Maximum likelihood estimation of generalized linear models for adaptive designs: Applications and asymptotics

Abstract

Due to increasing discoveries of biomarkers and observed diversity among patients, there is growing interest in personalized medicine for the purpose of increasing the well-being of patients (ethics) and extending human life. In fact, these biomarkers and observed heterogeneity among patients are useful covariates that can be used to achieve the ethical goals of clinical trials and improving the efficiency of statistical inference. Covariate-adjusted response-adaptive (CARA) design was developed to use information in such covariates in randomization to maximize the well-being of participating patients as well as increase the efficiency of statistical inference at the end of a clinical trial. In this paper, we establish conditions for consistency and asymptotic normality of maximum likelihood (ML) estimators of generalized linear models (GLM) for a general class of adaptive designs. We prove that the ML estimators are consistent and asymptotically follow a multivariate Gaussian distribution. The efficiency of the estimators and the performance of response-adaptive (RA), CARA, and completely randomized (CR) designs are examined based on the well-being of patients under a logit model with categorical covariates. Results from our simulation studies and application to data from a clinical trial on stroke prevention in atrial fibrillation (SPAF) show that RA designs lead to ethically desirable outcomes as well as higher statistical efficiency compared to CARA designs if there is no treatment by covariate interaction in an ideal model. CARA designs were however more ethical than RA designs when there was significant interaction. K E Y W O R D Sadaptive designs, clinical trials, consistency, generalized linear models, maximum likelihood estimation 630

Three influential design quantities on the power of Wald-type tests for treatment comparisons in clinical trials

“…Let θn be the maximum likelihood estimator of the vector of true parameters θ 0 , where θ 0 = (β A0 , γ 00 , γ 0 , δ 0 ) . Readers may refer to Selvaratnam, Yi and Oyet (2019) for detailed discussion and implementation of the maximum likelihood (ML) estimation of θ 0 with examples. We note that the non-random Fisher information matrix I(θ) associated with the model (2.1) can be partitioned, such that…”

“…The proof of Theorem 1 follows directly from Lemma 3.2 and Theorem 3 of Selvaratnam, Yi and Oyet (2019). Now, in general, the hypotheses for testing the significance of any of the effects in the model (2.1) can be written as…”

“…Though, the results in this paper apply to a general class of designs, we will assume that treatments were assigned to the 500 or 1000 patients based on RA or CARA or CR design. To generate the data for our simulation studies based on these 3 designs, we have followed the detailed process for treatment assignment and data generation discussed by Selvaratnam, Yi and Oyet (2019). The combination of true parameter values we have used for data generation in this paper were:…”

“…Previous authors have used only simulation studies to document the advantages RA and CARA designs have over CR designs in achieving this important ethical objective of any clinical trial. Recently, however, Selvaratnam, Yi and Oyet (2019) introduced a new approach for using data from an actual clinical trial to mimick the trial under a different design in order to obtain real data which can be used to compare the performance of designs. They implemented their approach on data obtained from a clinical trial on Stroke Prevention in Atrial Fibrillation (SPAF) (Hart et al, 2003), which was obtained based on a CR design.…”

“…Furthermore, we extend the results of Ma, Hu and Zhang (2015) and Selvaratnam, Yi and Oyet (2019) by constructing Wald-type test statistics for, (a) examining whether the effect of treatment-bycovariate interaction on the response is significant; and (b) comparing two treatments, say treatment A (new treatment) and treatment B (existing treatment), under a general class of adaptive designs when the true model is a GLM with treatment-by-covariate interaction. Let β A0 be the true effect of treatment A compared to treatment B.…”