What are the advantages and contributions of quadratic relationship in a model? I'm working on an article where investigators are looking for a relationship between mortality and individual-levels variables, area-levels variables in a population.
Area-levels variables are an index of precariousness in the living area of individuals, urbanity of the area, and others. The individual variables are among others, age, sex, marital status. Different models have been realized. In these models they have tried to find a quadratic relationship between age and mortality, without specifying why. I would like to know what could lead methodologists to opt for a quadradic relationship rather than a linear or categorical relationship.
What are the advantages and contributions of opting for a quadratic relationship, generally and potentially in this case?
 A: When we do linear regressions, we often add powers of variables to account for the non-linearity effect of that variable while still staying in the linear regression framework. As long as $X,X^2$ are not perfectly collinear (ie if X is binary), then I may want to consider
$$Y = \alpha_0 + \alpha_1X + \alpha_2X^2 + e_1$$
instead of
$$Y = \beta_0 + \beta_1X + e_2$$
Why would I do this? Well precisely to allow for a quadratic effect of $X$. Let's relate it directly to your question of outcome being mortality, and covariate being age. In the linear specification (the second one), let's suppose that $\beta_1 > 0$ is positive. That means that higher age always means higher mortality. Seems fine right? Older people die more. But wait!! Whenever researchers looked at mortality rate by age, they notice something curious: very young children have a very high mortality rate, then it dips in the teens up to the 50s, and then it goes up again. Why? Well infant mortality is a very big problem, especially in developing nations, but more generally, a 1 year old is more likely to unfortunately pass away than a 25 year old. One reason is that there are many diseases and complications that threaten a newly born, but conditional on making it past the first few years, they are no longer going to kill you (developed immunity, stronger body, etc). This evidence suggests that I want to somehow allow for young ages to have higher mortality, but then it slopes down to a low mortality rate, and then rises again later in maybe the 50s+. But that kind of curve is precisely a quadratic curve. For example, $x^2-40x+400$ is the curve where mortality is $0$ at age $20$, but increases on either side... by specifying a quadratic form as in the first form I wrote, you can estimate what these coefficients should really be according to the data.
There are many situations where a similar concept arises.
