Being in quite a rush to do your analysis is not a good place to be! It encourages you to take short cuts, advice from strangers and to leave model checking aside.

Your data won't follow a normal distribution because you have a data generation process that produces event counts. And no amount of wishful thinking (=data transformation) will change this process.

A Poisson or negative binomial regression model, with robust standard errors to account for overdispersion, would be my first approach to the data.

You may have an excess of zero values, corresponding to a data generation process that has two steps

- does the fish contain plastics

- if so, how many

And if both of these questions are interesting, your analysis gets a little more complex

Being in quite a rush to do your analysis is not a good place to be! It encourages you to take short cuts, advice from strangers and to leave model checking aside.

Your data won't follow a normal distribution because you have a data generation process that produces event counts. And no amount of wishful thinking (=data transformation) will change this process.

A Poisson or negative binomial regression model, with robust standard errors to account for overdispersion, would be my first approach to the data.

You may have an excess of zero values, corresponding to a data generation process that has two steps

- does the fish contain plastics

- if so, how many

And if both of these questions are interesting, your analysis gets a little more complex

If your count data fits a Poisson distribution, the square root (possibly adjusted by an additive constant) or the Freeman-Tukey transformation would do. The 2/3 power function might work even better. But, as Ronan has mentioned, if you have an access of zeros as compared to the Poisson, you have a mixed distribution after transformation and things are really complicated.

Maybe, you could estimate the probability of zeros in access of Poisson in each group and test for differences and analyse the rest by ANOVA.

Listen to Rónan, he gave expert advice!

A negative-binomial model should do. If excess zeros are still a problem, you can use a zero-inflated negative-binomial model.

I don't subscribe to Manfred. There is no need to transform count data, because there are appropriate models for counts and the coefficients from transformed values are often difficult to interpret. Removing the zeros from your data is also not a good idea because the analysis of the "rest" will give you distorted results (when the zeros are not "randomly" distributed, what is actually expected here!). Also, (zero-infalted) negative binomail models are not more difficult to fit than a standard linear model (like an "ANOVA" model) using computers and software ;)

I don't see why you are focussed on ANOVA. There are many types of model for the many types of data we have. Even if you could coerce your data into some kind of normal distribution, you would then have the problem that your ANOVA would give you effect sizes that were not on the original measurement scale, and did not correspond to arithmetic means.

Poisson regression, on the other hand, will give you expected mean values and measure effect sizes by incidence rate ratios, which give a scale-free and intuitive measure of the effect of a factor.

Why do you want the data normally distributed? That kind of thinking seems very outdated, given all the advances in statistical theory and computer-assisted statistical analysis techniques. Fit various distributions to the data and examine which one most closely approaches the distribution of your sample. Compare the goodness of fit statistics of different ways of modeling the data as they are (without any transformation). My first guess, without looking at your data, is that one of these will provide the best fit: (1) zero-inflated negative binomial regression, (2) zero-inflated poisson regression, (3) zero-inflated beta regression

http://onlinelibrary.wiley.com/doi/10.1002/bimj.200390024/abstract

http://math.usu.edu/jrstevens/biostat/projects2013/rep_ZIP.pdf

http://www.sciencedirect.com/science/article/pii/S0167947311003628

Anthony,

you are right to point out that one should think about the right tool for the data rather than torture the data until it seems to fit to the only tool one knows. However, I want to make a note of care to your statement "Fit various distributions to the data and examine which one most closely approaches the distribution of your sample":

From an inferrential perspective, this comes at a cost you neglect or don't mention. If you try different models, there is an increasinly large chance to find one that accidentally fits your data. Your data may contain "strange" features that will even drive you to select a "wrong" model. Selecting the model by the data is an "unfair practice". Actually, the model should be specified based on theoretical grounds. Often enough we lack such grounds, so the data is all we have to get an idea - but getting an idea is already part of the inference that has to be taken into account when the results of the fit are interpreted! And here is where the costs come into play:

The correct way to tackle this problem is to "play around" with only a part of the available data (call it the "training set"), and leave the rest ("test set") for later. After some hopefully appropriate model is identified based on the training set, the analysis of the actual scientific question is done only with the test set. You see: you use less data for the tests, you will have fewer degrees of freedom, implicating that the estimates will be less precise. This is the price you have to pay for "fitting various models"

Further to Jochen's point, what we need to do is not rely on transforning the data but make a pot of coffee and think about the data generation process. There are two reasons for this. The first is that it will lead us to a model that should fit both the data and the theory.

