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## Accuracy psychometric functions

The tasks that subjects perform in psychophysical experiments typically require to report whether a target is present or absent in a stimulus (detection task), or whether two stimuli presented during the trial are the same or different (discrimination task). Other discrimination tasks may require to assign stimuli to one of two (or more) categories. In all these tasks, the salience of sensory information (either, indicating the presence of the target, the presence of a change, or one of the stimulus categories) may be varied by the experimenter, and the probability that a subject answers correctly will depend on this salience. For an auditory detection task, for instance, salience depends, among other things, on the loudness of the target. For a discrimination task, the salience of a change between two stimuli depends on its amplitude, e.g. on the difference between the two tones’ frequencies in the case of pure tone frequency discrimination. The stimulus variable that, when manipulated, affects the salience of relevant information is often referred to as stimulus strength. When stimulus strength is infinitesimally small, the target stimulus or change cannot be perceived by the subject who is then expected to respond randomly, or according to some strategy in decision making (Abrahamyan et al., 2016). The probability that they give a correct response then depends on the number of response alternatives and may be referred to as chance level. On the opposite extreme, a very salient stimulus or change is unlikely to be missed, and the probability of a correct response is therefore close to 1 for very large stimulus strengths. The accuracy level associated with the threshold by definition is chosen so as to be intermediate between chance performance and maximal performance. For instance, in the case of a task with two possible responses, chance performance is at 50%-correct and the threshold may be chosen as corresponding to 75%-correct or nearby, possibly depending on the estimation method, as will be discussed later (2.1.1).

The relationship between success probability and stimulus strength is represented by the accuracy psychometric function. Apart from rare cases, the accuracy function is expected to be monotonic, the probability of a correct response increasing if stimulus strength is increased. This increase is generally continuous, unlike what would be assumed from the initial notion of threshold. However, the rapidity of the increase is generally unknown. A good representation of the psychometric function involves to choose an appropriate scale for the stimulus strengths on the horizontal axis, so that points corresponding to intermediate values of the correct response probability on the vertical axis take up a visually important part of the curve. In lots of cases, this is achieved by reporting on the horizontal axis the logarithm of the physical attribute of interest. E.g. in the case of tone frequency discrimination, correct response probability is generally reported as a function of = log(|Δ / |), where Δ is the difference between frequencies of the two tones to discriminate and is one of these frequencies or their average. In most cases, the resulting function has a sigmoid (i.e. S-like) shape (Figure 1.1, lower panels), the abscissa of the inflexion point being near the threshold of interest.

**Choice psychometric functions**

Some tasks require to assign a stimulus to one of two categories A and B (or more). For instance, in a frequency discrimination task, the subject may be presented with two successive pure tones with different frequencies 1 and 2 and have to report which one has the highest pitch, which amount to saying that the tone pair belongs to the “up” (or upward) category and the other to the “down” (or downward) category. This applies to our sliding 2-AFC task in which each tone of a continuous series is judged as higher or lower than the previous (Arzounian et al., 2017). In these cases, it is useful to describe the responses of the subject in terms of choice, rather than accuracy. In general, the probability of choosing one alternative versus the other is expected to vary monotonically with some attribute of the stimuli that are being manipulated by the experimenter. In the case of the pitch discrimination tasks, relevant variables can be chosen as the frequency difference = Δ = 2 − 1 or as the log interval = log( 2/ 1). In both cases, the quantity is negative when 2 < 1 and positive when 1 < 2. Consistently, the probability that the subject reports the second tone as being higher is low for extremely negative values of and increases when increases to zero and beyond.

It is common to define the point of subjective equality (PSE) as the value of the variable associated with equal probabilities of choosing A or B. In some cases, the PSE can be compared to some point of objective equality, meaning that the experimenter has an objective criterion for labeling a response as being correct or incorrect. In these cases it is in general possible to define as a variable whose sign depends on the correct response category, like the two variables proposed for the pitch discrimination task. The difficulty of the task is then linked to the absolute value of : if | | is large, the trial is easy, and the listener will give the correct answer with a large probability; if | | is small, the trial is more difficult, and the listener’s response is less certain. The PSE will be positioned at = 0 if the listener is unbiased. It can be shifted to the left or to the right if the listener has a choice bias, i.e. a tendency to report more often one category or the other regardless of the stimuli presented. In other situations, the category of the stimuli is by nature ambiguous and it is not possible to set an objective criterion regarding response accuracy; there is in fact no correct response.

