Equally Flexible and Optimal Response Bias in Older Compared to Younger Adults

Author Note Roderick Garton, Department of Psychology, University of Tasmania, Sandy Bay, Tasmania, Australia; Angus Reynolds, Department of Psychology, University of Tasmania, Sandy Bay, Tasmania, Australia; Mark R. Hinder, Department of Psychology, University of Tasmania, Sandy Bay, Tasmania, Australia; Andrew Heathcote, Department of Psychology, University of Tasmania, Sandy Bay, Tasmania, Australia.

Binary decisions require making a judgement about the relative likelihood of each option.Normatively, uncertain judgements require combining knowledge of the base rates for each option (i.e., knowledge about prior probabilities) with the probability of the observed evidence given each option (i.e., a likelihood ratio).In both behaviour and the brain, judgement and decision making has been conceptualized as a dynamic process (formally, as a weighted sum of the logarithm of the prior likelihood and gradually accumulating estimates of the logarithm of the likelihood of each option; Carpenter & Williams, 1995;Gold & Shadlen, 2001).A tendency to under-weight base rates (often termed the "base-rate fallacy" or "base-rate neglect") is well known in high-level decision making (e.g., Bar-Hillel, 1980;Kahneman & Tversky, 1973, 1982).Here we study the extent to which aging affects the weighting of prior evidence in simple perceptual decision making.
It is increasingly understood that older adults make slower decisions between stimulus alternatives not exclusively or necessarily because they process less stimulus information over time (i.e., have a slower information-processing rate), but because they accumulate more information before making a definitive decision about what response to make (i.e., have greater response caution).Formal "evidence accumulation" models of decision processing conceive of this greater caution as setting a higher threshold of evidence for a response.
Making use of base-rate bias is also conceived as a matter of threshold setting, only requiring lower response-specific thresholds for more likely options.Accordingly, if they retain their higher overall thresholds when base rates are unequal, older adults would have to make larger absolute changes to their decision thresholds to achieve the same facilitatory biasing effect as the less cautious younger adults.Together with neurological changes with age that are associated with threshold setting, this could make flexibly adapting thresholds to base-rate information more difficult with age, especially in situations where base-rate information changes rapidly.
In the first part of this report, we review evidence about the effects of aging on the setting of evidence thresholds in the context of evidence accumulation models of choice responding.We then draw out the implications of this evidence for the way in which normal aging affects how response bias is controlled in association with base-rate information.We then report an experiment in which older and younger participants made binary perceptual discriminations under two procedures of base-rate biasing-one in which the more likely target was constant across a block of trials, the other in which the more likely target randomly changed from trial-to-trial.We hypothesised that the need to adjust thresholds quickly and flexibly under the latter condition could make it harder for both older and younger participants to appropriately bias their response thresholds, and that this limitation could be greater in older participants due to a reduced ability to flexibly control threshold setting.We also benchmarked the biasing of response thresholds to base-rate information against the level of biasing required to optimise accuracy relative to given levels of response caution.

Response Caution and Age-Related Slowing
Slowing of choice responses is well known to occur with increasing adult age.
Although decreased central processing efficiency was for some time the dominant account of age-related slowing (e.g., Hartley, 2006;Salthouse, 2016), a long-standing alternative account attributes this to increased response caution with age (e.g., Craik, 1969;Welford, 1977).This hypothesis has received new impetus across the last two decades from analyses based on evidence-accumulation models of performance on speeded choice tasks.
Our analysis focuses on an evidence-accumulation model where accumulation occurs independently for each choice option, the Linear Ballistic Accumulator (LBA; Brown & Heathcote, 2008).The racing choice options are referred to as "accumulators".In an elementary binary choice task, the response is determined by the winner of the race between two accumulators to their own decision thresholds, and the decision time is determined by the finishing time of the winning accumulator.Response time (RT) is the sum of the decision time and the duration of non-decision processes (i.e., initial encoding of the stimulus into a form from which evidence relevant to the choice can be derived, and the production of a response corresponding to the choice once it is made).Modelling usually focuses on two core components: the placement of decision thresholds, and the rate at which evidence is accumulated.Models differ in how they describe these two components.Greater response caution is identified with the persistent use of higher thresholds across the two accumulators and slowed information processing is identified with a lower rate of evidence accumulation.
We also report subsidiary analyses based on another type of evidence accumulation model, the Diffusion Decision Model (DDM; Ratcliff & McKoon, 2008), which explains information processing rates and response caution in a comparable way.
Increased caution with age has been attributed to both strategically determined behavioural causes and obligatory neural causes.An early behavioural explanation held that this is a matter of preference, with greater caution applied with age as a risk-avoidant strategy even in tasks that do not show changes in processing efficiency with age (Ratcliff et al., 2001;Thapar, Ratcliff, & McKoon, 2003).This proposition has been refined by Starns and Ratcliff (2010), who studied threshold setting relative to optimizing the number of correct responses in a fixed time interval ("reward rate", see Bogacz et al., 2006).They found that younger participants were closer to the optimal reward rate than older participants, although both set higher than optimal thresholds that favoured accuracy over speed.They concluded that the two age groups differed in task goals, with younger participants focused on balancing speed and accuracy, and older participants focused almost exclusively on accuracy.Starns and Ratcliff (2012) acknowledged that reward rate may not have been an appropriate measure of optimality as their earlier work did not enforce a fixed time interval or instruct participants to maximize reward rate.However, when they did so in a subsequent study, the optimality difference persisted.They suggested that these age differences were not solely due to differences in goals "but also the ability to achieve those goals" (Starns & Ratcliff, 2012, p. 145; see also Smith & Brewer, 1995).
Age-related changes in the brain provide a potential explanation for this reduced ability to achieve the goal of balancing speed and accuracy.In particular, reduced white matter integrity in fronto-striatal tracts connecting the pre-Supplementary Motor Area (pre-SMA) to the striatum has been suggested to reduce flexibility in threshold setting with age.
Neurocognitive modelling has shown not only a correlation of fronto-striatal activity with LBA response caution, but a negative correlation between the white matter integrity of these tracts and both age and LBA thresholds (e.g., Forstmann et al., 2011).Building on a range of evidence linking degradation of white matter integrity and impairments in memory retrieval and cognitive control with age, Forstmann et al. (2011) suggested that "unwillingness of older adults to adopt fast speed-accuracy tradeoff settings may not just reflect a strategic choice that is entirely under voluntary control … it may also reflect structural limitations: age-related decrements in brain connectivity" (p.17242).

