We present a nonuniform multiphase (NUMP) method to construct a high-resolution time-todigital converter (TDC) for low-cost field-programmable gate array (FPGA) devices. The NUMP method involves a system clock being passed through a series of delay elements to generate multiple clocks with different phase shifts. The phases of the rising and falling edges of all the

9Motivated by, but not limited to, association analyses of multiple genetic variants, we propose here 10 a summary statistics-based regression framework. The proposed method requires only variant-11 specific summary statistics, and it unifies earlier methods based on individual-level data as spe-12 cial cases. The resulting score test statistic, derived from a linear mixed-effect regression model, 13 inherently transforms the variant-specific statistics using the precision matrix to improve power 14 for detecting sparse alternatives. Furthermore, the proposed method can incorporate additional 15 variant-specific information with ease, facilitating omic-data integration. We study the asymptotic 16 properties of the proposed tests under the null and alternatives, and we investigate efficient p-value 17 calculation in finite samples. Finally, we provide supporting empirical evidence from extensive 18 simulation studies and two applications. 19

Motivated by, but not limited to, association analyses of multiple genetic variants, we propose here a summary statistics-based regression framework. The proposed method requires only variantspecific summary statistics, and it unifies earlier methods based on individual-level data as special cases. The resulting score test statistic, derived from a linear mixed-effect regression model, inherently transforms the variant-specific statistics using the precision matrix to improve power for detecting sparse alternatives. Furthermore, the proposed method can incorporate additional variant-specific information with ease, facilitating omic-data integration. We study the asymptotic properties of the proposed tests under the null and alternatives, and we investigate efficient p-value calculation in finite samples. Finally, we provide supporting empirical evidence from extensive simulation studies and two applications.

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