September 2012
Volume 12, Issue 10
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Article  |   September 2012
Improving behavioral performance under full attention by adjusting response criteria to changes in stimulus predictability
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Journal of Vision September 2012, Vol.12, 1. doi:10.1167/12.10.1
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      Steffen Katzner, Stefan Treue, Laura Busse; Improving behavioral performance under full attention by adjusting response criteria to changes in stimulus predictability. Journal of Vision 2012;12(10):1. doi: 10.1167/12.10.1.

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Abstract
Abstract
Abstract:

Abstract  One of the key features of active perception is the ability to predict critical sensory events. Humans and animals can implicitly learn statistical regularities in the timing of events and use them to improve behavioral performance. Here, we used a signal detection approach to investigate whether such improvements in performance result from changes of perceptual sensitivity or rather from adjustments of a response criterion. In a regular sequence of briefly presented stimuli, human observers performed a noise-limited motion detection task by monitoring the stimulus stream for the appearance of a designated target direction. We manipulated target predictability through the hazard rate, which specifies the likelihood that a target is about to occur, given it has not occurred so far. Analyses of response accuracy revealed that improvements in performance could be accounted for by adjustments of the response criterion; a growing hazard rate was paralleled by an increasing tendency to report the presence of a target. In contrast, the hazard rate did not affect perceptual sensitivity. Consistent with previous research, we also found that reaction time decreases as the hazard rate grows. A simple rise-to-threshold model could well describe this decrease and attribute predictability effects to threshold adjustments rather than changes in information supply. We conclude that, even under conditions of full attention and constant perceptual sensitivity, behavioral performance can be optimized by dynamically adjusting the response criterion to meet ongoing changes in the likelihood of a target.

Introduction
One of the key features of active perception is our ability to predict critical events, such as the sudden onset of a visual stimulus. The benefits are clear: Being able to predict a stimulus improves behavioral performance. Less clear are the underpinnings of such benefits. Can stimulus predictability improve sensory processing, or does it mainly influence decision-making processes, thought to be subsequent to sensory processing? 
Accurate predictions do not require explicit temporal information as they can also rely on implicit representations of statistical regularities in the timing of events. A common experimental strategy has been to manipulate stimulus predictability through the hazard rate, which specifies the likelihood of a stimulus appearing, given it has not appeared so far. A long tradition of reaction time (RT) experiments has documented that RT decreases as the stimulus becomes more likely. These findings have been interpreted to show that observers implicitly learn to use changes in stimulus likelihood to reduce RTs (e.g., Näätänen, 1971; for a review, see Niemi & Näätänen, 1981). The question of how we benefit from implicit representations of changes in stimulus likelihood is the focus of the present investigation. 
Early behavioral studies in humans seem to indicate that stimulus predictability largely affects postperceptual processes rather than the quality of sensory processing. These studies have typically relied on RT measures, and they largely provided indirect evidence for postperceptual effects. A widely used strategy, for example, has been to investigate whether the effects of stimulus predictability can be modulated by concurrent manipulations of decision- or even motor-related variables (for a review, see Sanders, 1980). The results of these early studies have led to the general consensus that stimulus predictability mainly changes the willingness of the observer to respond (Luce, 1986). 
Studies in animals seem to indicate that stimulus predictability can have profound effects on both decision-making processes and early sensory activity. Janssen and Shadlen (2005), for instance, have documented a relationship between stimulus predictability and activity of single neurons in the monkey lateral intraparietal area (LIP), which is implicated in decision-making. Stimulus predictability has been shown to modulate responses of single neurons in macaque areas V4 and MT of the extrastriate visual cortex (Ghose & Bearl, 2010; Ghose & Maunsell, 2002). In addition to such modulations, behavioral data suggest that changes in stimulus probability can lead to changes in sensory integration (Ghose, 2006). Finally, stimulus predictability has also been shown to modulate firing rates of individual neurons in the primary auditory cortex of the rat (Jaramillo & Zador, 2011). 
Functional magnetic resonance imaging (fMRI) studies in humans have also raised the possibility that stimulus predictability can affect early sensory processing. Predictability-related changes of activity have been observed in widespread cortical regions, including the earliest stages of visual processing (Bueti, Bahrami, Walsh, & Rees, 2010). Stimulus predictability has also been assumed to reduce activity in the primary visual cortex (Alink, Schwiedrzik, Kohler, Singer, & Muckli, 2010). 
On the level of behavior, however, it remains less clear whether stimulus predictability can have a measureable influence on the quality of sensory processing. To address this issue, one needs to disentangle sensory from nonsensory components of perceptual decisions. One way to achieve this is by applying Signal Detection Theory (SDT) (Gescheider, 1997; Green & Swets, 1966). Most previous studies of predictability effects, however, cannot be easily subjected to SDT analyses. Some have used highly salient stimuli to prevent false alarms (e.g., Karlin, 1959; Klemmer, 1956). In others, the target event was a subtle change of a stimulus, which was continuously presented (e.g., Ghose & Maunsell, 2002; Janssen & Shadlen, 2005). With such a stimulus, assignment of hits and false alarms to certain stimulus events is not straightforward. 
To disentangle sensory from nonsensory components of perceptual decisions, we designed a discrete-interval, noise-limited motion detection task. We presented a sequence of random motion stimuli at regular intervals, one of which contained a threshold-level target signal. This design offers the possibility to determine the frequency of hits and false alarms for every single interval in the stimulus sequence. Between sessions, we manipulated the predictability of the target signal. We examined response accuracy within the SDT framework and asked whether predictability of target signals leads to changes in perceptual sensitivity or to criterion adjustments. Although our paradigm was not designed with a focus on RT measurements, we were able to use a simple rise-to-threshold model (Carpenter, 1981; Carpenter & Williams, 1995; Reddi, Asrress, & Carpenter, 2003) to test whether target predictability influences distance to threshold or rate of information supply. 
Methods
Subjects
Eight subjects (ages 21–28, two males and six females) participated in this study. All had normal or corrected-to-normal vision. They gave informed written consent, and all of them were naïve as to the purpose of the experiment. 
Stimulus
The stimulus was composed of 250 black dots moving within a virtual circular aperture of 5° in diameter at the center of gaze on a white background. Each dot subtended 0.075° of visual angle and moved at a speed of 5°/s. Dot lifetime was infinite; dots that left the aperture reentered at random locations. A noise stimulus was characterized by all dots moving in random directions (0% coherence). In contrast, a target motion signal contained threshold-level coherent motion, which was embedded in the noise. The stimulus was presented on a VGA monitor (Lacie, Electron22 Blue IV) operating at a refresh rate of 85 Hz and a resolution of 80 pixels per degree of visual angle. Presentation of the stimulus and recording of the responses was controlled by an Apple Power Mac G4 computer. 
Design and procedure
A stationary dot pattern of 235-ms duration indicated the start of a trial. After a blank screen of 1,500 ms, subjects viewed a temporal sequence of 1–7 brief presentations of a centrally displayed, moving random dot pattern (235 ms) followed by a blank screen (1,500 ms). The subject's task was to press a key (“H”) on a computer keyboard upon detection of a threshold-level coherent motion signal of a predefined direction, inserted at random into the sequence of motion noise stimuli (0% coherence). For half of the subjects, the target signal consisted of rightward motion; for the other half, leftward motion was used. Trials were terminated after a subject's response (hit or false alarm) or after the target had been presented and no response was given before the next stimulus in the sequence would have been presented (miss). Subjects received auditory feedback after each trial. 
Predictability of a target signal was varied by manipulating the hazard rate, given by In this expression, f(x) represents the probability density, and F(x) represents the cumulative density function of the random variable x
An increasing hazard rate was implemented by drawing the target signal interval for a given trial from a uniform distribution defined over the values 1–7. Thus, the hazard rate for each interval is given by where i is the index for interval and k is the total number of intervals. 
A constant hazard rate was implemented by drawing the target interval from a geometric distribution with a mean of 4. Here, the hazard rate for each interval is given by where i is the index for interval and p is the probability of a target signal. As we took the mean to be 4, p was 0.25. In case the draw from the geometric distribution produced a number larger than 7, no target signal was presented and the trial was considered a no-go, in which subjects had to withhold the response. 
The experiment was divided into six sessions completed on consecutive days. A single session was composed of three blocks consisting of 53 trials each. The first three trials served to allow participants to get acquainted with the strength of the target signal and were not included in any of the analyses. Because trials were terminated after correct and incorrect responses, the number of stimulus presentations necessarily decreased across intervals (Table 1). Half of the subjects had a constant hazard rate in the first nine blocks and an increasing hazard rate in the second nine blocks. For the second half of the subjects, this assignment was reversed. Subjects were not informed about predictability manipulations. 
Table 1
 
