Multiple Regression Analysis

MRA is used similarly to LDA. The chief difference is that MRA is appropriate for applications where the dependent variable is along a continuum and the measurement values are continua, whereas LDA is suitable for assigning a sample to a disjoint set of categories, ie, classes. In the typical regression application, flavor intensity or something that progressively changes is predicted from several measurement values that themselves can be any value along continua. The item being predicted is called the dependent value; the predictors are the independent values. A multiple regression equation takes the form:

Y = a + b^ + b2X2 + b3X3 + b4X4 + e where Y is the value of the dependent variable; a is the intercept of the regression line; the b's are the slopes, ie, the weighting values attached to each predicting variable; e is residual error; and the Xs are the measurement values. Stepwise regression analysis (SRA) can be applied just as can SDA. The difference is that an estimate is given for the intercept and for the b values (the weighting value) for regression of each of the independent values as they are added to the total number of independent regressors included in the equation.

side by side for five years to be sure that the first indications of effectiveness were not just a case of good luck at the first try. That kind of effort is not wasted. As data accumulates, the predicting equation can be refined to make it even more applicable and robust than it probably was originally. It has been pointed out that the process of validating an equation by accumulating more and more data is much like the building up of a delta (23). Eventually through accretion, the shifting sands of random happenings become firm earth on which one can build with confidence. The caveat is to be cautious at first, no matter how good an equation seems to be, until it has proved itself on repeated occasions.

There are some features of MVA that the user should be aware of. Novices to the use of SDA and SRA are sometimes disconcerted because the process of selecting good discriminators appears to be overlooking some that are good. The step process bypasses a measurement that by itself is moderately effective as a discriminator in favor of one less of a discriminator by itself, because the computer program is looking for a discriminator not correlated with one already selected. If discriminating measurements are correlated, use of both of them adds little to the discrimination. All that is being provided is redundant information. Table 7 provides an illustration of bypassing measurements that have a high F-value, ie, are effective discriminators, in favor of measurements that are not correlated to one already selected. A noncorrelated variable is adding more to the discrimination needed than would one merely providing redundant information. Note from Table 7 that often a term not appearing to be particularly discriminating by itself is selected ahead of terms appearing to be better discriminators because the one selected in conjunction with the terms already selected adds more to discrimination than its betters would have. Sometimes a

Premises to Use of SDA or MRA

The MVA procedures above are risky ventures unless relations among the variables are based on a firm foundation of replication of measurement and the initial set of samples is truly representative of those likely to be encountered in the future. If there is not adequate sampling and replication of measurements, there is the risk that relations that appear to exist may be spurious. The usual statistical risk is set at a probability level of 0.05. When there are 7 or 10 or 20 measurement values, each one of which will fluctuate some from determination to determination, there is no practical way to calculate the total risk involved in making a wrong decision. The only ways to overcome the uncertainty are to carry on determinations long enough in time, to evaluate a large number of samples, and to choose the sources of the samples carefully to be fairly sure of having selected a fair representation of nature's variability. Although on first use an equation may appear to be a good predictor of the class to which a product belongs or the degree of some quality it has, it should be looked at with suspicion. Some industrial firms that have replaced routine sensory evaluation with instrumental analysis conducted sensory and instrumental analyses

Table 7. Illustration of Order Variables Selected to Avoid Covariation"
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