**Abstract**:

**Abstract**
**In the perceptual sciences, experimenters study the causal mechanisms of perceptual systems by probing observers with carefully constructed stimuli. It has long been known, however, that perceptual decisions are not only determined by the stimulus, but also by internal factors. Internal factors could lead to a statistical influence of previous stimuli and responses on the current trial, resulting in serial dependencies, which complicate the causal inference between stimulus and response. However, the majority of studies do not take serial dependencies into account, and it has been unclear how strongly they influence perceptual decisions. We hypothesize that one reason for this neglect is that there has been no reliable tool to quantify them and to correct for their effects. Here we develop a statistical method to detect, estimate, and correct for serial dependencies in behavioral data. We show that even trained psychophysical observers suffer from strong history dependence. A substantial fraction of the decision variance on difficult stimuli was independent of the stimulus but dependent on experimental history. We discuss the strong dependence of perceptual decisions on internal factors and its implications for correct data interpretation.**

*Nature Neuroscience*) as well as an established specialist journal for visual psychophysics (

*Journal of Vision*) and found that only 1 out of 54 “candidate” articles (Meier, Flister, & Reinagel, 2011) checked and corrected for intertrial dependence (for details, see Appendix A1). The vast majority of articles did not explicitly state that they assume trials to be independent but analyzed their data as if they were. Thus, despite considerable evidence for intertrial dependencies in experiments that were designed to explicitly study them, these results are rarely applied in practice. This is potentially problematic, as nonstimulus determinants could lead to both spurious correlations between behavior and measurements of neural activity as well as (downward) biases in estimates of psychophysical performance.

*magnitude*of trial-by-trial, nonstimulus determinants of perceptual decisions in the context of psychophysics with trained observers. Therefore, it is unknown what percentage of variance in typical psychophysical data is caused by the stimulus and what percentage can be attributed to task-irrelevant experimental history. Third, although some early studies reported to find a weak or no effect of intertrial dependence on psychophysical thresholds (Senders & Sowards, 1952; Verplanck & Blough, 1958), there have been no generally applicable and practical methods for quantifying and correcting such biases on a wide range of psychophysical data. Thus, it is still an open question how strong and problematic serial dependencies are across typical psychophysical tasks.

*r*to the presented stimulus intensity

_{t}*s˜*. We used “signed” stimulus intensities

_{t}*s˜*:=

_{t}*s*here, which consist of the product of the absolute intensity of the stimulus

_{t}z_{t}*s*and an identity factor

_{t}*z*, which codes when or where the target was presented. For example, in the 2AFC task (Jäkel & Wichmann, 2006) considered below, we use the stimulus identity

_{t}*z*= 1 to indicate that the second of two presented stimuli contained a luminance increment (“target”) and set

_{t}*r*= 1 if the observer also chose the second interval (

_{t}*z*= −1 or

_{t}*r*= −1 otherwise, see below and Appendix A2 for details). Choice models in psychophysics usually have a bias term

_{t}*δ*, which captures a stimulus-independent tendency of observers to choose a particular response. To model sequential dependencies, we simply assume that

*δ*is not constant but may shift dependent on experimental history (Treisman & Williams, 1984). This is in accordance with a large number of experimental findings (Hock, Kelso, & Schöner, 1993; Lages & Treisman, 1998, 2010; Lages &Treisman, 2010) and previous modeling attempts (Green et al., 1977; Ward, 1979; Green et al., 1980; Lockhead & King, 1983; Corrado et al., 2005; Busse et al., 2011; Bode et al., 2012; Goldfarb, Wong-Lin, Schwemmer, Leonard, & Holmes, 2012; Raviv et al., 2012). Concretely, we assume that the bias term

*δ*can be written as a linear combination of “history features,” i.e., summary statistics of the events on preceding trials (Corrado et al., 2005; Busse et al., 2011):

^{1}

*h*can be taken to be any feature of the recent history that might potentially influence behavioral responses. We say that a data set exhibits history dependence if, given the current stimulus, the current response is statistically dependent on previous stimuli and previous responses, that is,

_{kt}*P*(

*r*|

_{t}*s˜*,

_{t}**h**

*) ≠*

_{t}*P*(

*r*|

_{t}*s˜*).

_{t}**h**

*to be a concatenation of the last seven responses and stimulus identities, that is,*

_{kt}*h*= (

_{t}*r*

_{t}_{−1}, …,

*r*

_{t}_{−7},

*z*

_{t}_{−7}), a vector of dimensionality

*K*= 14.

