Let's assume that age and running performance have an inverted U-shaped relationship. How do I know at which age performance will be best?
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1$\begingroup$ Set the first derivative to 0 and solve for age. $\endgroup$– Patrick CoulombeCommented Sep 27, 2014 at 5:41
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1 Answer
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There are an infinite number of U-shaped relationships, so it depends on your regression model.
If there's a lot of noise about the model, or few points you might fit something very simple like a quadratic function, and you can estimate the maximum from the fitted relationships. There are several ways you might form some kind of interval for the location of the maximum. See also here, and the discussion here is also largely relevant.
If there's at least a moderate amount of noise or a moderate amount of points, you may consider some kind of smooth function such as a local polynomial regression or one of a variety of spline models. There are some ways you might attempt to generate am interval for the maximum here also.
If there's very little noise or a very large number of points you can deal with rougher functions, such as ones containing some discrete jumps.
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$\begingroup$ +1 It's valuable to look for an interval instead of just a point estimate. $\endgroup$– rolando2Commented Sep 27, 2014 at 12:41