**Abstract**:

**Abstract**
**Adaptive testing methods serve to maximize the information gained regarding the values of the parameters of a psychometric function (PF). Such methods typically target only one or two (“threshold” and “slope”) of the PF's four parameters while assuming fixed values for the “nuisance” parameters (“guess rate” and “lapse rate”). Here I propose the “psi-marginal” adaptive method, which can target nuisance parameters but only when this is the most optimal manner in which to reduce uncertainty in the value of the parameters of primary interest. The method is based on Kontsevich and Tyler's (1999) psi-method. However, in the proposed method a posterior distribution defined across all parameters of unknown value is maintained. Each of the parameters is specified either as a parameter of primary interest whose estimation should be optimized or as a nuisance parameter whose estimation should be subservient to the estimation of the parameters of primary interest. Critically, selection of stimulus intensities is based on the expected information gain in the marginal posterior distribution defined across the parameters of interest only. The appeal of this method is that it will target nuisance parameters adaptively and only when doing so maximizes the expected information gain regarding the values of the parameters of interest. Simulations indicate that treating the lapse rate as a nuisance parameter in the psi-marginal method results in smaller bias and higher precision in threshold and slope estimates compared to the original psi method. The method is highly flexible and various other uses are discussed.**

*intensity*of the stimulus, symbolized by

*x*. In models of observer performance, a specific shape of the function is typically assumed and is almost invariably some sigmoidal function such as the cumulative Gaussian distribution or the Weibull function. In the remainder, I will use

*F*(

*x*;

*α*,

*β*), or simply

*F*, to refer to the function that characterizes performance of an underlying sensory mechanism. Two parameters characterize the function, its threshold and its slope. The threshold parameter (symbolized by

*alpha*:

*α*) specifies the function's location and is typically defined as that stimulus intensity at which a specific level of performance is achieved. The slope parameter (symbolized by

*beta*:

*β*) determines the function's rate of change. In this paper I use the Gumbel (or log-Weibull) function throughout as the model of

*F*and use the subscript “

*G*” to specify this: In practice, we cannot measure

*F*directly. In psychophysical studies, we instead infer

*F*from the pattern of responses (e.g., proportion correct in a multiple alternative forced-choice [mAFC] task or proportion endorsement [e.g., “I perceived the stimulus”] in a “yes/no” task) observed across trials in which stimuli of varying intensity

*x*are used. The probability of observing a positive response as a function of stimulus intensity

*x*is usually modeled as:

*F*

_{G},

*x*,

*α*, and

*β*are as defined above. The parameter

*γ*corresponds to the lower asymptote of

*ψ*, while the parameter

*λ*determines the upper asymptote. Neither

*γ*nor

*λ*characterizes the performance of the underlying sensory mechanism, which is our primary interest. In a “yes/no” task, the lower asymptote,

*γ*, corresponds to the false alarm rate and characterizes the decision process. In the more common mAFC task,

*γ*is determined by the task and is assumed to equal 1/

*m*in a task with

*m*response categories. Parameter

*λ*determines the upper asymptote of

*ψ*. The upper asymptote often deviates from unity due to so called lapses, which are negative responses (“no” or incorrect) that are stimulus independent (for example, “finger errors”). Thus, parameter

*λ*also does not characterize the underlying sensory mechanism but rather is a function of observer characteristics such as attention or vigilance. Since

*γ*and

*λ*do not describe the underlying sensory mechanism but nevertheless do affect our observations (which we must model to recover the underlying function

*F*), they are considered to be “nuisance parameters”.

*x*do we place stimuli in order to derive parameter estimates in a manner that is optimally efficient (i.e., requiring as few trials as possible in order to obtain parameter estimates of desired precision)? The answer to this question will depend on which of the PF's four parameters we are interested in. In order to estimate the stimulus intensity at which performance reaches some specific level, stimuli are most optimally placed at the intensity corresponding to the targeted performance level (e.g., Harvey, 1986; Levitt, 1971). In order to optimize slope estimation, stimuli should be placed at intensities that are some distance away from threshold intensity on either side (e.g., King-Smith & Rose, 1997; Levitt, 1971). If one is interested in estimating both threshold and slope, Levitt (1971) suggests using a compromise between the optimal placement for threshold estimation and optimal placement for slope estimation.

