Presented to the Society for Chaos Theory in Psychology and the Life Sciences, Berkeley, CA; June 1996

In this paper, nonlinear dynamics is used as a means of describing and testing data sets exhibiting sequence effects. The term sequence effects describes the phenomena in which the dependent measure appears to be systematically influences by prior events such as the stimulus from the previous trial. For example, a particular type of sequence effect known as a repetition effect has faster response latencies to the letter X when the letter X was preceded by another letter X. Although sequence effects have been observed over a wide range of paradigms, the discussion will focus on sequence effects in choice reaction time experiments. There are a number of experimental variables known to systematically influence sequence effects. The question we are examining is whether or not these mean differences in the experimental variables reflect differences in the number of nonlinear dynamic variables (i.e., a change in the dimensionality).

To examine the plausibility of a nonlinear dynamical account of trial-to-trial variability, we chose a data set that appears to have trial-to-trial dynamics. This data set exhibits sequence effects (i.e., behavior that appears to be influenced by prior trial events). An example of a sequence effect is a speeded response when the stimulus is the same as the previous stimulus (e.g., responding to an X preceded by an X would be faster than an X preceded by an O).

Figure 1 depicts the sequence effect data from twelve participants who were instructed to make simple X-O classifications (i.e., when the stimulus X is presented, press the X key and when the stimulus O is presented, press the O key). Although neither the task instructions nor the task required the subjects to use past trial events, subjects appeared to use past trial events from at least four trials back. The sequence patterns (e.g., OXOXX) correspond to the temporal order in which the stimuli were presented. For the sequence pattern, OXOXX, the right most stimulus, X, corresponds to the stimulus most recently presented (see Table 1). As shown in Figure 1, response times (RT) decrease with increasing number of past repetitions (cf. OOOOO) and recent alternations admits a history of repetitions slows response latencies (cf. OOOXO). Sequence effects are not unique to this simple X-O classification task. Sequence effects have been observed in a wide range of paradigms and dependent measures (see for a review; Kirby, 1980; Kornblum, 1973; Luce, 1986).

     Figure 1.  Stimulus Sequence plot for O-X classification task (n=12).  Stimuli O 
     and X are collapsed.  For simplicity, the graph is labeled using sequence patterns 
     corresponding to the O stimulus. (data discussed in detail in Frey, 1995; Clayton & 
     Frey, 1995).  Order corresponds to the amount of past history used to 
     partition latencies (see Table 1).

     Figure 2.  Response Time (RT) - time series plot for subject 1, O-X classification 
     task.

     Figure 3.  Response latencies to the stimuli O (left) and X (right).

The data from participant 1 was obtained from Frey (1995). This data file (Obs. TS) was treated as a single time series of length 500. The participant was instructed to press the key that corresponded to the presented stimulus (an X or an O). The experiment started with 100 practice trials in which the participant received both accuracy and response time (RT) feedback. The practice trials were followed by 500 experimental trials in which the participant received no feedback. Each trial was initiated by the participant. Both the trial initiation times and the response times were recorded. However, only response times are discussed in this paper.

To examine the assumptions concerning the variability in the observed time series (Obs. TS), two other time series are analyzed: a shuffled version of the observed time series (Shuf. TS) and the observed times series with the sequence effects removed (Obs. TS-SEr). The observed time series was shuffled so that the dynamical structure in the time series was destroyed while other characteristics of the data set (e.g., mean and standard deviation) were maintained.

The sequence effects were removed from the observed time series in the following manner. Suppose the observed time series is describe by the following linear notation:

  (Response Time_t )= (Grand Mean) + (Sequence Effect_c) + (Random Error_t), 
          where t= index for a particular trial,
          c = index for a particular fifth-order condition (e.g., OOOOO, see Figure 1)

     For example:
          Grand mean = 495
          OOOOO condition mean = 418
          Adjustment that removes SE = 77 = 495 - 418
          RT  = 403
          RT SEr = RT  + (adjustment that removes SE)
          RT SEr = 480 = 403 + 77

Table 2. Summary of statistics.

