**The staircase method has been widely used in measuring perceptual learning. Recently, Zhao, Lesmes, and Lu (2017, 2019) developed the quick Change Detection (qCD) method and applied it to measure the trial-by-trial time course of dark adaptation. In the current study, we conducted two simulations to evaluate the performance of the 3-down/1-up staircase and qCD methods in measuring perceptual learning in a two-alternative forced-choice task. In Study 1, three observers with different time constants (40, 80, and 160 trials) of an exponential learning curve were simulated. Each simulated observer completed staircases with six step sizes (1%, 5%, 10%, 20%, 30%, and 60%) and a qCD procedure, each starting at five levels (+50%, +25%, 0, −25%, and −50% different from the true threshold in the first trial). We found the following results: Staircases with 1% and 5% step sizes failed to generate more than five reversals half of the time; and the bias and standard deviations of thresholds estimated from the post hoc segment-by-segment qCD analysis were much smaller than those from the staircase method with the other four step sizes. In Study 2, we simulated thresholds in the transfer phases with the same time constants and 50% transfer for each observer in Study 1. We found that the estimated transfer indexes from qCD showed smaller biases and standard deviations than those from the staircase method. In addition, rescoring the simulated data from the staircase method using the Bayesian estimation component of the qCD method resulted in much-improved estimates. We conclude that the qCD method characterizes the time course of perceptual learning and transfer more accurately, precisely, and efficiently than the staircase method, even with the optimal 10% step size.**

*d*′ (Ball & Sekuler, 1982, 1987), or with adaptive procedures that estimate either contrast thresholds or difference thresholds in blocks of trials (T. Bi, Chen, Zhou, He, & Fang, 2014; Donovan & Carrasco, 2015; Liang, Zhou, Fahle, & Liu, 2015; Polat et al., 2004; Wang et al., 2016; Xiao et al., 2008) with various forms of the staircase procedure (Cornsweet, 1962; Watson & Pelli, 1983) as the most frequently used method (Leek, 2001; Meese, 1995; Monsen & Horn, 2007).

*m*AFC procedure strongly affects the performance of the adaptive procedures (J. Bi, Lee, & O'Mahony, 2010; Hall, 1983; Hou, Lesmes, Bex, Dorr, & Lu, 2015; Shelton & Scarrow, 1984). Forced-choice procedures with three or four alternatives provide more satisfactory measurement of psychometric performance (Leek, 2001). Hou et al. (2015) found that increasing the number of alternatives in a forced-choice task greatly improved the efficiency of assessing a contrast sensitivity function in both simulation and psychophysical studies. Previously, we have used 4AFC and 8AFC tasks in comparing the staircase and qCD methods (Zhang, Zhao, et al., 2018, 2019; Zhao et al., 2017, 2019).

*n*without a lapse rate and

*n*with a lapse rate,

*λ*values (from 0.1 to 0.4), 50 log-linearly spaced

*γ*values (from 20 to 200), and 50 log-linearly spaced

*α*values (from 0.05 to 0.2). Both

*λ*and

*α*are in the units of the threshold measurement, while

*λ*values are in the unit of trial number. The joint prior distribution

*T*(

*n*) was calculated using Equation 1. The expected probability of a correct response was calculated using Equation 2b. To determine whether the observer's response was correct, we first drew a random number

_{10}unit) measures were used in the following analyses because this allowed us to compare the quality of measures in different dimensions.

_{10}scale) of the estimated threshold of the

*n*th simulated trial is defined as

*n*th trial in the

*j*th simulation and

*n*th trial.

*N*is the total number of trials in each run.

_{10}scale) of the estimated threshold of the

*b*th simulated block is defined as

*b*th block of the

*j*th simulation,

*b*th block—that is, the threshold in trial

*B*is the total number of blocks.

*n*th trial in the

*j*th simulation,

*f*= 1, 2, 3.

*SD*) of repeated measures. For the qCD method, the

*SD*of the estimated threshold of the

*n*th simulated trial is defined as

*SD*of the estimated thresholds in the

*b*th block is defined as

*SD*of the estimated threshold of the entire learning curve from the staircase method is defined as

*B*is the total number of blocks in each run.

