Dear All,
I am writing a paper on an ecological study and I am not really sure about the best way of presenting the statistical models.
Since the models proposed involve many covariates and interaction terms, I assume that a non-expert reader would be a bit confused by looking at a mathematical formula or just a paragraph explaining the predictors used. Following the indications from Zuur et al 2016 DOI: 10.1111/2041-210X.12577 I was thinking to use a mixed approach, and therefore include a small paragraph explaining my variables and a "simplified" mathematical explanation of the model.
Let's use a simple scenario (just to explain my idea): "In my study I measure the HEIGHT of many student (response variable) and I want to find a relationship with their AGE (dependant variable, continuous) and GENDER (dependant, fixed, categorical), but I also want to include the potential effect of their POST-CODE of origin (stupid example, just to use a random term)”.
(assuming that Gender is categorical with “j” levels, Post-Code with ”i” levels and I have “k” individuals)
So If in my model I have all the terms + interactions between gender and age, the mathematical formula would be :
Heightijk = αj + βj Ageik + Post-codei + εijk
εijk~N(0;σ2)
Post-codei ~N(0;σpost-code2) my random term
But I think this is not really “clear” at a first glance. So I was thinking to explain the model showing individual terms, interactions and random terms involved, excluding intercept and error (which are implied) such as
Heightijk = Ageik + Genderij + Ageik x Genderij + Post-codei
Post-codei ~N(0;σpost-code2)
This would be presented with a paragraph stating which the response and covariate involved are + the meaning of each subscript.
Would this be a clear way of presenting a model? Any advise would be much appreciated.
Thank you in advance for your help!
Hi Luca,
I apologise in advance for my English. In brief, although you first approach (i.e. to show the mathematical formulas) would theoretically be the most correct one, I think that specifying any model structure in an ecological article may be superfluous, leading to a potentially harder interpretation by a non-expert reader.
Moreover, GLMMs and GLMs are well-known by ecologists. Thus, I think that their structure does not need to be specified (if you want, you could quote a book/paper, e.g. Zuur et al., 2009; or you might consider to show it in a "Supplementary material" section).
Therefore I would use the second approach, i.e. specifying clearly your:
i) response variable;
ii) covariates;
iii) random terms
in the Methods section, without writing any model structure.
In the majority of papers I have read (in turn, papers on zoology), mathematical formulas of models are not usually specified. I think that keeping the things simpler and clearer as possibile leads to better results.
The S (or R) statistical language has a nice way to symbolically represent model formulae, for a short overview see here:
http://faculty.chicagobooth.edu/richard.hahn/teaching/formulanotation.pdf
You could write down your model symbollically as
Height ~ Age*Gender + (1|Postcode)
what means:
Height is the response variable which is modelled by Age and Gender (including their interaction) with fixed effects and postcode with a random effect on the intercept.
You may additionally specify that the conditional distribution of Height is assumed normal with variance σ²: (Height | Age*Gender + (1|Postcode)) ~ N(0, σ²). The only drawback is that the tilde (~) is used as the "model operator" and also as the "distribution operator", but the meaning is evident from the context.
Your simple scenario has one variable, location (post code), that is ecologic. There are known responses for age and gender under various conditions. One would presume that something distinguishes location but the distinction is currently poorly quantified. Jochen gave a good recommendation for presenting the model. You also have the problem of presenting the data.
A suggestion is ranking the data by location and plotting the data by rank. The data could also be ranked against a possible quality of the location. Explanation of the data would be how the ranking followed the location quality ranking. Model fits against the rankings should be avoided. Discuss methods of improving the qualitative data or investigating an alternate explanation. You should include information that quantifies the importance of the ranking, i.e., why should we care.
Dear Nicolò, Jochen and Joseph, thank you so much for your useful advises!
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