I am trying to make a correlation between phosphorus (P) and yield of wheat grown in sandy calcareous soils. I have three parameters (concentrations of P in plant, soil P and yield of wheat) and I need your help to make a curve then asses the critical value of P which gave the maximum yield. I want to asses this point if the correlation non liner. Could I asses the change point through the program (SAS)? How?
Please if you can inform me the details to make the curve and asses the critical value.
I want to know which phosphorus concentration in soil and plant gave the best crop yield. Thus, grain yield is the dependent variable.
I need to :
1- make the curve between P (in soil and plant) and yield
2- asses the critical P which gave the maximum yield
Some details about the data:
I collected wheat and soil samples from 12 fields (differs in their soil phosphorus concentrations), three samples from each field, then I measured the concentrations of phosphorus in soil and in plant tissues. At the end of growth season I recorded the grain yield. I collected this data for three years but I want to assess the optimum P in soil and plant overall years (n=108) ( 12 locations x 3 replicates x 3 years).
From OP's description, this is a very complex panel data, because he/she collected 3 samples from 12 fields over 3 years. Therefore, you have to use simultaneous equation system (SES) for best efficiency and accuracy, since OLS is definitely biased for heteroskedasticity.
Mamdouh, Please run SES for 3 replicates (samples) over 3 years for each field, so you will have twelve 3-equation models. Below, I will demonstrate how to model this correctly.
(L1 to L12): Yit=constant + X1it + X2it + error terms
Where, Y=yield, X1=P in soil, X2=P in plant, i=sample (1 to 3), t=time (1 to 3), L=Location
Notes: You might also consider running SES for each different sample of all 12 fields, because they might be correlated by unobservable factors. For example, three 12-equation models are shown below.
(S1, S2, S3): Yit=constant + X1it + X2it + error terms
Where, Y=yield, X1=P in soil, X2=P in plant, i=location (1 to 12), t=time (1 to 3), S=Sample
Please also refer to the SAS manual here: http://support.sas.com/documentation/cdl/en/etsug/60372/HTML/default/viewer.htm#etsug_syslin_sect017.htm
Before dealing with SAS, lets worry about the experimental design. I can see Juehui's take on the problem. However, I would suggest that what you have are 12 observations, 3 subsamples, and 3 years. Depending on how things were done, the 3 years are either repeated measures or replicates. That depends on the year-to-year correlation between field conditions.
So your fields give you a range of different soil P levels. Your subsamples let you estimate the within field variability (if you care). The years let you estimate year-to-year variation in outcome. To simplify things I would average the subsamples. I would see if repeated measures analysis is necessary. With a small bit of luck, it will not be. I can then run a standard OLS multiple regression model.
Now, all of this depends on there not being any missing details in the description of the problem. This will be a difficult study to get published. I could measure P in two fields and find that grain yield increases with decreasing P. What I failed to note was that the field with low P had high N, and nitrogen drives crop yield. Simpson's paradox: see http://optimalprediction.com/files/pdf/V3A25.pdf and http://optimalprediction.com/files/pdf/V1A13.pdf. However, maybe the fertilizer treatment supplied all other nutrients in abundance, so P was the only variable.
Once you have the regression equation, you can look at the equation and see if there will be any curvature (polynomials, log transformation, maybe others). If the range in your data is sufficient there will be curvature, but the curvature in your data may be so slight that it is undetectable. If there is curvature, then take the derivative of the regression line. If it is complex, you might have to take the second derivative or more. See Wikipedia for a brief discussion http://en.wikipedia.org/wiki/Maxima_and_minima. Look under the section on "finding local maxima and minima" (second paragraph) and the following "examples." Just dust off your old calculus book.
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