How should I analyze this design with no replication?

Question

I have data from an experiment where I measured the $CO_2$ from:

• 4 different dissolved organic carbon (DOC) solutions
• each at 4 different concentrations
• each in light and in dark.

Each combination of DOC source, concentration, and light occurred only once (e.g., only one jar each of DOC-A: conc-A: Dark, and one jar for DOC-B: conc-A: Dark, etc...) but a series of 4 measurements was taken from each combination (i.e., jar).

The production of $CO_2$ was linear over the 4 time points so, the response variable can be represented as a single rate from each jar.

I am interested in the effect of light, DOC source, and concentration on $CO_2$ production, plus interactions. I am not necessarily interested in the effect of the different time pointsGiven that each experimental unit was unreplicated, how can I best analyze this experiment?

Answer 1

I agree with your assessment that the 4 measurements of a jar are pseudo-replicates, and the easiest way of dealing with it is to combine them into one value (I think you are creating a rate, but any meaningful combination could be used). Let's concentrate on modeling one $Y$ value per jar. Consider the ANOVA table for the sources of variability:

Source       Degrees of freedom
 DOC             4-1=3
 Conc            4-1=3
 Light           2-1=1
 DOC*Conc        3*3=9
 DOC*Light       3*1=3
 Conc*Light      3*1=3
 DOC*Conc*Light  3*3*1=9
 Residual        0
-------------------------
 Total           32-1=31 

As you can see, if you include all 2- and 3-way interactions, then the observed value for each jar is predicted perfectly, and there are no degrees of freedom left to estimate the residual variability. The only alternative to giving up at this point is to assume that there is no 3-way interaction, and use the corresponding term as the residual. The exact implementation depends on the software, but it could be done by hand from the ANOVA table. If this assumption is not reasonable a-priori, then you have to replicate the experiment.

Answer 2

I think you're believing "replication" in this instance to mean sampling a unit with the exact same concentration and light which is inherently continuously valued. Obviously, this is impossible to do with any moderately sensitive equipment. It's also something that arises in real life, no two units are the same, but intuitively one can pool adjacent data to make apples-to-apples comparisons between units.

You should handle this effect using methods for continuously valued data. This sounds to me like a repeated measures ANOVA problem or a linear mixed model, accounting for random effects due to "jar". If you adjust continuously for darkness and concentration, the model naturally borrows information across groups and estimates a consistent trend, regardless of whether there were replications within any one continuous measure.