Average number of stimulus presentations (targets and nontargets) encountered by an individual observer in each interval, separately for both hazard rate conditions (mean across n = 8 subjects).
Table 1
 
Average number of stimulus presentations (targets and nontargets) encountered by an individual observer in each interval, separately for both hazard rate conditions (mean across n = 8 subjects).
Interval
1 2 3 4 5 6
Increasing hazard rate 450 357 265 179 111 55
Constant hazard rate 450 282 174 102  58 34
For each subject and before each single experimental block, a two-interval, forced-choice algorithm was run to determine the individual, practice-dependent level of motion coherence required to achieve 75% correct responses. Here, subjects had to indicate which of two successive stimulus presentations contained the target motion signal, adjusted according to a 1-down-2-up procedure (Kaernbach, 1991) across 50 trials. This up-to-date estimate of motion coherence threshold was then used for the subsequent experimental block. Motion coherence threshold declined from 10.8 ± 0.7% (mean across subjects) in the first block to 5.6 ± 0.4% in the last block. This calibration procedure was effective in preventing practice-dependent improvements in the main task, where the percentage of correctly completed trials remained constant across blocks (44.4 ± 0.7%, mean across subjects). Throughout, accuracy was emphasized over speed. 
Analysis of accuracy data
We used Signal Detection Theory (Green & Swets, 1966) to compute measures for perceptual sensitivity (d′) and response criterion (C). We consider the point at which an observer's responses are neither biased toward “yes” nor “no” as the zero bias point. C gives the number of standard deviation units that the criterion is above or below the zero bias point. Defined in this way, the criterion C is expressed in the same units as the sensitivity d′ (Gescheider, 1997). These two measures were determined for every combination of subject, signal interval, and hazard rate condition. Statistical significance was evaluated with a two-way Analysis of Variance (ANOVA), separately for d′ and C, involving the within-subject factors hazard rate condition (increasing vs. constant) and stimulus interval (1–6). Interval 7 was excluded from the analyses because, with an increasing hazard rate, false alarms cannot be made here. The resulting p values were adjusted, whenever appropriate, for violations of the sphericity assumption using the Greenhouse-Geyser correction. To further check the validity of our results, we analyzed our data using nonparametric methods based on marginal effects analysis (Brunner & Pur, 2001) or Monte Carlo simulations (Anderson, 2001). Both nonparametric variants of the ANOVA yielded very similar results leading to identical conclusions. 
Analysis of reaction times
Approximately 0.1% of trials with responses to targets were excluded because RTs were less than 150 ms. The remaining RTs ranged from 300 to 1,540 ms (mean = 727, SD = 190) under an increasing hazard rate and from 450 to 1,800 ms (mean = 799, SD = 195) under a constant hazard rate. To assess statistical significance, we performed an ANOVA on RT analogous to the one on accuracy, except for the inclusion of the additional within-subject factor target presence (present vs. absent). 
LATER model
To test whether stimulus predictability affected distance to threshold or rate of information supply, we performed standard reciprobit analyses (Carpenter & Williams, 1995; Reddi et al., 2003) on responses to targets, separately for each hazard rate condition. RTs from individual observers were normalized to the population average and standard deviation before being pooled. We considered a LATER model (linear rise to threshold with ergodic rate) consisting of a single process rising toward threshold (Figure 5A). We fitted two variants of such a LATER model using maximum-likelihood estimation of their parameters. Both of these variants modeled the distributions of RT across six presentation intervals. In the swivel variant, the only parameter allowed to vary between intervals was the distance to threshold. Thus, the swivel model had eight parameters: the mean of the slope r, the standard deviation of the slope r, and six independent parameters for distance to threshold (STS0), one for each presentation interval. In the shift variant, we only allowed the slope r to vary between intervals, resulting in the same number of parameters: the distance to threshold (STS0), the standard deviation of slope r, and six independent parameters for mean slope r, one for each interval. As these two variants of the LATER model have identical degrees of freedom, model superiority was determined simply on the basis of the difference in log likelihoods (Reddi et al., 2003). 
Results
In the experiment reported here, we studied the effect of stimulus predictability on behavioral performance. More specifically, we asked whether the predictability of an upcoming target signal can lead to changes in perceptual sensitivity or rather influences a response criterion, i.e., an overall tendency to report the presence of a target. 
Eight human subjects performed a threshold-level motion detection task. We presented a sequence of brief random motion stimuli at regular intervals, one of which contained a designated, threshold-level target signal (Figure 1A). This design allowed us to determine the frequency of hits and false alarms for every single interval in the stimulus sequence. 
Figure 1
 
