In genome-wide association studies of binary traits, investigators typically use logistic regression to test common variants for disease association within studies, and combine association results across studies using meta-analysis. For common variants, logistic regression tests are well calibrated, and meta-analysis of study-specific association results is only slightly less powerful than joint analysis of the combined individual-level data. In recent sequencing and dense chip based association studies, investigators increasingly test low-frequency variants for disease association. In this paper, we seek to (1) identify the association test with maximal power among tests with well controlled type I error rate and (2) compare the relative power of joint and meta-analysis tests. We use analytic calculation and simulation to compare the empirical type I error rate and power of four logistic regression based tests: Wald, score, likelihood ratio, and Firth bias-corrected. We demonstrate for low-count variants (roughly minor allele count [MAC] < 400) that: (1) for joint analysis, the Firth test has the best combination of type I error and power; (2) for meta-analysis of balanced studies (equal numbers of cases and controls), the score test is best, but is less powerful than Firth test based joint analysis; and (3) for meta-analysis of sufficiently unbalanced studies, all four tests can be anti-conservative, particularly the score test. We also establish MAC as the key parameter determining test calibration for joint and meta-analysis.