Hi there,
I need a bit help in statistics.
I developed an analytical method, which relies on calibration curves. For calibration, the measured signal value of a calibrator is plotted against their corresponding concentration and a sigmoidal curve is fitted to this data. (See picture attached).
To show, that the calibration works reliable we use the parameter R² (coefficient of correlation) , besides some other quality cirteria that must be full-filed. For my method, it is published, that the R² should be in the range between 0.990 an 1.000 (which is the maximum anyway)
I have made 16 calibration curves and now I want to give a value what the typical R² is and what distribution of R² can be expected. Can I simply calculate the Mean and standard deviation of the 16 R² values?
For example: I have 16 R² Values, spanning from 0.990 to 1.000 with a mean of 0.998 and a Std. of 0.003 But: 0.998 + 0.003 is bigger than 1.000 so I cannot write 0.998 ± 0.003 ?
What kind of mean, median value etc should I use in this case and what kind of variance descriptor?
Thanks!
Peter
Find the distribution of these 16 observations by:
Z = (X - X^) / S
where X^ = 0.998 and S = 0.003. Once you get Z for each, your can get the corresponding F(Z) from the Z table. Now you should have the percentage probability (F(Z)) for all 16 observations. You can now report R2 in its probability term, none will not exceed 100%.
In this way, you do not use R2 directly, but using the percentage probability for each R2 and the distribution of the R2 to report the mean, variance, S.
I don't think that Z is standard normal distributed.
The approximate distribution of R² is described for linear regression models, but here it can be different.
I would try to bootstrap the distribution of R² from your data.
Maybe another option is to show that R² is larger than 0.990. Without knowing the distribution, you can use Chebyshev's inequality.
Thank all of you for your help!
I never did bootstapping before, I try to find out, how it works!
Best regards
Peter
Please be in touch with me on my email. [email protected]. I will answer there.
I would just say R^2 was >=0.99. If you want to show the distribution of the values put them in a table or draw a box plot (probably into the SI).
Anyways, R^2 is usually used as a measure for the linearity of a function. Since you're not having a linear function ,but a sigmoid one, I'm a bit sceptic whether R^2 is a good measure for the quality of your fit? I would have rather looked at the residuals..
Yes Konrad you are certainly right. This is just one criterium out of some others. R^2 is not a "hard" variable when it come to sigmoidal curves with heteroskedastic distributed residuals... anyway I need to compare them. A Box plot will certainly do it!
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