A Maximum Likelihood Mixture Approach for Multivariate Hypothesis Testing in case of Incomplete Data

Multivariate hypothesis testing becomes more and more necessary when data is in the process of changing from scalar and univariate format to multivariate format, especially financial and biological data is often constituted of n-dimension vectors. Likelihood ratio test is the best method that applies the test on mean of multivariate sample with known or unknown covariance matrix but it is impossible to use likelihood ratio test in case of incomplete data when the data incompletion gets popular because of many reasons in reality. Therefore, this research proposes a new approach that gives an ability to apply likelihood ratio test into incomplete data. Instead of replacing missing values in incomplete sample by estimated values, this approach classifies incomplete sample into groups and each group is represented by a potential or partial distribution. All partial distributions are unified into a mixture model which is optimized via expectation maximization (EM) algorithm. Finally, likelihood ratio test is performed on mixture model instead of incomplete sample. This research provides a thorough description of proposed approach and mathematical proof that is necessary to such approach. The comparison of mixture model approach and filling missing values approach is also discussed in this research.

Measuring community shifts in a weed seedbank study with the use of distance-based redundancy analysis

Distance-based redundancy analysis (db-RDA), a recently developed ordination technique useful for both multivariate hypothesis testing and data interpretation, was used to evaluate treatment effects on weed communities in a long-term study of alternative potato cropping systems. The experiment consisted of a factorial arrangement of three pest management systems, conventional (CON), reduced input (RI), and biointensive (BIO), two soil management systems (amended vs. unamended), and two crop-rotation entry points. Soil samples collected in the spring of 1998 were subjected to exhaustive germination as a means of characterizing the weed community. Using partial ordinations, each factor in the factorial treatment structure was tested separately, revealing a significant interaction between pest and soil management systems. An ordination diagram of the pest by soil management interaction was used to interpret the results. Weed species that were highly correlated with the first two ordination axes included: common lambsquarters, broadleaf plantain, oakleaf goosefoot, common hempnettle and a complex of the Brassicaceae that included wild mustard, birdsrape mustard, and wild radish. Univariate analyses confirmed the response of these species to the factors examined. The BIO pest management system showed a different response to soil amendments than the other systems. Soil amendments caused an increase in the total weed density in the CON and RI systems, but caused a decrease in the BIO system. Given the need for better multivariate hypothesis testing and data interpretation in many types of weed science research, the use of db-RDA is expected to grow.