The psychometric function: the lapse rate revisited

In their influential paper, Wichmann and Hill (2001) have shown that the threshold and slope estimates of a psychometric function may be severely biased when it is assumed that the lapse rate equals zero but lapses do, in fact, occur. Based on a large number of simulated experiments, Wichmann and Hill claim that threshold and slope estimates are essentially unbiased when one allows the lapse rate to vary within a rectangular prior during the fitting procedure. Here, I replicate Wichmann and Hill's finding that significant bias in parameter estimates results when one assumes that the lapse rate equals zero but lapses do occur, but fail to replicate their finding that freeing the lapse rate eliminates this bias. Instead, I show that significant and systematic bias remains in both threshold and slope estimates even when one frees the lapse rate according to Wichmann and Hill's suggestion. I explain the mechanisms behind the bias and propose an alternative strategy to incorporate the lapse rate into psychometric function models, which does result in essentially unbiased parameter estimates.