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I'm looking for some help devising a hypothesis test for the following situation.

  1. I have a radioactive source that spits out a particle every so often.

  2. Also, I have two particle detectors: a red particle detector, and a green particle detector. Whenever the red particle detector detects a particle, it flashes a red light; let $R$ denote the event that the particle was detected by the red detector, and $r$ the complement event that the particle was not detected by the red detector. Whenever the green particle detector detects a particle, it flashes a green light; let $G$ be the event that the green detector detects the particle, and $g$ that it does not. Thus, each emitted particle falls into one of four categories:

    • detected by both detectors ($RG$),
    • detected by the red detector but not the green detector ($Rg$),
    • detected by the green detector but not the red detector ($rG$), or
    • not detected by either detector ($rg$).
  3. Each time a particle is emitted, the red detector has some probability of detecting the particle, and the green detector has some probability of detecting the particle. (They will never trigger a false detection when no particle is present.) I know that each particle is handled identically and independently of all other particles, but I don't know whether the two detectors are independent of each other. It's possible that they are independent (i.e., $\Pr[RG] = \Pr[R] \Pr[G]$), or that they are correlated (i.e., $\Pr[RG] \ne \Pr[R] \Pr[G]$); I don't know which is the case, a priori.

  4. I keep a count of the number of $RG$-detections (i.e., number of times when both detectors detected something), number of $Rg$-detections (i.e., number of times when the red detector detected something, but not the green one), and the number of $rG$-detections. Unfortunately, I have no way to measure the number of $rg$-situations, since those particles aren't detected by either of the detector. At the end of the experiment, I have three non-negative integers, representing these counts.

I want to test the hypothesis $H$ that the two detectors are independent, i.e., that event $R$ is independent of event $G$. Can anyone help suggest a way to compute the $p$-value of this hypothesis, given 3 numbers from such an experiment?

I'd be perfectly satisfied with a computer algorithm/procedure to compute the $p$-value. I don't need a simple formula; something that could be computed by a computer would be sufficient.


Here's another way to view this. We could form a 2x2 contingency table, such as this one:

     G |  g
  ---------
R | 17   22
r | 12    ?

recording that we saw 17 $RG$-events, 22 $Rg$-events, and so on. Unfortunately, the lower-right cell is empty, since we don't know how many $rg$-particles were emitted. If we had counts for all four cells, presumably we could use Fisher's exact test, but we don't. Also, we aren't given $\Pr[R]$ or $\Pr[G]$ (I guess they are nuisance parameters) or the total number of particles emitted.

Any suggestions?

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If the missing count were close to 22*12/17, the table would appear independent. This is consistent with your observations. If the missing count is far from this value, the table would exhibit strong lack of dependence. This, too, is consistent with your observations. Evidently your data cannot discriminate the two cases: independence or lack thereof are unidentifiable. Therefore, your only hope is to adopt additional assumptions, such as a prior for the missing count (equivalently, for the total number of emitted particles).

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  • $\begingroup$ Ohmigosh, you are so right! I'm embarrassed I missed this. Thank you so much for your help: it is much appreciated. $\endgroup$ – D.W. Jan 27 '11 at 5:46
  • $\begingroup$ @D.W. There's no cause for embarrassment. In fact, your question was worded in an unusually clear manner. $\endgroup$ – whuber Jan 27 '11 at 7:24

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