[R] Quadratic regression: estimating the maximizing value

[gheine at blm.gov]

A bioligist colleague sent me the following data.

x     Y
3     1
7     5
14    8
24    0

(Yes, only four data points.)  I don't know much about the
application, but apparently there are good empirical
reasons to use a quadratic model.

The goal is to find the X value which maximizes the
response Y, and to find a confidence interval for this X
value.

Finding the maximizing X value is pretty straightforward:

>Ldat <- data.frame("X"=c(3,7,14,24), "Y"=c(1,5,8,0))
>(LM<-lm(formula = Y ~ X + I(X^2), data = Ldat))
Call:
lm(formula = Y ~ X + I(X^2), data = Ldat)

Coefficients:
(Intercept)            X       I(X^2)
   -3.86978      1.77762     -0.06729

> DZ<-function(B,C) { (-B)/(2*C) } # Solve  d/dx(A + Bx + Cx^2) = 0
> DZ(LM$coefficients[2],LM$coefficients[3])
       X
13.20961


To find a confidence interval, I used "confint()".
Default confidence level of 95% was not useful; used 80% instead,
and then computed DZ with the extreme X and I(X^2) coefficients:

>(CI80<-confint(LM,level=0.8))

                  10 %        90 %
(Intercept) -5.6147948 -2.12476138
X            1.4476460  2.10759306
I(X^2)      -0.0790312 -0.05553898

> DZ(CI80[2,1],CI80[3,1])
[1] 9.1587
> DZ(CI80[2,2],CI80[3,2])
[1] 18.97400

Conclusion: the 80% confidence level for the maximizing X value is
included in the range (9.158, 18.974)

#################
Questions:

1) Is this the right procedure, or totally off base?

2) The coefficient of the "Intercept" is irrelevant to calculating
the maximizing value of X.  Is there a way to reduce the size of
the confidence interval by doing a computation that leaves out this
parameter?

3) I believe that confint() indicates the axes of an ellipsoid,
rather than the corners of a box, in parameter space;
so that the above procedure is (slightly) too conservative.

4) Where are the calculations for confint() documented ?

Thanks,
George Heine

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 George Heine, PhD                                                         
                                                                           
 Mathematical Analyst                                                      
                                                                           
 National Operations                                                       
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