*ψ*in Equation 1).

*any*of the PF's parameters based on the information gained on previous trials. Including a fixed proportion of free trials among trials otherwise controlled by an adaptive method of course adds a distinct nonadaptive component into the stimulus selection procedure. Note that not only is the proportion of free trials not adjusted based on information gained on previous trials, the adaptive method in control of placement of the trials that are not free will also not modify its behavior based on information gained from the free trials.

*F*(e.g., Weibull, Logistic, etc.), a fixed small value for the lapse rate (such as 0.03) as well as a value for the guess rate dictated by the experimental method (e.g., for a 2-AFC task a guess rate equal to 0.5 would be assumed). After each trial, the psi method updates a Bayesian posterior distribution across a range of possible values for the threshold and slope parameters of the PF. Based on this posterior distribution, the method determines for each of a limited number of possible stimulus intensities the probabilities of observing a positive and negative response. For each possible stimulus intensity, it also determines the entropy in the posterior distribution that would result for either of the two possible responses. The stimulus intensity chosen on the next trial is that intensity which minimizes the expected entropy in the posterior distribution. In other words, on each trial that stimulus intensity is selected, which is associated with the greatest expected information gain in the threshold and slope estimate.

*x*. The proportions correct shown in Figure 1d correspond to the most likely to be obtained for this placement pattern and generating PF, except that one of the five trials presented at the highest intensity was set to be incorrect here (with this generating function it is in fact more likely for all five responses to be correct at this intensity rather than four).

*λ*= 0.01,

*λ*= 0.03, and

*λ*= 0.05). Clear from these figures is that at high lapse rates, the posterior distribution peaks at lower thresholds and higher slopes than it does at low lapse rates. Using a fixed value of the lapse rate (as the psi-method does) corresponds to selecting one horizontal slice of the parameter space shown in Figure 1a. Our particular choice of value for the lapse rate will systematically affect our threshold and slope estimates. In other words, selecting the wrong value for the fixed lapse rate in the psi-method would lead to systematic bias in the threshold and slope estimates. Figure 1d shows the best-fitting PFs that assume the fixed values for the lapse rates shown in Figure 1c. While these functions are virtually identical with regard to modeled values of

*ψ*at the intensities

*x*at which observations are concentrated, the models differ with regard to the values of the parameters of interest (i.e.,

*α*and

*β*). Modeled values of

*ψ*for these three functions differ substantially only at high stimulus intensities, higher than those targeted by the psi-method.

*F*in Equation 1) in all simulations was a Gumbel (“log-Weibull”) function with threshold

_{G}*α*= 0 and slope

*β*= 1. Note that these specific values were chosen purely for mathematical convenience but that this choice of particular values has no consequence whatsoever for the conclusions drawn in this paper. The guess rate

*γ*was set to 0.5 (effectively simulating a 2-AFC task; results for other values of

*γ*are presented in the Supplementary materials). Three generating lapse rates were used:

*λ*= 0,

*λ*= 0.025, and

*λ*= 0.05. Prior distributions in all simulations were uniform across a constrained 2-D (threshold × [log] slope) parameter space. The choice of limits on the parameter space will have a significant effect on Bayesian parameter estimates at the start of a run for as long as the posterior distribution would otherwise have mass beyond the edges of the prior distribution. However, as the run progresses the posterior distribution will become narrower and eventually will, effectively, be fully contained within the limits of the prior distribution. In order to control for the effect of the choice of limits of the prior, four different choices for limits on the prior across values for

*α*and log(

*β*) were used. These are illustrated in Figure 2. Each contained 21 threshold × 21 slope values. The threshold limits for the red and gray priors in the figure were [

