Selection bias may arise when data have been chosen in a way that subsequent analysis does not account for. Such bias can arise in climate event attribution studies that are performed rapidly after a devastating "trigger event'', whose occurrence corresponds to a stopping rule. Intuition suggests that naively including the trigger event in a standard fit in which it is the final observation will bias its importance downwards, and that excluding it will have the opposite effect. In either case the stopping rule leads to bias recently discussed in the statistical literature (Barlow et al., 2020) and whose implications for climate event attribution we investigate. Simulations in a univariate setting show substantially lower relative bias and root mean squared error for estimation of the 200-year return level when the timing bias is accounted for. Simulations in a bivariate setting show that not accounting for the stopping rule can lead to both over-and under-estimation of return levels, but that bias can be reduced by more appropriate analysis. We also discuss biases arising when an extreme event occurs in one of several related time series but this is not accounted for in data analysis, and show that the estimated return period for the "trigger event'' based on a dataset that contains this event can be both biased and very uncertain. The ideas are illustrated by analysis of rainfall data from Venezuela and temperature data from India and Canada.