I have rapid light curve data (ETR for each PAR value) for 24 different specimen of macroalgae. The dataset has three factors: species (species 1 and species 2), pH treatment (treatment 1 and treatment 2) and Day (day 1 of the experiment and day 8 of the experiment).
I have fitted a model defined by Webb 1974 to 8 subsets of the data:
species 1,pH treatment 1, day 1
species 1, pH treatment 1, day 8
species 1, pH treatment 2, day 1...etc.
I have plotted the curves of the data that is predicted by the model. The model also gives the values and standard error of two parameters: alpha (the slope of the curve) and Ek (the light saturation coefficient). I have added an image of the scatterplot + 4 curves predicted by the model for species 1 (so each curve has a different combination of the factors pH treatment and Day).
I was wondering what the best way would be to statistically test if the 8 curves differ from each other? (or in other words: how to test if the slopes and Ek of the models are significantly different?). When googling for answers, I found many ways to check which models with your data better, but not how to test if the different treatments also cause differences in rapid light curves.
Any help would be greatly appreciated.
Cheers,
Luna
Hi
I think the best way is to fit your nonlinear model with the drm function of the drc library. In this library you can fit simultaneously multiple curve by groups. Afterward you'll be able to use the glht function of the multcomp library to test wether there is differences between you cruves for each model parameter. For more information see the publication by Ritz et al. 2015.
Let me know if it helped.
Cheers
https://cran.r-project.org/web/packages/drc/drc.pdf
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0146021#sec022
Hi Denis,
Thank you for your answer. Is it also possible to define your own function with this package? I can only find the predefined functions with the getMeanFunctions list.
Cheers,
Luna
yes it is. You simply have to define your function within the fct argument. But then you'll have to give starting values to the function. And the difficulty arise when you fit multiple curves at once because they may not all have the same starting values...
So 3 options :
- you're lucky and a common set of starting values is enough to fit your different curves
- you use a built-in function. Advantage of built-in function is that they are selfstarter (i.e. you do not need to give starting value). There is a great diversity and you may find something similar to yours
- you define your own self-starter function. this is a bit more tricky but still possible.
Dear Luna:
Using the Webb's formula, you can linearize the model using the following equation:
LN (1 - P/Pmax) = -alpha (I / Pmax)
So, You can compare slopes using a covariance test. Is easier than a non-linear comparison.
Dear Denis and Fernando,
Thank you both for your useful answers! With this, I can definitely solve it. Much appreciated.
Cheers,
Luna
Although the question is few years old, I’ll add an answer just in case someone finds this relevant.
For comparing between treatments, you need to fit single curve for each individual light curve measured, not a single curve for a treatment, as your graph above implies. Then you can extract the parameters for all the curves and compare these between the treatments, as suggested by Denis.
There is a nice R package called “phytotools” (Silsbe & Malkin 2015) which I have used and found very convenient. It has four different light curve models: Eilers & Peeters 1988, Jassby & Platt 1976, Platt, Gallegos & Harrison 1980 and Webb et al. 1974. The manual also provides the equations for the models.
As the functions for fitting are provided in the package, fitting the light curves is straightforward. The package has basic programming examples also. One needs to do a bit of coding to loop through the data containing the curves and fitting a model. Each of the measured light curves needs to have a unique id. When looping through the ids, you need to extract the parameters (e.g. alpha, beta, so on) from the model fitted, along with the info of your treatments for that particular curve into a data frame.
After this it should be easy to perform the analyses e.g. with ANOVA or something else, depending whether your data shows heterogeneity etc...
I am going to focus on the analysis of the coefficients. I would strongly recommend the methodology prescribed by Schabenberger to compare the coefficients statistically. "Statistical tests for hormesis and effective dosage in herbicide dosage responses" J Agron 1999, 91:723. I used this method in my study to evaluate the treatments. A sas procedure is used to compare the results. You can refer the "intrinsic changes in photosynthetic parameters of carrot leaves under increasing CO2 concentrations and moisture regimes" publication where this method is using for gas exchange curves. Good luck with your analysis.
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