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I am trying to find the correct way to assess a trend for significance over time. This is the data I have: for a 15 year period I have the fate of all nests in a colony as either failed for fledged. So as an example:

2000 - 25 out of 50 nests successful, 2001 - 35 out of 60 nests successful, and so on for the 15 years.

What I hope to do is to show whether or not there is a significant trend, either positive or negative, for numbers of fledged nests over this 15-year period. I don't think linear regression or Poisson error regression is a way to go since the response isn't really a normal distribution. However, I admit I don't really know. Can someone suggest an appropriate approach if indeed there is one? Thanks.

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Here are three suggestions. If your total counts are known, then this sounds like binomial regression to me. Say you have for observations $i=1,\ldots, n$ the response count $y_i$ (in this case it's your number of successful nests), and your predictor $x_i$ (in this case it sounds like it's time, so $t$). Binomial regression assumes that each $Y_i$ follows a binomial distribution with known $n_i$ and a parameter $p_i(x_i)$ that depends on your input. Typically people use logistic regression, which is a special case of this, where $n_i = 1$ for all $i$. But this doesn't have to be so. In your case $n_i$ is changing and seems to be around $50$ or $60$.

You can also use a beta regression if you don't want to assume that your total nests $n_i$ are known. In this case your response variables would be proportions.

Finally, if you are interested in modeling just the number of total nests, then you can run a Poisson regression. Poisson random variables don't have an explicit upper bound

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