Optimal Treatment Assignment Rule With Multiple Treatments In Personalized Medicine

Patients with different characteristics have significant heterogeneity in their responses to treatments.Personalized medicine provides the right treatment to the right individual patient according to patient characteristics.Thus,it becomes an increasingly important research topic among clinical and intervention scientists in establishing an evidence-based personalized treatment assignment rule.The traditional approach is finding an optimal treatment assignment rule through the estimation of the response expectation.Efforts have been made to separate the main effect from the covariate-treatment interaction effect,either through multiple testing strategies[82]or through prediction[33]in this approach.A more recent alternative approach is to by-pass the estimation of the covariate-treatment interaction and directly search an optimal treatment assignment rule by maximizing the expected clinical outcome.This approach is named as outcome weighted learning[94,96].However,almost all existing outcome weighted learning methods are for the case of two treatments,which can not directly extend to the case of multiple treatments.The purpose of this PhD thesis is to develop an outcome weighted learning method for three or more treatments,and to study its theoretical and empirical properties.The main works are as follows:The expected clinical outcome involves the 0-1 loss,which is difficult to maximize due to its discontinuity and nonconvexity.In the case of two treatments,the 0-1 loss is replaced by a convex surrogate loss,the hinge loss.We utilize a vector hinge loss[54]in multicategory support vector machine and prove that maximizing the expected clinical outcome is equivalent to minimizing the risk under the convex vector hinge loss weighted by clinical outcomes.This is called Fisher consistency and justifies the validity of using the vector hinge loss.Since the minimization of the risk function corresponding to the vector hinge loss is still quite hard in application,we use the skill of Reproducing Kernel Hilbert Space(RKHS)[54]to turn the original minimization problem into a quadratic programming problem with some equality and inequality constraints.In practice,the quadratic programming problem can be easily carried out via available software packages.Naturally,an important theoretical question is,as the sample size of the training data increases to infinity,whether the risk evaluated at the solution in RKHS converges to the risk of the unknown optimal rule,which is called risk-consistency.We first establish a relationship between the excess risk under the vector hinge loss and the excess risk under the expected clinical outcome.Using this result,we prove the risk-consistency of our proposed solution under some minor conditions.Our proof is more rigorous than some existing similar proofs in the following aspects.The first one is the technical requirement that the range of covariate values is compact.We show that this requirement can be relaxed by applying a one-to-one bounded transformation and our solution is invariant with this transformation.The second one is that the RKHS we use needs to be dense in the space of continuous functions,which can be achieved by using the so-called universal kernel in RKHS.All existing approaches so far are limited to equal losses,i.e.,the misclassification costs are equal.The case of unequal losses may be encountered in real world problem-s,especially in medical applications.We develop a framework to extend the outcome weighted learning to unequal loss case,and establish the Fisher consistency together with the risk consistency when a vector hinge unequal-loss function is used.Due to the poor interpretability of traditional SVM,we introduce a structured SVM to further enhance the interpretability of the optimal assignment rule and identify impor-tant predictors,which is often of crucial importance in practice.Based on the original risk function,we added an extra l1 penalty,which contributed to the sparsity of the final solution.At the same time,we give a simple and easy handling one-step update process to solve the optimal treatment assignment rule.With technology advances,the number of measured covariates nowadays is often very large,even comparable with the sample size.However,the number of covariate actually related with the response is usually small.This suggests the necessity of covariate selection or screening,or dimension reduction in the process of constructing the optimal rule.Hence,we propose an adaptive covariate screening procedure that can actually handle the situation where the covariate sets in different treatments are different,and it remains to be risk-consistent if the covariate screening or the dimension reduction is consistent.Besides the theoretical derivation,we carry out many simulation studies to evaluate the fixed sample performance of the proposed method in both low dimensional covariate and high dimensional covariate cases.Our proposed method is compared to two other methods,the one-versus-others and weighted tree method.We consider many scenarios in low dimensional case.The performance of our proposed method is better in terms of the misclassification error rate or the magnitude of the excess risk.The improvement by using our approach can be tremendous in some cases.In high dimensional case,we study the performance of the proposed method after covariate screening and compare it with the Oracle method.The performance of the proposed method with screening is close to the Oracle method when the sample size is large.Finally,we apply the proposed method to a real data set from a breast cancer mam-mography behavioral study with four treatment arms.We define a comparative treatment effect to measure the increase of average outcome when assigning patients according to the proposed rule.The m-out-of-n bootstrap method is used to construct a confidence interval for the quantity.The results show that the estimated treatment assignment rule by using our proposed method seems to be able to enhance the treatment effect.The achievements and methodologies in our study enrich the theory to estimate the optimal treatment assignment rule,which also help to analyse the complicated and volatile problems in personalized medicine.