Bhattacharya, BhaskarAlmazmomi, Afnan2023-07-092023-07-092023-07-05https://hdl.handle.net/20.500.14154/68537One of the most common tools used in statistical methodology is the regression analysis which assumes independence of data and uses the standard F-statistic to make decisions. However, for two-stage cluster samples data, the independence assumption fails, and consequently, the test statistic does not follow the F-distribution. It has been shown that when testing a null hypothesis with equality constraints the use of an F-distribution leads to in inflated type I error for two-stage cluster samples data. When testing hypotheses with inequality constraints, the standard F-statistic is updated as the   F-statistic. So far the effect of two-stage cluster samples on the   F-statistic remained unexplored, and is the topic of this dissertation. We  rst investigate under a normal model the effect of misspecified covariance matrix on the chibar test statistic for inequality constrained hypotheses. We proposed an adjusted chibar distribution which corrects the type I error. For the regression setup, we proposed a two-step generalized least square F test statistic and have shown its distribution to be of the F type. We show through simulation that the proposed test statistic performs better than the corresponding unrestricted F-test statistic in terms of type I error under the null hypothesis. It also achieves power gains under the alternative with a variety of different types of misspecification of the covariance matrix compared with the unrestricted F-test statistic.53en-UStwo-stage cluster samplesinequality constrainedgeneralized least squareThesis