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 est
Factorial treatment designs are popular, due to advantages of research on multiple treatment factors and how they interact. But if the design includes a control treatment that is not part of the factorial, problems occur in estimation of least squares means. A typical example is shown here, with 2 fertilizer and 3 irrigation treatments, giving 6 factorial treatment combinations, plus a control that is defined by a 3rd level of fertilizer, and a 4th level of irrigation: Fert1:Irrig2 Fert2:Irrig1 Fert1:Irrig1 Fert2:Irrig3 Fert1:Irrig3 Fert2:Irrig2 Control Other situations might have the control sharing a level of one of the factors, for example the control might be defined as Fert2:Irrig4. But this still causes problems with estimation of least squares means due to the levels of one factor not occurring with all levels of the other factor. Let's jump into a SAS example, but using random numbers to allow easy creati