The graphical comparison between the FFTs for the three file types is displayed in Figure 4. The shuffled TS exhibits a flat or zero slope typical of white noise. Both the Obs. TS and the Obs. TS-SEr appear to have similar spectra. If the variability in the observed TS is the result of white noise, the Obs. TS-SEr should have a similar spectrum to the Shuf. TS, but this does not appear to be the case (see Table 3). Rather the Obs. TS and the Obs. TS-SEr both have non-zero slope near -0.25 (for similar findings see Gilden, et al, 1995).

     Figure 4.  Fast Fourier Transform for participant 1, X-O classification task.  See 
     Clayton & Frey (1995) for FFT data for each task collapsed across 12 subjects.

The linear account of variability as white noise is the assumption of IID. To test this assumption the data sets were submitted to the BDS statistic (Brock, Hseih, & LeBaron, 1991). The results are presented in Table 3. In both the Obs. TS & the Obs. TS-SEr the assumption of IID can be rejected. This is not the case for the Shuf. TS. The results of both the FFT and the BDS analyses do not provide support for an account advocating that the variability is due to white noise.

Table 3. Results from the FFT analysis and the BDS statistic.

The previous two tests argue against the white noise account of variability. The next two tests examine the low-dimensional nonlinear dynamical account. The first test examine the number of dynamical variables needed to produce the time series (i.e., dimensionality). Two different estimates of dimensionality are used: Grassberger & Procaccia (1983) and Judd (1992, 1994). The Grassberger & Procaccia (1983) algorithm is widely used, but when the sample size is small as in this case it tends to under-estimate the dimensionality. JuddŐs estimate has been reported to give reasonable estimates with small data sets.

The dimensionality is estimated via the point at which the estimated dimensionality stops increasing as the embedding dimension increases. Since white noise is assumed to have an infinite number of factors, the estimated dimensionality should not level off, but rather continue to increase as embedding dimension increases. As seen in Figure 5, neither estimate suggests that the dimensionality is less than 6. For both dimensionality estimates, the Obs. TS & the Obs. TS-SEr appear to have similar estimates. Both of these time series also appear to differ from the Shuf. TS, particularly in JuddÕs estimate. The closeness of dimensionality estimate for the Shuf. TS to the other two time series and the ambiguity in whether the estimated dimensionality is leveling off, leaves these estimates of dimensionality tenuous.

     Figure 5.  Dimensionality Estimates using Grassberger & Procaccia (1983)Judd (1992, 1994) algorithm (right).  Thin solid line 
     corresponds to the expected dimensionality estimate of white noise.

The final test (Sugihara & May, 1990) provides a means of examining how predictability changes as prediction time increases. The predictability of white noise should not change as prediction time increases. A signature of chaotic systems, on the other hand, is high predictability that drops off as prediction time increases. As presented in Figure 6, the Shuf. TS behaves very similar to what is expected from white noise (i.e., low-predictability that remains constant as prediction time increase). Both the Obs. TS and the Obs. TS-SEr exhibit non-zero predictability that drops off as prediction time increases. This suggests that the Obs. TS and the Obs. TS-SEr might be the result of a chaotic system.

     Figure 6.  Predictability as prediction time increase, E=11, lag=1, Sugihara & May 
     (1990).

The random error account of trial-to-trial variance via independently distributed white noise is not supported by the statistics. The spectral analysis and the BDS test suggest that the shuffled time series exhibits characteristics of white noise and both the observed time series and the observed time series with the sequence effects removed appear to differ from the shuffled time series. The estimates of dimensionality are less clear. Two different estimates of dimensionality suggest that the dimensionality of the observed times series and the observed times series with the sequence effects removed might be less than the dimensionality of the shuffled time series. The estimates also suggest that the dimensionality for the observed time series and the observed time series with the sequence effects removed might be as low as 6-9 dynamic variables. The observed time series and the observed times series with the sequence effects removed have the signature characteristic of chaotic dynamics in which the prediction accuracy decreases as prediction time increases. The convergent results provide support for the nonlinear dynamical account and these results suggest that other data sets exhibiting sequence effects might be excellent candidates for nonlinear analysis.

Table 4. Summary of Results.

During the intervening time, I have had the opportunity to apply the discussed tests to the remaining 11 participants' data sets. All but the dimensionality conclusions appear to be supported. The results from the dimensionality tests do not suggest a difference between the three file types and neither observed time series appears to have a dimensionality less than 7. Click here to see the corresponding data figures for n=12.

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