Multiple Regression model from experimental design

I'm trying to figure out optimal values for the variables pH, air velocity and light intensity for eight growth chambers. The factorial design is as follows:

The weight (gram) for each chamber is:

How do I go from experimental design to a linear regression model?

Is it correct to use the same values for the dependent variables in the experimental design as in the model i.e. -1,1 etc? How do I make use of replicates in the model (each growth chamber has several response values)?

A = matrix(c(-1,    1,  -1,-1,  -1, -1,-1,  1,  1,1,    1,  1,-1,   -1, 1,1,    1,  -1,1,   -1, 1,1,    -1, -1), nrow=8, ncol=3)
df=data.frame(A)
y=c(0.5705, 0.4358, 0.5295, 0.1256, 0.4193, 0.5303, 0.289, 0.2818)

fit <- lm(y ~ X1 + X2 + X3, data=df)

• What do 3 plants mean? Are they the same kind or different kinds? May 18, 2017 at 17:35
• Its plants grown in a growth chamber. It's three different plants, except for in chamber 7 which only has 1 plant and chamber 8, which has 2 plants. May 18, 2017 at 18:12
• Given they are 3 different kinds of plants, it seems my model does not work for your data. Sorry. May 19, 2017 at 3:22
• Well, they are replicates. And some outliers can be taken out due to clogged machinery. Should be fine with using the highest weight for each chamber. What model did you think of using? May 19, 2017 at 11:42

It seems likely that each of the three variables (pH, air velocity, light intensity) has optimum values giving maximum weight, with weight decreasing away from those optima on either side - that is a graph of weight against each variable would be curved, decreasing on each side of a maximum. Indeed your question refers to "optimal values".

Your experimental design, with only two values for each variable, only allows fitting straight lines. In such a model weight increases indefinitely as the variable increases (or decreases, depending on the data).

A commonly used and parsimonious design giving three values in each variable is a "central composite design". For a three variable experiment such as yours this would usually imply at least 16 data rows ("chambers" in your parlance)

• 8 corner points; more or less corresponding to your 2-level factorial design, say +/- 1 in each variable.
• 6 axial points; for each variable +/- 1.4 (typically) in that variable and zero in the other two.
• 2 (duplicate) centre points; zero in all three variables.

This would allow fitting a "quadratic response surface" model with quadratics (linear and square terms) in each variable and all six 2-way interactions.

Such a simple model is quite limited, but has been found to be quite effective in many biological applications, and is hard to better for very small numbers of data rows ("chambers"),

Given they are replicates, here is the model:

$Y_{ij} = \beta_0 +\beta_1X_{i1} + \beta_2X_{i2} + \beta_3X_{i3} +\gamma_i +\epsilon_{ij}$

where $i = 1,2,...,8$ index the chamber, $j = 1,2,3$ index the replicate. $X_{i1}$ is pH, $X_{i2}$ is air velocity nad $X_{i3}$ is light intensity as displayed in you data table. $\gamma_i$ ~ $N(0,\sigma^2_c)$ is random intercept and $\epsilon_{ij}$ ~ $N(0,\sigma^2)$ is the error term and $\gamma_i$, and $\epsilon_{ij}$ are independent.

It is a mixed model. Next you need to find a way to estimate 6 parameters (4 $\beta$s and 2 $\sigma$s). Seems you use R, but I am not familiar with R. If you can run SAS, I can give you SAS code in 3 minutes.

BTW, by "replicate" I means you have no idea which one will be heavier among 3 plants in the same chamber when you plant them. Otherwise, this model is not good.