Why would one use age-squared as a covariate in a genetic association study? Why would one use age and age-squared as covariates in a genetic association study? I can understand the use of age if it has been identified as a significant covariate, but I am at a loss as to the use of age-squared.
 A: Keeping it simple: adding the square of the variable allows you to model more accurately the effect of age, which may have a non-linear relationship with the independent variable. For instance, the effect of age could be positive up until, say, the age of 50, and then negative thereafter. 
Adding the age squared to age, allows you to model the effect a differing ages, rather than assuming the effect is linear for all ages.
See my blog post for a simple step by step guide and how to interpret the age & age squared variable.
http://www.excel-with-data.co.uk/blog-1/how-to-regression-analysis-in-excel/
A: Taylor series approximations tell us that pretty much any smooth function can be approximated by a polynomial, so including terms like $x^2$ or $x^3$ (where x is age for your example) let us estimate the coefficients for the approximation for a known or unknown non-linear function of $x$, or age in your case.  Testing these coefficients is also a simple way to test if the relationship is reasonably linear or if non-linear terms will give a better fit.
Depending on the ultimate goal of the analysis the non-linear terms can be kept for prediction, or plots of the prediction can be used to suggest the actual functional relationship.  There are other tools, such as cubic splines, that can be used instead of polynomial terms to accomplish similar goals, but adding a squared term is a quick and easy way to do this.
A: It might be possible that a transformation was made in order to satisfy model assumptions.  It may have also been done because of the presence of some sort of quadratic relationship.