Experimental design. (A) Discrete-interval, noise-limited motion detection task. Subjects viewed a temporal sequence of 1–7 brief presentations of a centrally displayed, moving random dot pattern. Timing is indicated below the sequence. The task was to press a button upon detection of a designated, threshold-level coherent motion signal, which was inserted into a sequence of otherwise random noise stimuli (0% coherence). Filled symbols in (A) indicate the target signal consisting of rightward motion presented in the fifth interval (in the actual experiment, target signal dots and noise dots were indistinguishable). Trials were terminated after a response (hit or false alarm) or if the target had been presented and no response was given (miss). Dashed lines in (A) indicate parts of the trial sequence that were not shown as the trial had already been terminated. (B–C) Manipulation of target predictability. In different blocks of trials, the stimulus interval containing the target was either drawn from a uniform distribution yielding an increasing hazard rate (B) or from a geometric distribution for which the hazard rate is constant across intervals (C).
Figure 1
 
Experimental design. (A) Discrete-interval, noise-limited motion detection task. Subjects viewed a temporal sequence of 1–7 brief presentations of a centrally displayed, moving random dot pattern. Timing is indicated below the sequence. The task was to press a button upon detection of a designated, threshold-level coherent motion signal, which was inserted into a sequence of otherwise random noise stimuli (0% coherence). Filled symbols in (A) indicate the target signal consisting of rightward motion presented in the fifth interval (in the actual experiment, target signal dots and noise dots were indistinguishable). Trials were terminated after a response (hit or false alarm) or if the target had been presented and no response was given (miss). Dashed lines in (A) indicate parts of the trial sequence that were not shown as the trial had already been terminated. (B–C) Manipulation of target predictability. In different blocks of trials, the stimulus interval containing the target was either drawn from a uniform distribution yielding an increasing hazard rate (B) or from a geometric distribution for which the hazard rate is constant across intervals (C).
Between sessions performed on different days, we manipulated the predictability of the target signal (Figure 1B and C). Predictability was manipulated through the hazard rate, which specifies the probability that a target signal will come next, given it has not been shown yet. In half of the sessions, the target signal interval was drawn at random from a uniform distribution. Under this distribution, the hazard rate increases from interval to interval such that the likelihood of target presentation grows with every interval (Figure 1B). In the other half of the sessions, the target signal interval came from a geometric distribution. Under this distribution, the hazard rate remains constant across intervals such that the likelihood of a target signal coming next cannot be predicted by the number of past intervals (Figure 1C). 
To distinguish changes in sensory processing from changes in response criterion, we applied Signal Detection Theory and measured perceptual sensitivity (d′) and response criterion (C). Whether stimulus predictability affects sensitivity, the criterion, or both can most easily be seen in ROC curves. An ROC curve commonly assumes a constant sensitivity and describes how hit and false alarm rates change with changes in criterion. Thus, if stimulus predictability only influences the criterion, plotting hit and false alarm rates across intervals will resemble an ROC curve. 
We found that stimulus predictability indeed affected the criterion while leaving sensitivity unchanged; this result was evident in individual subjects (Figure 2). Consider, for example, subject cti (Figure 2A through F). As the target signal became more and more predictable, the hit rate strongly increased (Figure 2A, filled symbols). This increase in hit rate, however, was accompanied by an increase in the number of false alarms (Figure 2A, open symbols). Plotting hit rate versus false alarm rate across intervals (Figure 2C, numbers 1–6) yields a series of data points that nearly fall along a theoretical ROC curve (Figure 2C, gray curve), suggesting a constant sensitivity and a variable criterion. Conversely, in the condition in which the likelihood of a target signal coming next did not increase, hits and false alarm rates did not change as much with passing intervals (Figure 2B). As a consequence, rather than describing an ROC curve with a constant value of d′, these data points tend to form a cluster (Figure 2D). To assess the reliability of these findings, we compared effects of interval on sensitivity and on criterion between the hazard rate conditions. Effects of hazard rate on the criterion were systematic (Figure 2E): At a criterion of zero (grey horizontal line), the subject is equally likely to respond with yes or no regarding the presence of a target. As the target signal became more and more likely (solid line), the subject's tendency for no responses decreased, culminating in a bias toward yes responses. Under the condition in which the target likelihood did not change, however, the tendency for no responses remained on a fairly constant level (dotted line). Sensitivity, in contrast, seemed to fluctuate unspecifically both within and between hazard rate conditions (Figure 2F). 
Figure 2
 
Effects of target predictability on response accuracy for two example subjects. (A–B) Changes of hit rate (filled symbols) and false alarm rate (open symbols) across the sequence of stimulus presentations under an increasing hazard rate (A) or constant hazard rate (B). (C–D) Relation between hit and false alarm rate across presentation intervals (numbers) for increasing (C) and constant hazard rate (D). For plotting the theoretical ROC curves (grey line), d′ was taken to be the average across all intervals. (E) Changes in criterion C across presentation intervals under increasing (solid line) and constant hazard rate (dotted line). (F) Same, for sensitivity d′. (G–L) Same, for a second observer. In all panels, error bars give the standard error of the mean as determined by bootstrap tests.
Figure 2
 
Effects of target predictability on response accuracy for two example subjects. (A–B) Changes of hit rate (filled symbols) and false alarm rate (open symbols) across the sequence of stimulus presentations under an increasing hazard rate (A) or constant hazard rate (B). (C–D) Relation between hit and false alarm rate across presentation intervals (numbers) for increasing (C) and constant hazard rate (D). For plotting the theoretical ROC curves (grey line), d′ was taken to be the average across all intervals. (E) Changes in criterion C across presentation intervals under increasing (solid line) and constant hazard rate (dotted line). (F) Same, for sensitivity d′. (G–L) Same, for a second observer. In all panels, error bars give the standard error of the mean as determined by bootstrap tests.
A very similar pattern of results was obtained for subject kao (Figure 2G through L). With an increasing target signal likelihood, hit and false alarm rates change concurrently, and these changes are well described by a theoretical ROC curve (Figure 2G and I). This well-ordered relation breaks down when target signal likelihood does not increase with passing intervals (Figure 2H and J). Indeed, effects of interval and hazard rate on sensitivity (Figure 2L) are rather variable, when compared to those on the criterion (Figure 2K). 
The observation that stimulus predictability affects the criterion but not the sensitivity was consistent across our eight subjects (Figure 3). With increasing target likelihood, changes in accuracy can be well explained by assuming a constant sensitivity and a decreasing criterion (Figure 3A and C). Conversely, with constant target likelihood, changes in accuracy are essentially absent (Figure 3B and D). We evaluated statistical significance with overall analyses of variance (ANOVAs). These ANOVAs were performed separately for sensitivity and criterion (Figure 3E and F), and they included the within-subjects factors of hazard rate condition (increasing vs. constant) and target signal interval (1–6). 
Figure 3
 