Second, when this model fails (and it will, from time to time) then we learn that our ideas about the data generation process were wrong. When this happens, you can begin considering protocol violation (I've detected two serious occurrences this way) bias in the conduct of the study or – and here is where it gets interesting – that something new and unexpected is driving the data.

We all play with data, but it's equally important to take a walk around the block, drink a cappuccio, think about where it came from

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This has been a very educational discussion! I completely agree with Ronan's stmt: "We all play with data, but it's equally important to take a walk around the block, drink a cappuccino, think about where it came from." 

It is great to see that the distribution of data at hand has been checked. I have seen ANOVA or Linear logistic regression analyses being done without even looking at the distribution of the continuous outcome. 

Or a glass of wine, as Guven elegantly demonstrates in his paper.

A "trick" many applied statisticians use is to set your zero values to a small positive value, such as 0.5 and then you can log transform. And contrary to what some others have said, a Poisson distribution will go to normal in the long run (when you get a large enough sample). However, I agree that if you can use models made for count data that is always best. But you want a comparative design and not regression so I can see why you want the ANOVA.

Also, the assumption of normality is robust to deviations as long as you have equal variances between your groups. Good luck! :)

Hi Ronán Michael Conroy

I have a similar problem and would appreciate your advice.

I have a dataset of 3 biological repeats (experiments) with each 3 groups but with varying N per group and per biological repeat. Hence, I wanted to use the Mixed Model option in JMP (with condition as Fixed effect and experiment as Random Effect). However, the count data are not normally distributed, so I transformed them before doing the mixed model. The problem here is I had 3x zero counts (3 on a total of 18 only in 1 group) so I left them out of the analysis.

Alternatively, I could choose Generalized Mixed Model with Poisson distribution and Log Personality (JMP) but then this does not account for the fact my 3 experiments have different N per group.

What would be the best way to do the analysis?

Hi Marlies, check out Generalized Estimating Equations. They don't require a normal distribution and work with missing data on repeated measures. Here is a little reference, with more references at the end, that I found online:

https://rlbarter.github.io/Practical-Statistics/2017/05/10/generalized-estimating-equations-gee/

Hope this helps!

Hi Elaine Eisenbeisz

Thank you. Do you think the GEE would be a better approach than taking the square root of the count numbers (so that assumptions for normality and homoscedacity are met) and doing the Mixed Model?

If I take the square root I can include the zero values. For reporting the data (graph) I would do a backtransformation.

Marlies, you could run a sensitivity analysis and try both ways and see if you get similar results. If you could use the GEE, then you most likely wouldn't need to transform the data. I always like to use the actual units of the data for ease in interpretation if nothing else. So, I'd give the GEE a try at least.

Great link to the blog post, Elaine Eisenbeisz – very useful and clear. Marlies Boeren – I'd go with Elaine here. Try not to obsess about making your data fit your model by some kind of transformation, and instead look for a model that fits your data. People get into contortions trying to log-transform data with zeros. Clearly, if your data have zeros it isn't generated by a log process.

Thank you Elaine Eisenbeisz and Ronán Michael Conroy for your advice!

As a postscript, I am profoundly suspicious of changing zeros to some (arbitrary) small value and taking a log. If you use 0.5, the log is -.69314718. However, if you use 0.1, then the log is –2.3. Worse, choosing 0.001 gives you a log value of –6.9.

So depending on what you choose, your zeros will have very different effects on your regression model.

My feeling, as I say, is that if your data cannot be put into your model, then the model needs changing, not the data.