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I am attempting to standardize recreational fishery CPUE data. I am using a delta approach, with a binomial model fit to the presence/absence data and a lognormal model fit to the positive observations. When I test the log(positives) data for normality, I get a plot that looks almost normal, but with a heavy left tail. I am not quite sure why. My question is, how heavy tailed is too heavy tailed for a normal distribution to be appropriate? Are there alternative distributions you would consider, and how would I test which one is most appropriate? I have over 80,000 observations so a shapiro wilk test here is useless. I have included an image of my density plot and qqplot.

diagnostic plot

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    $\begingroup$ Please say why you care about having a normal distribution for these data. If you are simply describing the data there's no need. If this is for use in something like a regression model, what matters is the distribution of residuals around the model fit, not the data values per se. Even then, much can be accomplished without normally distributed residuals. See this page, for example. $\endgroup$ – EdM Feb 13 at 20:11
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I cannot speak to how heavy a tail is too heavy but I can recommend some readings and other distributions. I highly recommend first reading Maunder and Punt (2004) on approaches for standardizing fisheries CPUE data. In addition to the lognormal, I would consider a Poisson or negative binomial using your effort as an offset in the model as these models are based on counts. You can also consider the gamma or Tweedie distributions. See Shono (2008) for an example application of the Tweedie distribution for CPUE standardization. If you have a lot of zeros, you may want to consider zero-inflated or hurdle models. As determining the most appropriate model, you want to look at plots of residuals and predictions of observed data. I often calculate the model dispersion as an initial indication of model fit. Refer to Zuur et al. (2009) for excellent guidance on model selection. Hinton and Maunder (2003) also provide guidance on model tests and comparisons.

Hinton, M.G., and M.N. Maunder. 2003. Methods for standardizing CPUE and how to select among them. ICCAT Collective Volume of Scientific Papers 56:169-177.

Maunder, M.N., and A.E. Punt. 2004. Standardizing catch and effort data: a review of recent approaches. Fisheries Research 70(2-3):141-159.

Shono, H. 2008. Application of the Tweedie distribution to zero-catch data in CPUE analysis. Fisheries Research 93(1-2):154-162.

Zuur, A.F., E.N. Ieno, N.J. Walker, A.A. Saveliev, and G.M. Smith. 2009. Mixed effects models and extensions in ecology with R. Springer-Verlag, New York. 574 p.

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