Average effects of target predictability on response accuracy (n = 8 subjects). Conventions as in Figure 2. Error bars show 95% confidence intervals after between-subject variability had been removed (Loftus & Masson, 1994). In (A) and (B), error bars are smaller than marker size.
Figure 3
 
Average effects of target predictability on response accuracy (n = 8 subjects). Conventions as in Figure 2. Error bars show 95% confidence intervals after between-subject variability had been removed (Loftus & Masson, 1994). In (A) and (B), error bars are smaller than marker size.
The analysis of the criterion (Figure 3E) revealed a significant main effect of target signal interval (p = 0.04) and of the interaction between target signal interval and hazard rate condition (p = 0.006). Post-hoc comparisons indicated that the criterion dropped when the hazard rate increased (p = 0.01), but did not change when the hazard rate remained constant (p = 0.32). Thus, the main effect of target signal interval arose because of the relatively strong decrease of the criterion with an increasing hazard rate. From this we can conclude that subjects adjust their decision criterion over time, and they only do so when the target signal becomes more and more predictable. 
Conversely, the analysis of sensitivity (Figure 3F) did not reveal significant effects. Neither the main effects of hazard rate condition (p = 0.37) or target signal interval (p = 0.41), nor their interaction (p = 0.77) approached statistical significance. From this we can conclude that, across intervals, sensitivity remains unchanged, irrespective of the predictability of the target signal. 
Apart from these effects on accuracy, target signal predictability also had a sizeable effect on RT (Figure 4). Consistent with the pioneering studies (e.g., Klemmer, 1956; Requin & Granjon, 1969), an increase in the target signal likelihood lead to a decrease in RT. This effect can be seen in individual subjects (Figure 4A and B, solid lines) as well as in the average across subjects (Figure 4C). In contrast, when the target signal likelihood remained constant, RT did not decrease (Figure 4A through C, dotted traces). A very similar pattern of RTs was observed in the absence of a target, i.e., when observers made a false alarm (Figure 4D through F). To assess statistical significance of these findings, we performed an ANOVA on RT, involving the within-subject factors hazard rate condition (increasing vs. constant), interval (1–6), and target presence (present vs. absent). This analysis revealed a significant main effect of target presence (p < 0.0001), of hazard rate condition (p = 0.01), and a strong trend toward a significant interaction between interval and hazard rate condition (p = 0.06). Observers were faster, overall, for correct responses than for false alarms and for an increasing compared to constant hazard rate. The interaction effect arose because, from interval to interval, RTs strongly decreased under an increasing hazard rate (post-hoc comparison, p = 0.02), but did not change under a constant hazard rate (p = 0.5). None of the other interactions approached statistical significance, indicating that the relation between hazard rate condition and interval did not depend on whether a target was present or absent. Thus, irrespective of the presence of a target signal, increasing stimulus predictability decreases RTs, which is consistent with adjustments of the response criterion, rather than changes in sensory processing. 
Figure 4
 
Effects of target predictability on reaction times. (A–C) Changes in reaction time for correct responses under increasing (solid line) or constant hazard rate (dashed line). (A) and (B) Single subject data. (C) Average across eight subjects. (D–F) Same, for false alarms. Conventions as in Figures 2 and 3.
Figure 4
 
Effects of target predictability on reaction times. (A–C) Changes in reaction time for correct responses under increasing (solid line) or constant hazard rate (dashed line). (A) and (B) Single subject data. (C) Average across eight subjects. (D–F) Same, for false alarms. Conventions as in Figures 2 and 3.
Figure 5
 
RT analysis within the framework of the LATER model. (A) The LATER model, adapted from Reddi et al. (2003). (B) Illustration of a reciprobit plot. The abscissa represents the reciprocal of RT with reversed direction, terminating at infinity. The ordinate is a probit scale, i.e., the inverse of the normal cumulative distribution function. Under these transformations, a Gaussian distribution will be turned into a straight line; the RT corresponding to a z-score of 0 is the median of the distribution. (C) Swivel variant of the LATER model. Changes in RT distributions between conditions are explained by adjustments of distance to threshold, evident in a swivel around the infinite time intercept. (D) Shift variant. Changes in RT distributions are explained by changes in information supply, evident in a horizontal shift. (E) Effect of target signal likelihood on RT distributions for an increasing hazard rate. Presentation interval is coded in color. Dots correspond to measured RT distributions. Solid lines represent maximum-likelihood estimates of the LATER model under the swivel-constraint. (F) Same, for constant hazard rate. For purpose of illustration, the abscissa is clipped at 350 ms omitting a total of five data points from being shown in (E). Fits are based on all data points. (G) Relation between median RT and log hazard rate for increasing hazard rate condition. Measured median RTs (open circles) are shown along with the linear regression line. Closed circles represent maximum-likelihood estimates of median RT obtained from the swivel variant of the LATER model.
Figure 5
 
