1.1 Fixed Clock-speed Pacemaker-accumulators

We now consider two different types of timing model and ways to simulate them, thereby producing fake rt data. We can then apply the rt analysis methods described in the final section of this chapter to test whether these methods can properly determine which model best fits the fake data. This is an important step to undertake before applying the methods to real rt data from an experiment. If these methods cannot accurately determine which model generated the data even when we already know the answer, then they will not be useful in the real world. For this reason, model simulation should always be the first step in vetting a model-fitting procedure. We will restrict our attention in this chapter to the most basic and most classical type of timing model: the pa.

pas are just about the simplest possible model of timing. They operate like a stopwatch, which counts up regular or irregular clock ticks until hitting some threshold either for making a decision (such as, whether the current interval is longer or shorter than another interval in memory) or for producing a response (such as pressing a lever to earn a reinforcement). We will refer to these two possibilities as decision and production tasks respectively. There are many possible variations on the basic pa idea, however, and different variations make different predictions, especially about production times. rt analysis techniques will therefore be useful in teasing these models apart.

Creelman (1962) developed an early pa model in which a source of clock pulses emits pulses randomly, at a fixed rate. The time between pulses was exponentially distributed, making the process of counting them up into a Poisson counting model. Since this basic process is central to several of the models for which we provide Matlab code, we begin with it. Poisson_countermodel.m contains the code in its entirety (see book’s GitHub repository). It generates exponentially distributed random inter-pulse intervals, then adds them up, and checks when their sum has exceeded a threshold.

An important variation on this model allows for changes in pacemaker rate across trials, as well as variability in the pulse count threshold across trials. Gibbon and Church (1990) considered how these variations could account for a problem with Creelman’s model, which is its conflict with a widely observed phenomenon in timing known as ‘scalar invariance’. This benchmark phenomenon in interval timing research is one in which the standard deviation of remembered interval durations appears to grow linearly with the timed duration. This pattern yields a constant ratio of the standard deviation divided by the mean – a ratio known as the coefficient of variation (cv). Indeed, in production tasks, the production time distributions for different durations frequently superimpose when the data from different duration-conditions are divided by their means. Creelman’s model, in contrast, predicts decreasing cvs, and, thus, increased relative precision, as durations increase. Gibbon (1992) referred to this pattern as ‘Poisson timing’ (though as we shall see below, a Poisson timer that works on different principles yields perfect scalar invariance).

Treisman (1963) developed an early variant of this pa model that did not make specific assumptions about the distribution of clock pulses except for a key constraint on their statistics. The constraint is that the inter-pulse intervals are mostly relatively short in some trials, compared to the global average over trials, and mostly relatively long in others. That is, the clock speed varies from trial to trial (but always around a fixed average). Another way to say it is that, across trials, the ith inter-pulse interval following the clock-start is correlated with the jth inter-pulse interval in the same trial, for all intervals i and j occurring prior to the end of the timed duration. Simulations of this model demonstrate that it takes only a remarkably small amount of such correlation (i.e., variation in the pacemaker rate) across trials to recover the pattern of constant cvs. Treisman63.m contains code (see book’s GitHub repository) that allows any desired distribution of inter-pulse intervals to be simulated while still observing the constraints. Part of what the code produces is a correlation matrix for selecting random inter-pulse intervals with the specified level of correlations. As users can see for themselves, correlations can be reduced nearly to 0 in a Creelman-style pa model but still obtain constant cvs.

1.2 Variable Clock-speed Pacemaker-accumulators

The models in the preceding section used a pacemaker that emits random pulses at a constant rate on average, and stores different pulse-count totals to encode different durations T. In contrast, the models in this section use a fixed pulse-count threshold (call it θ) and a variable pacemaker rate, A = θ/T. When the pulses are purely excitatory, the model is equivalent to Killeen and Fetterman’s (1988) Behavioral Theory of Timing (BeT) model (although the “pulses” in that case are interpreted as transitions between states of behavior). When we add negative pulses emitted at a rate proportional to the positive pulse emission, the resulting process closely approximates a process known as a diffusion process (Simen et al., 2013). Such processes are idealized, one-dimensional Brownian motion systems, in which a particle (equivalent to a pulse-count) drifts upward while being continuously perturbed by noise at every moment. This kind of perturbation (in three dimensions) accounts for the diffusion of particles in a liquid or gas over time. One-dimensional drift-diffusion models (ddms) are leading models of two-alternative perceptual decision making, where the “particle position” represents something like the log odds ratio of one hypothesis vs. the alternative. In either case, BeT or the time-adaptive, opponent Poisson ddm (TopDDM) of Simen et al. (2011, 2013), scalar invariance falls out of the models automatically. The code in opponent_Poisson_Appendix3.m explicitly compares the simulation of actual pulse-counting to a diffusion approximation of pulse-counting (specifically, the Euler-Maruyama method, in which spike counting per se does not occur). The primary computational difference between these approaches is that in the former, a sequence of pulse-times is generated and then summed; in the latter, properly scaled deterministic and random noise increments are added to a running sum at every one of a sequence of time steps. The primary theoretical difference is that the pulses are finite in number and spaced out at separate points in time, in the former, while they are idealized as occurring infinitely often at every moment in the latter. Since the latter theoretical assumption of the diffusion model is not likely to be true in reality, it is worth noting that this assumption allows some very simple approximate mathematical expressions to be used to fit rt distributions.