Response Bias and Aging
In speeded choice tasks, responses are faster and more accurate for options with a higher base rate.In the context of the accumulator model described above, the effects of biased base rates can be modelled by setting the decision threshold for the more likely option lower than for the less likely option.In this way, the accumulator for the more likely option wins the decision race more frequently and in less time-so producing greater accuracy and shorter RT for the more likely option.Whether this occurs by reducing the threshold associated with the more likely option and/or increasing the threshold associated with the less likely option, studies modelling choice performance with both the DDM and racing accumulator models such as the LBA have repeatedly confirmed that this effect is well described by response-specific threshold adaptation.These studies have mostly involved a block-wise bias where the base rates for each option were held constant across a block of trials and might only vary in direction (which target was the more likely one) between blocks (e.g., Arnold, Bröder, & Bayen, 2015;Criss, 2010;Leite & Ratcliff, 2011;Ratcliff, 1985;Ratcliff & McKoon, 2008, Exp. 3;Simen et al., 2009, Exp. 2;Van Ravenzwaaij, Mulder, Tuerlinckx, & Wagenmakers, 2012;Van Zandt, Colonius, & Proctor, 2000;White & Poldrack, 2014).Other studies have involved a trial-wise bias in which there were equal base rates across the block but targets were sampled on each trial such that the direction of bias could change between trials, the more likely target being cued ahead of each stimulus presentation (e.g., Dunovan & Wheeler, 2018;Forstmann, Brown, Dutilh, Neumann, & Wagenmakers, 2010;Mulder, Wagenmakers, Ratcliff, Boekel, & Forstmann, 2012).
It is commonly observed in these studies that, for the more likely option, error responses tend to be slower than correct responses, whereas, for the less likely option, error responses can be as fast, or faster, than correct responses (see, e.g., Mulder et al., 2012, Fig. 4).This pattern in the behavioural data is supportive of a relative threshold model of bias effects given that, with equal accumulation rates per option, this model predicts that (1) for the more likely option, the threshold for a correct response is lower than that for an error response, leading to faster correct than error responses, but (2) for the less likely option, the threshold for the correct response is higher than that for the error response (i.e., in this case, the more likely option), leading to faster errors relative to correct responses.
Bias in the setting of response thresholds induced by base-rate information has been shown to be partly mediated by a fronto-striatal network.However, this network differs slightly from that which mediates responses during the speed/accuracy manipulations that involve response caution.In particular, Forstmann et al. (2010) found that bias as measured by LBA thresholds was associated with differential fMRI activation in the orbitofrontal cortex, hippocampus and the side of the striatal putamen contralateral to the cued direction (see also Mulder et al., 2012, for a similar finding using the DDM but with stronger parietal involvement; and Forstmann, Ratcliff, & Wagenmakers, 2016, for a review).Given the pervasiveness of the association between age-related degradations in white matter integrity and impairments in cognitive control, and similarities between networks controlling caution and bias, it seems reasonable to hypothesise that the control of thresholds necessary to set an appropriate response bias might decline with age.
Consistent with this hypothesis, there is evidence of limited trial-by-trial adjustment of response caution among older adults.Specifically, Karayanidis et al. (2011) found that, in a task-switching paradigm, younger adults flexibly adjusted response thresholds on a trial-bytrial basis so as to maintain greater caution on the harder switch trials relative to easier repeat trials (replicating earlier findings), whereas caution was held uniformly high by older participants over switch and repeat trials.Also, Forstmann et al. (2011) found larger variability in the startpoints of evidence accumulation for older than younger adults in their LBA fits of data from a trial-wise caution (speed/accuracy) manipulation, concluding that older adults are "less proficient at controlling their bias or response caution settings" (p.