RT analysis within the framework of the LATER model. (A) The LATER model, adapted from Reddi et al. (2003). (B) Illustration of a reciprobit plot. The abscissa represents the reciprocal of RT with reversed direction, terminating at infinity. The ordinate is a probit scale, i.e., the inverse of the normal cumulative distribution function. Under these transformations, a Gaussian distribution will be turned into a straight line; the RT corresponding to a z-score of 0 is the median of the distribution. (C) Swivel variant of the LATER model. Changes in RT distributions between conditions are explained by adjustments of distance to threshold, evident in a swivel around the infinite time intercept. (D) Shift variant. Changes in RT distributions are explained by changes in information supply, evident in a horizontal shift. (E) Effect of target signal likelihood on RT distributions for an increasing hazard rate. Presentation interval is coded in color. Dots correspond to measured RT distributions. Solid lines represent maximum-likelihood estimates of the LATER model under the swivel-constraint. (F) Same, for constant hazard rate. For purpose of illustration, the abscissa is clipped at 350 ms omitting a total of five data points from being shown in (E). Fits are based on all data points. (G) Relation between median RT and log hazard rate for increasing hazard rate condition. Measured median RTs (open circles) are shown along with the linear regression line. Closed circles represent maximum-likelihood estimates of median RT obtained from the swivel variant of the LATER model.
To further test whether the pattern of RTs does indeed support criterion adjustments, we considered the LATER model proposed by Carpenter and coworkers (Carpenter, 1981; Carpenter & Williams, 1995; Reddi et al., 2003). In the model (Figure 5A), stimulus onset (bottom) triggers a decision signal S (middle), which rises linearly from start level S0 to a threshold ST. The crossing of the threshold initiates a behavioral response (top). The slope r of the decision signal S varies randomly from trial to trial following a Gaussian distribution, which produces the commonly observed skew in the distribution of RTs (shaded areas). The LATER model predicts that the reciprocal of RT is normally distributed, which can be verified in a reciprobit graph (Figure 5B). 
We asked how stimulus predictability affects the distribution of RTs. As the target signal likelihood increases, observers could achieve reduction of RTs by changes in distance to threshold (STS0), rate of information supply (slope r), or both. Pure changes in either parameter would be particularly simple to describe. If observers adjusted distance to threshold, it would effectively rotate the reciprobit line around the infinite time intercept (swivel variant, Figure 5C). In contrast, changes in slope r would shift the line along the horizontal axis (shift variant, Figure 5D). 
We examined the RT distributions and found strong evidence in favor of threshold adjustments. We constructed reciprobit plots separately for each interval under both hazard rate conditions (Figure 5E and F). With increasing target signal likelihood, all RT distributions seem to converge at the faster end, while fanning out at the slower end of the distribution (Figure 5E, dots). The successive displacement of these distributions is well structured, corresponding to the step-wise increases in target signal likelihood. In contrast, in the absence of changes in target signal likelihood, all RT distributions are superimposed (Figure 5F, dots). We fitted a simple swivel variant of the LATER model (Figure 5E and F, solid lines), in which distance to threshold was the only parameter allowed to vary between presentation intervals. All individual RT distributions conformed to the LATER model (one-sample Kolmogorov-Smirnov tests, p > 0.12 and p > 0.3 for increasing and constant hazard rates). We also fitted a shift variant of the model, in which only slope r was allowed to vary. Under an increasing hazard rate, the swivel model outperformed the shift model (log likelihood difference of 6.71), indicating that the observed distributions of RT can largely be accounted for by adjustments of the distance to threshold. 
The observed adjustments of distance to threshold with increases in stimulus predictability can be understood as a strategy to optimize behavioral performance. Previous work has established a close link between the LATER model and the Bayesian framework of decision-making (Carpenter & Williams, 1995; Reddi et al., 2003). Within this framework, the start level S0 is taken to represent the prior log likelihood of a target signal being present. The decision signal S represents the posterior log likelihood which, when high enough, triggers a response. According to Bayes' rule, the posterior log likelihood, and hence RT, should be linearly related to variations in prior log likelihood. Experiments manipulating prior target likelihood have documented this relationship (Carpenter & Williams, 1995), showing that, under such conditions, observers seem to behave in the Bayesian optimal way: Prior knowledge of target likelihood is incorporated by setting the appropriate starting level S0. In our paradigm, as the hazard rate increases, the prior likelihood of a target being shown changes from interval to interval, and Bayesian optimal performance would therefore require adjusting the distance to threshold accordingly.Indeed, we observed a linear relation between median RT and the logarithm of the hazard rate (Figure 5G). From this we can conclude that our observers optimize behavioral performance by taking into consideration step-wise changes in the prior likelihood of a target signal. 
Discussion
In the experiment reported here, we investigated the influence of stimulus predictability on behavioral performance in a visual detection task. We analyzed response accuracy in the context of Signal Detection Theory and found that stimulus predictability does not affect perceptual sensitivity; the changes in response accuracy we have observed can be solely attributed to a lowering of the response criterion. We complemented this analysis by an examination of RTs and found them to be well described by a simple rise-to-threshold model. Consistent with the results based on SDT, stimulus predictability in high-attention tasks predominantly leads to threshold adjustments rather than changes in information supply. These adjustments conform to Bayesian optimal decision-making. 
Compared to previous studies of predictability effects, our paradigm has a number of advantages. First, by using a regular sequence of brief stimuli, separated by appropriate gaps, we were able to unambiguously assign hits or false alarms to certain elements of the stimulus sequence and relate perceptual decisions to the current value of the hazard rate. With rapid streams of stimuli or continuous presentations, it becomes difficult to tell what exactly triggered a behavioral response. Second, by using a discrete stimulus sequence, the hazard rate (which determines the predictability of the target signal) can easily be represented by an observer, as they only need to count the stimuli from 1 to N, where N is the end of the sequence. With continuous stimuli, in contrast, an observer's prediction will become less precise with passing time (Bausenhart, Rolke, Seibold, & Ulrich, 2010) leading some researchers to approximate an observer's presentation of the actual hazard rate by a subjective hazard rate (Bueti et al., 2010; Janssen & Shadlen, 2005). Though we instructed our subjects to aim for accuracy rather than speed, the pattern of RTs is very consistent with the results of pioneering RT studies. A reduction in RT with an increasing hazard rate has been taken as evidence that observers can anticipate the appearance of the target signal (Niemi & Näätänen, 1981). The fact that we can replicate this basic pattern of RT shows that our manipulation of expectancies was successful, indicating that effects of stimulus predictability can also be studied with classic signal detection paradigms. 
The question of how we benefit from being able to predict important sensory events has a long tradition; over the years, a range of approaches has provided a variety of results. Early behavioral studies (e.g., Hohle, 1965) have relied on measures of RT, and their results have been interpreted to show that stimulus predictability mainly influences postperceptual processes (for reviews, see Luce, 1986; Müller-Gethmann, Ulrich, & Rinkenauer, 2003; Sanders, 1980). This interpretation has received support from studies that have documented effects of stimulus predictability on motor-related variables, such as spinal cord reflexes (Brunia & Van Boxtel, 2000) and response force (Mattes & Ulrich, 1997). The same conclusions have been reached by studies measuring event-related potentials in humans. Here, temporal expectancies have been found to modulate common indices of motor-related processes such as the contingent negative variation (e.g., Loveless, 1973) and the lateralized readiness potential (Hackley & Valle-Inclán, 2003), but seem to have less influence on common indices of early sensory processing such as the visually evoked potentials N1 and P1 (e.g., Miniussi, Wilding, Coull, & Nobre, 1999). Only recently, the pendulum has swung toward perceptual improvements. A few behavioral studies have documented effects of temporal expectancies at the perceptual level (Bausenhart, Rolke, & Ulrich, 2007, 2008; Correa, Lupiáñez, & Tudela, 2005; Rolke & Hofmann, 2007). Recent measurements of event-related potentials in humans have localized predictability effects closer to the sensory than to the motor end of processing (Hackley & Valle-Inclán, 2003; Müller-Gethmann et al., 2003). The most direct evidence has come from single-cell recording studies measuring neural responses at identified stages of the cortical processing hierarchy. These studies have shown that stimulus predictability can modulate responses in the monkey extrastriate visual cortex (Ghose & Bearl, 2010; Ghose & Maunsell, 2002) and rat primary auditory cortex (Jaramillo & Zador, 2011). 
Given these recent demonstrations of predictability effects on sensory processing, one might wonder why we fail to notice changes in perceptual sensitivity measured by d′. Indeed, predictability-related changes in d′ have been reported in two recent behavioral studies (Correa et al., 2005; Rolke & Hofmann, 2007). How can this apparent discrepancy be explained? 