*ω*in Equation 1 indicate how much the respective response/stimulus influences the current response. For example,

_{k}*ω*

_{1}> 0 indicates that the observer tended to repeat the previous response and

*ω*

_{1}< 0 that there was a tendency to switch responses. Our model captures covariations between the observer's responses and previous responses or stimuli. These covariations could increase the variance of the resulting responses in a block

^{2}but could also lead to a decrease in variance or even leave it unchanged.

*s˜*=

_{t}*z*is the signed stimulus intensity,

_{t}s_{t}*λ*and

*γ*describe the probabilities of stimulus-independent responses to the right (

*γ*) or to the left (

*λ*), and

*g*(

*x*) is a sigmoid function. In the following, we chose the logistic function

*g*(

*x*) = 1/(1 + exp(–

*x*)), and

*δ*′ and

*α*are the offset and slope of the stimulus-dependent part of the psychometric function. We note that an alternative view of the model is to assume that observers implicitly combine the stimulus with a trial-specific Bayesian prior assumption about whether the next target is in the first or second stimulus and that this prior probability

*P*(

*z*|

_{t}**h**

*) depends on the recent stimulus history (Yu & Cohen, 2008; Wilder, Jones, & Mozer, 2010).*

_{t}*α*to be different across conditions but assumed that the history couplings

*ω*were constant across conditions. As our model describes the probability of particular responses (not the probability of the response being correct), we need to introduce a sensory threshold

_{k}*ν*, which accounts for the fact that observers perform at chance level whenever the stimulus has an intensity less than some sensory threshold

*ν*. We use the nonlinear threshold function

*u*, which maps all stimuli with an intensity of less than

_{ν}*ν*to zero (for details see Appendix A2 and Figure A1).

*δ*′ that can capture potential within-trial biases that are unrelated to the experimental history (see, for example, Ulrich & Vorberg, 2009, and Garcia-Perez & Alcala-Quintana, 2011, for more detailed treatments of these effects). In some cases, these within-trial biases are also associated with differences in the observers' sensitivity. In that case, it would be possible to apply the model separately to trials of each presentation order, which is equivalent to the approach advocated in Garcia-Perez and Alcala-Quintana (2011). Our formulation could also be used to include additional covariates in

**h**

*, which describe the current trial and which, thus, could be used to capture more complex within-trial effects, but this is not pursued here.*

_{t}*slope*of the psychometric function. To model effects on slope, one could include features that are proportional to

*s*. For example, by including a feature of the form

_{t}*s*

_{t}z_{t}_{–1}, one could make the slope dependent on the position of the target in the previous trial.

*y*∈ {

*z, r*} and the

*b*=

_{kt′}*η*∈ {0,1/2,1/4}, which are sensitive to fluctuations at different time scales. Each of these filters was applied to the response sequence and to the stimulus sequence. As these basis functions are strongly correlated, they would result in strongly correlated history features, which can result in numerical problems when trying to identify their parameters from data. We, therefore, orthogonalized the basis functions with respect to each other, i.e., we ensured that they are of unit-norm and mutually orthogonal (Paninski, Pillow, & Simoncelli, 2004), resulting in new basis functions

_{k}**h**

*thus lives in a six-dimensional subspace with its first three components given by the projections of the previous stimulus identities onto the basis functions and the last three components given by the projections of the previous responses. Although our algorithm identifies the coefficients of these basis functions, we report the effective history filters, which can be reconstructed by multiplying the basis functions with their matching coefficients. We find the parameters*

_{t}*α*,

*δ*′,

*γ*,

*λ*, the sensory threshold

*ν*, and the history weights

*ω*by maximizing the log-likelihood of the data under the response probabilities predicted by the model

*L*= ∑

*log*

_{t}*P*(

*r*|

_{t}*s˜*,

_{t}**h**

*). This likelihood can have multiple local maxima, and there are multiple constraints on parameters (e.g., 0 ≤*

_{t}*λ*≤ 1 –

*γ*), which renders naive gradient-based approaches problematic. We, therefore, used the expectation maximization algorithm (Bishop, 2006), an iterative algorithm that is guaranteed to find a (local) optimum for mixture models (see Appendix A3 for details). Although this algorithm cannot guarantee convergence to a global optimum, we have found empirically that, by using a modest number of restarts combined with heuristic starting values, the algorithm typically converged to parameters that explained our data well and were close to the true parameters on simulations with known ground truth.