17246).
Unlike control of caution, control of bias does not necessarily relate to response speed.
That is, response bias can be modulated by balancing a reduction in one threshold with an increase in the other threshold, leaving no net effect on speed.Furthermore, there is no tradeoff between speed and accuracy in bias setting because a lower threshold for the more likely option increases both speed and accuracy.Hence, bias control is compatible with older adults' focus on accuracy, which, in turn, could enable them to perform as well as, or better than, younger adults.Consistent with this hypothesis, older adults have shown speed facilitation at levels equivalent to, or greater than, younger adults in association with response precuing in spatial choice (reviewed in Proctor, Vu, & Pick, 2005) and task-switching (Mayr, 2001), and with sequential biases over trial history in elementary choice response (e.g., Fozard, Thomas, & Waugh, 1976;Melis, Soetens, & van der Molen, 2002;Rabbitt & Vyas, 1980).These effects are indicative of flexible trial-by-trial adjustments of response-specific thresholds, which manifest as greater startpoint variability (Heathcote et al., 2015).As such, it is possible that greater startpoint variability among older relative to younger participants is indicative of greater trial-wise adjustment of response-specific thresholds to prior experimental or incidental cues rather than lack of control of overall threshold setting.

Experiment
To investigate age-related differences in bias control, we ran a binary perceptual discrimination experiment using both the block-wise and trial-wise biasing procedures described above.One response was always correct with a probability of 0.7, and participants were reliably cued as to the more likely option ahead of each stimulus presentation.Under block-wise biasing, designed to make threshold control relatively easy, the more likely option remained the same over a block of trials, whereas, under trial-wise biasing, designed to make threshold control relatively hard, the more likely option changed randomly from trial to trial.Furthermore, the effect of biased base rates under the block-wise procedure is at least partly mediated by sequential contingencies, with more frequent events occurring in longer runs and so with responses reliably facilitated by repetition; sequential effects that are reduced under trial-wise biasing, which depends more on learning the base-rate validity of cues (LaBerge, Van Gelder, & Yellott, 1970).Altogether, a weaker effect from trial-wise than block-wise biasing was likely, and, to the extent that control of response bias is reduced with age, the difference between block-wise and trial-wise biasing might be expected to increase with age.
To measure evidence thresholds while accounting for possible effects on nonthreshold parameters, we fit our data with the LBA (Brown & Heathcote, 2008) using Bayesian methods (Heathcote et al., 2018).Just as is the case for RT measurement, where the equivalence or otherwise of the effects of a manipulation must be judged relative to baseline differences between older and younger participants (Verhaeghen, Steitz, Sliwinski & Cerella, 2003), higher overall thresholds for the older group raise the question of whether to use an additive or multiplicative measure of bias effects.We circumvented this issue, and benchmarked the normative performance of both groups, by determining the level of bias that optimized accuracy based on the distribution of posterior parameter estimates of the LBA model.This afforded us tests of the absolute and relative optimality of older and younger participants that take account of both the speed and accuracy of decisions as well all sources of uncertainty in our data.
Given that our choice task did not require discrimination of fine visual detail or task switching, and did not make difficult memory demands, we did not expect older participants to have accumulation-rate deficits.Indeed, Forstmann et al. (2011) interpreted their LBA rate estimates as consistent with older participants engaging more carefully with their dot-motion task.In an LBA for a binary-choice task there are two rates, one for the accumulator that matches the stimulus and one for the mismatching accumulator, and they found both rates to be significantly higher in their older than younger participants.However, there was an advantage, albeit non-significant, for the younger participants in terms of the difference between matching and mismatching rates, which largely controls accuracy.Forstmann et al. also replicated, in their LBA fits, the commonly observed age difference in non-decision time, with younger participants showing a small advantage over older participants (~0.025s).
Given similarities of task and analysis between our study and that of Forstmann et al., and that our sample of older participants was also healthy and highly motivated, we expected similar results.
Several aspects of our design were motivated by minimizing potential confounds that occur under bias manipulations, and to improve measurement of LBA parameters.Simply responding with the more likely correct or rewarded option as soon as a stimulus is detected (fast guessing) occurs with a short response-stimulus interval (Simen et al., 2009) and when fast responding is encouraged by instruction (Noorbaloochi, Sharon & McClelland, 2015).
We therefore used a long response-stimulus interval, a fast-guess warning, and a moderate base-rate manipulation to minimize fast guessing.Furthermore, base-rate bias can be less effective in producing prospective shifts of response thresholds when, with highly practiced participants, difficulty varies widely and unpredictably (including choices with no correct answer) between trials.Under these conditions, thresholds might nevertheless be retrospectively adjusted in relation to base-rate bias on more difficult trials when decision time is slow and stimulus evidence is less reliable (Hanks, Mazurek, Kiani, Hopp, & Shadlen, 2011;Malhotra, Leslie, Ludwig, & Bogacz, 2017;but see Van Ravenzwaaij et al., 2012).In this way, as the effect of discrimination difficulty maps to rates (e.g., Voss, Rothermund & Voss, 2004), an indirect effect of base-rate bias on accumulation rates is likely.Because we were interested in the interaction of age and bias-rate bias on thresholds, we used a moderate level of discrimination difficulty under which performance was well above chance without requiring extensive practice.This also enabled us to acquire a reasonable level of error trials in order to identify model parameters (Heathcote et al., 2018), and to make a selective-influence assumption that this manipulation was explained only by differences in rate parameters (Voss et al., 2004).To account for effects of base-rate bias on both thresholds and accumulation rate, we performed model selection among three LBA parameterizations that assumed a selective influence of the bias manipulation on either thresholds or rates alone, or on both thresholds and rates.For generalizability across model architectures, we also performed a post hoc modelling of the data with the DDM (as encouraged by a reviewer).

