# Optimal Tests for Combining \(p\)-Values
## 1 Introduction  
In statistical inference and decision making, it is critical but challenging to appropriately aggregate information from different sources. \(p\)-value combination approaches provide possible solutions. A \(p\)-value combination method usually combines the transformed statistics via the original individual \(p\)-values, and then, an overall \(p\)-value is obtained. The development of combining \(p\)-value has a long history. Many pioneer statisticians, including [PERSON] [1] and [PERSON] [2], had important contributions in this area. Their methods (e.g., Fisher test), along with others, such as the z-test [3] and the minimal \(p\)-test [4], are still widely used in today's statistical practice. Many studies have been conducted to compare the performances among those \(p\)-value combination tests [5; 6; 7]; it turned out that no test is uniformly most powerful although some methods may perform better than others under certain conditions. Combining dependent \(p\)-values is another research direction, as many robust and powerful methods have been proposed in the literature, including a recently proposed test based on Cauchy distribution (CCT) [8]. Although the CCT can be applied to both independent and dependent \(p\)-values, it has been shown that this test can never obtain a \(p\)-value less than the smallest \(p\)-value to be combined, and therefore, is not recommended for combining independent \(p\)-values [9]. In this paper, we focus on the situation where we have independent \(p\)-values to be combined. It is well known that combining \(p\)-value methods have important applications in meta-analysis [10; 11]. However, it is less recognized that combining \(p\)-value methods are more frequently but implicitly used in statistical testing. For instance, the commonly used chi-square tests, including the likelihood ratio test, the score test, and the Wald test, with degrees of freedom (df) greater than one can be viewed as \(p\)-value combination methods, which are special cases of our proposed gamma distribution-based tests (see Section 2). In other words, the popular chi-square tests only provide possible and special ways to combine \(p\)-values that are not necessarily optimal; more powerful methods for combining \(p\)-value may be found and used instead. With recent technical developments, larger volume data, such as genome-wide genomic data, are generated more easily and rapidly. Consequently, advanced statistical methods, including \(p\)-value combination tests, are highly desirable [12; 13; 14; 15; 16; 17; 18; 19; 20]. For instance, meta-analyses, which combine information from different genome-wide association studies (GWAS), have identified many associated genetic variants that could not be identified from a single GWAS [21; 22]. It is expected that with more powerful \(p\)-value combination tests being developed and available, more and more significant associated genetic variants will be discovered in cancer genomics. Unfortunately, as [PERSON] [23] already noticed, there exists no uniformly most powerful (UMP) \(p\)-value combination test for all alternative hypotheses. However, it is possible that a method is UMP under a certain condition. Moreover, if the true condition is unknown, it is desirable to choose a robust \(p\)-value combination method in the sense that it has reasonable detection power under many conditions. In this paper, we first propose a class of \(p\)-value combination methods based on gamma distribution. We show that several existing popular methods are equivalent to certain special cases of this class of tests. Then, we show that the proposed tests are UMP when the \(p\)-values to be combined are from certain distributions. When the \(p\)-values to be combined are from certain type of distributions whose parameters are partially or fully unknown, asymptotically UMP tests based on constrained likelihood ratio test (CLRT) are proposed and studied.