*L*= ∑

*log(*

_{t}*P*(

*r*|

_{t}*h*,

_{t}*s˜*)) across all trials indexed by

_{t}*t*. In each trial, the choice is influenced by the effect of the stimulus

*δ*

_{stim}(

*s*) =

_{t}*a*·

*u*(

_{v}*s˜*) as well as the effect of the history,

_{t}*δ*

_{hist}(

*h*) (Equation 2). We thus quantified the relative influence of the history as the ratio between the variance of the history influence and the sum of the variances, where Var

_{t}*indicates that the variance is determined across all trials*

_{t}*t*. This measure quantifies to what extent fluctuations in the internal decision variable can be attributed to history, and we, therefore, refer to this measure as “history contribution to variance in the decision variable.” This measure quantifies the

*relative*contribution of the stimulus and the experimental history to fluctuations of the decision variable but does not model additional noise in the decision process. Thus, if the stimulus intensity is 0, the internal variance of

*s˜*will be zero as well, and HistCont will be 100%. We excluded blocks with performance <55%.

*absolute*measure of the influence of history dependence on behavioral choice, we computed the accuracy of different models in predicting behavior: In every trial

*t*, the model provides two probabilities,

*P*(

*r*= 1) and

_{t}*P*(

*r*= −1). In order to quantify how well a model predicted behavioral choice, we said that the model predicted

_{t}*r*= 1 whenever

_{t}*P*(

*r*= 1) >

_{t}*P*(

*r*= −1), i.e., in which

_{t}*P*(

*r*= 1) > 0.5, and said that the model predicted

_{t}*r*= −1 otherwise. The percentage of correct predictions was calculated by counting the number of correct predictions. For the “history only” model, the parameter

_{t}*α*was set to zero. To correct the psychometric function for the effect of errors attributable to serial dependencies, we first fitted the full model (Equation 2) to data and then extracted a psychometric function by setting the history couplings

*ω*to zero. This model was then compared to a conventional psychometric function (Equation 1). Confidence intervals for history kernels were determined using a bootstrap procedure. After the values of the history features in each trial had been calculated, 2,000 bootstrap data sets of the same size were sampled from the data with replacement (Efron & Tibshirani, 1993). Confidence intervals were defined as the 2.5 and 97.5 percentiles of this distribution. As the full model has six more parameters than the history-free model, bootstrap samples cannot be used to evaluate the statistical significance of history features (as, on each bootstrap set, the full model would have a higher likelihood than the history-free one). We therefore used a permutation test: We permuted the sequence of trials randomly such that the history features could contain no information about the response but that the association between stimulus and response was left intact. Thus, the permuted data sets yielded an approximation of the distribution of likelihoods that would be expected under the null hypothesis of no serial dependence. We fit the model to 2,000 permutations of the original data sets and compared performance measures and other parameters to the 95th percentile of this permutation distribution. In cases in which the effect of history dependence is nonlinear, our model will capture a linear approximation to this nonlinear system and will detect the presence of history dependence provided that its linear term is not negligible. Of course, weak history dependence on small data sets might not reach the level of statistical significance and might therefore not be detected by the statistical tests described here.

*l*

_{full}= −911.7) was substantially larger than the one of the independent-trial model (

*l*

_{0}= −941.2, corrected for difference in parameters using Akaike's information criterion, AIC), and a permutation test against random trial sequences revealed that this performance benefit was significant at level

*p*< 0.0005 (Figure 1b, see section “Performance measures, correcting for history effects and statistical tests” for details; all

*p*values in the following are derived from this test unless stated otherwise).

*p*< 0.0005, Figure 1d). As expected, behavior was primarily explained by the stimulus on easy trials (for which performance >75% correct), on which 98% of the variance of the observer's decision variable was explained by the stimulus, and only 1.6% of the variance of the observer's decision variable was still history-dependent (

*p*< 0.0005, Figure 1e).

*p*< 0.0005, chance level is 50%, Figure 1g). Our analysis shows that, on difficult stimuli, which are of relevance for measuring psychophysical performance, perceptual decisions of this observer are almost as strongly influenced by internal decision variable fluctuations induced by experimental history as they are by the experimental stimulus. Nevertheless, the fact that the prediction power was far away from 100% implies that much of the variability in subjects' responses could not be accounted for by the stimulus or experimental history.

*p*< 0.03, Figure 1h).

*p*< 0.0005, average across conditions 97.2%, Figure 2c). Thus, the fact that conventional analyses cannot discount the errors induced by dependence on previous trials leads to a slight underestimation of the stimulus sensitivity of this observer.

*any*degradation in performance as stimulus-independent lapses in one direction and in the other direction would cancel perfectly. However, the psychometric function is a nonlinear model. We found that a history-induced

*standard deviation*of

*σ*= Std(

*δ*

_{hist}(

*h*)) leads to a quadratic reduction in the slope of the psychometric function, i.e., one that is proportional to the history-induced

_{t}*variance σ*

^{2}. Thus, for typical history dependence (

*σ*< 1), the reduction in the slope is even smaller (as

*σ*

^{2}<

*σ*). Furthermore, the proportionality factor of this relationship is also small, namely

*π*/16 ≈ 0.2. For example, for a standard deviation of

*σ*= 0.3, one would only expect a reduction of the slope by 0.09 ×

*π*/16 ≈1.18%.

