# Average the confidence intervals for multiple predictions

I've been doing some mixed modelling and obtaining model predictions for the size of a population through time. Each of these predictions comes with a confidence interval calculated from the various components of the model. I would now like to present an average abundance and an associated confidence interval for the entire time span of the study. I can easily average the point estimates of daily abundance to achieve a mean population size through time, but it's not immediately obvious how to calculate the confidence interval.

Is it as easy as averaging the confidence intervals from my daily abundance estimates to achieve an "average" confidence interval, or is this bad form statistically?

Thanks, Kevin

Update: I'm using GAMs in the mgcv package by way of the dsm package in R. dsm is two part modelling that incorporates a detection function to model survey count data in a GAM. negative binomial response distribution, REML fitting, mix of smooths and parametric terms, double penalty approach to fitting the smooths. i'm not sure how much more information is relevant, but i can certainly provide more.

I'm thinking one way to do this that may be more statistically valid is to predict while omitting Julian day from the model, thereby getting an abundance estimate (and confidence intervals) that is essentially averaged over my entire survey period. This solves that problem, but what if I want an estimate that covers a narrower selection of dates (say, average abundance for a month or something). I'm unsure how I might do that with the predict function or otherwise.

• I've never heard of a scenario where it's valid to average confidence intervals, but I also doubt averaging the point estimators is statistically rigorous either. If you could post a little bit more about the model you're fitting and the questions you are trying to answer you might get better suggestions about how you could approach the problem using more sound statistics. – jntrcs Aug 22 '18 at 23:24
• Appreciated. It made me nervous also. Will update with more info. – ice_hawk10 Aug 23 '18 at 3:00
• Not adding this as an answer as I'm unsure about the dsm part of this, but, you could simulate from the model, for each simulation average the daily data to give you the mean abundance you want, this yields a posterior distribution for the mean abundance you calculated, take the 2.5th and 97.5th probability quantiles of this distribution as the 95% confidence interval on the computed mean abundance. My issue with the detection part of the model is that if you just simulate from the GAM part you're not including the detection model in the uncertainty estimate/posterior. – Gavin Simpson Aug 29 '18 at 17:20
• That said, Dave Miller and colleagues have a new paper out (Bravington et al) on how to do the detection part directly in mgcv via random effects, so what I suggested might be easier than I think? – Gavin Simpson Aug 29 '18 at 17:35
• when i estimate abundance using my models, uncertainty from the gam and from the detection function are simply mashed together using the delta method, so perhaps the simulation issue is not as difficult as it might be. Will take a look at that paper. – ice_hawk10 Aug 30 '18 at 18:51