Abstract
In their influential paper, Wichmann and Hill (2001, Perception & Psychophysics, 63, 1293–1313) have shown that the threshold and slope estimates of a psychometric function may be severely biased when we assume that the lapse rate equals 0 but lapses do, in fact, occur. Wichmann and Hill also claim that threshold and slope estimates are essentially unbiased when we allow the lapse rate to vary within a rectangular Bayesian prior during the fitting procedure. Here I fail to replicate Wichmann and Hill's results. Instead, I show that both threshold and slope estimates are biased when we follow Wichmann and Hill's suggestion to estimate the lapse rate. I explain the mechanism behind the bias in detail. Fortunately, since we are rarely interested in the absolute value of a threshold or slope, bias in the estimates thereof need not necessarily be a problem. Instead, we are generally concerned with whether differences among parameter values exist between experimental conditions. I demonstrate that, unlike estimates of the absolute value of thresholds, estimates of differences between thresholds values obtained in different conditions are unbiased even if an assumed, fixed lapse rate does not match the true, generating lapse rate. Similarly, results of statistical model comparisons are virtually unaffected by a mismatch between an assumed, fixed lapse rate and the generating lapse rate. In contrast, severe negative consequences for statistical model comparisons can be obtained when we allow the lapse rates to vary during fitting. I explain the mechanism behind these negative consequences in detail. Overall, my results indicate that unless we are interested in the absolute values of the parameters of a psychometric function, we should not, and need not, estimate the lapse rate.