*σ*= 0.52), this simplified analysis predicted a reduction of threshold by 5.3%, which slightly overestimates the empirically measured reduction of 4%. Across observers, the threshold changes predicted by this simplified model were correlated with those of the full model (c = 0.73) and did not differ in their mean level (paired samples

*t*test:

*t*(21) = 0.57,

*p*= 0.57).

*p*< 0.05) in 19 out of 22 observers, and modeling the nonstimulus determinants of the behavioral choices led to an average increase in log-likelihood of 0.009 ± 0.0020 per trial (

*SEM*across observers, Figure 3a, see Appendix A7 through A12 for detailed results of two further observers). In other words, a data set of 500 trials would be (on average) 77 times more likely under our model than under a conventional model assuming independent trials. Significant history dependence was found in all four experimental paradigms we investigated (see Figure 3a), and its strength varied considerably across observers.

*SEM*) of variance of the decision variable on difficult stimuli was determined by the experimental history (Figure 3b) and not by the presented stimulus with values for individual observers as high as 48.2%. Experimental history was a meaningful predictor of behavioral choices in individual trials. On average, the model based on previous trials predicted 56.5% ± 1.0% (

*SEM*, chance level 50%, significant for 16 out of 22 observers) of responses on difficult stimuli correctly, compared to 64.8% ± 0.5% for the stimulus and 65.5% ± 0.4% for the combined model (Figure 3c). Across observers, the prediction performance of the full model was significantly better than for a model that only contained stimulus terms (

*p*< 10

^{−3}, permutation test of paired differences). For one observer, the history was, in fact, a better predictor of the behavioral choice than the presented stimulus.

*shifts*in the psychometric functions and not changes in the slope. We fit four conventional psychometric functions separately to trials that followed a left response and trials that followed a right response as well as trials that followed a left stimulus and trials that followed a right stimulus. Differences in horizontal shifts were, on average, 2.3 times larger than differences in slopes (after normalizing with the estimates' standard errors). In 35 out of 44 cases, the effect of the previous trial on the horizontal position of the psychometric function was larger than the effect on the slope of the psychometric function (

*p*= 0.0001 binomial test).

*SEM*) per trial across all data sets (corrected using AIC,

*p*= 0.00001 binomial test).

*SEM*), reflecting a tendency to switch responses from trial to trial. Weights associated with previous stimuli were heterogeneous and, on average, a factor of 3.85 ± 1.98 times smaller than the response weights. In contrast, most observers from two masking experiments (plaid mask in the first experiment and a sine grating mask of the same spatial frequency and orientation in the second experiment) are near the upper left diagonal of this plot, reflecting the fact that these observers primarily changed their response criterion after errors, which is likely to be a consequence of the fact that they received trial-by-trial feedback. In contrast, subjects in a yes-no audio experiment had weak weights associated with the previous stimulus (average weight −0.11) and stronger weights associated with previous responses (average weight 0.33). Thus, these observers showed slow fluctuations of their decision variable, which manifests itself in a tendency to repeat their previous responses. Although the overall influence for longer trial lags was substantially weaker, a clear clustering of weights according to experimental paradigm was still evident (Figure 4d).

*direct*influence is negligible. Thus, with our method, we can disentangle serial dependence on previous responses from a dependence on previous stimuli or feedback.

*not*to the probability of a

*correct*response.) Yet it is possible that history has an influence on the slope of the psychometric function as well (see Lages & Treisman, 2010, for some indication of this). Our model is not able to capture these kinds of sequential dependencies, and thus, the history dependence reported here should be treated as a lower bound.

*individual*choices (Lau & Glimcher, 2005; Corrado & Doya, 2007; O'Doherty et al., 2007) and thus has the potential to reveal a rich source of information that had previously been buried by trial averaging. Given that time series of behavioral observations are ubiquitous in neuroscience and related fields, our methods will be applicable to a wide range of experimental or clinical paradigms that measure human or animal performance. Combined with methods for single-trial analyses for neurophysiological recordings (Churchland, Yu, Sahani, & Shenoy, 2007), they thus have the potential to contribute to a more realistic understanding of perceptual decision-making.

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^{1}Letting the bias term vary dynamically from trial to trial complicates the interpretation of the model in terms of a signal and a decision criterion as defined in signal-detection theory. We here restrict ourselves to finding a statistical description of history effects, and we do not attempt to disentangle whether they influence the criterion or the signal in the decision process.