Discussion On Improving The Test Efficiency Of Multivariate Analysis Of Variance By Generalized Eigenvalue

Multivariate analysis of variance is a very important multivariate statistical analysis method.The main tasks include testing the impact of each factor on the experimental data is significant or not,estimating the effect values of the different levels of each factor,estimating the interaction effect between the various factor levels,estimating the covariance matrix and so on.The primary task is to test the impact of various factors on the experimental indicators is significant or not.So,we need to carry out hypothesis testing,there are a lot of often used test statistics such as wilks test statistic,hotelling test statistic,pillai-Bartlett criterion test statistic,and Roy maximum eigenvalue test statistic and so on.These test statistics are properly deformed and can be converted into F test statistics.In order export these test statistics,the calculation process is relatively large and transforming them into F test statistics is difficult to understand.In order to overcome these problems,high-dimensional data can be reduced to one-dimensional data by using projection techniques.It can be shown that the projected data still satisfy the same variance and obey the normal distribution.We can base on the projected data to construct F test statistics for multivariate analysis of variance.This method of analysis greatly improves the efficiency of multivariate analysis of variance.The F test statistic is different from normal F test statistic,because the projection direction is uncertain,so it is impossible to calculate the specific value of the test statistic.Projection techniques can maximize differences between groups and can also minimize differences between groups,in order to eliminate the interference of different projection directions to the test results,we can project the date to minimize the difference between groups and then calculate the specific value of the test statistic.If the value falls into the deny field,we project the data to any direction and the value are calculated will fall into the denial field,so the mean vector is really different between groups,according to the small probability event in an experiment does not occur in principle,there are enough reasons to reject the original hypothesis.In the other hand,if the value does not fall into the reject domain.It is possible when the data project in other directions,the values fall into the denial field.In order to rule out this situation,we can calculate the maximum value of the F test statistic.If the F test statistic's maximum value still falls into the accepted domain,the data project in any direction and the F test statistic's value still falls into the accepted domain,so we accept the original hypothesis.On the contrary,we can adjust significance level until it satisfies the first case or the second case.