*x*at which the generating function

*F*(

_{G}*x*;

*α*= 0,

*β*= 1) evaluates to

*p*). The (log) slope limits for the gray and blue priors were [log

_{10}(0.0625), log

_{10}(16)] – 0.15, those for the red and green priors were [log

_{10}(0.0625), log

_{10}(16)] + 0.15. For each of the three generating lapse rate values considered, 500 simulations were performed within each of the four groups of priors. For each of the 2,000 resulting simulations per generating lapse rate value, some random jitter (sampled from a rectangular distribution on [-0.15, 0.15]) was applied to the limits of the prior distribution. While mean parameter estimates will systematically differ between the four priors at a low number of trials, eventually the parameter estimates will converge when posterior distributions are (effectively) contained within the limits set by the prior. From that point forward, parameter estimates based on the constrained uniform priors will be, for all intents and purposes, equal to those based on priors which are truly uniform. The range of possible stimulus intensities the psi-method could select from included 21 equal-spaced values between

*N*= 120, 240, 480, 960, and 1,920 trials were also derived. The ML estimates correspond to those parameter values that maximize the likelihood function and were determined by an iterative simplex search (Nelder & Mead, 1965). Like the Bayesian estimates, ML estimates were derived using a fixed value for the lapse rate (0.03). Note that the maximum likelihood estimates were not constrained to lie within the limits placed on the prior. The standard error of threshold estimates for both Bayesian and ML estimates was derived as: where

*α*is the generating value of the threshold (i.e.,

*α*= 0),

*a*is the threshold estimate after Trial

_{i,j}*j*in simulated run

*i*, and

*I*is the number of simulated runs (i.e.,

*I*= 2,000). The standard error of the slope was derived in log units: where

*β*is the generating value of the slope (i.e.,

*β*= 1),

*b*is the slope estimate after Trial

_{i,j}*j*in simulation

*i*, and

*I*is as above.

*α*and log

_{10}(

*β*) were equal to 0, the mean bias is simply the mean parameter estimate itself. When the generating and assumed lapse rate coincide, I replicate Kontsevich and Tyler's (1999) result that, at least within a reasonable number of trials, the psi method is virtually free from bias. However, when the generating lapse rate deviates from that assumed by the psi-method, a systematic bias is obtained. Not surprisingly, the observed bias does not appear to resolve itself asymptotically. The degree of bias is relatively severe when compared against the standard error of estimate. The symbols in Figure 3 show obtained biases and standard errors in ML estimates. At high numbers of trials, Bayesian and ML estimates converge, as do their SEs. The counterintuitive initial rise in SE for slope was noted also by Kontsevich and Tyler (1999). At the start of a run, the posterior is flat with respect to slope and the slope estimate will simply be the center of the range of slope values contained within the prior. The initial rise in SE of slope occurs because at the start of a run the psi-method concentrates on the threshold rather than the slope. Low

*N*datasets are often consistent with extremely steep or shallow slopes and this will initially increase SE. As was found by Kontsevich and Tyler, only after about 30 trials or so does the psi-method begin to address the slope and only at this point does SE of slope begin to decrease.

*N*, threshold and slope estimates are positively correlated (cf. Figure 1c).

*F*and, as a result, the data are fit about equally well by a family of functions which have different values of threshold, slope, and lapse rate (cf. Figure 1d). In other words, the posterior density is elongated with respect to the lapse rate dimension (see Figure 1a). Note from Figure 6 that the Bayesian lapse rate estimate starts to deviate from the mean of the prior distribution (i.e., 0.05) only after a relatively large number of trials have been collected and even then changes only at a very slow rate. Bayesian and ML estimates fail to converge even at

*N*= 1,920 trials. This is primarily because very little information regarding the value of the lapse rate is available in the data. As a result, the Bayesian lapse rate estimates remain near the mean of the prior distribution on the lapse rate (i.e., 0.05; see the lapse rate histograms in Figure 4b). The ML lapse rate estimates tend to equal 0 (Figure 5b). Due to the dependency between lapse rate estimates and estimates for the threshold and slope parameters, systematic differences between ML and Bayesian lapse rate estimates are associated with systematic differences in threshold and slope estimates as well. Results for simulations using different values for the guess rate was equal to 0 and 0.25 are included in the Supplementary materials.