Method Participants
All participants were self-screened for any relevant neurological, visual or motor deficits, provided prior written consent to participate and to make their de-identified triallevel data publicly accessible, and were informed that the research procedures were approved by the UTAS Human Research Ethics Committee.Two participants (both in the younger agegroup) were replaced given non-compliance that was either self-reported or ostensible in performance (e.g., by near-chance response accuracy, and inordinately large proportions of both fast guesses (RTs  .2s)and response timeouts).Data collection continued until the intended sample size of 40 participants was achieved, half younger than 30 years (range of 18-28, M = 21.4,SD = 3.2, excepting four participants whose age was verified as less than 30 but whose date of birth was not obtained, 70% female), and half older than 60 years (range of 63-76, M = 68.3,SD = 3.7, 65% female).Younger participants were University of Tasmania students, 11 of whom were first-year psychology students who received course-credit for participation; the remainder received AU$30 in shopping vouchers.Older participants were community-dwelling local residents who were recruited from a pool of respondents to advertisements for participants in age-related research and received AU$40 in vouchers.
Those participating comprised 51% of those in the pool who were approached to participate in the current study.

Design and Stimuli
All participants completed one session in which bias direction was manipulated in a block-wise manner, and another session in which bias direction was manipulated in a trialwise manner, with session order counter-balanced within each age group.Each stimulus colour was presented as the more likely (cue-congruent) target and the less likely (cueincongruent) target in equal proportions across blocks under both bias-types, and within blocks under trial-wise biasing.Within blocks, easy vs. hard target discriminations occurred equally often in a random order.
All displays were presented against a black screen background.Participants were seated approximately 50 cm from the monitor such that the centre of the screen was at eyelevel.The choice stimulus subtended approximately 6.2 of visual angle and consisted of a square 5.4 cm centre-aligned grid of 20 × 20 cells that were either blue (RGB = 0, 65, 255) or orange (RGB = 255, 127, 0); see the inset to Figure 1.Either 216 (54%, easy condition) or 208 (52%, hard condition) of the cells were filled by the target colour, with the remaining cells being filled by the alternative colour.Within each trial, while keeping these proportions constant, the blue/orange colour of each cell was pseudo-randomly shuffled every .050sor .067s(per technical limits, but subjectively transitioning at a constant rate).This dynamic display served to minimize focus on a small subset of cells and counting as the basis of responding.

Procedure
After introductory verbal and on-screen instructions, each session commenced with a practice block of 20 trials, all without biasing and with easy discriminations where the target colour filled 60% of the cells of the stimulus array.This was followed by 10 blocks of 60 trials each, the first two blocks of which were designated practice and for which the data were not analysed.Participants self-initiated each block by pressing a space-bar.This firstly produced an instruction screen that reminded participants of the type of bias for that session-specifically, that in the block-wise bias session, 70% of the stimuli across the block would be of a specified target colour, and that this would be cued ahead of each trial; and, alternatively, that in the trial-wise bias session, there were equal proportions of the target colours across the block but there was a 70% chance that the target colour that was cued ahead of each trial would be congruent with that colour.Participants were instructed to try to use this information to help them make the response as quickly as possible while being correct.The assignment of targets to trials according to these probabilities was made by sampling without replacement, where there were always 42 targets of one colour and 18 targets of the other colour within each block-wise bias block, and 30 targets of each colour within each trial-wise bias block.
Each trial commenced with the bias-cue-the statement "70% blue" or "70% orange"-displayed for 0.5s slightly above the central stimulus region.This was followed by a blank interval of 0.5s, a small white fixation square in the centre of the screen for 0.5s, a blank interval of 0.5s, and then the stimulus.Using a standard USB keyboard, participants were instructed to press the 'z' or '/' key to indicate the predominant colour within the stimulus.The mapping of response keys to target colour was counterbalanced across participants.To remind participants about the mapping, the onset of the fixation square coincided with the display of single blue and orange squares on the relevant side at the bottom of the screen.Responses had to be made within 2s.The stimulus and the mapping cues were offset upon registration of a response or passing of the RT deadline.Onscreen feedback was then presented for 0.5s: responses were followed by a "too fast" message if RT was less than 0.2s, or by a "too slow" message if no response was made within 2s; otherwise, if the response was correct, the RT was presented, or, if the response was incorrect, the word "incorrect".This was followed by a final 0.5s blank interval.Accordingly, the responsestimulus interval was 3s, including a cue-to-stimulus foreperiod duration of 2s.The entire sequence of events in each trial is illustrated in Figure 1.Elements within each image are not to scale with respect to the enclosing display region or to each other; for example, in the experiment, the width of the stimulus grid was only 10% of the width of the screen.The inset presents a static sample of the stimulus grid where orange (light grey) cells predominate relative to blue (dark grey) cells under both easy (54%) and hard (52%) levels of target discrimination difficulty.
At the end of each block, participants received performance feedback with an onscreen display of their mean RT in milliseconds for correct, time-valid responses, and their accuracy as a percentage of all trials within that block.Participants were instructed (both verbally ahead of the session, and by on-screen text during the session) to take breaks between each block.Each session lasted 45-60 minutes.After these blocks, within each session, a stop-signal task (not analysed further here), of the same length, was administered.
Finally, in debriefing, verbal feedback about task performance was elicited and noted.

