# Experimental Design: Number of treatments and replications

I am learning about experimental design. I have been provided the following definitions:

• Factors: Variable(s) manipulated by researchers in order to observe response(s)
• Levels: Values of factors
• Treatments: Combinations of level of factors, applied to experimental units

I am trying to apply this knowledge to the following problem:

To study how different nutrients effect plant growth, 120 seedlings were placed in individual pots. The 120 pots were randomly divided into 3 groups of 40 each and a different plant nutrient (A, B, or C) was applied to the soil in the pots. After 10 days of growth, 10 pots were randomly selected from each group and the plants weighed. The same was done at 20 days, 30 days and 40 days of growth and the weight of each plant measured and recorded.

I am trying to determine the number of factors and treatments. Here is my thinking:

• There are 2 factors ("Plant Nutrient" and "Growth Time"). There are 12 Treatments (A10, A20, A30, A40, B10, B20, B30, B40, C10, C20, C30, C40). In this case, there are 10 replications in each treatment.

However, I wonder if instead I should see it this way (because our main goal is to see the differences between the plant nutrient, and we are simply taking into account different times within that):

• There is 1 factor ("Plant Nutrient"). There are 3 Treatments (A, B, C). In this case, there are 40 replications in each treatment.

Which of these approaches may be more sound? And if neither, how should I approach it differently?

You could start by writing down a model, which could be: $$Y_{ijk} = \mu +\alpha_i+\beta_j + (\alpha\beta)_{ij}+\epsilon_{ijk}$$ where $i=A,B,C$, $j=10,20,30,40$ and $k=1, \dots, 10$. $(\alpha\beta)$ represents the interaction.
It could well be that interest does not only center on the nutrient, but also in differences in the growth curves, so it could be of interest to parametrize the model differently, with a covariate $x$ measuring days of growth (with values 10, 20, 30, 40) like this: $$y_{ijk} = \mu +\alpha_i +\beta x_{ik} +\epsilon_{ik}$$ (version without interaction ... ). Here $i=A,B,C$, $k=1,2\dots,40$. In this version we only assume linear growth, you can augment it with a quadratic term, or replace it with a spline model.