In this paper, the theory for the application of the Merge and Chop Algorithm are defined and proven. The algorithm is used to find isotonic regressions in more situations than comparable methods. A program is included with this paper for computing the desired isotonic regressions through the algorithm. These isotonic regressions can then be used to perform more accurate and powerful hypothesis tests. This paper also defines these possible areas of utilization with use of restricted maximum likelihood estimators and likelihood ratio tests.