Next-generation sequencing of DNA provides an unprecedented opportunity to discover rare genetic variants associated with complex diseases and traits. However, when testing the association between rare variants and traits of interest, the current practice of first calling underlying genotypes and then treating the called values as known is prone to false positive findings, especially when genotyping errors are systematically different between cases and controls. This happens whenever cases and controls are sequenced at different depths or on different platforms. In this article, we provide a likelihood-based approach to testing rare variant associations that directly models sequencing reads without calling genotypes. We consider the (weighted) burden test statistic, which is the (weighted) sum of the score statistic for assessing effects of individual variants on the trait of interest. Because variant locations are unknown, we develop a simple, computationally efficient screening algorithm to estimate the loci that are variants. Because our burden statistic may not have mean zero after screening, we develop a novel bootstrap procedure for assessing the significance of the burden statistic. We demonstrate through extensive simulation studies that the proposed tests are robust to a wide range of differential sequencing qualities between cases and controls, and are at least as powerful as the standard genotype calling approach when the latter controls type I error. An application to the UK10K data reveals novel rare variants in gene BTBD18 associated with childhood onset obesity. The relevant software is freely available.