Goodness-of-Fit Tests for Several Models
Open AccessThis thesis contains three projects in the Goodness-of-Fit test.In the first project, we propose a test for assessing nonlinear dose-response models based on a Cramer-von Mises statistic. We establish the asymptotic distribution of the test and demonstrate that the test can detect the local alternative converging to the null at the parametric rate. We provide a Wild bootstrap resampling technique to calculate the critical values. It is observed that the test has good power performance in small sample sizes. We apply the proposed method to analyze 250 datasets from a pharmacologic study and conduct two small simulation experiments to explore the numerical performance of the proposed test and compare one commonly used test in practice.In the second project, we propose a robust projection-based test to check linear regression models when the dimension may be divergent. The proposed test can maintain the type I error rate effectively when outliers are present, achieve dimension reduction as if only a single covariate was present, and inherits the robustness of the M-estimation. The test is shown to be consistent and can detect root n local alternative hypotheses. We further derive asymptotic distributions of the proposed test under the null hypothesis and analyze asymptotic properties under the local and global alternatives. We evaluate the finite-sample performance via simulation studies and apply the proposed method to analyze a real dataset as an illustration.In the third project, we propose a test for assessing Stukel's generalized logistic model for binary dose-response data. Even though two additional shape parameters are shown in the link function, we establish the asymptotic distribution of the test and demonstrate that the test can detect the local alternative converging to the null at the parametric rate 1 over square root of n. We provide a bootstrap resampling technique to calculate the critical values. It is observed that the test has good power performance with balanced and unbalanced data. We compare the performance with the Hosmer-Lemeshow test, Osius and Rojek Test, and Stukel test and show the superiority of the proposed test under the standard logistic model setting. We apply the proposed method to analyze age at menarche in Warsaw girls data and conduct two small simulation experiments to explore the numerical performance.
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