It is common in experimental design to consider a treatment as factor an make a factorial analysis (ANOVA). For instance, to simulate N deposition, it is frequent to fertilize with one or few quantities of nitrogen and make replicates (like 1, 2, 3, 4, with 6 replicates: total 24 quadrants/pots).
Is it sound to change this design by a non factorial design and, while excluding replicates, simulate a continuous variation of the independent variable and analyse data by means of a GLM? Being ANOVA a particular case of LM, and LM a particular case of GLM I guess it is sound, but I would appreciate an expert opinion. In example, instead of planning the experiment described above, using the same ammount of quadrants/plots, i.e. 24 (same power, I think), fertilize with 0, 0.2, 0.4, 0.6, ... , 4.6 each.
Thank you.
Hey Ciro,
your question sounds like a regression model to me. Remember that the ANOVA is a member of this family. If you have a linear predictor, use it in a regression. If you have interacting factors on different levels, multi-level models would be the next suggestion.
Best, René
Dear Ciro, as René mentioned in the second experiment you would be doing a Regresión Analysis. It has some advantages and some weaknesses . Because you have 24 differents levels of X you could adjust a polynomial of greater order (or some non linear model) than in the first experiment where you only could adjust a polynomial of order 3 (4 levels of N), but the first experiment admits the compute of a lack of fit test that may detect a gross inadecuacy of the model. To produce such a test you need replications of the levels of N. It would be better to use 6 of 12 different levels of N with respectively 4 or 2 replications for level. The first experiment is not a factorial because you have only one factor.
Yes!
If your aim is to find/investigate the relationship between the amount of fertilizer and the response variable, your suggested design is the best (given you have 24 plots). It is the best design to find a possible linear range of the relationship, or if the relationship may rather be described by some other function like an exponential or logistic function. It is the best you can do to identify an optimum for the fertilizer concentration.
The used concentrations do not even need to be equally spaced. You may use smaller steps in regions where you expect faster changes in the response.
In many cases using a LM (a regression model assuming normally distributed residuals) will work. Exceptions are cases where the response is dichotomous, ordinal, counts, proportions, rates, concentrations or any other bounded variable with values relatively close to a boundary. For concentrations using log-transformed response values in the LM is usually appropriate. For variables like proportions and counts there are binomial and Poisson GLMs, for rates there are gamma-GLMs available (that would also work on concentrations when the response is not transformed).
Whenever it is experimentally feasable to make measurements for many different values of a predictor (like your fertilizer conc.) then use as many different values as possible and treat the variable (fertilize conc.) as a quantitative variable in your model! The only reason against such a design are experimental limitations. For instance, if you would analyze the impact of CO2 in chambers over a couple of months, and you have only a limited number of chambers. Then you can analyze several different plant in each chamber and get several (replicate) values for each of the few fixed CO2 concentrations. But if you had as many chambers as plants I would put a single plant in each chamber and use a (slightly) different CO2 conc. in each chamber!
I concur with Jochem - please do treat it as continuous! You can then explore different functional relationship with the DV. Good luck!
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