A mean shift can cause tests for a unit root to erroneously fail to reject the null hypothesis of the existence of a unit root. Perron (1990) and Hendry and Neale (1991) provide simulation evidence of this for (augmented) Dickey-Fuller tests in models without a time trend. This paper extends these analyses by considering a wider range of test statistics (including statistics proposed by Bhargava (1986)) applied to models (possibly including a time trend) subject to a shift in mean. Our simulation results show that, at least for alternatives close to the unit root, either an appropriate Bhargava test statistic or the suitably normalised OLS estimator of the unit root has higher power than the Dickey-Fuller or augmented Dickey-Fuller t-tests. In particular, in models with a trend, an increase in the mean shift does not reduce the power of Bhargava's R2 test as much as it reduces the power of the other tests. Estimated response surfaces summarise the likely power loss due to any particular mean shift.