Results
Prior to analysis, response timeouts and implausibly fast responses were removed (0.3%, see Supplemental Materials for details).A detailed analysis of observed performance measures (correct and error RTs, and error rates) using linear mixed models is reported in Supplemental Materials.These results are summarized in the next section along with graphical summaries of performance measures, and the subsequent sections define the LBA model (Brown & Heathcote, 2008) and report the results of fitting the model and the optimality analysis.

Performance Measures
Observed RT distributions are presented in for older participants, and in Figure 3 for younger participants.Error rates are presented in Figure 4 for older and younger participants.
The figures also represent LBA posterior predicted values (described below).Filled circles with 95% credible intervals represent LBA predictions.
Older participants had slower overall responses, but slightly greater accuracy, than younger participants.Older participants were faster when correct than incorrect, whereas younger participants tended to be faster when incorrect than correct.Older participants also had fewer errors of omission than did younger participants.These results are consistent with increased caution in older relative to younger participants.
Overall, correct responses were faster, and accuracy was greater, for the more likely option.Consistent with a relative-threshold account of this effect, error RTs were, conversely, slower for the more likely option.These effects of base-rate bias were stronger under blockwise than trial-wise biasing, and they were at least as large for older as for younger participants, older participants particularly showing a greater benefit in accuracy for more likely options under block-wise biasing.Consistent with our methodological expectations, sequential facilitation (speed-up by first-order target repetition) was larger under block-wise than trial-wise biasing for both age groups, and it was larger and more reliable on trials involving repetition of more likely options.This repetition effect was nevertheless smallerand even absent under trial-wise biasing-among younger relative to older participants.

LBA Modelling
Figure 5 illustrates the LBA model, with one accumulator for orange responses and one accumulator for blue responses.Each accumulator can have different parameters, but we begin by describing the simplified case where they are the same.On each trial the starting evidence level is sampled independently for each accumulator from a uniform distribution on 0-A; A  0 is called the startpoint noise parameter.Evidence accrues linearly (the slanting dashed lines in Figure 5); the first accumulator to reach its threshold (b) determines the response (orange in this case); the time taken to reach the boundary represents the decision time (td, Figure 5).The total response time is then calculated as td plus non-decision time (t0), where t0 constitutes stimulus encoding and response production processes.We report threshold results in terms of B = b -A, enforcing b  A (i.e., accumulation cannot begin above the response threshold) by requiring B  0. We report response caution in terms of the average per bias-type of the B values that were estimated for each combination of blue/orange response and cue.
Figure 5.An LBA model that incorrectly makes an orange response when deciding on a blue stimulus.Available at https://tinyurl.com/ybpgwn84under a Creative Commons CC-BY license, https://creativecommons.org/licenses/by/2.0/The stimulus that is presented on a particular trial determines accumulation rates, which are sampled independently for each accumulator on each trial from positive (i.e., truncated below at zero, Heathcote & Love, 2012) Gaussian distributions, one for the accumulator that matches the stimulus (blue in the case illustrated in Figure 5) and the other for the accumulator that mismatches the stimulus, with each distribution having mean v and standard deviation sv.When accuracy is above chance, v is greater for the matching accumulator than the mismatching accumulator and so sampled rates tend to be higher for the matching accumulator (e.g., the slanting dashed line is steeper for the blue than orange accumulator in Figure 5).One accumulator parameter has to be fixed to make the model identifiable (Donkin, Brown, & Heathcote, 2009); accordingly, we fixed sv = 1 for the mismatching accumulator and estimated it for the matching accumulator.
In Figure 5, an incorrect orange response is made to a blue stimulus, despite the higher rate of the blue accumulator in this example, because startpoint noise gives the orange accumulator a "head-start", which in this example outweighs the higher rate of the blue accumulator.If the threshold for the orange response had been set higher (for example, based on base-rate bias), the correct blue accumulator might reach threshold first (triggering a correct response) because its greater rate might have overcome the head-start for the orange accumulator.This illustrates the way in which threshold adjustment can trade off accuracy for speed (i.e., higher thresholds slow down responding but increase accuracy).Errors can also occur due to noise in rate sampling (i.e., the mismatching accumulator gets a greater rate than the matching accumulator).Such errors represent genuine misperceptions and cannot be ameliorated by an increased threshold.Critically for the current analysis, bias in the LBA has been conventionally assumed to be mediated by the relative levels of the thresholds for each accumulator.For example, a lower threshold for the blue than orange accumulator produces a bias towards blue responses, speeding blue responses and making them more accurate (e.g., in Figure 5, a sufficiently lowered blue threshold would result in a correct response).
However, in the current experiment, we also investigated whether bias was mediated by higher rates for the more likely option (producing faster responses), and/or a larger difference favouring the more likely option between matching and mismatching rates (producing greater accuracy).

