My hypothesis is that the functional diversity of fish (responsible variable) increases until an optimum level of a gradient of habitat structural complexity (predictor variable), but decreases after that (which I have noticed in the graphic inspection). Then, I am really interested in this hump-shaped relationship.
To test this relationship, I will run a beta regression. In R, I have found two ways to include a second-order term in the model: I(x^2) and poly(x,2). The first one does not include the lower-order term, but the 'poly' function does.
According to Cohen, Cohen, West, & Aiken (2003), in order that the higher order terms have meaning, all lower order terms must be included, since higher order terms are reflective of the specific level of curvature they represent only if all lower order terms are partialed out.
First, I would like to know if this is a consensus and if I really cannot use only the second order term as a predictor.
Second, if I use a likelihood ratio test to compare models (e.g., only first order term vs. first and second order term) and the result is not significant, how can I choose the best model?
Hi Bárbara,
To describe the hump-shaped relation you have to model your response with both the first and second-order polynomials (otherwise I think that you would be describing only the "negative" part of the relation).
However, both methods you use in R should give the same result. Note that (with a linear model ("lm"); the formula structure will be similar in beta regression):
res1
Hi Bárbara,
To describe the hump-shaped relation you have to model your response with both the first and second-order polynomials (otherwise I think that you would be describing only the "negative" part of the relation).
However, both methods you use in R should give the same result. Note that (with a linear model ("lm"); the formula structure will be similar in beta regression):
res1
Hi, Barbara
First of all, I appreciate your way of previously analyzing data and I endorse all Jean's suggestions. The way of including second-order terms are also the best I know so far.
For clarification, I have some minor perspectives.
First, to understand the statement of Cohen et al. to justify the inclusion of the lower-order terms when dealing with quadratic (or another polynomial) equations, it is easier to drawback to their basic form. Let's take a look at quadratic ones:
f(x) = a + bx + cx²
The lower-order term, in this case, is b, which is, by definition, the coefficient that defines the position of the curve on the X (horizontal) axis. If you remove it from your model fitting, you will declare that it is equal to zero and will force the inflection to occur at (0,y). In practice, your hump-shaped equation will peak when 'habitat complexity' is zero, which may be unrealistic. Thus, it is fundamental for model fitting that your b should be non-zero, and I must say negative, so the curve will trace rightwards.
Second, the likelihood approach actually provides you with some useful guidance. If not sure, confirm it through the AIC.
Best wishes,
Matheus
Hi Jean Carlo Gonçalves Ortega and Matheus T. Baumgartner
Thank you so much for your answers. Both explanations clarified a lot. Now I am really convinced and I understand why it doesn’t make sense not to include the lower-order term in a quadratic equation. In fact, when I plot the graph using “y ~ x” or “y ~ x + I(x^2)” the curves are very similar and the second one does not show a quadratic trend. Actually, it shows just a negative relation, as you both said.
About the second question, choosing the less complex model makes total sense, since it is the null hypothesis of the likelihood test. In such cases, I will also try to use AIC to confirm.
Cheers,
Bárbara
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