A preliminary exploratory factor analysis was performed with SAS using principal factors and considering all 31 10nm bands to be manifest variables. This solution was rotated to simple structure using an oblique rotation method (Promax) and the factor loadings were plotted on the same scale as the color matching curves. This factor solution was stable over the natural versus man--made subsets of the data and as illustrated in Figure 2-B, the factor loadings appeared very similar to the empirical color matching curves shown in Figure 2--A.
Maximum likelihood estimation of the common factor model allows the calculation of a goodness of fit index when parameters in a factor model are constrained. In this analysis each factor loading was constrained to be equal to the corresponding mean color matching coefficient from the Guild and Wright experiments. Then the factor model was fit to the 15 odd--numbered 10nm wavelength bands of reflectance spectra data using SAS's PROC CALIS. In this model the only values which were allowed to be free were the factor variances , factor covariances and the unique variances of each wavelength band.
Figure 2: (A) Mean color matching curves; data from 3 studies reported in Wright (1928). Each line expresses the amount of light of the appropriate primary wavelength (nm, nm, nm) required to match a sample of wavelength on the X axis. (B) Oblique (Promax) rotation of three principal factors solution using 31 10nm wavelength band (395nm to 705nm) measurements of reflectance spectra from 354 natural and man-made objects.
Table 1: Goodness of fit indices from confirmatory maximum likelihood factor analysis of a three factor model with loadings constrained to psychophysical color matching coefficients. Note: RMSEA refers to Root Mean Square Error of Approximation, a combined measure of goodness of fit and parsimony.
Table 1 presents the results of testing the goodness of fit of the factor solution obtained by the adaptation of the eye to the photic environment under the assumption that the perception of color could be modeled as a linear combination of three wavelengths. At a cost of only six degrees of freedom, the of the three factor model has been reduced to nearly one third of of the Null Model. However, there are two potential sources of misfit in this model: (1) the constrained loadings may not represent a good fit to the spectral distributions, or (2) no three factor model may be a good fit these spectral distributions. By fitting unconstrained maximum likelihood factor models we can disentangle these two sources of misfit.
Unconstrained factor models with 3, 4 and 5 factors were fit using maximum likelihood estimation so that and RMSEA goodness of fit indices could be computed. Table 2 presents the results of these analyses. At a cost of another 36 degrees of freedom, the unconstrained three factor model achieves a 30% reduction in the over the constrained model. According to the the RMSEA 95% confidence intervals, the constrained model marginally achieves a significantly better fit than the unconstrained model. In fact, the RMSEA of the constrained model in Table 1 is the same as the unconstrained four factor model. This suggests that the psychophysical color matching coefficients provide a near to optimum fit to a three factor model of the covariances of spectral distributions from the environment.
Table 2: Goodness of fit indices from unconstrained maximum likelihood factor analyses for the null model and three, four and five factors.
Table 2 shows slight improvements in fit for the four and five factor models over the three factor model. Figure 3--A plots the factor patterns for the four factor unconstrained maximum likelihood model. The first three factors in Figure 3--A show a similar factor pattern to the three factors plotted in Figure 2-B. The fourth factor appears to provide a contrast between wavelengths centered around 510nm and wavelengths centered around 590nm.
Figure 3: (A) Factor loadings from a four factor unconstrained maximum likelihood model fit to 15 10nm bins of reflectance spectra of 354 natural and man-made objects. (B) Factor loadings from a three factor unconstrained model plus a fourth factor calculated as a nonlinear combination of the first three factors.
Could a fourth factor be perceived when the human retina only contains cones with three primary wavelength sensitivities? A nonlinear combination of the three primary factors such that F4 = FM * (FL - FS) will produce a calculated fourth factor which has a pattern of loadings very similar to the pattern of loadings observed in the four factor unconstrained model (compare Figure 3--B and Figure 3--A). The visual system may exploit this nonlinear dependency in the spectral environment in order to obtain a fourth factor without the biological cost of another receptor.