*SD*s of the estimated parameters from the qCD method are defined as

*SD*s of the estimated parameters from the staircase method are defined as

*SD*from repeated experimental runs is not possible.

_{10}units, respectively, for Observer 1; 0.015, 0.004, 0.007, 0.015, 0.030, and 0.092 log

_{10}units for Observer 2; and 0.018, 0.002, 0.003, 0.015, 0.030, and 0.093 log

_{10}units for Observer 3. The qCD method was much more accurate than the staircase method. The RMSEs of the estimated thresholds with starting levels of ±25% and ±50% are given in Table 3; more details are provided in Supplementary Figure S6.

*SD*s of the estimated thresholds from the qCD and staircase methods with the 0% starting level are shown in Figure 3B. The

*SD*s of the estimated thresholds from the qCD trial-by-trial, qCD post hoc segment-by-segment, SC10%, SC20%, SC30%, and SC60% were 0.029, 0.017, 0.045, 0.052, 0.059, and 0.084 log

_{10}units, respectively, for Observer 1; 0.032, 0.018, 0.044, 0.052, 0.059, and 0.085 log

_{10}units for Observer 2; and 0.032, 0.017, 0.043, 0.052, 0.059, and 0.085 log

_{10}units for Observer 3. For the staircase methods, the

*SD*increased with step size. The

*SD*s of the estimated thresholds from the qCD method were always considerably smaller than those from the staircase methods. In summary, the precision of the estimated thresholds from the qCD method was much higher than that of those from the staircase methods. The

*SD*s of the estimated thresholds with five starting levels are listed in Table 4; more details are provided in Supplementary Figure S7.

_{10}units, respectively, after 100, 200, 400, and 800 trials; for the post hoc segment-by-segment analysis across the three simulated observers they were 0.025, 0.018, 0.013, and 0.016 log

_{10}units. The 68.2% HWCIs with all five starting stimulus levels are listed in Table 5; more details are provided in Supplementary Figure S8.

*SD*s of the estimated parameters from the qCD and staircase methods with 0% starting levels are plotted in Figure 4. For the qCD method, the bias was computed from the post hoc segment-by-segment analysis. For the staircase method, we calculated the bias and

*SD*of the best-fitting parameters of the exponential model to the estimated learning curves. The biases of the estimated parameters from the qCD method were smaller than those from the staircase method, especially for simulated Observer 1, who has the fastest learning rate among the three observers.

*λ*,

_{10}units, respectively, from the qCD method and 0.149, −0.029, and −0.004 log

_{10}units from the staircase method; the

*SD*s of the estimated

*λ*,

_{10}units from the qCD method and 0.489, 0.213, and 0.017 log

_{10}units from the staircase method. For Observers 2 and 3, with slower learning, the biases and

*SD*s of the estimated parameters from the staircases were not as large as for Observer 1. However, qCD still provided more accurate (less biased) and precise estimates. In addition, different starting stimulus levels affected the accuracy of the estimated parameters from the staircase method but not the qCD method (see Supplementary Figure S9 for details).

*λ*were 0.111, 0.087, 0.072, and 0.069 log

_{10}units, respectively, after 100, 200, 400, and 800 trials; for the estimated

*γ*they were 0.196, 0.158, 0.105, and 0.074 log

_{10}units; and for the estimated

*α*they were 0.169, 0.117, 0.051, and 0.018 log

_{10}units (see Supplementary Figure S10 for details on ±25% and ±50% starting levels). The results indicate that the qCD method can estimate the parameters of the learning curve with high precision.

*λ*+

*α*) and percentage of improvements (PI = [

*λ*+

*α*]/

*α*) are critical for understanding characteristics of perceptual learning, such as specificity, transfer, and retention. As shown in Figure 6A, the histogram of the estimated ITs from the qCD method are distributed symmetrically and tightly around the true IT (= 0.358). However, the histograms of the estimated IT from the staircase method have long tails in one direction, indicating systematic and sometimes large biases, with corresponding effects on the

*SD*. For example, when the time constant was 40 trials (Observer 1), the estimated IT was 0.344 ± 0.041 (

*M*±

*SD*) from the qCD method and 0.847 ± 1.608 from SC10%.

