Analysis of Variance for Three-Way Unbalanced Mixed Effects Interactive Model

In the study, a method of solving ANOVA problems based on an unbalanced three-way mixed effects model with interaction for data when factors A and B are fixed, and factor C is random was presented, and the required EMS was derived. Under each of the appropriate null hypotheses, it was observed that none of the derived EMS was unbiased for the other. Unbiased estimators of the mean squares were determined to test hypotheses. With the unbiased estimators, appropriate F-statistics as well as their corresponding pseudo-degrees of freedom were obtained. The theoretical results presented in the paper were illustrated using a numerical example.


Introduction
The role of multi-factor experiments in agriculture, engineering and other fields cannot be overemphasized. Through a multi-factor experiment, it is possible to test the interaction effect of two or more factors. Sometimes, a multi-factor experiment conducted to compare factor levels and factor level combinations results in unbalanced data. Often, in the case of the analysis of variance (ANOVA) for unbalanced data, an exact F-test does not exist. As a remedy to this problem, authors have recommended some methods of testing effects in various multi-factor ANOVA problems. Consequently, [1] proposed an exact permutation test for fixed effects ANOVA based on balanced and unbalanced data. [2] derived expected mean squares for the unbalanced two-way random effects model with integer degrees of freedom. The F-test statistics for testing main effects as well as

Model Specification and Restriction
The three-way unbalanced mixed effects cross-classification model with interaction terms, in which factors A and B are fixed while factor C is random is given by [4] and [5] as where: ijkl X , denotes the 1st observation at the ith level of factor A, the jth level of factor B, and the kth level of factor C, µ denotes the overall mean, i A denotes the effect of the ith level of factor A. j B denotes the effect of the jth level of factor B, k C denotes the effect of the kth level of factor C, (  Under the assumptions above, we consider the following notations so as to derive the requisite expected mean squares.

Main Results
The mean squares due to the three main effects and four interaction terms for . This completes the proof.
Similarly, if MS B and MS C denote the mean squares due to factor B and factor C respectively. Then A major step in the derivation of the F-statistics is to find the unbiased estimates of the mean squares due to the main factors and the interactions. Therefore, the unbiased estimates are presented as follows.
The F-statistics for the main effects and interactions effect are given below: and where: F A is the F-statistic for factor A, F B is the F-statistic for factor B, F C is the F-statistic for factor C, F AB is the F-statistics for the interaction factors A and B, F AC is the F-statistics for the interaction factors A and C, F BC is the F-statistics for the interaction factors B and C, and F ABC is the F-statistics for the interaction factors A, B and C. Proof: Recall that ( ) Recall that e f N abc = − Extending our idea of (33) into (31), leads to where: The degree of freedom associated with F A is The degree of freedom associated with F B is The degree of freedom associated with F C is  In the experiment, the sample (lemon grass) was dried in the oven at 45˚C for 1440 mins. The dried sample was pulverized and 1 g of pulverized sample was used for each solvent in a typical extraction 1g of sample was dissolved in 25 ml of water for 40 mins. At the end of the time, the solute was filtered using a suitable filter paper (Whatman). The solution was then vaporized at 105˚C for 720 mins leaving the remaining extract which was weighted in an analytical balance. The process was repeated and replicated three times for 50, 60 and 70 mins. A similar procedure was done using different volumes of ethanol and ether as extracts at different durations of 40, 50, 60 and 70mins. Results of the extraction are shown in Table 2.
Using the information in Table 2   The ANOVA table for the data is shown in Table 3.
Our hypothesis for factor A shall be 0 Similarly, our hypothesis for factor B shall be 0 Our hypothesis for factor C shall be Our hypothesis for factor A and C shall be Our hypothesis for factor B and C shall be

Conclusion
In this study, we have presented a hypothesis testing procedure based on an unbalanced three-way cross-classification mixed effects model with interaction when factors A and B are fixed while factor C is random. From the theoretical results obtained in this study, it was observed that exact F-tests do not exist for any of the hypotheses to be tested. As a consequence, approximate F-tests were considered. A numerical example was given to illustrate theoretical our results.