Possibly, this discrepancy can be explained by differences in attentional demands. When stimulus predictability is manipulated by providing an explicit temporal cue at the beginning of each trial (Correa et al., 2005) or by presenting early versus late targets in separate blocks of trials (Rolke & Hofmann, 2007), attention can be directed to particular moments in time, which can improve perceptual sensitivity (for a review, see Nobre, Correa, & Coull, 2007). Studies that report changes in perceptual sensitivity typically compare conditions in which a stimulus is either expected (likely under full attention) or unexpected (likely in the absence of attention). Here, improvements in perceptual sensitivity might actually be related to differences in the attentional state. In our paradigm, in contrast, there is no such condition in which the target signal appears unexpectedly. We assume that full attention is directed to each stimulus in the sequence and compare conditions that differ in the likelihood of the target signal being presented. Under such a constant focus of attention, stimulus predictability might not affect perceptual sensitivity, possibly because it has already been maximized. 
Alternatively, the inconsistency between these studies might be explained by specific challenges imposed by these different perceptual tasks. In Rolke and Hofmann (2007), for instance, observers were required to discriminate two possible orientations of a briefly presented Landolt square. Under such conditions, task performance might strongly benefit from a sharpening of feature-selective channels, which might underlie the improved perceptual sensitivity as measured by d′. In contrast, in our task, observers were required to detect a weak coherent motion signal, which was embedded in noise. The neuronal mechanisms underlying stimulus detection might differ from those mediating discrimination (Hol & Treue, 2001; Jazayeri & Movshon, 2007) and might not leave a trace in common measures of perceptual sensitivity. 
Although our study was not designed to focus on RT, the application of the LATER model revealed predictability effects on RT consistent with criterion adjustments. With increasing stimulus predictability, the pattern of RT distributions was best explained by successive adjustments of the distance to the decision threshold, rather than by changes in information supply. These results are in accordance with a recent fMRI study in which predictability was manipulated through dependencies within a stream of sequentially presented stimuli (Domenech & Dreher, 2010). The model terms of LATER, distance to threshold and rate of information supply, can be related to the criterion and sensitivity of SDT (Lauwereyns & Wisnewski, 2006). While none of these two frameworks can offer a joint account of accuracy and RT, such as achieved by sequential sampling models (e.g., Palmer, Huk, & Shadlen, 2005; Ratcliff & McKoon, 2008), our separate analysis of accuracy and RT converge to the same basic conclusion. 
The successful application of the LATER model to our data indicates that observers adopt an optimal strategy. Previous work has documented that the LATER model implements an ideal Bayesian decision-making process (Carpenter & Williams, 1995; Reddi et al., 2003). This process is understood to accumulate evidence until the certainty about the presence of the target is high enough. Consistent with this previous work, we find that increases in prior target likelihood can lead to adjustments of the distance to threshold, thereby reducing the amount of evidence required to reach a decision. We conclude that, under conditions where full attention is directed to the stimulus, observers can use temporal expectancies to adjust the distance to threshold. Such adjustments optimize behavioral performance by speeding up the detection of critical events in the face of changes in their likelihood. 
Acknowledgments
This work was supported by the German Ministry for Education and Science grants BMBF 01GQ0433 and 01GQ1005C to the Bernstein Center for Computational Neuroscience, Göttingen, Germany; by the Deutsche Forschungsgemeinschaft (DFG) Collaborative Research Center 889 “Cellular Mechanisms of Sensory Processing”; by funds awarded to the Centre for Integrative Neuroscience, Tübingen, Germany (DFG Exec 307); and by a Starting Independent Researcher grant from the European Research Council (project acronym: PERCEPT) awarded to Steffen Katzner. 
Commercial relationships: none. 
Corresponding author: Steffen Katzner. 
Email: steffen.katzner@uni-tuebingen.de. 
Address: Centre for Integrative Neuroscience, University of Tübingen, Germany. 
References
Alink A. Schwiedrzik C. M. Kohler A. Singer W. Muckli L. (2010). Stimulus predictability reduces responses in primary visual cortex. Journal of Neuroscience, 30(8), 2960–2966. [CrossRef] [PubMed]
Anderson M. J. (2001). A new method for non-parametric multivariate analysis of variance. Austral Ecology, 26, 32–46.
Bausenhart K. M. Rolke B. Seibold V. C. Ulrich R. (2010). Temporal preparation influences the dynamics of information processing: Evidence for early onset of information accumulation. Vision Research, 50(11), 1025–1034. [CrossRef] [PubMed]
Bausenhart K. M. Rolke B. Ulrich R. (2007). Knowing when to hear aids what to hear. Quarterly Journal of Experimental Psychology, 60, 1610–1615. [CrossRef]
Bausenhart K. M. Rolke B. Ulrich R. (2008). Temporal preparation improves temporal resolution: Evidence from constant foreperiods. Perception & Psychophysics, 70, 1504–1514. [CrossRef] [PubMed]
Brunia C. H. M. Van Boxtel G. J. M. (2000). Motor preparation. In Cacioppo J. T. Tassinary L. G. Berntson G. G.(Eds.), Handbook of psychophysiology, 2nd ed. (pp. 507–532). Cambridge, UK: Cambridge University Press.
Brunner E. Pur M. L. (2001). Nonparametric methods in factorial designs. Statistical Papers, 42, 1–52. [CrossRef]
Bueti D. Bahrami B. Walsh V. Rees G. (2010). Encoding of temporal probabilities in the human brain. Journal of Neuroscience, 30(12), 4343–4352. [CrossRef] [PubMed]
Carpenter R. H. S. (1981). Oculomotor procrastination. In Fisher D. F. Monty R. A. Senders J. W.(Eds.), Eye movements: Cognition and visual perception. (pp. 237–246). Hillsdale, NJ: Lawrence Erlbaum.
Carpenter R. H. Williams M. L. (1995). Neural computation of log likelihood in control of saccadic eye movements. Nature, 377(6544), 59–62. [CrossRef] [PubMed]
Correa A. Lupiáñez J. Tudela P. (2005). Attentional preparation based on temporal expectancy modulates processing at the perceptual level. Psychonomic Bulletin & Review, 12(2), 328–334. [CrossRef] [PubMed]
Domenech P. Dreher J. C. (2010). Decision threshold modulation in the human brain. Journal of Neuroscience, 30(43), 14 305–14 317.
Gescheider G. A. (1997). Psychophysics: The fundamentals (3rd ed.). Hillsdale, NJ: Erlbaum.
Ghose G. M. (2006). Strategies optimize the detection of motion transients. Journal of Vision, 6(4):10, 429–440, www.journalofvision.org/content/6/4/10, doi:10.1167/6.4.10. [PubMed] [Article] [CrossRef]
Ghose G. M. Bearl D. W. (2010). Attention directed by expectations enhances receptive fields in cortical area MT. Vision Research, 50(4), 441–451. [CrossRef] [PubMed]
Ghose G. M. Maunsell J. H. (2002). Attentional modulation in visual cortex depends on task timing. Nature, 419(6907), 616–620. [CrossRef] [PubMed]
Green D. M. Swets J. A. (1966). Signal detection theory and psychophysics. New York, NY: Wiley.
Hackley S. A. Valle-Inclán F. (2003). Which stages of processing are speeded by a warning signal? Biological Psychology, 64(1–2), 27–45. [CrossRef] [PubMed]
Hohle R. H. (1965). Inferred components of reaction times as functions of foreperiod duration. Journal of Experimental Psychology, 69, 382–386. [CrossRef] [PubMed]
Hol K. Treue S. (2001). Different populations of neurons contribute to the detection and discrimination of visual motion. Vision Research, 41(6), 685–689. [CrossRef] [PubMed]
Janssen P. Shadlen M. N. (2005). A representation of the hazard rate of elapsed time in macaque area LIP. Nature Neuroscience, 8(2), 234–241. [CrossRef] [PubMed]
Jaramillo S. Zador A. M. (2011). The auditory cortex mediates the perceptual effects of acoustic temporal expectation. Nature Neuroscience, 14(2), 246–251. [CrossRef] [PubMed]
Jazayeri M. Movshon J. A. (2007). A new perceptual illusion reveals mechanisms of sensory decoding. Nature, 446(7138), 912–915. [CrossRef] [PubMed]
Kaernbach C. (1991). Simple adaptive testing with the weighted up-down method. Perception & Psychophysics, 49(3), 227–229. [CrossRef] [PubMed]
Karlin L. (1959). Reaction time as a function of foreperiod duration and variability. Journal of Experimental Psychology, 58, 185–191. [CrossRef] [PubMed]
Klemmer E. T. (1956). Time uncertainty in simple reaction time. Journal of Experimental Psychology, 51(3), 179–184. [CrossRef] [PubMed]
Lauwereyns J. Wisnewski R. G. (2006). A reaction-time paradigm to measure reward-oriented bias in rats. Journal of Experimental Psychology: Animal Behavior Processes, 32(4), 467–473. [CrossRef] [PubMed]
Loftus G. R. Masson M. E. (1994). Using confidence intervals in within-subjects designs. Psychonomic Bulletin & Review, 1(4), 476–490. [CrossRef] [PubMed]
Loveless N. E. (1973). The contingent negative variation related to preparatory set in a reaction time situation with variable foreperiod. Electroencephalography and Clinical Neurophysiology, 35, 369–374. [CrossRef] [PubMed]
Luce R. D. (1986). Response times. New York, NY: Oxford University Press.
Mattes S. Ulrich R. (1997). Response force is sensitive to the temporal uncertainty of response stimuli. Perception & Psychophysics, 59, 1089–1097. [CrossRef] [PubMed]
Miniussi C. Wilding E. L. Coull J. T. Nobre A. C. (1999). Orienting attention in time: Modulation of brain potentials. Brain, 122(8), 1507–1518. [CrossRef] [PubMed]
Müller-Gethmann H. Ulrich R. Rinkenauer G. (2003). Locus of the effect of temporal preparation: Evidence from the lateralized readiness potential. Psychophysiology, 40(4), 597–611. [CrossRef] [PubMed]
Näätänen R. (1971). Non-aging fore-periods and simple reaction time. Acta Psychologica, 35(4), 316–327. [CrossRef]
Niemi P. Näätänen R. (1981). Foreperiod and simple reaction time. Psychonomic Bulletin & Review, 89(1), 133–162. [CrossRef]
Nobre A. Correa A. Coull J. (2007). The hazards of time. Current Opinion in Neurobiology, 17(4), 465–470. [CrossRef] [PubMed]
Palmer J. Huk A. C. Shadlen M. N. (2005). The effect of stimulus strength on the speed and accuracy of a perceptual decision. Journal of Vision, 5(5):1, 376–404, www.journalofvision.org/content/5/5/1, doi:10.1167/5.5.1. [PubMed] [Article] [CrossRef] [PubMed]
Ratcliff R. McKoon G. (2008). The diffusion decision model: Theory and data for two-choice decision tasks. Neural Computation, 20, 873–922. [CrossRef] [PubMed]
Reddi B. A. Asrress K. N. Carpenter R. H. (2003). Accuracy, information, and response time in a saccadic decision task. Journal of Neurophysiology, 90(5), 3538–3546. [CrossRef] [PubMed]
Requin J. Granjon M. (1969). The effect of conditional probability of the response signal on the simple reaction time. Acta Psychologica, 31(2), 129–144. [CrossRef] [PubMed]
Rolke B. Hofmann P. (2007). Temporal uncertainty degrades perceptual processing. Psychonomic Bulletin & Review, 14(3), 522–526. [CrossRef] [PubMed]
Sanders A. (1980). Stage analysis of reaction processes. In Stelmach G. E. Requin J.(Eds.), Tutorials of motor behavior (pp. 331–354). New York, NY: North-Holland.
Figure 1
 
