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Estimation of the Peak in Quadratic Regression

 Problem:  You are running a standard quadratic (polynomial) regression analysis, and are specifically interested in the X and Y values at the peak.  If you use standard regression software, typically there will be no option that allows the peak to be estimated, with standard errors.

Example:  You are studying Growth as a function of Age.  Of particular interest is when maximum Growth occurs, and at what Age.

SAS code to generate artificial data, and run the analysis is: 

data one;
 do Age=1 to 20;
 Growth=95 + 2.7*Age - .3*Age*Age + 5*rannor(22);
 end;
proc nlin plots=fit;
 parms int=2 lin=1 quad=1;
 model Growth = int + lin*Age + quad*Age*Age;
 estimate 'Age at peak' -lin/(2*quad);
 estimate 'Growth at peak' int + lin*(-lin/(2*quad)) + quad*(-lin/(2*quad))*(-lin/(2*quad));
run;

The standard quadratic regression model with intercept, linear and quadratic slopes, is coded into Proc NLIN which has the ability to estimate any function of the parameters.  The peak estimates are obtained using standard calculus, set the first derivative to zero to estimate Age at peak, then put that estimate into the quadratic regression equation to estimate Growth.
Output from the example is copied below, and you can visually verify that the peak estimates match the fitted curve.  Without using a statistically based process like this, you will not have standard errors or associated confidence intervals, which definitely help assess the confidence you should have in the estimated values.  Here we conclude that maximum growth occurs at 5.4 years, and maximum growth is 97.1 mm.


Additional Estimates
Label Estimate Standard
Error		 DF t Value Approx
Pr > |t|		 Alpha Approximate Confidence
Limits
Age at peak 5.4310 0.7727 17 7.03 <.0001 0.05 3.8007 7.0613
Growth at peak 97.1044 1.7731 17 54.76 <.0001 0.05 93.3634 100.8

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