Model selection
Model-selection methods were used to determine which model parameters were influenced by experimental factors.Candidate models were constructed where bias (i.e., cuecongruence) could affect (1) only thresholds (i.e., B varied with congruence of cue and accumulator); (2) only mean accumulation rate (i.e., v varied with congruence of cue and stimulus); or (3) both B and v.In all models both B and v were allowed to vary with biastype.Mean accumulation rate was also allowed to vary with discrimination difficulty and matching versus mismatching accumulator.Difficulty was not set to affect thresholds (given its unpredictability) or non-decision time (t0) (given that it does not affect this parameter in this task; Voss et al., 2004).Further models also explored the effect of bias-type on t0, startpoint variability (A) and/or rate variability (sv) for the matching accumulator; otherwise, estimates for the latter two parameters were assumed to be the same across all within-subject conditions.Because older and younger groups were fit separately, all parameters were allowed to vary with age-group in all model fits.
Model comparison (as detailed in Supplemental Materials) strongly supported the bias-on-threshold-only model relative to the bias-on-rate-only model, and the bias-onthreshold-and-rate model produced implausible parameters estimates.Hence, we focus on the bias-on-threshold-only model in further analyses.Allowing bias-type to affect startpoint variability and non-decision time improved model predictions.For this model, Figures 2-4 present posterior-predictive fits of this model.These demonstrate generally good fits that captured all the trends evident in the data.Some misfit was evident for slower and incorrect responses, particularly those of older participants, as has been observed in other studies, and as naturally occurs at ends of the distribution that are subject to less reliable decision processing and produce too few observations to appropriately constrain estimation (e.g., Ratcliff, Thapar, & McKoon, 2006;Thapar et al., 2003).This was confirmed by measuring goodness-of-fit in terms of root mean square deviations (see Supplemental Materials for details).These indicated a typical range of error of .02s-.05s for RTs, and less than 20% for ERs; better estimation was observed for older than younger participant correct RTs and ERs, as well as for error RTs when taking account of variability in the observed data.

Parameter tests
We report parameter estimates, and differences in parameter estimates (denoted d), as the medians of posterior samples averaged over participants along with 95% credible intervals (CIs).We used Bayesian p-values (e.g., Klauer, 2010) to perform fixed-effects inferences about differences in these average parameter estimates 1 between conditions and groups, and 95% CIs to quantify uncertainty about the values of parameters and differences between parameters.The p-values correspond to tail areas in the distributions of differences between posterior parameter estimates from different conditions or groups.The 95% CIs correspond to the locations of the 2.5 and 97.5 percentiles of the parameter or difference distributions.These values are directly interpretable as the probability that the true difference falls in the tail (e.g., the probability that it is greater or less than zero) or in the interval.For ease of interpretation we report p-values corresponding to the smallest of the areas above or below a difference of zero.For example, a small p-value for a positive average difference indicates the probability that the difference is negative is correspondingly small.Because they are of primary interest, we present details of the analysis of threshold parameters below but only summaries of the results for other parameters, with details provided in Supplemental Materials.
Thresholds.Table 1 shows that we replicated many previous findings of older participants responding with greater caution, as compared with younger participants.This was true for both block-wise and trial-wise biasing.For older participants, the average threshold under trial-wise biasing was slightly larger than under block-wise biasing [d = 0.07 (0.01, 0.13), p = .008].For younger participants, there was no difference in average thresholds between bias-types [d = 0.01 (-0.02, 0.04), p = .259].Table 2 shows that the manipulation of bias was successful, with thresholds for the cue-congruent accumulator having been set lower than for the cue-incongruent accumulator.
Furthermore, the difference in thresholds between cue-incongruent and cue-congruent accumulators was greater for older as compared to younger participants under both biastypes.Between bias-types, the difference between cue-incongruent and cue-congruent thresholds (right-most column of Threshold change can also be measured as a proportion of the overall level of caution.
We defined proportional bias as the threshold for the cue-congruent accumulator divided by the sum of thresholds for both accumulators: In this way, values less than 0.5 indicate a successful bias manipulation.Consistent with the absolute bias effects in Table 2, relative bias was less than 0.5 in every case (ps < .001).
Accumulation rates.Performance in terms of rates was better for older than younger participants.Matching rates were higher for older than younger participants for both levels of difficulty and bias-type.The difference between matching and mismatching rates (an index of the quality of stimulus evaluation) was larger for older than younger participants in all conditions.We also calculated a measure of sensitivity that considers the effect of rate variability.Sensitivity was better for the older than younger participants for all conditions except for hard trials under trial-wise biasing.As expected, the difference between matching and mismatching rates was greater for easy than for hard trials in all cases, and this effect was larger for older than younger participants under both bias-types.Sensitivity was always better for easy than for hard trials, and the sensitivity advantage for easy trials was larger for older than younger participants under both bias-types.
Startpoint noise (A) was larger for older than younger participants under both biastypes.For older participants, startpoint noise was larger under trial-wise than block-wise biasing, whereas for younger participants, startpoint noise was larger under block-wise than trial-wise biasing.
Non-decision (t0) was longer for older than younger participants under both bias-types.
For older participants, non-decision time was longer under block-wise than trial-wise biasing whereas, for younger participants, it was slightly longer under trial-wise than block-wise biasing.
Almost all of these results were replicated in an analysis of the data using the DDM: The corresponding bias-on-threshold-only model fitted the data better than a corresponding bias-on-rate-only model, and older participants were more cautious overall, were additionally cautious under trial-wise biasing, and the block-wise biasing effect was greater among older than younger participants (see Supplemental Materials for details).However, the overall goodness-of-fit, and the parameter differences, were weaker for the DDM than the LBA model.Consistent with other DDM analyses (e.g., Dirk et al., 2017;Ratcliff et al., 2001;Ratcliff et al., 2006;Thapar et al., 2003), there was no age-related difference in startpoint the extent that older participants achieved greater accuracy than younger participants, further improvement (by more optimal bias setting) would be more difficult to attain.If older and younger participants had the same optimal bias curves (i.e., optimal bias as a function of caution), then given that they had the same proportional bias values, the advantage for the older participants might just have been due to them having a higher level of caution, and hence a bias curve that was closer to asymptote.However, as indicated in Figure 6, the bias curves differed between bias-types and age groups due to differences in the nonthreshold parameters (see also Figure 2 in Supplemental Materials).In particular, the asymptotic optimal bias was closer to 0.5 for older participants, but the bias curve for younger participants approached asymptote more quickly.This raises the question as to how each non-threshold parameter affected the bias curves so as to produce the optimality advantage for older participants.The results of an analysis reported in Supplemental Materials indicate that larger overall rates and larger differences between matching and mismatching rates improved the optimality of bias for the older group by increasing their asymptotic bias.However, the asymptotic optimal bias values given in Table 3 indicate that this difference in rates explained only a relatively small portion of the bias advantage for older participants.The older group was disadvantaged by the slower approach to asymptote, and a shift to the right, of their bias curves, as caused by higher levels of startpoint noise, and it appears that the particularly low level of startpoint noise for younger participants in the trial-wise condition was responsible for their gain in optimality relative to older participants.However, under both block-wise and trial-wise biasing, having greater startpoint noise was more than overcome by older participants having a higher level of caution, and hence their bias curves being closer to asymptote.
The DDM analysis replicated most of these results (see Supplemental Materials for details), including the findings that (1) both groups were above the optimal curve, and so could, theoretically, further bias their responses toward cue-congruent targets so as to increase their accuracy, and that (2) the difference between observed and optimal bias was smaller for older than younger participants under block-wise biasing.However, the latter difference was not as strong as for the LBA, and under trial-wise biasing there was a weak tendency for younger participants to be closer to optimal than older participants.

