If you already have the structural definition of the response eg Emax model or a sigmoidal or whatever, than you have two alternatives.
First your dataset is suitable created (it contains two additive 0-1 coded collumn) to be implementd into a simple nonlinear least sqares method. Than you pick the appropriate parameter(s) from your model (eg. ED50, Emax etc) and you describe it as a linear function of your additive collumn (Emax=a+b*x where ix is your 0-1 coded collumn)
Second you use nonlinear mixed effects modeling ie nlme package in R or PROC NLMIXED in SAS
Hope it helps,
I
I do not have the definition of the response or a standard model for the dose-volume curve... their shapes are all depend on how the treatment plan being generated. You can imagine they are only two curves which should be quite similar to each other (doesn't matter what exactly the X and Y represent), but I would like to quantify how much they look alike in a statistical way. Of course comparison by selecting some points in each curve can be done but I would like to compare the curve as a whole.
Sorry I am a statistics geek, thanks Lstvan and Jochen!
Winky, if you want to "quantify how much they look alike in a statistical way" then you will need a well-defined statistical model. It is a bit like the wish to express an idea in a language when it is not absolutely clear in *which* language (note that in this allegory the language compares to the statistical model).
To my opinion, it will be most instructive to compare the curves visually.
If you vould like to compare you have to have a model which describes the relationship between X and Y, otherwise what will you compare?? Probably you missunderstand my point. For example you have an Emax model: E = (Emax*Dose)/(ED50*Dose) in this function Emax and ED50 is the parameters of your response-curve. Your goal is to detect difference (or be dare to confirm similarity) in one or both of these parameters. Its pretty easy...Emax is expressed as a+b*x or ED50 is expressed as a+b*x or both. Emax model was just an example you also can chose another model eg three parameter logistic, four parameter logistic and so on...but to compare you need to chose for both treatment the same structural part. So, you cant choose Emax for treatment A and 4 parm logistic for treatment B.
@Winky Wing Ki Fung
All the above scholarly suggestions are quite valid, however, your problem is not well-defined. Your latter response, "You can imagine they are only two curves which should be quite similar to each other" suggests that dose-volume in the two cases should be similar. In this case, tow methods:one parametric and a non-parametric may be used. In the first case differences in response variable in two curves may be considered and their significance may be tested as in bi-variate regression models. In the second case Chi-Square test may be applied.
I do not think there is a common approach that does not depend on what kind of curves You compare. But there are two approaches to solving similar problems that can be used.
1. The Kolmogorov-Smirnov and omega-squared tests, which are used for comparison of sample homogeneity. The essence of the criteria is to compare the distribution functions, which is a particular case of your problem.
2. Methods for analysis of residues in checking of the models adequacy. They suggest verification of the hypothesis that the model residuals have a normal distribution with zero mean and are independent random variables. In your case, as the residues You can used the differences of the functions for a set of the independent variable values.
In my case, I have several curves and I need to quantify their shapes. So, I am trying with statistical parameters like Kurtosis, RMS, Peak, etc.
My goal is to propose a simple identification method. I know that there are sophisticated methods in the literature, but... for now, it is not my way...
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