*SD*for the estimated PI. These results demonstrate that the accuracy and precision of the estimated IT and PI from the qCD method were much higher than those from the staircase method. The means and

*SD*s of the estimated IT and PI of the three observers with five starting levels are summarized in Table 6.

_{10}units, respectively, for Observer 1; 0.009, 0.003, and 0.005 log

_{10}units for Observer 2; and 0.015, 0.005, and 0.004 log

_{10}units for Observer 3. The qCD method was much more accurate than the staircase method when the learning rate was more rapid. The RMSEs of the estimated thresholds with starting levels of ±25% and ±50% are listed in Supplementary Table S2; more details are provided in Supplementary Figure S14.

*SD*s of the estimated thresholds from the qCD and staircase methods with the 0% starting level in the transfer phase are shown in Figure 8B. The

*SD*s of the estimated thresholds from the qCD trial-by-trial, qCD post hoc segment-by-segment, and SC10% were 0.027, 0.016, and 0.045 log

_{10}units, respectively, for Observer 1; 0.030, 0.017, and 0.044 log

_{10}units for Observer 2; and 0.031, 0.018, and 0.044 log

_{10}units for Observer 3. For the staircase methods, the

*SD*increased with step size. The

*SD*s of the estimated thresholds from the qCD method were always considerably smaller than those from the staircase methods. In summary, the precision of the estimated thresholds from the qCD method was much higher than from the staircase methods. More details are provided in Supplementary Table S3 and Supplementary Figure S15.

_{10}units, respectively, after 100, 200, 400, and 800 trials; from the post hoc segment-by-segment analysis they were 0.024, 0.017, 0.013, and 0.015 log

_{10}units. More details are provided in Supplementary Figure S16.

*SD*s of the estimated parameters from the qCD and staircase methods with 0% starting level (relative to the true threshold in the beginning of the training phase) in the transfer phase are plotted in Figure 9. The biases of the estimated parameters from the qCD method were smaller than those from the staircase method, especially for simulated Observer 1, who has the fastest learning rate among the three observers. For Observer 1, with the starting stimulus level at 0%, the biases of the estimated

*λ*,

_{10}units, respectively, from the qCD method and 0.302, −0.042, and −0.028 log

_{10}units from the staircase method; the

*SD*s were 0.087, 0.107, and 0.013 log

_{10}units from the qCD method and 0.978, 0.453, and 0.470 log

_{10}units from the staircase methods. For Observers 2 and 3, with slower learning, the biases and

*SD*s of the estimated parameters from the staircases were not as large as for Observer 1. However, qCD still provided more accurate and precise estimates. In addition, different starting stimulus levels affected the accuracy of the estimated parameters from the staircase method but not the qCD method (see Supplementary Figure S17 for details).

*λ*were 0.125, 0.109, 0.099, and 0.095 log

_{10}units, respectively, after 100, 200, 400, and 800 trials; for the estimated

*γ*they were 0.276, 0.242, 0.150, and 0.109 log

_{10}units; and for the estimated

*α*they were 0.151, 0.098, 0.040, and 0.017 log

_{10}units (see Supplementary Figure S18 for details on ±25% and ±50% starting levels). The results indicate that the qCD method can estimate the parameters of the learning curve with high precision.

*SD*s. Consistent with the results of Simulation Study 1, the qCD method produced estimated ITs and PIs with higher accuracy and precision than the staircase method. The means and

*SD*s of the estimated ITs and PIs of the three observers with five starting levels are summarized in Supplementary Table S4.

*M*±

*SD*) and −121% ± 491% for Observer 1, 43% ± 14% and 25% ± 219% for Observer 2, and 47% ± 9% and 46% ± 48% for Observer 3. The results indicate that the qCD method provided more accurate and precise measures of transfer than the staircase method, especially when the learning is more rapid.