2.1 Using Moments to Characterize the Data and Evaluate Conformance to the Scalar Invariance Pattern

cvs are often found to be roughly constant as a function of T across groups of participants. Violations of cv constancy are often observed if multiple durations T are mixed in a single experiment, with both human and non-human participants (Bizo, Chu, Sanabria, & Killeen, 2006).

2.2 Skewness and Model Selection

Skewness, the third standardized moment, captures how symmetric a distribution is. Gaussian distributions (a shape that is often used to fit production time data) have zero skewness. Positive skewness, in which the mean is greater than the mode, and the tail on the right is heavier than on the left, is widely observed in two-alternative perceptual decision rts, but is less often observed in timing tasks, which typically yield skewness levels closer to 0. Nevertheless, skewness may provide a fairly robust test of the predictions of a timing model, because models that predict a Gaussian distribution of rts can be tested by verifying whether skewness equals 0. Simple Poisson counter models, such as BeT, predict skewness equal to twice the cv. Diffusion models such as TopDDM predict skewness equal to three times the cv (see Simen et al., 2011, 2013, for discussion of proofs; these properties can be verified by running the model simulation code provided in the GitHub).

It should be noted that when the cv is small, skewness is, therefore, also expected to be small. Thus, if conditions can be created that generate large cvs, such as by driving attentional resources away from the primary task, then skewness should be expected to increase for models such as BeT and ddm, but not for models that predict a normal distribution of rts, such as the information processing implementation (Gibbon, Church, & Meck, 1984) of scalar expectancy theory (set; Gibbon 1977).

In summary, skewness is worth computing as it helps in model selection. Matlab has a built-in skewness function that can be used in this way. The Matlab Statistics Toolbox also includes helpful functions for assessing normality or deviations from normality in a set of rt data in variable rt (e.g., normplot(rt)).

2.3 Quantiles and Timescale Invariance

In Section 2.1, we have discussed how the cv tends to remain the same for different base intervals. Another famous feature of timing data that implies constant cvs, but is a stronger form of invariance, is called scalar invariance or timescale invariance. This is a phenomenon in which the entire distribution of rts can be rescaled, through division by the mean, so that the shape of the rt distributions with different means superimpose on each other perfectly after rescaling.

A good test of this form of invariance is to examine a feature of the rt data that is widely used in 2AFC research. To obtain a compact empirical description of the distribution, the rts are divided into quantiles. For example, after the rts are sorted from fastest to slowest, it is possible to calculate the 10th, 30th, 50th, 70th, and 90th percentile by taking the 10% fastest, 30% fastest, 50% fastest, 70% fastest, and 90% fastest rt, respectively. If there is scalar invariance, the plot of the quantiles from one duration condition should line up with the corresponding quantiles from another condition.

These plot points should form nearly a straight line, with slope equal to the ratio of the average of the timed durations in the two different conditions. Additional analyses based on linear regression can be computed and the best fitting line can be superimposed on the data (a quick but incomplete way of doing this is to go to the Tools menu of the figure window in Matlab, select the Basic Fitting menu option, and then select “linear” in the popup dialog box that appears).

2.4 Maximum Likelihood Fitting of Distributions to rt Data

The most sophisticated method for testing model predictions is by fitting model parameters to empirical rt distributions. In two-alternative perceptual decision research, this approach is powerful and widely used. Despite having a similar goal, different fitting methods greatly vary in terms of their computational speed, their robustness (resilience from data not actually produced by the process under investigation, but instead delayed by distraction, for example), and the amount of data they require for fitting accurately. For this reason, a large amount of work is devoted to comparing the fitting methods and trying to find new ones.

