Why are you using the z-transformation??? The link function is what allows a linear model to predict the counts (and log is one choice for this). I am not sure what you think the purpose of standardization is for. Maybe you can expand on why you think it is appropriate.

Daniel Wright I wanted to standardize my data because I want to compare the different coefficients in an effect size comparison. In the end I chose a min-max-normalization because values end up being between 0 and 1.

Paul Amos Scheytt I am afraid you do not know what you are doing..... as Daniel Wright already said, the link function connnects the linear combination of the predictors with the mean values of the dependent variable. If you have count data, a Poisson or negative binomial distribution could be suitable. But counts cannot be negative (what are negative counts?!?), therefore, the model cannnot predict negative values. This does not make sense. And standardization on the range between 0 and 1 wont help either to get unseful estimates with a Poisson model.

Maybe you could elaborate, which effect sizes you are looking for and with which you want to compare your results. If other studies did also use count data and a Poisson model, I would assume that they did not use a z-transformation.

Rainer Duesing , Daniel Wright ok, thanks for your answers! I'm doing this for the first time, so it's within the bounds of possibility, that I'm a bit confused at some point.

Indeed I have count data, let's say the number of species for different observations. I want to run a statistical model with several predictors (different environmental variables) and one dependent count-data-variable (the species number). This is why I chose poisson distribution for my multivariate glm.

Based on this glm with several environmental variables as predictors, I want to know: How much is every predictor influencing the species number. Therefore I wanted to do a bar plot representing the different coefficients β1, β2, β3 and so on. I guess it would be "comparing apples with oranges" to plot there coefficients for predictors like "distance", "pH-Value" and "canopy closure degree" all together without having them standardized/normalized. This is why I first wanted to do a z-Transformation (-> but this makes negative data between -1 and 1) and then decided to do a min-max normalization (X - min (X)) / (max (X) - min(X)) . The normalized data would then be the input for my glm. From the glm result I would extract the coefficients and compare them.

In the end I guess I only did the fault to standardizing my dependent variable (=counting data) too. I guess leaving it un-standardized, even the z-Transformation would work fine.

Best

Paul

I think you should consult a local statistician for your problem, if you are very unexperienced. Poisson or negative binomial models are typically used exactly for counts, which can't be negative, therefore are bounded at 0 and have only integer values, no need to transform your dependent variable (that is the whole point of GLM, that do NOT need to transform the data, which makes interpretation much more complicated).

But also the idea that standardization of your variables will give you beta values that indicate the "best" predictor is misleading, even in OLS regression models (for a discussion see Darlington & Hayes, 2016, chapter 8). A better approaches might be Dominance Analysis, for example.

But in GLM, if you do not use the identity link, things get much more complicated. For example, in a log link model, the relationship of the predictors to the dependend variable is not additive, but multiplicative. That means, how much the a predictor influences the DV depends on the value of the DV you are starting at to calculate the in-/decrease. You could view it as a change in percentage, which depends on the base value.

Therefore, I think this topic is to intense to be discussed here in detail. A local statistician might help with your request. All the best!

Chapter 5 of Mosteller and Tukey's (1977) "green" book is I think a good discussion of what to do with different types of data. GLMs have taken off since then, so any of these books will cover these well.

@book{MostellerTukey1977,

title = {Data Analysis and Regression: A Second Course in Statistics},

author = {Frederick Mosteller and John W. Tukey},

year = {1977},

address = {Reading, MA},

publisher = {Addison-Wesley},

}

Hi Paul Amos Scheytt as I see it, it is a field biologist asking the question: which "predictor" variable has the highest "relative effect size" (for Lichens I presume)? It is something typical to do for a biologist/ecologist to look at these "relative", "standardized effect sizes" and regression coefficients. In the past I did this with random forest models and their "relative importance". Most biologist think that the highest "relative importance" also directly represents the strongest "effect". I learned from my mistakes, I will never do this againg unless people force me, which is very often. I think this is not the most valid approach.

First of all, by standardizing the environmental gradients (ecologist jargon for a IV vector) you lose all information. Hence, comparing apples with pears is indeed impossible, but but at least you know you have apples and pears. You can reason why these pears are more tasty than apples, but why you have apples in your kitchen, as they last longer.

By standardizing each environmental gradient you start compering something with something, of which the "relative importance" is entirely dependent on the lenght of the gradient you sampled. Hence, the results are impossible to compare among studies, and we can not repeat it, not to say that it is not generalizable. For example, sulphate in forest areas are ofcourse less "relative" important than in coastal areas.

Second, after publishing your standardize coeficient (from the forest area) there is no one who can compare your coeficient obtained to theirs, After standardization mu = 0 and sd = 1 no matter the lenght of your gradient. More importantly, these coefficients are not in any way pragmatical and informative if one wants to use these models, e.g., formulation for a prior or future prediction which are actually quite exciting.

Article THE FALLACY OF EMPLOYING STANDARDIZED REGRESSION COEFFICIENT...

One approach would be to log transform your predictor gradients, not the target variable. Select a count model i.e., Poisson and the "log" link. Considering, the linear component is now on the "log" scale one could see the estimates as close to an elasticity coefficient. If you have zeros in the gradient then a square root of cube root transformation is possible, but you lose the interpretation. It is much more meaningful in my opinion if sulphate has a coefficient of -0.7. Given the "log" link and the gradient is log transformed, this would be exp(-0.7*log(sulphate concentration mg/L) + Intercept). Unfortionately I have no knowledge about what a wide gradient of sulphate for lichens would be. But, for example, an estimate loss -0.7 of aquatic plants species by log(Nitrogen mg/L) is a pretty strong loss in species.

Hope this helps

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