*SD*for the estimated learning curve parameters in some cases. Furthermore, when learning is more rapid, the staircase method is less able to track the detailed time course of perceptual learning. This is not surprising, since there is an insufficient number of block measurements to track reductions in threshold, especially in the early rapid-changing phase of learning.

*SD*s of the estimated parameters from the staircase method depended more on the starting levels, while the qCD method was much less dependent on the deviation of the starting value from the true value. The staircase method performed worse when the starting levels were set nonoptimally.

*SD*) of the rescored thresholds were comparable to those of the direct estimates with the qCD method, except that the accuracy of the rescored thresholds of the simulated observer with the fastest learning rate, 5% step size, and +50% starting level were worse than those obtained with the qCD method directly (RMSE: 0.024 vs. 0.006 log

_{10}units; Supplementary Table S5; Supplementary Figures S20 through S23). These results are different from those of Zhao et al. (2019), who found that even after rescoring with the qCD method, the accuracy and precision of the estimated thresholds from the staircase methods were worse than those obtained directly with the qCD method. The difference between the two studies is that Zhao et al. investigated faster perceptual-sensitivity changes than we did (time constants of 20 vs. 40 trials).

_{10}units, respectively, in the 4AFC task and 0.001 and 0.008 log

_{10}units in the 8AFC task (Supplementary Table S6).

_{10}units); in the 8AFC task they were −0.003, −0.004, and −0.004 for the qCD method and −0.043, −0.038, and −0.032 for the rescored staircase data (Supplementary Figure S24). The same pattern of results was also true for the HWCIs of estimated thresholds. In the 4AFC task, the HWCIs of the estimated threshold from the qCD method were 0.050, 0.040, and 0.032 log

_{10}units, respectively, after 1, 10, and 20 trials, and for the rescored staircase data they were 0.063, 0.052, and 0.042 (Supplementary Figure S25). To examine the effect of the number of alternatives on the efficiency of the qCD method, we also evaluated the efficiency of the qCD method in 2AFC, 4AFC, and 8AFC tasks for Observer 1 (time constant = 40 trials). Defining efficiency as the number of trials for the procedure to reach criterion accuracy (RMSE) and precision (

*SD*) levels (Zhao et al., 2019), we found that the qCD procedure required 93, 84, and 76 trials to reach 0.02 log

_{10}-unit accuracy in 2AFC, 4AFC, and 8AFC tasks, respectively, and 677, 379, and 291 trials to reach 0.02 log

_{10}-unit precision (see Supplementary Figures S26 and S27). Putting Zhao et al. (2019) and the current study together, we conclude the following: Rescoring the staircase-method data with the qCD method could lead to comparable accuracy and precision with direct qCD estimates when perceptual sensitivity changes relatively slowly, but not when it changes fast; the qCD method holds advantages in early phase of learning; and the qCD method is more efficient with more alternatives in forced-choice tasks.

*SD*= 0.020; all in log

_{10}units) were much higher than those obtained by 10 runs of the qFC procedure (RMSE = 0.020; HWCI = 0.032;

*SD*= 0.031).

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*n*without a lapse rate, and

*d*

*h*is the information entropy of the distribution

*p*.

*n*th trial of the learning curve,

*n*− 1). The prior distribution

*n*th trial in the learning curve is used as the prior of trial

*n*+ 1:

*n*th trial in the learning curve:

*f*= 1, 2, 3, respectively. (These are cast as summations rather than integrals because the parameter values for which these quantities are computed are quantized in the qCD method and in the simulations; see main text.)

*f*= 1, 2, 3, respectively;

*b*= 1 or 2) is the

*i*

*MD*

_{0}is a predetermined criterion.

*l*and

*l*th segment and the

*l*th segment, respectively;

*l*th segment; and

*L*is the number of segments of the entire dark-adaptation curve. To partition the dark-adaptation curve into segments, we start from the last trial of the entire experiment,

*MD*between

*L*− 1, is found:

*u*and

_{L}*U*(last trial) in the

_{L}*L*th segment (last segment).

*l*− 1.