[R] Confidence Intervals on Standard Curve

[Ben Ward]

Hi, I wonder if anyone could advise me with this:

I've been trying to make a standard curve in R with lm() of some 
standards from a spectrophotometer, so as I can express the curve as a 
formula, and so obtain values from my treated samples by plugging in 
readings into the formula, instead of trying to judge things by eye, 
with a curve drawn by hand.

It is a curve and so I used the following formula:

model <- lm(Approximate.Counts~X..Light.Transmission + 
I(Approximate.Counts^2), data=Standards)

It gives me a pretty decent graph:
xyplot(Approximate.Counts + fitted(model) ~ X..Light.Transmission, 
data=Standards)

I'm pretty happy with it, and looking at the model summary, to my 
inexperienced eyes it seems pretty good:

lm(formula = Approximate.Counts ~ X..Light.Transmission + 
I(Approximate.Counts^2),
     data = Standards)

Residuals:
    Min     1Q Median     3Q    Max
-91.75 -51.04  27.33  37.28  49.72

Coefficients:
                           Estimate Std. Error t value Pr(>|t|)
(Intercept)              9.868e+02  2.614e+01   37.75 <2e-16 ***
X..Light.Transmission   -1.539e+01  8.116e-01  -18.96 <2e-16 ***
I(Approximate.Counts^2)  2.580e-04  6.182e-06   41.73 <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 48.06 on 37 degrees of freedom
Multiple R-squared: 0.9956,    Adjusted R-squared: 0.9954
F-statistic:  4190 on 2 and 37 DF,  p-value: < 2.2e-16

I tried to put some 95% confidence interval lines on a plot, as advised 
by my tutor, to see how they looked, and I used a function I found in 
"The R Book":

se.lines <- function(model){
b1<-coef(model)[2]+ summary(model)[[4]][4]
b2<-coef(model)[2]- summary(model)[[4]][4]
xm<-mean(model[[12]][2])
ym<-mean(model[[12]][1])
a1<-ym-b1*xm
a2<-ym-b2*xm
abline(a1,b1,lty=2)
abline(a2,b2,lty=2)
}
se.lines(model)

but when I do this on a plot I get an odd result:


They looks to me, to lie in the same kind of area, that my regression 
line did, before I used polynomial regression, by squaring 
"Approximate.Counts":

lm(formula = Approximate.Counts ~ X..Light.Transmission + 
I(Approximate.Counts^2), data = Standards)

Is there something else I should be doing? I've seen several ways of 
dealing with non-linear relationships, from log's of certain variables, 
and quadratic regression, and using sin and other mathematical devices. 
I'm not completely sure if I'm "allowed" to square the y variable, the 
book only squared the x variable in quadratic regression, which I did 
first, and it fit quite well, but not as good squaring Approximate 
Counts does:

model <- lm(Approximate.Counts~X..Light.Transmission + 
I(X..Light.Transmission^2), data=Standards)


Any advice is greatly appreciated, it's the first time I've really had 
to look at regression with data in my coursework that isn't a straight line.

Thanks,
Ben Ward.