This paper addresses the question of how a minimum wage increase affects the wages of low-wage workers. Most studies assume that there is a simple mechanical increase in the wage for workers earning a wage between the old and the new minimum wage, with some studies allowing for spillovers to workers with wages just above this range. Rather than assume that the wages of these workers would have remained constant, this paper estimates how a minimum wage increase impacts a low-wage worker's wage relative to the wage the worker would have if there had been no minimum wage increase. The method allows for the effect to depend not only on the initial wage of the worker, but also nonlinearly on the size of the minimum wage increase. Using Current Population Survey data from 2005 to 2008, a period with a large number of U.S. state-level minimum wage increases, this paper finds that low-wage workers who experience a small increase in the minimum wage tend to have lower wage growth than if there had been no minimum wage increase. A large increase to the minimum wage increases the wages of not only those workers who previously earned less than the new minimum wage, but also spill over to workers with moderately higher wages. Finally, this paper finds little evidence of heterogeneity in the effect by age, gender, income, and race.