For instance, certain types of artificial sounds, like Shepard tones, contain a superposition of several pure tones spaced by octaves, so that their pitch is ambiguous (Chambers et al., 2017; Chambers and Pressnitzer, 2014). Deciding which of two Shepard tones with different frequency contents is higher in pitch depends on how one binds their frequency components.

The function that relates the probability of choosing one alternative, e.g. A, to the value of can be called a choice psychometric function. A typical psychometric function has the shape of a sigmoid function that goes from 0 for extreme negative values of to 1 for extreme positive values of (Figure 1.1, upper panels). The slope of the curve reflects the subject’s sensitivity: if the subject is able to correctly discriminate stimuli even when they are close to the PSE, the curve will be steep; if not, the curve will be shallower.

### Relationship between the accuracy and choice psychometric functions in the case of 2 response alternatives

The accuracy and choice psychometric functions differ in several respects. First, the accuracy function can only be defined when there is a way to objectively label a choice as being correct or incorrect. This is the case of the sliding 2-AFC pitch discrimination task we will consider in this thesis, the objectively correct report being an upward pitch change when the frequency of the last tone is higher than that of the previous tone and vice-versa. Second, these functions take as inputs different variables: a signed variable that can be both positive and negative in the case of the choice function and a positive variable signifying “stimulus strength” in the case of the accuracy function. In many cases, data from the same task can be analyzed with either psychometric function. In fact, for any choice function where one stimulus category corresponds to > 0 and the other to < 0 , the most straightforward accuracy function will relate the success probability to | |. However, the best approach is not necessarily to define = | |. Indeed, commonly used accuracy functions typically differ from choice functions in the scale chosen for the horizontal axis, so that it is more common to define as being a logarithmic transform of | |, either log(| |).

**Experimental estimation of psychometric functions and thresholds**

Psychometric functions can be estimated empirically by reporting the proportion of one type of responses (correct responses in the case of the accuracy function, choice A responses in the case of the choice function) observed for several values of the relevant stimulus variable sampled along the horizontal axis. The precision of the estimated response probabilities will depend on the number of stimuli per sampled value that were presented in the experiment.

If knowing the actual entire shape of the accuracy psychometric function, or even just its slope, might be of interest to assess some mechanisms of perceptual decision making (Gold and Ding, 2013), it is most of the time not essential in order to get a useful indication of performance in the task and of sensitivity. In particular, the values of performance when approaching the lower or upper asymptote are not very informative about the listener’s sensitivity. In contrast, the steep part of the function is of much interest and the threshold to measure is typically located near the inflection point of the curve. Adaptive procedures were therefore developed to speed up the estimation of the threshold by targeting stimulus strength values that have a high chance to be in the vicinity of this threshold, according to the responses made by the listener in earlier trials (Leek, 2001; Macmillan and Creelman, 2005, chap. 11; Treutwein, 1995). These methods will be discussed in more details in a future chapter (see 2.1.1).

To extract features of interest from these sampled, estimated psychometric functions, like the subject’s PSE, sensitivity or threshold, some analyses rely on specific assumptions regarding the shape of the function. They assume the mathematical function ( or ℎ ) belongs to some parametric family. Fitting such a function to the observed behavior allows to make use of the entire dataset in a more efficient way, increasing the reliability of the probability estimates. Reliable estimation of the many parameters implicit in a psychometric function sampled at discrete values requires large number of trials. A parametric function involves fewer parameters, making it easier to derive a meaningful fit. Several parametric families have been proposed both for the choice and accuracy functions, among which logistic functions, cumulative normal distribution functions, and cumulative Weibull distribution functions (Gold and Ding, 2013; Klein, 2001; Macmillan and Creelman, 2005, chap. 11). Note that despite the theoretical link between the choice and accuracy functions, they tend to be modeled independently, the analyses generally being based on either the one or the other. A common feature of the previous families of parametric functions is that they have two main parameters that determine (1) the position of the function center along the (horizontal) axis and (2) the slope of the curve. The position of the curve relates to the threshold in the case of the accuracy function, to the PSE in the case of the choice function. The slope relates to sensitivity in the case of the choice function; it characterizes how fast accuracy varies with the absolute stimulus strength in the case of the accuracy function. The reliability of measured thresholds varies inversely with the slope of the accuracy function. In both cases, curve fitting methods allow to determine the set of function parameters that leads to the best match to the experimental results (Klein, 2001; Wichmann and Hill, 2001). These optimized parameters may then yield estimates of the corresponding features.

