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We have observed four phone booths by recording, at a given time point, whether each phone booth is occupied (0 or 1). We have approximately 50 such observations (i.e. at 50 different time points, so each phone booth was recorded 50 times). Observations were pre-planned and spaced far enough apart so as to be independent -- the data were recorded over a period of about a month. How can I compute a confidence or credible interval for how often all four phone booths were occupied?

My thought process and ideas:

Observing each phone booth at a given time point is a Bernoulli trial. So we can model a phone booth as a binomial random variable with probability $p$ of being occupied. There are several ways to compute intervals for binomial variables (see this question). The trick is we want when they are all full.

Ideas:

  1. Model each phone booth as a binomial. Using a $\text{Beta}(a, b)$ prior, the posterior is $\text{Beta}(y+a, n-y+a) = \text{Beta}(y+a, 50-y+a)$ where we observed $y$ times it was occupied. Then, draw posterior samples of all four phone booths separately and approximate $p(\text{all four are occupied})$ by counting the proportion of samples for which all four are occupied. Cons: assumes the four phone booths are independent, when really there is a such a thing as rush hour vs. down times, so there should be some correlation, perhaps even more relevant since we are interested in the "surge" periods when all are full.

  2. Like the above, but as a multivariate normal so that a covariance matrix can be used so that observations are not assumed to be independent. Cons: doubtful that normal model is a good approximation for binomial data on small $N$.

  3. At any given time, either 0, 1, 2, 3, or 4 phone booths is/are occupied. So model these as a multinomial distribution (either frequentist, or with Dirichlet prior). Cons: Not sure if this captures the ordered structure 0 < 1 < 2 < 3 < 4, or if it needs to. My understanding is that multinomial is intended for separate categories, like political candidates in an election and you want to see voting percentages of each based on a sample of voters; not sure it applies to this case.

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    $\begingroup$ Is your assumption of independence between phone booths (i.e.the spacing argument is about distance between phone booths) or independence between trials (and the spacing argument is about being far apart in time)? $\endgroup$ – Glen_b Jul 23 '16 at 3:56
  • $\begingroup$ Exactly how were the times chosen? Are they random or not? If they are not random, we have to be concerned about the possibility that phone booth occupancy and observation time are dependent. $\endgroup$ – whuber Jul 23 '16 at 12:55
  • $\begingroup$ @Glen_b the spacing argument is about spacing in time, so that we are not "observing" say the phone booths 50 times in one hour (or to the extreme, 50 times in one minute, which is not indicative at all of overall usage, and we didn't have resources to observe for long stretches and use Poisson for the counts, that's why we have "instant" binary observations). The observations on the same day are typically hours-apart. $\endgroup$ – alexperrone Jul 23 '16 at 15:50
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    $\begingroup$ @whuber, we have a planned protocol for the times so that each top-of-the-hour of the business day (8am-5pm) has an equal chance of being observed. Thus, the observations are counterbalanced by hour of the day (and also by day of the week). $\endgroup$ – alexperrone Jul 23 '16 at 15:53
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If I understand correctly, at each of 50 total observation periods, you observed all four booths at once. So just recode each observation as 1 if all four booths were occupied and 0 otherwise. Now you have 50 draws from a Bernoulli distribution and can calculate confidence intervals and credible intervals the usual way.

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  • $\begingroup$ Let's say in the data 5/50 observations have all four full. But couldn't you imagine two different scenarios from that: Scenario 1: 5/50 have exactly 3 full, 5/50 have exactly 2 full, 5/50 have exactly 1 full, and 30/50 have 0 full. Scenario 2: 45/50 have all 3 full (again, we assume 5/50 have all four full). Shouldn't those give two scenarios give different probabilities (particularly the upper bound on the interval) of all four being full, even though they yield the same encoded data (and thus intervals) you propose (forty-five 0's and five 1's)? $\endgroup$ – alexperrone Jul 23 '16 at 19:47
  • $\begingroup$ In other words, in the case that 45/50 are 3 full, shouldn't it be more likely to "transition" (thinking of a Markov transition matrix as an analogy) from 3 -> 4, than in the case of 5/50 that have 3 full? It's true, in the stated question which I do not wish to alter, I'm most interested in only all four being full, but perhaps I don't see how one can not model each of the values, and then look at the particular value for 4 being full as a special case of that. The distribution of other values seems relevant given the "filling up" nature of the phone booths. $\endgroup$ – alexperrone Jul 23 '16 at 19:49
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    $\begingroup$ I agree that it's possible to model the booths individually, and that doing so would cause scenario 1 and scenario 2 to lead to different interval estimates. But doing so is likely to require committing to a model of how phone booths fill up, particularly of the dependence (or lack thereof) between the booths. Your estimates could be foiled by complex dependencies. My method doesn't even require thinking about these issues, let alone deciding how to solve them, so it seems a better approach for your situation. $\endgroup$ – Kodiologist Jul 23 '16 at 20:10

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