^{+}and psi-marginal methods

^{+}-method (Prins, 2012b). An obvious drawback of this strategy is that reducing uncertainty in the lapse rate parameter is made to be an explicit goal of the method. However, estimation of the lapse rate, being a nuisance parameter, should play a subservient role only. That is, our method should gather information regarding the lapse rate only insofar as this is the optimal strategy to reduce uncertainty in our parameters of primary interest. A second strategy then is to keep track again of a three-dimensional posterior distribution but to select stimulus intensities that will minimize expected entropy in the

*marginal threshold x slope posterior distribution*. The marginal threshold × slope posterior distribution is derived simply by summation across the lapse rate dimension: where

*p*(

*α*=

*a*,

*β*=

*b*,

*λ*=

*l*) is the full posterior distribution defined across the threshold values

*a*, slope values

*b*and lapse rate values

*l*that are contained in the parameter space. Consider again Figure 1. Figure 1(a) shows the full posterior distribution. The top panel under (b) is the marginal threshold × slope posterior derived by Equation 4. One may loosely conceive of this marginal distribution as the projection of the full posterior in (a) unto a threshold × slope plane (as if one were to look straight down on [a]). Since the high-density, cigar-shaped region is tilted, the marginal posterior displays more uncertainty regarding the threshold and slope values compared to a posterior that assumes a specific value for the lapse rate (i.e., one of the horizontal slices through the posterior shown in Figure 1c). In other words, the marginal threshold × slope distribution has incorporated the uncertainty regarding the value of the lapse rate. Since threshold, slope, and lapse rates show some degree of dependency, uncertainty with regard to the lapse rate contributes to the uncertainty in threshold and slope parameters. Thus, reduction of the entropy in the threshold × slope marginal posterior can be accomplished by gathering information regarding the value of the lapse rate. The appeal of this strategy is that all stimulus placements are motivated entirely by the criterion to maximize expected information gain regarding the values of the threshold and slope parameters only. In the process, the method may place stimuli at intensities at which information is gained primarily regarding the lapse rate, but only if that is the best placement to reduce uncertainty regarding the threshold and slope parameters. I refer to the general strategy of marginalizing nuisance parameters as the “psi-marginal method” and will use the shorthand notation psi

_{αβ}_{(λ)}to specify the particular implementation mentioned above. In this shorthand notation, the subscripted symbols indicate which parameters are included in the posterior distribution and the parentheses indicate the marginalization of a parameter. In this shorthand notation, “psi

*” would correspond to the original psi-method and “psi*

_{αβ}*” would correspond to the psi*

_{αβλ}^{+}method.

*and the marginal psi*

_{αβλ}

_{αβ}_{(λ)}methods, additional simulations were run. The procedures were identical to that described above for the psi-method, except for the following. For both the psi

*and the psi*

_{αβλ}

_{αβ}_{(λ)}method a range of values for the lapse rate was included in the posterior distribution. Values for the lapse rate that were included ranged from 0 to 0.1, spaced in intervals of 0.01. In the psi

*method, after each trial the stimulus intensity associated with the lowest expected entropy in the full 3-D posterior was selected for the next trial. In the psi*

_{αβλ}

_{αβ}_{(λ)}method, the stimulus intensity associated with the lowest expected entropy in the marginal posterior across threshold and slope values was selected for the next trial. In very loose terminology, whereas the goal of the psi

*method is to shrink the 3-D high-density “blob” in the posterior (see Figure 1a) in all three dimensions, the goal of the psi*

_{αβλ}

_{αβ}_{(λ)}method is to shrink the blob's projection unto the threshold × slope plane (see Figure 1b).

