Traditionally, average causal effects from multiple-baseline data are estimated by aggregating individual causal effect estimates obtained through within-series comparisons of treatment-phase trajectories to baseline extrapolations. Concern that these estimates may be biased due to event effects, such as history and maturation, motivates our proposal of a between-series estimator that contrasts participants in the treatment to those in the baseline phase. Accuracy of the new method was assessed and compared in a series of simulation studies where participants were randomly assigned to intervention start points. The within-series estimator was found to have greater power to detect treatment effects but also to be biased due to event effects, leading to faulty causal inferences. The between-series estimator remained unbiased and controlled the Type I error rate independent of event effects. Because the between-series estimator is unbiased under different assumptions, the two estimates complement each other, and the difference between them can be used to detect inaccuracies in the modeling assumptions. The power to detect inaccuracies associated with event effects was found to depend on the size and type of event effect. We empirically illustrate the methods using a real dataset and then discuss implications for researchers planning multiple-baseline studies.