## Factor Analysis

Factor analysis (FA) has been left to the last because it is a method about which an unusual amount of controversy swirls. Some statisticians look on all the internal analysis methods with some suspicion on the grounds that the results are high provincial simply because only one data set is involved. Others frown on FA particularly because of indeterminacy in factor solutions (19). There is no doubt that different solutions can be obtained according to the particular method of factor analysis employed. Notwithstanding such criticisms, FA is a procedure that is used more and more in sensory-instrumental analysis. FA is different from PCA in that PCA is concerned with variance only. FA requires a statistical model and is concerned with explaining the covariance structure as well as variance. Unlike PCA where there is no particular assumption about possible underlying structure among the variables, FA is based on the faith that the observed correlations are mainly the result of some underlying regularity in the data. The various factoring techniques initially extract common factors, but generally they are uninterpretable. To enhance interpretability, there are several models for rotation and determination of the number of factors. The ideal situation is to have a variable highly loaded on one factor and scarcely loaded on the others.

There are three kinds of matrices involved in FA. One is the factor pattern. It provides the weights to estimate variables from the factors. A second one is the factor score. That matrix provides weights to estimate factors from the variables. The third kind of a matrix is the factor structure. It gives the correlations between the factors and the variables. The pattern matrix delineates more clearly the groupings or clustering of variables than the structure matrix.

If two rotations give rise to different relationships, the two interpretations should not be looked on as conflicting. They are two different ways of looking at the same thing, ie, they are two different points of view in the common-factor space. The basic impetus to employing any rotational method is the same: somehow to achieve simpler and theoretically more meaningful factor patterns. Orthogonal factors are mathematically simpler to handle; oblique factors are empirically more realistic.

Although FA is frowned on by some statisticians, some of their reasons are highly suspect because provinciality does not apply to many sensory-instrumental data sets. Most of the applications of FA in the past dealt with such things as determining people's opinion or some similar data where there was no opportunity to replicate responses. Sensory panel results are different. Often there is replication; thus the analyst has the opportunity to determine whether the panelist is responding randomly or consistently for the same sample given at different times. The objection to FA that it is likely to be provincial may still apply to the factoring of consumer responses, because in most instances there is no replication. If, however, the survey is large enough, that objection loses some of its validity.

Whether or not replication is involved, there is one phase of FA that has been overlooked by most investigators. As is true for all methods of analysis where a mean is calculated, the following question should be asked. Is the mean merely the average of a group of disparate responses or do the responses tend to agree with each other? To illustrate, if a particular perfume receives scores of 7, 7, 6, 8, 8, 6, and 6 out of a possible score of 10, or an average score of 6.86, and another perfume receives scores of 10, 10, 4, 5, 10, 6, and 3, or an average score of 6.86, the conclusion should be entirely different although the means are the same. The first perfume is likely to have general acceptance whereas the second will appeal to some and be scorned by others. In the same fashion, rather than just generate a factor pattern for the whole sensory panel and assume that it represents the pattern for all members of the panel, the patterns yielded by the different panelists should be examined to learn if they agree in general or the pattern for the panel is merely the mean for a group of disparate patterns. Rarely, have factor patterns been examined in that light. Early studies began to look at the homogeneity of the factor patterns of a panel in that manner. One study visually examined the eigenvalues to learn if they were similar (46). Another study applied statistical tests to learn objectively how well they agree (38). Later other studies (13,47,48) refined the process even more with the result that today factor patterns should be examined as most other sensory-instrumental data are, to learn if the individual patterns are quite homogeneous or if they are heterogeneous. If they are heterogeneous, then, of course, considerably less credence should be put into their value and representativeness of how things really are. Figure 8 shows the extent of agreement between a panel of 18 and a subset of 8 from the same panel. Often it is difficult to compare the pattern for an individual member versus the panel because of lack of degrees of freedom to carry on the analysis for the individual's responses. Tb apply some of the FA methods there must be at least limited pooling of responses to provide sufficient degrees of freedom to enable a comparison to be made. Figure 9, which is a little easier to interpret than Figure 8, shows one comparison (45). While a little harder to visualize the agreement, Figure 8 is more realistic of the pattern that exists. Note that the cube standing for total flavor fits in factor 1 or 2 as does sour flavor. That is as it should be. Pointed out in the beginning is the fact that an attribute may be partly in one factor and partly in another, because FA involves both variance and covariance. The same sort of standing in one factor with one foot and in another with the other foot applies to chemical data as well. A chemical may have more than one type of functional group; consequently its covariance may be shared by two different factors reflecting the two kinds of properties caused by different functional

Figure 8. Comparison of factor patterns exhibited by a panel of 18 and a subset consisting of 8 of the panel members. The numbers in the opening of each cone stand for the factor pattern, the symbols within a cone represent the different attributes evaluated, pyramid = rye flavor; spade = rye aroma; circle = total aroma; cube = total flavor; cross = sour aroma; club symbol = sour flavor; triangle = sweet aroma; flower = sweet flavor; heart = saltiness; diamond = bitter flavor; square = flour flavor; cylinder = desirability.

Figure 8. Comparison of factor patterns exhibited by a panel of 18 and a subset consisting of 8 of the panel members. The numbers in the opening of each cone stand for the factor pattern, the symbols within a cone represent the different attributes evaluated, pyramid = rye flavor; spade = rye aroma; circle = total aroma; cube = total flavor; cross = sour aroma; club symbol = sour flavor; triangle = sweet aroma; flower = sweet flavor; heart = saltiness; diamond = bitter flavor; square = flour flavor; cylinder = desirability.

groups. A major use both of PCA and FA is to discern redundancies in the choice of descriptive terms. Table 10 shows a factor that was derived in one study. Panelists evaluated canned and frozen green beans for 27 attributes. The seven terms comprising Factor 4 reveal that the panelists had been put to a lot of unnecessary work. In theory, any one of the 7 terms could substitute for all the others. In practice, a sensory technologists might not want to go that far and might still want to use three or four different terms in case some panelists are more adept at distinguishing among products based on one descriptor whereas a different one is more meaningful to other panelists. It can readily be understood that crispness and tenderness are the complements of each other, but it is not as clear that slimy, juicy, and soggy represent nearly the same characteristic.

WHITHER SENSORY-INSTRUMENTAL CORRELATION?

The tools to carry on sensory-instrumental analysis are pretty much in place. Some industrial firms have made good use of the tools available by working out for themselves beneficial applications involving the complementary uses of sensory and instrumental analysis. There has been little full substitution of instrumental methods for sensory ones because chemical and other analytical methods themselves are not complete. There are instances of as many as 400 or more peaks being discernible on gas chromato-grams, but that number often falls short of the number of compounds comprising the sensory character of a food product; thus no chemical analysis is probably ever really

Figure 9. Comparison of factor patterns: O, panel results pooled; •, two of the panelists.