*SE*s are shown in Figure 7, while relative placement of stimuli averaged across simulations is shown in Figure 8. Data shown for the original psi method (psi

*) in Figure 7 were also shown in Figure 3 and are included here again in order to allow easy comparison. Scatterplots of parameter estimates are shown in Figures 4c and 5c for the psi*

_{αβ}*method and in Figures 4d and 5d for the psi*

_{αβλ}

_{αβ}_{(λ)}method. Results indicate that Bayesian estimates of threshold and slope parameters asymptote towards their true values much faster when data are collected using the psi

*and psi*

_{αβλ}

_{αβ}_{(λ)}methods compared to data collected using the traditional psi-method. The psi

*and psi*

_{αβλ}

_{αβ}_{(λ)}methods perform about equally well, with the latter having a slightly lower SE in threshold estimate. Not surprisingly, both psi

*and psi*

_{αβλ}

_{αβ}_{(λ)}lead to much more accurate and precise estimates of the lapse rate compared to psi

*. ML estimates and Bayesian estimates appear to converge at high*

_{αβ}*N*for data collected using psi

*and psi*

_{αβλ}

_{αβ}_{(λ)}methods while they do not for data collected using the psi

*method.*

_{αβ}*and psi*

_{αβλ}

_{αβ}_{(λ)}methods appear to do the same. The behavior of the psi

*and psi*

_{αβλ}

_{αβ}_{(λ)}methods with regard to targeting of the lapse rate is most clearly seen in Figure 8. Not surprisingly, psi

*places many trials at the highest stimulus intensity available to it even at*

_{αβλ}*N*= 120: Reducing uncertainty in the lapse rate is an explicit goal of psi

*. While reducing uncertainty in the lapse rate is not an explicit goal of psi*

_{αβλ}

_{αβ}_{(λ)}, it places a relatively large number of stimuli at the highest possible intensity also (though far fewer than psi

*does). Note, however, that this is not true at low*

_{αβλ}*N*. Apparently, at low

*N*, including stimulus placements at very high intensities is not the most efficient manner in which to reduce uncertainty in threshold and slope parameter values. Note also that even though psi

_{αβ}_{(λ)}does not place more stimuli at high intensity than the original psi-method does (fewer even) at

*N*= 120 and

*N*= 240, it far outperforms the psi-method in terms of bias and

*SE*of the threshold and slope parameters.

*places more stimuli at the highest intensity during the first 120 trials than psi*

_{αβ}

_{αβ}_{(λ)}does. At first glance, this seems counterintuitive. After all, psi

*only has uncertainty regarding the threshold and the slope and very little information regarding either of these parameters is to be gained at this highest intensity. However, we need to realize that such placement is only unexpected to one who knows that the generating PF is near its asymptotic value at this intensity. The adaptive method does not know this. In fact, both psi*

_{αβ}*and psi*

_{αβ}

_{αβ}_{(λ)}only place stimuli at the highest intensity this early in a run when responses thus far collected are consistent with a PF that has a very shallow slope. In other words, either method is under the impression (if you will) that it is placing a stimulus at a much lower intensity than it in fact is. Similarly, the apparent greater precision of placements of psi

*compared to the other methods is not a result of strategy, rather it results from psi*

_{αβ}*having more information regarding the generating PF compared to the other methods (i.e., it knows the value of the lapse rate).*

_{αβ}

_{αβ}_{(γλ)}. By way of demonstration, 2,000 simulations were run which were identical to the psi

_{αβ}_{(λ)}runs above except for the following. The generating guess rate was arbitrarily set to 0.1. Guess rates were included in the posterior, which was now 4-D (threshold × slope × guess rate × lapse rate). The values for the guess rates included were 0 through 0.3 in steps of 0.03. The values included for the other parameters were as above. A stimulus of zero intensity (i.e., log(

*x*) = -∞) was added to the 21 stimulus intensities described above. After each trial, the stimulus intensity to be used on the next trial was selected based on the criterion that it minimizes the expected entropy in the marginal threshold × slope posterior. Results are shown in Figure 9.

