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I am trying to go from a functional relationship between a predictor variable and a probability of an event occurring ($P = f(X)$) to an overall rate at which the event occurs. My scenario is estimating annual rates of fish escapement from a reservoir given a relationship between daily mean discharge $Q$ and probability of a fish escaping given said discharge, i.e. $P = f(Q)$. Let's assume that $P = f(Q)$ is a discrete relationship, i.e. I have discretized the continuous function $f(Q)$.

I am getting very low values for my annual escapement rate compared to what empirical data suggests (~3 % vs over 13%), so I am concerned that my methodology for upscaling from daily to annual escapement probability is flawed. My original thinking for calculating the annual probability of escapement is the formula

$$ P_{annual} = \sum_i 1 - [1 - P(Q_i)]^{d_i} $$

where $Q_i$ is a given discharge from the discretization of $f(Q)$ and $d_i$ is the total number of days in a year that reservoir discharge was held at that rate. Without any better information and given a fairly large fish population in the reservoir, I assume that the annual rate of escapement is equal to the annual probability of escapement, i.e.

$$ N_{escaped} = N*P{annual} $$ where $N$ is the starting population of fish that year and $N_{escaped}$ is the number of fish that escaped that year.

Looking around the site, there seems to be some debate on whether my definition of $N_{escaped}$ is acceptable, and I'm not sure I am correctly defining $P_{annual}$ or that it doesn't translate to $N_{escaped}$ the way I have defined. Can any probability gurus clarify things for me?

EDIT

Thanks to @Creosote's answer the correct formulation of the annual escapement probability should actually be $$ P_{annual} = 1 - \prod_i \left[1 - f(Q_i) \right]^{d_i} $$ However, this changes my results on the order of 0.01%, so my problem is not solved (or perhaps the issue is not with the formulation). I obtained the discharge-escapement curve from this paper. The 13% annual escapement estimate for years 2009--2011 from the paper is based on mark-recapture studies, and the curve (with 95% confidence interval) is based on a multi-state model the authors generated from the same data (I have obtained the curve generated by the discharge-only model, which predicts slightly higher escapements that the figure in the paper). I use the reservoir release record to calculate the number of days that a release is within a given "flow band", e.g. the number of days in the year the flow is between 1000 cfs and 1250 cfs. The flow bands are actually defined such that the bounds of each band span a change in the escapement probability of 0.005%.

I don't get much above 4% for any of the years even when using the upper confidence limit of the curve. I recognize that there should be some discrepancy between the two results, but I would actually expect my escapement rates to be higher because the model accounts for tag loss and mortality. Any ideas?

EDIT 2

I'm accepting Creosote's answer because it provides the correct way to scale up from daily to annual probability. EdM's answer was also very helpful in pointing out that the issue probably has more to do with the underlying the escapement-discharge model than with the upscaling procedure. I'll post another edit if I get more information on my problem, but the daily-->annual probability question seems solved.

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A randomly-chosen fish on a randomly-chosen day with discharge $Q_i$ will remain in place with probability $1-f(Q_i)$. So it'll remain in place for the whole year with probability $\prod_i \left( 1-f(Q_i) \right)^{d_i}$, and so your $P_\textrm{annual}$ is one minus that. Assuming I've understood your notation, that is.

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  • $\begingroup$ Thanks @Creosote, this yields some slight differences at the 100th of a percent (for the final annual percent chance of escapement, 100*P_annual), but for practical purposes this doesn't change my results. I appreciate the correct specification however and will update my analysis to reflect your formulation. $\endgroup$
    – mikeck
    Oct 6 '15 at 19:20
  • $\begingroup$ Glad to have been of (some) help. So ... I wonder if the problem is with your $f(Q)$. You don't say how you derived it. Could you expand on that? $\endgroup$
    – Creosote
    Oct 6 '15 at 19:29
  • $\begingroup$ thanks for the continued interest. I have updated the post to provide information on the curve. $\endgroup$
    – mikeck
    Oct 8 '15 at 15:35
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As a sanity check, consider a constant daily "escapement probability" of 0.0004 (0.04%), without replacement of escaped fish back into the lake. After 365 days, the probability that an individual hasn't escaped is $(1-0.0004)^{365}$, 0.864, or a 13.6% probability of having escaped. That's about equivalent to the annual escape probability based on tagging and recapture in the cited paper. So you should be looking for a typical daily escape probability of 0.04% to account for an annual rate of about 13%.

From page 646 of the cited paper: "Model averaging indicated that daily escapement probability was approximately 0.01%"; that's only a 3.6% annualized rate, similar to what you are finding. You should note, however, that the authors state earlier on that page, at the end of Methods: "Escapement estimates obtained from Program MARK ... only provided a daily estimate of escapement that could not be extrapolated to an annual escapement rate." (Emphasis added). I am not familiar with the analysis program, but evidently its estimates as provided in this paper cannot reliably be used in the way that you intend. You might want to check with the authors of the study (or of the analysis program) for why this is the case.

Also, you should note that for 3 months during the study "daily discharge rates increased to 85 $m^3/s$" (page 649), above the upper limit of the plot in Figure 5 showing the relation deduced between "emigration probability" and discharge. "Escapement increased exponentially as mean daily discharge increased from 8 to 61 $m^3/s$" (page 646), so it's maybe not surprising that daily escapement estimates based mostly on low-discharge conditions would underestimate the annual escape rate.

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  • $\begingroup$ Thanks @EdM---I've contacted the authors to request more information on why the daily estimates can't be extrapolated to annual rates. My original assumption was that they did not have data/time to perform the flow analysis I am doing. Based on their curve, a daily escapement probability of 0.0004 is well outside the range of their observed flows and much higher than the normal operating range of the reservoir. I have extrapolated their curve to higher flows using a few different methods, but it doesn't make a big difference because those flows only account for < 1% of the historical record. $\endgroup$
    – mikeck
    Oct 15 '15 at 18:08

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