Discussion
We examined differences between older and younger participants in the way they used base-rate information when making simple perceptual decisions.Recent studies of agerelated increase in overall response thresholds suggested that-distinct to earlier theorizing about response caution as a preferential and altogether compensatory factor in age-related slowing-there are increasing constraints with age on the ability and neural substrates to flexibly adjust response thresholds in relation to speed/accuracy goals.Our study was designed to examine how this might constrain adjustment of response-specific thresholds to cued base-rate biases, and so to selectively increase speed and accuracy for the more likely target, while minimising the possible role of non-threshold factors in the biasing process.We used the LBA model (Brown & Heathcote, 2008) to measure how much and how well evidence thresholds were set for more and less likely options, benchmarking participants' performance relative to threshold settings that optimize accuracy.
In both age groups, our LBA-based analysis found base-rate neglect, but less so in the older group, so that they were closer to optimal in their use of bias to maximise accuracy.
This occurred despite older participants having to make larger absolute adjustments in the threshold amount of evidence required to trigger a response (given that they are more cautious in making decisions).These results were largely replicated with the DDM, albeit with a reduced difference in optimality between the groups.Given that the LBA fit provided a better tradeoff between fit and complexity than the DDM, we have focused on the LBA results.However, an appropriately cautious conclusion acknowledging uncertainty related to the choice of model is that older participants are at least as close to optimal as younger participants.
Our results are perhaps surprising in light of findings of reduced ability of threshold setting with age, and of reduced white matter integrity with age in the fronto-striatal tracts thought to mediate threshold setting-factors that could potentially make it more difficult for older adults to flexibly adapt their evidence thresholds to base-rate biases.We did find evidence of reduced flexibility in our older adults in that their advantage relative to younger adults was reduced when base-rate direction varied from trial to trial, and so when rapid threshold-adjustment was required.This condition could also be expected to reduce sequential facilitation as-unlike under block-wise biasing-integrating trial history into the decision process could compromise optimal use of immediate target cues; but, under trialwise biasing, older participants continued to show sequential facilitation (in association with cue validity), whereas (replicating LaBerge et al., 1970) it fell away among younger participants.Consistent with reduced control of thresholds but persistent integration of trial history, older participants had greater noise in the startpoints of evidence accumulation (replicating Forstmann et al., 2011), especially under trial-wise biasing; and this factor largely explained how younger participants could make up some of their optimality deficit relative to older participants under trial-wise biasing.A more direct test would be afforded by a model-based neuroscience approach (Forstmann & Wagenmakers, 2015) that assesses the association between white-matter integrity in relevant cortico-striatal tracts and differences in bias optimality when base-rate information is provided in a block-wise versus trial-wise manner, while also studying how older adults can persist in using sequential information when it compromises optimality.
Our older participants displayed greater processing efficiency in terms of both the overall rate of evidence accumulation and the quality of the information they obtained from the stimuli as measured by their ability to discriminate between correct and incorrect responses.As described above, this result was not unexpected given that loss of efficiency, in terms of lower evidence accumulation rates, is only found for tasks with strong perceptual and memory retrieval demands.Forstmann et al. (2011) also found an advantage for older participants in terms of overall rates but not in terms of discrimination.It is possible that these findings were due to greater motivation, and consequently attentional focus, among our older than younger participants, but it is important to note that this factor did not explain their more optimal bias settings.Rather, it appears that this was associated with their higher level of caution, because, at these higher levels, less bias is required in a relative sense to achieve optimality.In this way, our results are not inconsistent with evidence that, under some conditions, efficiency is lost with age, or even that the effect of bias can be mediated by accumulation rates.A more direct test of the role of rate-related factors would be to use a choice task where older adults are known to have reduced processing efficiency, such as discrimination of fine visual detail (e.g., Thapar et al., 2003), or demanding memory or cognitive control tasks (e.g., Karayanidis et al., 2011, Ratcliff et al., 2011).Under such conditions, it might also be seen that accumulation rate has a role in the effect of base-rate bias (as per Dunovan & Wheeler, 2018), and, because this could involve a fronto-parietal pathway (Mulder et al., 2012), the concept of compensatory neural (parietal) recruitment with age (Grady, 2008) could become relevant to interpretation of age-related bias effects.
A question that arises from our results is why all participants were less biased by baserate information than is optimal.One possibility is that our definition of optimality is too narrow as it only takes account of accuracy, unlike the reward-rate definition that has been used to assess the optimality of caution setting (Bogacz et al., 2006).However, any criterion that values speed must predict bias that is greater than optimal, because that results in faster responses for the more likely stimulus, and so faster responses overall.A second possibility is that participants simply lacked sufficient experience, and that they would have become more optimal with practice.Practice has been found to improve the reward-rate optimality of caution in younger but not older participants (Starns & Ratcliff, 2010).Future experiments might examine how practice, and appropriate guidance (Evans & Brown, 2017), affects the superiority of older participants in terms of bias optimality.
It is also possible that participants were acting optimally, but relative to a base-rate estimate that differed from the true base rate.This possibility was suggested by Hanks et al. (2011), who also found that participants were less biased than optimal and suggested an explanation in terms of the idea put forward by Good (1983) of a "prior on a prior probability".The idea here is that participants treat information about base rates in a Bayesian manner, integrating it with their prior beliefs in order to come up with a posterior estimate that is then used as a prior in the task.2On this logic, by finding the base rate that results in the optimal bias that matches the observed bias, we can infer that older participants had an effective base rate of 0.62 under block-wise bias and 0.56 under trial-wise bias, whereas younger participants used a base rate of 0.55 under both bias-types.
More generally, Good's (1983) explanation is related to findings in high-level decision making that probability information is not treated veridically (Tversky & Kahneman 1992).
This link also suggests it would be interesting to contrast the manipulation we used, where base-rate information is given by description, with a manipulation where it is learned from experience.In higher-level decision making there is a gap between the effects of description and experience (Hertwig & Erev, 2009), so it is possible that a different pattern of results may be found in the present paradigm with an experience manipulation.
Finally, the superior performance of older relative to younger adults could have applications in the detection of cognitive decline from conditions such as dementia.Because most cognitive abilities diminish gradually during the latter part of life in a healthy population, it can be difficult to detect the acceleration caused by disease processes.For disease processes that impact cognitive control, and in particular the control of thresholds, the preserved and even enhanced ability of healthy older adults to take account of base-rate information suggests that measuring bias optimality may provide a sensitive early-onset test.