_{α}_{(βλ)}which maintains a posterior distribution across threshold, slope and lapse rates (as above) but selects stimulus intensities such as to optimize estimation of the threshold only. By way of demonstration, 2,000 simulations were performed exactly as those performed in Experiment 2 except that now only the threshold was regarded as a parameter of primary interest while the slope and lapse rate were regarded as nuisance parameters by marginalizing them. Resulting Bayesian and ML estimates are shown in Figure 10. From Figure 10, it is clear that the method has adjusted its placement and that this has resulted in a higher precision of the threshold estimate while precision in the slope has decreased.

_{(α)β(λ)}. In words, only the slope parameter is considered to be of interest, the threshold and slope are considered to be nuisance parameters. Note that while it is common practice to fix the slope when one is interested in the threshold (e.g., Pentland's [1980] Best PEST and Watson & Pelli's [1983] Quest do this), fixing the value of the threshold when one is interested in the slope would be a grave error indeed. Again, 2,000 simulations were performed exactly as those performed in Experiment 2 except that the threshold and the lapse rate were regarded as nuisance parameters. As it turns out, this condition benefits greatly from having some information regarding the location of the PF at the beginning of the run. For that reason, the threshold parameter was marginalized only after 30 trials. That is, for Trials 1 through 30 the method was psi

_{αβ}_{(λ)}, for the remaining trials it was psi

_{(α)β(λ)}. Note that any of the parameters in the posterior may be marginalized or demarginalized at any point during a run. For example, another possibility would have been to start the run off as psi

_{α}_{(βλ)}, then switch to psi

_{(α)β(λ)}. Resulting Bayesian and ML estimates are shown in Figure 11, from which it is clear that the slope estimate has benefitted from being the only parameter of primary interest while the estimate of the threshold, being defined as a nuisance parameter, has become less precise.

_{αβ}_{(λ)}method this is not an issue at values of

*N*up to 240. As a matter of fact, at

*N*= 120, psi

_{αβ}_{(λ)}places

*fewer*trials near upper asymptotic levels of performance compared to the original psi-method (Figure 8) and also has fewer lengthy consecutive series of near asymptotic performance placements. At larger

*N*, however, lengthy series of consecutive placements at near-asymptotic performance tend to occur. For example, in the psi

_{αβ}_{(λ)}simulations, a run of 480 trials contained on average about one series of eight or more consecutive stimulus placements at the highest intensity available to the method (Figure 12). Such lengthy series of “free trials” will almost certainly affect an observer's level of vigilance. A few strategies may be used to avoid such consecutive series of high-intensity placements. For example, the problem will be lessened when multiple conditions, each controlled by its own psi-marginal run, are randomly interleaved in a testing session. Any series of consecutive high intensity placements that may occur in any given run will then be interrupted by trials from the other runs.

_{αβ}_{(λ)}method in Experiment 2, except that after any trial in which placement was at the highest stimulus intensity available to the method, the selection criterion was temporarily changed to that of the original psi-method. The value at which the lapse rate was fixed on such trials corresponded to the current ML estimate of the lapse rate. After a “wait time” that was randomly drawn from an exponential mass function (so as to maintain “constant hazard”) the method resorted back to selecting the stimulus intensity that maximized expected information gain in the marginal threshold × slope posterior. Three different mean wait times (

*τ*) were used: two, four, or eight trials. The frequency of occurrence of consecutive placements at near-asymptotic levels for these simulations is included in Figure 12. Note that the strategy to suspend the psi-marginal selection criterion temporarily to favor that of the original psi-method did not affect bias or precision of threshold and slope estimates significantly (bias and standard errors shown in Supplementary materials).

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