It’s been accepted that the Delta method is the best way to go, and you haven’t yet convinced me – or apparently, very smart guys like Phil or Tango – that your method is more correct.

Well, if it’s been accepted, then clearly it must be superior. No explanation, reasoning or facts necessary once a group of “very smart guys” has been designated to think for the rest of us.

Calculating the exact confidence interval around the maximum or minimum from quadratic regression estimates is difficult. I don’t know of a text that discusses this specific issue. If you would like to take a crack at it, here’s a paper that discusses two methods. If you want to see a rough measure how the peak might differ, you could take the extreme bounds for each variable’s confidence interval (positive for age and negative for age², and vice versa) the peak age estimates range from 29.28 to 29.5. But, you can’t hold one constant and push the other to its extreme, because that would generate a shape far different from best estimate.

You’re right. Not even close. I should have realized that.

If you do know a way to calculate the SE of the peak age, and you have a reference, I’ll look it up.

Fair enough … those confidence intervals look too wide. Is there a specific section of either (or any) textbook that talks about calculating the standard error of a function of two of the coefficients?

Were you able to calculate a standard error for peak age yourself? It seems like a wild goose chase hunting down textbooks if there’s no obvious way to answer the question.

I’ll illustrate for your LWTS estimate of 29.41 years (first column of Table III).

Holding the estimate for age (beta1) constant, and bumping the coefficient for age^2 (beta2) up by 2 standard errors (as obtained from Table III), you get a peak age of about 25. If you bump the coefficient *down* by 2 standard errors, your new estimate for peak age is about 36.

(Using -beta1/(2 * beta2) as stated in the text.)

So not even considering a confidence interval for beta1, using the 95% confidence interval for beta2 gives an interval of (25, 36).

That doesn’t seem like it contradicts the other studies putting peak age at 27.

I mean the model does not assume that every player has a random trajectory (and therefore peak age) chosen from some distribution. It assumes that there is one fixed trajectory and peak age (because beta1 and beta2 in equation 1 are not indexed by player).

I was mistakenly thinking that every player can have his own trajectory and therefore his own peak age, and you were estimating the mean of those peak ages. But it looks like your model assumes that peak age is the same for all players, and you’re finding an estimate of that single peak age.