The path diagram in Figure 1 presents a schematic representation of the common factor model used in these analyses. The variance of each component wavelength band can be recovered as a linear combination of the three factors , and plus a unique variance . A factor loading is the proportion which the factor contributes to the variance of the wavelength band .
Figure 1: Path diagram of a three common factor model. , and represent the three factors (latent variables). represent the fifteen measured wavelength bands (manifest variables). represent the fifteen unique components to the variance (uniquenesses). represent the factor loadings, the proportion of the variance of each wavelength band which can be accounted for by each of the common factors.
Other researchers have analyzed spectral distributions using principal components methods [5,6]. Principal components extracts the largest source of common variance as the first component, extracts the second component from the residuals left over from the first component, and so on. There are two problems with using this method. The first problem is strictly mathematical; since principal components focuses on fitting the larger sources of common variance it tends to overestimate the goodness of fit and underestimate the number of required components . The second problem is that in order to measure the correspondence between human vision and the spectral environment, it is important to extract components which have the potential of a psychophysical interpretation. Factor analysis with an oblique rotation extracts sources of common variance (factors) without requiring that these factors be orthogonal or that one factor be as large as possible with respect to the others. There is no theoretical reason why it is to be expected that the visual system would operate under the same constraints as principal components analysis, so by relaxing these constraints the factors extracted from spectral distributions are more likely to have correspondence with psychophysical measurements.
Two forms of factor analysis were used in the present work. Exploratory factor analysis was used as a preliminary step in order to examine the structure of the common variance in the spectral distributions. Maximum likelihood latent variable analysis using a factor model was used to test the goodness of fit between the color matching data and the factors extracted from the spectral distributions. Maximum likelihood factor analysis was also used to test whether the addition of more than three factors would provide a significantly better fit to the spectral distributions.