# What are the theoretical assumptions of a power curve versus an exponential or quadratic one?

I have a set of bivariate biological data that has a clearly non-linear distribution. I have found that a power curve best fits the data under several metrics (e.g., AIC, residuals versus fits plot, etc.). Specifically, I've noticed that a fractional power curve of $$f(x)=x^{b}$$ (where b is some fraction) seems to reliably model the data but a quadratic function $$f(x)=x^{2}+x$$ ends up being heavily influenced by the most extreme values on the x axis and is not robust to subsetting the data (i.e., subsetting the data at random produces a dramatically different curve). Overall the function seems mostly linear but higher values of x produce slightly higher than predicted values of y. I cannot log-transform the data, as the data are already log-transformed (and log-transformation normalizes their distribution) and this non-linearity is present even after log-transformation. When I submitted the research for consideration the reviewers wanted me to provide a theoretical justification as to why a power curve is preferable over a quadratic or exponential model.

I know that a quadratic equation, i.e., $$f(x)=x^2+x$$, is one where the best-fit line is nonlinear and the slope $$dx/x$$ changes at a constant rate. I also know that an exponential function, such as $$f(x)=n^x$$, is used when the slope of the best fit line is expected to change at a constant rate in proportion to x. I also know that exponential functions do not have constant elasticity, unlike polynomial or power curves.

What I'm having more trouble finding is what are the theoretical expectations that would lead one to assume a power function versus a quadratic or exponential one? I've seen a lot of questions talking about the difference between quadratic and exponential models (e.g., this question on Quora), but not so much how either relates to a power model. I.e., I know an equation of $$f(x)=x^{1/2}$$ would have a constant $$dx/x$$ of 0.5, but I don't know what theoretical expectations would make somebody consider that a more reasonable approximation of the relationship than quadratic model (or vice versa).

• You could answer to the reviewers with some theoretical statistical considerations for the use of the power law fit, and it would be something like 'I tried a few functions and the power law had the least residual errors'. What the reviewers are probably hoping for is that you did not just try a large gamut of different curves untill there was one that fitted well. And instead of statistical theoretical justification they are probably expecting some biological theoretical justification. Are there underlying mechanisms in the biological model that make a power law plausible? Oct 14, 2021 at 17:15
• So what biological data are we talking about? Is it some sort of growth? Oct 14, 2021 at 17:17
• @SextusEmpiricus It's a scaling relationship between a skeletal measurement versus size. There's a good reason to believe that it scales non-linearly with respect to size because it's a weight-bearing structure (i.e., square-cube law). You are right that I think they are expecting a biological justification. However, I've read through most of the literature addressing non-linearity and nobody seems to have proposed a biomechanical model that could explain this. Most papers either just use a quadratic model without explaining why or suggest different allometries at different sizes. However... Oct 14, 2021 at 18:29
• @SextusEmpiricus ...the data suggests a single non-linear allometric curve for all species controlled by some unknown factor. So the problem I am facing is that this pattern is known, but proposed biological explanations do not exist yet. This is why I am trying to figure out the rationale behind a quadratic versus power function, as the results of the log-quadratic function are a bit suspicious (only the extreme values in any subset have any kind of leverage, for one). Oct 14, 2021 at 18:31

You could answer the reviewers with something like:

"It is expected to observe a scaling law that is close to a quadratic power because it is a weight bearing structure. We chose an allometric law with an exponent determined by fitting because it is close to such quadratic law but still allows some flexibility due to unforeseen deviations from the ideal behaviour."

• An answer with a more rigid justification could be given if a more clear description of the background is given.

• One general principle that I know of is the Buckingham-$$\pi$$-theorem. which you could use to argue that the scaling law must consist of a function with two dimensionless numbers of the form $$\pi_1 = \left(\frac{\text{length}_1}{\text{length}_3}\right)^{n_1}\\ \pi_2 = \left(\frac{\text{length}_2}{\text{length}_3}\right)^{n_2}$$ where $$n_1$$ and $$n_2$$ are some exponents. And that a law $$\pi_1/\pi_2 = \text{some constant}$$ is an expression with these dimensionless numbers that is practical to apply (because it is simple). It will result in a power law like $$\text{length}_1 = \text{constant} \times \text{length}_2^{n_2-n_1}$$ where the constant absorbs the $$\text{length}_3$$ which could be some distance length scale like the force between cells in bones per surface area.

• But, an answer based on biology might be better digestible than the stuff above.

• Is there ever a scenario in which a power law is preferred over a quadratic one? E.g., as you mention "deviations from ideal behavior"? I'm wondering when one is preferred over the other. Oct 14, 2021 at 19:56
• @use42352714 I know that in engineering you find extremely many scaling laws and only few cases have a theoretical justification (often those ones have simple rational numbers as exponents). Many of the laws are based on experimental correlations and the power laws are popular because they are pragmatic and resemble the theoretical laws. A particular advantage of power laws is that they can deal with relationships over a large range while having a simple form. But there is no real theoretical justification. Oct 14, 2021 at 21:19
• "Is there ever a scenario in which a power law is preferred over a quadratic one?" Nice relationships with simple exponents like cubes, squares or fractional powers occur in idealistic relationships. They occur in situations where an exact relationship can be computed. When an exact relationship can not be computed, when we do not know for sure that the relationship is quadratic or cube, then a power law is preferred $y = a_1x^{n_1} + a_2x^{n_2}+ \dots$. The coefficients, exponents and number of terms can be determined experimentally. Oct 14, 2021 at 22:00
• An answer with a more rigid justification could be given if a more clear description of the background is given. I was hesitant to give more background because I had already asked a similar question here and was worried it might be a duplicate. I can give more information either there or here if it would help. Oct 15, 2021 at 21:58