# Optimal Concordant Tests
## 1 Introduction  
Due to recent technical developments, larger volume data, including genome-wide genomic data, can be generated rapidly and at a low cost. As a consequence, powerful statistical approaches, including \(p\)-value combination tests, are highly desirable for analyzing these data sets. For instance, meta-analyses, when applied to genome-wide association studies (GWAS), have discovered many associated genetic variants that could not be identified from a single GWAS. With the development and application of advanced powerful \(p\)-value combination tests, we expect that many more casual genetic variants will be identified through existing data.  
In meta-analyses, \(p\)-value combination methods are important alternatives to model-based techniques [1, 2], especially when the fixed- or random-effect models have a lack of fit [3]. In the literature, many \(p\)-value combination methods, although first proposed a long time ago, are still widely used nowadays. These methods include popular ones: Fisher test [4], Pearson test [5], minimal p-test [6], z-test [7], and chi-square test [8, 9]. Recently, we studied a class of \(p\)-value combination tests based on gamma distribution, which include those popular tests as special cases [10]. However, as noticed by [PERSON], there exists no uniformly most powerful (UMP) \(p\)-value combination test for all conditions [11]. Therefore, robust methods that have good power under many conditions are desirable [12].  
Under some conditions, we may know or reasonably assume that all or most of the true effects under the individual alternative hypotheses have the same but unknown direction. In this case, the concordant tests which use one-sided \(p\)-values have been shown to be more powerful than those methods based on two-sided \(p\)-values, such as the chi-square test [1]. For instance, in GWAS, studies with the same phenotype are usually conducted among several different subpopulations, from which the effect of a causal genetic variant may have the same direction but various sizes. Then, if the commonly used fixed- and random-effect meta-analyses are applied to combine the results from individual studies, they may lose the power to detect the associated genetic variants. To improve the detecting power, inthis paper, we study a class of concordant tests based on gamma distribution, and then an optimal concordant test based on constrained likelihood ratio test (CLRT) is developed.  
The rest of the manuscript is organized as follows. In Section 2, we first introduce the [PERSON]'s concordant test, which was studied by [PERSON] [1] (Section 2.1). In Section 2.2, we describe our proposed generalized concordant tests based on gamma distribution and study their properties as UMP tests under some conditions. The asymptotically UMP concordant test based on CLRT is proposed and studied in Section 2.3. In Section 3, we compare the performances of the proposed concordant tests with some existing methods through a simulation study. In addition, an example of a real data application is illustrated to demonstrate the desired performance of the proposed tests. This paper concludes in Section 4 with a discussion and conclusions.