## Multivariate Analysis Of Variance

Multivariate analysis of variance (MANOVA) is another of the MVA procedures of use in analyzing multicomponent data. It is akin to analysis of variance (ANOVA). In ANOVA, the total sum of squared deviations about the grand mean, SS (total), is partitioned into a sum of squares due to one or more sources and a residual sum of squares. In MANOVA, the p-dimensional multivariate analysis is based on the same design, there are pSS (total)'s to partition, one for each component measured. In addition, there are the measures of covariance between pairs of components presented as sums of products. MANOVA is concerned with partitioning of both of these measures, the variance and covariance, collected in a matrix of sums of squares and of products. The result is that MANOVA yields a global estimate as to whether there are any significant differences among the different variables or their correlations. MANOVA can be separated into ANOVAs yielding the same information. The number of ANOVA calculations required is p(p + l)/2. To illustrate, if there are seven dimensions, 28 ANOVAs would yield the same information as the one MANOVA.

The effect of the above is that ANOVA for one type of measurement acts as if the others did not exist whereas MANOVA takes all of the different kinds of measurements made into account in one operation. The global aspect means that if the multivariate F-value is not significant, there is not a significant difference anywhere among the variables. If the multivariate F-value is significant, there is statistical significance somewhere. For most purposes, the analyst then must apply other tests to ascertain where significance lies.

A use of MANOVA that has value in sensory analysis is to ascertain just how effective a judge has been at performing several tasks. When a prospective sensory judge has been trained to assign scores to several different attributes of the sensory material, there is a problem at the end of the training period to decide just how effective the judge should be for the various tests. Must each judge be significant for each one of the attributes examined, or must the sensory leader be willing to accept the fact that not every judge can be statistically significant for each of the tasks. The usual way is to look at all the probability values that go with the F-test (ANOVA) for each of the tasks, then make a judgment based on a composite opinion of overall performance. MANOVA puts a little objectivity into the process. The thing to do is to calculate the multivariate F-value; from it, some of the tediousness of examining a whole matrix of F-values or probability values for each judge can be eliminated. If a judge exhibits a multivariate F that has a high degree of probability, it is almost certainly possible to eliminate that judge as a suitable candidate for appointment to the sensory panel. If the probability is very low, the judge should be retained. Only for judges intermediate in performance need the sensory leader go back to the ANOVA matrix to make the final decision. Perhaps the judge is a poor performer for only one attribute in which case the leader might decide to accept the judge in spite of a questionable multivariate F-value. If the ANOVA documents that performance has been spotty across the board, the sensory leader more likely will reject the judge. Illustrations of the joint use of the frequency of significant probability according to ANOVA calculations and by MANOVA calculation have been given (15,38,39). MANOVA's merit is chiefly one of telling the analyst to stop at a particular point, because significance doesn't exist, or to go on, because significance does exist, but other methods of analysis must be applied to determine where or how often.

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