The next section will expose why, in the rest of this dissertation, choice functions will be assumed to belong to the family of cumulative normal distribution functions. The underlying model will be explained and the form of the theoretically associated accuracy function will be described.

#### Signal Detection Theory

**Modeling of the decision process**

Signal Detection Theory (SDT; Green and Swets, 1966; Macmillan and Creelman, 2005) provides a model to account for the variability of subjects’ responses to stimuli, interpret them in terms of sensitivity, and predict the shape of the psychometric function. In this model, the responses of a subject are assumed to derive from an internal variable encoding the relevant stimulus attribute . This internal variable is affected both by the presented stimulus and by internal factors, so that repetitions of the same stimulus may lead every time to a different value of the internal variable. It is assumed that the value of varies along a one-dimensional axis and results from the summation of a deterministic term given by some transduction function of and some random variable called internal noise.

The subject is assumed to split the internal representation axis into intervals, each assigned to one of the stimulus categories. If there are two categories A and B for instance, this means the subject compares to some boundary criterion and decides, e.g., that the stimulus belongs to A if > and that it belongs to B if < . If the task is to detect the presence of a target, one category may be “target absent” and the other “target present”. Some tasks require to compare two internal representations associated to different stimuli. In the pitch discrimination task for instance, each decision relies on the comparison of the internal representations of two tones’ pitches. The situation is however comparable to the previous if we consider that the relevant internal representation is that of the difference between the stimulus pitches. The noise that disturbs the internal representation of this difference may result from the summation of the noise in their individual representations and, possibly, of processing noise that adds when forming a representation of the distance between the stimuli.

For a fixed stimulus (or stimulus difference), i.e. for a fixed value of , the statistics of the subject’s choices across trials are determined by the statistical distribution of the internal variable, i.e. by the distribution of the internal noise . The psychometric function describes how response statistics vary in more complex situations where varies along a continuum and each category may contain a variety of stimuli. To predict the shape of the psychometric function from the SDT model, one needs to know how the center of the distribution of the internal representation is displaced when the physical attribute of the stimulus is changed. In other words, one needs a model for the transduction process that maps the physical attribute to a location on the axis of the internal representation, or psychological dimension (Macmillan and Creelman, 2005, chap. 11). This process is described by the function .

**Table of contents :**

**Introduction **

**Part I. Characterization of sensory variability**

1.1 Psychometric functions

1.2 Signal Detection Theory

2.1 Proposed method for threshold tracking

2.2 Assessment of tracking performance

2.3 Proof of concept: psychophysical experiment

2.4 Optimization of the method: simulation study

2.5 Threshold tracking in a real experiment

2.6 General discussion on threshold tracking

3.1 Introduction

3.2 Methods

3.3 Results

3.4 Discussion

3.5 Summary

**Part II. Relating sensory variability to brain state fluctuations **

4.1 Auto-predictive patterns in EEG signals

4.2 Brain oscillations and brain state

5.1 Literature about the modulations of perception by brain states

5.2 Modulation of auditory perception by ongoing brain oscillations?

5.3 Predicting threshold variations from ongoing EEG

**General discussion **

**References**

Appendices

Appendix A. A sliding two-alternative forced-choice paradigm for pitch discrimination

Appendix B. Explanation for reduced variance of weighted-mean threshold estimators when independent tracks are interleaved

Appendix C. Statistics for nested model comparisons