Mathematics and Statistics, Department ofQueen's University Informationhttp://hdl.handle.net/1974/62017-05-22T21:20:30Z2017-05-22T21:20:30ZStatistical Methods For Biomarker Threshold Models in Clinical TrialsGavanji, Parisahttp://hdl.handle.net/1974/153452017-03-30T14:26:56ZStatistical Methods For Biomarker Threshold Models in Clinical Trials
Gavanji, Parisa
In clinical trials, the main objective is to investigate the treatment effects on patients. However, many molecularly targeted drugs or treatments tend to benefit a subset of patients more, identified by a certain biomarker. The cut-point value defining patient subsets is often unknown. For this situation, the ordinary likelihood ratio test cannot be applied for testing treatment-biomarker interaction because of the model irregularities.
We develop a residual bootstrap method to approximate the distribution of a proposed test statistic to test for treatment-biomarker interaction in survival data. Simulation studies show that the residual bootstrap test works well. The proposed method is applied to BIG 1-98 randomized clinical trial of breast cancer with Ki-67 as biomarker to consider the treatment effects on patients in two subsets. We also extend the residual bootstrap method to clustered survival data with an application to data from the I-SPY 1 clinical trial with the estrogen receptor total score as a biomarker.
Another research topic of the thesis is deriving the asymptotic distribution of a penalized likelihood ratio test statistic for testing biomarker effect and treatment-biomarker interaction in binary data. The model can be viewed as a mixture of logistic regression models with unknown cut-point for which the regularity conditions of ordinary likelihood methods are not satisfied. We first approximate the indicator function defining biomarker subgroups by a smooth continuous function. To overcome irregularities, we develop a penalized likelihood method, introducing a new idea of using random penalty term. Proposing a new set of regularity conditions helps us to study the properties and limiting distributions of the maximum penalized likelihood estimates of the parameters. We further prove that the penalized likelihood ratio test statistic has an asymptotic $\chi^{2}_{3}$ distribution under the null hypothesis. Extensive simulation studies show that the proposed test procedure works well for hypothesis testing. The proposed method is applied to a clinical trial of prostate cancer with the serum pro-static acid phosphatase (AP) as a biomarker.
Degeneracy of velocity constraints in rigid body systemsBrggs, Jonnyhttp://hdl.handle.net/1974/149752017-03-30T14:25:37Z2016-09-27T00:00:00ZDegeneracy of velocity constraints in rigid body systems
Brggs, Jonny
The equations governing the dynamics of rigid body systems with velocity constraints are singular at degenerate configurations in the constraint distribution. In this report, we describe the causes of singularities in the constraint distribution of interconnected rigid body systems with smooth configuration manifolds. A convention of defining primary velocity constraints in terms of orthogonal complements of one-dimensional subspaces is introduced. Using this convention, linear maps are defined and used to describe the space of allowable velocities of a rigid body. Through the definition of these maps, we present a condition for non-degeneracy of velocity constraints in terms of the one dimensional subspaces defining the primary velocity constraints. A method for defining the constraint subspace and distribution in terms of linear maps is presented. Using these maps, the constraint distribution is shown to be singular at configuration where there is an increase in its dimension.
2016-09-27T00:00:00ZRepresentation of numbers by quaternary quadratic formsKar, Arpitahttp://hdl.handle.net/1974/149742017-03-30T14:25:41Z2016-09-27T00:00:00ZRepresentation of numbers by quaternary quadratic forms
Kar, Arpita
2016-09-27T00:00:00ZAn R-package for the Estimation and Testing of Multiple Covariates and Biomarker Interactions for Survival Data Based on Local Partial LikelihoodZhang, Siweihttp://hdl.handle.net/1974/149732017-03-30T14:25:40Z2016-09-27T00:00:00ZAn R-package for the Estimation and Testing of Multiple Covariates and Biomarker Interactions for Survival Data Based on Local Partial Likelihood
Zhang, Siwei
When we study the variables that a ffect survival time, we usually estimate their eff ects by the Cox regression model. In biomedical research, e ffects of the covariates are often modi ed by a biomarker variable. This leads to covariates-biomarker interactions.
Here biomarker is an objective measurement of the patient characteristics at baseline.
Liu et al. (2015) has built up a local partial likelihood bootstrap model to estimate
and test this interaction e ffect of covariates and biomarker, but the R code developed
by Liu et al. (2015) can only handle one variable and one interaction term and can
not t the model with adjustment to nuisance variables. In this project, we expand
the model to allow adjustment to nuisance variables, expand the R code to take
any chosen interaction terms, and we set up many parameters for users to customize
their research. We also build up an R package called "lplb" to integrate the complex
computations into a simple interface.
We conduct numerical simulation to show that the new method has excellent fi nite
sample properties under both the null and alternative hypothesis. We also applied the
method to analyze data from a prostate cancer clinical trial with acid phosphatase
(AP) biomarker.
2016-09-27T00:00:00Z