as many points as possible. In statistical terminology we call this simple structure, which will be explained later.
Equation	In regression the relationship between the outcome variable and the predictor variables can be expressed in a weighted linear combination such as Y = a + b1 X1 + b2 X2 + e.	In factor analysis the relationship between the latent variable (factor) and the observed variables can also be expressed in a weighted linear combination such as Y = b1 X1 + b2 X2. Note that there is no intercept in the equation.

Figure 2. Example of a Coneplot.

Figure 3. Example of a Biplot.