I have a set of univariate data, where the response variable in a trial is a sequence of time intervals between button presses (my experimental subjs have to press a button intermittently to react to some task conditions). Eventually I have to run a within-subject regression to see if my hypothesized predictors are indeed significant. However, the first thing I want to do is to show that the subjects didn't just press the button randomly. In other words, I need to make sure that the person didn't just press a button at a regular pace, (averaged response times + random variability).

One idea I have is to check if my data resemble a Poisson process, in which arrival times occur randomly at a certain rate. Based on what I've looked up on using Poisson process so far, there are two approaches to fit the poisson process to my time interval data: 1) count how many times the person pressed a button in each trial of fixed time period. Then I can follow the Poisson distribution fitting process to count data.
2) the wait times between events in a poisson process is said to follow an exponential distribution. So I can try to fit "exponential" distribution to the time intervals. In that case, should I try to fit a bunch of different distributions such as Exponential, Gamma, and Weibull?

In either case, I would then use a chi-square goodness of fit test and/or a QQ-plot to assess whether the distribution fit can be rejected.

Any advice would be greatly appreciated. Is fitting to Poisson Process to my data even the right way to go? I would be happy to hear about any other suggestions. Thanks in advance.

  • $\begingroup$ Are participants only required to press a button following an event and only required to press the button once? If so, why not model reaction time and also count and get the proportion of the number of events that received no response and the number of events that received two or more responses? (in general, I think I need more information about the exact experimental protocol in order to answer this question) $\endgroup$ – Jeromy Anglim Jul 17 '12 at 4:45
  • $\begingroup$ Are the button presses contingent to some experimental events? in this case the question reminds me on an event related potential design from EEG studies. I would slice the behavioral streams into blocks with t=0 fixed on each triggering event and plot the distribution of reaction times after the event. But as Jeromy said: more details are needed. $\endgroup$ – Felix S Jul 17 '12 at 10:17
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    $\begingroup$ @JeromyAnglim Thanks for your comments. This is related to visual sampling of an information processing task. Each button press is a request for visual sample of the world/system/monitor, and the subject is told to sample only when they have to (minimal information allowed to complete a task). There are multiple button presses in each trial, and it's related to the task going on, but not exactly 'triggered'. So here I am trying to show that people aren't just taking a visual sample at a regular pace, regardless of what is going on in the world. $\endgroup$ – Wynn Jul 17 '12 at 12:49
  • $\begingroup$ Thanks for your comment, @Felix. Someone referred me to the Integral Pulse-Frequency Modulation in neural spike train, and part of the idea is that the frequency of the pulse train is varied in accordance with the instantaneous amplitude of the modulating signal at sampling intervals. In my case, the "signal" is what the person perceived during a visual sample, and we have our own theory of how that works. My null hypothesis: the differences in the time intervals (time between samples) are random. I am still looking for a way to test this, thanks. $\endgroup$ – Wynn Jul 17 '12 at 16:48

To test for Poisson you can either test whether the number of events per unit time fits the Poisson or test that time between event fits the exponential distribution. This can be accomplished with a goodness of fit test. The chi square test is one Lillefors test of fit to the exponential is another that is especially powerful for exponential distributions. The Poisson process is not the only process where outcomes occur at random. A process with Gamma distributed interarrivals that are not exponential is also a random process. Poisson processes satisfy additional conditions including the fact that for any time interval of length T the event rate is a constant times t where the constant is the rate parameter of the exponential.

  • $\begingroup$ Thanks for your answer! May I ask if you know of any implementation for the Lillefors test in R? Or should I try a single sided ks test? I considered using chi square test, but I am not sure how to determine the appropriate bin size! $\endgroup$ – Wynn Jul 18 '12 at 0:47
  • $\begingroup$ I am not familiar with what is available in R but I think there are ways to search for such routinesusing the R platform. $\endgroup$ – Michael R. Chernick Jul 18 '12 at 1:04

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