Repeated measures are increasingly collected in a study to investigate the trajectory of measures over time. One of the first research questions is to determine the correlation between two measures. The following five methods for correlation calculation are compared: (1) Pearson correlation; (2) correlation of subject means; (3) partial correlation for subject effect; (4) partial correlation for visit effect; and (5) a mixed model approach. Pearson correlation coefficient is traditionally used in a cross-sectional study. Pearson correlation is close to the correlations computed from mixed-effects models that consider the correlation structure, but Pearson correlation may not be theoretically appropriate in a repeated-measure study as it ignores the correlation of the outcomes from multiple visits within the same subject. We compare these methods with regard to the average of correlation and the mean squared error. In general, correlation under the mixed-effects model with the compound symmetric structure is recommended as its correlation is close to the nominal level with small mean square error.

At present, climate and other environmental problems are arising because of the development of the industrial sector at a large level. The industrial sector is supposed to be a major cause of climate change problems that lead to global warming. Therefore, corporate social responsibility (CSR) with the help of corporate governance is an imperative approach to control these social problems. Consequently, in the context of the organizational and management theory, agency theory, and the stakeholder theory, this study focuses on important factors of internal corporate governance such as chief executive officer (CEO) power, the board size, independence, ownership concentration, managerial ownership, and audit quality for improving the profitability of firms. Moreover, this study considers corporate social responsibility as a controlling and moderating factor for firm performance and internal corporate governance. We employed ordinary least square (OLS) for endogeneity testing, fixed effect (FE), generalized method of moments (GMM), and feasible generalized least square (FGLS) on data of Pakistani firms for the period of 2010–2019. The results of this study demonstrate the following outcomes: firstly, all internal corporate governance factors are positively linked with firm performance; secondly, corporate social responsibility (CSR) is the most valuable tool for improving profitability. Importantly, this study suggests that all internal corporate governance factors are positively linked with firm performance because of the interactive role of corporate social responsibility (CSR). This study practically contributes to the literature by suggesting the imperative role of corporate social responsibility (CSR) for internal corporate governance, which may help to reduce climate and social problems.

Simon’s two-stage design is the most widely implemented among multi-stage designs in phase II clinical trials to assess the activity of a new treatment in a single-arm study. In this two-stage design, the sample size from the second stage is fixed regardless of the number of responses observed in the first stage. We develop a new minimax adaptive design for phase II clinical trials, by using the branch-and-bound intelligent algorithm based on conditional error functions. We compare the performance of the proposed design and competitors, including Simon’s minimax design, and a modified Simon’s design that allows early stopping for futility or efficacy. The maximum sample size of the proposed minimax adaptive design is guaranteed to be less than or equal to those from other existing designs. When the proposed design has the same maximum sample size as others, it always has the smallest expected sample size. In addition to the minimax adaptive design, we also introduce admissible adaptive designs determined from a Bayesian perspective. The proposed adaptive minimax design can save sample sizes for a clinical trial. The minimum required sample size is critical to reduce the cost of a project.

Adaptive designs play an important role in contemporary clinical trials to make designs flexible and efficient. In cancer clinical trials, given a relatively small sample size, it is important to obtain as much information as possible during this phase. We propose a new adaptive optimal design that stops for futility only in the first stage as Simon’s two-stage design. The existing adaptive two-stage designs are often allowed to be stopped for futility or efficacy due to computational advantage. It is difficult to search for an optimal design with futility stopping only in the first stage by using efficient search algorithms; for example, the branch-and-bound algorithm. We have to use multiple computational techniques to search for the optimal design. The proposed adaptive design meets the important property of the monotonic property that the second stage sample size is a nonincreasing function of the number of responses from the first stage. In this article, we show that the proposed adaptive design always has a smaller expected sample size than Simon’s optimal design. We recommend it for use in practice as an alternative to the commonly used Simon’s design.