In this paper, we derive an expression for the signal to interference-plus-noise ratio of a multistage parallel interference cancellation receiver. We focus on a linear multistage receiver employing partial parallel interference cancellation which converges to the linear minimum mean squared error receiver as the number of stages increase. The signal to interference-plus-noise ratio is given in terms of the system loading, the partial cancellation factor, the number of stages and the background noise variance. The expression allows a simple approximation for the bit error rate at each stage. Finally, we verify and apply our results to optimise the signal to interference-plus-noise ratio of the resulting linear multistage receiver.