Experimental design. (A) Discrete-interval, noise-limited motion detection task. Subjects viewed a temporal sequence of 1–7 brief presentations of a centrally displayed, moving random dot pattern. Timing is indicated below the sequence. The task was to press a button upon detection of a designated, threshold-level coherent motion signal, which was inserted into a sequence of otherwise random noise stimuli (0% coherence). Filled symbols in (A) indicate the target signal consisting of rightward motion presented in the fifth interval (in the actual experiment, target signal dots and noise dots were indistinguishable). Trials were terminated after a response (hit or false alarm) or if the target had been presented and no response was given (miss). Dashed lines in (A) indicate parts of the trial sequence that were not shown as the trial had already been terminated. (B–C) Manipulation of target predictability. In different blocks of trials, the stimulus interval containing the target was either drawn from a uniform distribution yielding an increasing hazard rate (B) or from a geometric distribution for which the hazard rate is constant across intervals (C).
Figure 1
 
Experimental design. (A) Discrete-interval, noise-limited motion detection task. Subjects viewed a temporal sequence of 1–7 brief presentations of a centrally displayed, moving random dot pattern. Timing is indicated below the sequence. The task was to press a button upon detection of a designated, threshold-level coherent motion signal, which was inserted into a sequence of otherwise random noise stimuli (0% coherence). Filled symbols in (A) indicate the target signal consisting of rightward motion presented in the fifth interval (in the actual experiment, target signal dots and noise dots were indistinguishable). Trials were terminated after a response (hit or false alarm) or if the target had been presented and no response was given (miss). Dashed lines in (A) indicate parts of the trial sequence that were not shown as the trial had already been terminated. (B–C) Manipulation of target predictability. In different blocks of trials, the stimulus interval containing the target was either drawn from a uniform distribution yielding an increasing hazard rate (B) or from a geometric distribution for which the hazard rate is constant across intervals (C).
Figure 2
 
