[R] Confidence Intervals on Standard Curve

[David Winsemius]

On Feb 19, 2011, at 1:08 PM, Ben Ward wrote:

> Hi Graham,
>
> Thanks, that does explain lots. I've been playing with making log's of
> data in models to make the relationship linear, which it does, which
> suggests to me that lm() is the right way to go, however, after if   
> try
> to predict after y values after about 60% on the x axis for light
> transmission, the y value, for bacterial numbers, crosses the axis and
> gives me negative values for y, which on a practical level isn't
> possible, as one can't have less than no bacteria in a culture.

Once you have your estimated parameter in the transformed "analysis"  
scale, you need to apply the inverse transformation, in this case  
exp,   to return the estimate to the measured scale. You also need to  
consider that in the process of transforming the values, performing an  
additive estiamtion, and transforming back to the "natural" scale, you  
will have estimated a multiplicative model, since exp(log(x)+log(y)) =  
x*y.

Had you used log(x) inside the lm call and then used predict, the  
predictions would be correct.

You might want to consider glm models with family="poisson".

-- 
David.

> On a
> practical level, when I include the cirve in my appendix I could say
> anything above around 60% is 0, and mention the negative results from
> the standard curve's prediction capabilities are not literal, and to  
> say
> turn any negative bacterial count obtained as a result of the curve to
> 0.

I wouldn't. The informed observer would know you were flailing around.

> I've not had to deal with such pleatauing curves before. The values I
> have for the curve don't go above 50%, so anything above it is
> prediction, and my experiment probably won't result in x values above
> 50% as the death of the culture proceeds slowly, but that depending on
> relative amounts of culture and antimicrobial I use the rate 'could'  
> go
> faster or slower, so could go above 50%. I was wondering if non-linear
> regression is better for such a thing, but I'm hesitant to go into  
> it in
> more detail for now because of the danger of drastically increacing
> complexity, if on a practical level, what I currenly have, works and  
> is
> very accurate, within the range I will most likely be using it.
>
> Thanks,
> Ben.
>
>
> On 19/02/2011 15:39, Graham Smith wrote:
>> Ben,
>>
>> Does this help.
>>
>> http://r-eco-evo.blogspot.com/2011/01/confidence-intervals-for-regression.html
>>
>> Not sure if it will work with your particular model, but may be worth
>> a try.
>>
>>
>> Graham
>>
>> On 18 February 2011 23:29, Ben Ward <benjamin.ward at bathspa.org
>> <mailto:benjamin.ward at bathspa.org>> wrote:
>>
>>    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.

>

David Winsemius, MD
West Hartford, CT