Maximum likelihood fitting is a standard technique for fitting models to data in many areas. The likelihood function is the point-by-point product of a candidate probability density function’s values evaluated at each of the observed rts. For example, a Gaussian density has two parameters, a mean, and a variance. To calculate the likelihood of the rt data for a given value of mean and variance, we multiply the probability density value at each rt point to get the overall likelihood of the data, given the particular parameter values used. If a new set of values for the mean and variance parameters gives a higher likelihood, then we prefer that set of parameters to the previous set. Intuitively, those parameters that make the data appear most likely are themselves likely to be the parameters that generated the data in reality. Thus, an iterative procedure can be implemented, wherein a parameter value is chosen (or multiple values are chosen, for models with more than one parameter), the likelihood of the data is computed, and then a new parameter is chosen to see if it gives a higher likelihood. This could be done purely randomly, but search procedures in Matlab’s Statistics Toolbox, and also in its Optimization Toolbox, can be used to do much more intelligent searching over parameter space. The code for implementing such a search is shown in fit_models.m (see book’s GitHub repository). It is relatively concise, because the fitdist function hides a large degree of complexity.

The maximum likelihood method is particularly useful in timing research, because pa models produce rt distributions with simple, closed-form expressions, such as the normal, gamma, or inverse Gaussian distributions. The most common distributions are built in to Matlab’s Statistics Toolbox, allowing samples to be drawn easily. Moreover, because of the closed-form expression, the fit to the data can be done by simply modifying the parameters of the distribution and finding the best fit rather than by sampling the data and simulating the model output as in the case of models with no closed-form expression. Instead, a computation-intensive process is required to evaluate the likelihood function for the data of models without a known, closed-form RT distribution, so that maximum likelihood methods are frequently outperformed by methods such as the chi-square method (Ratcliff & Tuerlinckx, 2002). Furthermore, new methods such as hierarchical Bayesian methods (e.g., Wiecki, Sofer, & Frank, 2013) are becoming increasingly interesting to researchers, given their ability to fit small amounts of data and to infer population-level parameters efficiently. For example, they can infer a set of parameter values that represents patients with attention deficit hyperactivity disorder versus a set of values that represents neurotypical control participants. We do not go into those methods here, since they are at the forefront of development in the two-alternative decision domain, and have not yet (to my knowledge) been widely used in fitting timing data.

2.5 Model Complexity

Maximum likelihood fitting offers one other useful property, which is that likelihood methods can easily be adapted so as to penalize for model complexity. Occam’s Razor is the principle that the simplest explanation of real data should be preferred, all else being equal. When one statistical model has more parameters than another, it has more flexibility to fit a wider range of data patterns. In the extreme, a model that has as many parameters as data points is likely to fit the data perfectly. However, such a model is not at all likely to generalize well to data that has not been fit. Such models are said to overfit observed data, at the risk of failing to fit unobserved data. To combat this risk, tests such as the Akaike Information Criterion (Akaike, 1974), and the Bayesian Information Criterion (Schwarz, 1978), can be used to rule out overly complex models. Both of these methods simply add a penalty to the logarithm of the computed likelihood function, with the amount of the penalty depending on the number of parameters. Thus, a model A that fits the data less well, but with fewer parameters than model B, may in the end have a higher likelihood score. In such a case, we select model A over model B.

Given that the log likelihood is subtracted from the parameter penalty in the standard formulation of the aic, the goal is to select the model with the minimum aic score.

2.6 Outlier Treatment

One factor that bedevils rt research is the presence of contaminated data. If a participant does not pay attention during a trial of an experiment, they may issue a response that is far later, or far earlier, than would normally occur. There is a host of different approaches to removing outliers from data, though none can be assured of doing it correctly (see, e.g., Ratcliff, 1993). After all, a very long rt may just be … a very long rt. Still, the presence of just one unusually long rt that results from inattention can throw off maximum likelihood fitting methods, because the likelihood of such a data point is so low according to the true parameter values.

Here is a simple technique for eliminating outliers that is by no means guaranteed to work perfectly, but in any case can be adapted by users to be more or less conservative as they see fit. It is included here primarily to emphasize an incredibly useful technique for indexing rt data that is outside an outlier cutoff range.

The syntax inside the outer parentheses creates a vector of logical 1s and 0s. Only those elements of the rt array that have a logical 1 in the corresponding array created by the <= operation will be assigned to rt. The result is that any rt greater than 3 standard deviations above the mean will be deleted from the array rt.