Figure 1 .
Figure 1.Trial sequence and stimulus difficulty samples.Events within each trial are shown in their temporal sequence from left to right.The duration of each event is shown below each event image (in ms), with the stimulus presentation remaining on-screen until a response was made, up to a maximum of 2s.See the text for definition of each event and its elements.

Figure 2 .
Figure 2. Observed and LBA-predicted 10th, 50th and 90th RT percentiles (corresponding to lower, middle and upper lines respectively) for older participants between cue-incongruent (low probability) and cue-congruent (high probability) targets per bias-type, discrimination difficulty and response accuracy.Open and line-joined circles represent observed data.Filled circles with 95% credible intervals represent LBA predictions.

Figure 3 .
Figure 3. Observed and LBA-predicted 10 th , 50 th and 90 th RT percentiles (corresponding to lower, middle and upper lines respectively) for younger participants between cue-incongruent (low probability) and cue-congruent (high probability) targets per bias-type, discrimination difficulty and response accuracy.Open and line-joined circles represent observed data.Filled circles with 95% credible intervals represent LBA predictions.

Figure 4 .
Figure 4. Observed and LBA-predicted error rates (ER) for older and younger participants per bias-type and discrimination difficulty.Open and line-joined circles represent observed data.

Figure 6 .
Figure 6.Optimal bias curves (dark lines), asymptotic optimal bias levels (grey horizontal lines), and observed bias under block-wise (B) and trial-wise (T) biasing for older and younger participants.

Table 1 .
Caution (Average B) Posterior Parameter Estimates: Medians of means over cue conditions, accumulators and participants (and 95% credible intervals).
Table2) was greater for block-wise than for trial-wise < .001]participants.Although, as these descriptives indicate, there was a slightly greater effect of block-wise relative to trial-wise biasing on the amount of threshold bias among older participants, this age difference was not itself reliable [d = 0.02 (-0.01, 0.06), p = .098].