Effects of target predictability on response accuracy for two example subjects. (A–B) Changes of hit rate (filled symbols) and false alarm rate (open symbols) across the sequence of stimulus presentations under an increasing hazard rate (A) or constant hazard rate (B). (C–D) Relation between hit and false alarm rate across presentation intervals (numbers) for increasing (C) and constant hazard rate (D). For plotting the theoretical ROC curves (grey line), d′ was taken to be the average across all intervals. (E) Changes in criterion C across presentation intervals under increasing (solid line) and constant hazard rate (dotted line). (F) Same, for sensitivity d′. (G–L) Same, for a second observer. In all panels, error bars give the standard error of the mean as determined by bootstrap tests.
Figure 2
 
Effects of target predictability on response accuracy for two example subjects. (A–B) Changes of hit rate (filled symbols) and false alarm rate (open symbols) across the sequence of stimulus presentations under an increasing hazard rate (A) or constant hazard rate (B). (C–D) Relation between hit and false alarm rate across presentation intervals (numbers) for increasing (C) and constant hazard rate (D). For plotting the theoretical ROC curves (grey line), d′ was taken to be the average across all intervals. (E) Changes in criterion C across presentation intervals under increasing (solid line) and constant hazard rate (dotted line). (F) Same, for sensitivity d′. (G–L) Same, for a second observer. In all panels, error bars give the standard error of the mean as determined by bootstrap tests.
Figure 3
 
Average effects of target predictability on response accuracy (n = 8 subjects). Conventions as in Figure 2. Error bars show 95% confidence intervals after between-subject variability had been removed (Loftus & Masson, 1994). In (A) and (B), error bars are smaller than marker size.
Figure 3
 
Average effects of target predictability on response accuracy (n = 8 subjects). Conventions as in Figure 2. Error bars show 95% confidence intervals after between-subject variability had been removed (Loftus & Masson, 1994). In (A) and (B), error bars are smaller than marker size.
Figure 4
 
Effects of target predictability on reaction times. (A–C) Changes in reaction time for correct responses under increasing (solid line) or constant hazard rate (dashed line). (A) and (B) Single subject data. (C) Average across eight subjects. (D–F) Same, for false alarms. Conventions as in Figures 2 and 3.
Figure 4
 
Effects of target predictability on reaction times. (A–C) Changes in reaction time for correct responses under increasing (solid line) or constant hazard rate (dashed line). (A) and (B) Single subject data. (C) Average across eight subjects. (D–F) Same, for false alarms. Conventions as in Figures 2 and 3.
Figure 5
 
RT analysis within the framework of the LATER model. (A) The LATER model, adapted from Reddi et al. (2003). (B) Illustration of a reciprobit plot. The abscissa represents the reciprocal of RT with reversed direction, terminating at infinity. The ordinate is a probit scale, i.e., the inverse of the normal cumulative distribution function. Under these transformations, a Gaussian distribution will be turned into a straight line; the RT corresponding to a z-score of 0 is the median of the distribution. (C) Swivel variant of the LATER model. Changes in RT distributions between conditions are explained by adjustments of distance to threshold, evident in a swivel around the infinite time intercept. (D) Shift variant. Changes in RT distributions are explained by changes in information supply, evident in a horizontal shift. (E) Effect of target signal likelihood on RT distributions for an increasing hazard rate. Presentation interval is coded in color. Dots correspond to measured RT distributions. Solid lines represent maximum-likelihood estimates of the LATER model under the swivel-constraint. (F) Same, for constant hazard rate. For purpose of illustration, the abscissa is clipped at 350 ms omitting a total of five data points from being shown in (E). Fits are based on all data points. (G) Relation between median RT and log hazard rate for increasing hazard rate condition. Measured median RTs (open circles) are shown along with the linear regression line. Closed circles represent maximum-likelihood estimates of median RT obtained from the swivel variant of the LATER model.
Figure 5
 
RT analysis within the framework of the LATER model. (A) The LATER model, adapted from Reddi et al. (2003). (B) Illustration of a reciprobit plot. The abscissa represents the reciprocal of RT with reversed direction, terminating at infinity. The ordinate is a probit scale, i.e., the inverse of the normal cumulative distribution function. Under these transformations, a Gaussian distribution will be turned into a straight line; the RT corresponding to a z-score of 0 is the median of the distribution. (C) Swivel variant of the LATER model. Changes in RT distributions between conditions are explained by adjustments of distance to threshold, evident in a swivel around the infinite time intercept. (D) Shift variant. Changes in RT distributions are explained by changes in information supply, evident in a horizontal shift. (E) Effect of target signal likelihood on RT distributions for an increasing hazard rate. Presentation interval is coded in color. Dots correspond to measured RT distributions. Solid lines represent maximum-likelihood estimates of the LATER model under the swivel-constraint. (F) Same, for constant hazard rate. For purpose of illustration, the abscissa is clipped at 350 ms omitting a total of five data points from being shown in (E). Fits are based on all data points. (G) Relation between median RT and log hazard rate for increasing hazard rate condition. Measured median RTs (open circles) are shown along with the linear regression line. Closed circles represent maximum-likelihood estimates of median RT obtained from the swivel variant of the LATER model.
Table 1
 
Average number of stimulus presentations (targets and nontargets) encountered by an individual observer in each interval, separately for both hazard rate conditions (mean across n = 8 subjects).
Table 1
 
Average number of stimulus presentations (targets and nontargets) encountered by an individual observer in each interval, separately for both hazard rate conditions (mean across n = 8 subjects).
Interval
1 2 3 4 5 6
Increasing hazard rate 450 357 265 179 111 55
Constant hazard rate 450 282 174 102  58 34
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