A likelihood ratio test for correlated paired multivariate samples

Many laboratory experiments in the fields of biological sciences usually involve two main groups say the healthy and infected subjects. In one of these kind of experiments, each specimen from each group can be divided in two portions; one portion is stimulated while the other remains unstimulated. Consequently resulting into two main groups with paired measurements that are correlated. For all the groups, p genes are measured for expression. The stimulation in this case can be done by introducing a known infection causing micro-organism like the group A streptococcus which is usually associated with the acute rheumatic fever. An important question in such experiment would be to statistically test for the differences in the differences in means for the healthy and the infected groups. That is, the difference in the means of the healthy group (stimulated and unstimulated) is tested against the difference in the means of the infected (stimulated and unstimulated) group. In this paper, a likelihood ratio test statistic is developed for such kind of problems. The developed statistics and the Hotelling T2 statistic are both applied to the data are simulated from real biological situations and their performances are compared. The simulated data exhibit the correlation structure similar to that of real biological data obtained from experiments involving the milliplex analyst biomarker data sets. The results indicate that the proposed test statistic give the same conclusions for the hypotheses tested as those of the Hotelling T2 test. However, the proposed test is intuitively more appealing since it takes care of the correlations between the pairs in the data. The simulation study confirms that the test statistics follow a chi-square distribution. This research contributes a theoretical analysis of paired correlated samples motivated by a practical problem for which the existing statistical methods in use have seldomly taken into account the correlation structure of the data.