When can a linear regression be used?
Anytime a continuous function can be “linearized”.
Beware of inappropriate or excessive inferences from a linear regression in all cases! Mathematics can do amazing things, and is always internally consistent. But that does not necessarily create value or validity in conclusions that misuse the math.
Linearization is a method to help us more easily visualize a correlation. It necessarily incorporates approximations to the reality, which may have other inherent error sources itself. A linearization is never as good as a model that fits the data directly to some function. For example, quadratic and third-order equations can be “linearized”, but the results are no where near the accuracy or validity of the appropriate primary function.
Similarly, “regression” statistics are useful to show how close your approximation is to the actual data. Do NOT use them as a justification for acceptance, but rather as a warning of the degree of inherent error represented by the linearized function. It is also extremely important to remember that interpolation includes the inherent error, but extrapolation (prediction) from a linear regression increases error dramatically as it moves further away from the actual data set.