Cohen's d in split-plot experiments

by rvl   Last Updated October 09, 2019 17:19 PM

There is some demand for effect-size measures in doing post hoc analyses, and I am trying to decide whether to provide for this or not in an R package of mine. If I do, I would concentrate on Cohen's $$d$$-style measures, which in the context of comparing the $$i$$th and $$j$$th means in a simple one-way experiment is defined as $$d_{ij} = \frac{\mu_i - \mu_j}{\sigma}$$ Here, $$\mu_i$$ and $$\mu_j$$ are the means, and $$\sigma$$ is the error standard deviation of the data, assumed common for all treatments. I can figure out how to estimate $$d_{ij}$$ from observed data and construct a confidence interval for it that takes into account the uncertainty of estimates of all three parameters.

However, my questions have to do with extending these ideas to split-plot experiments. There is some discussion of this here, but that focuses on pre-post testing rather than a more general case. For concreteness, I'll focus on Yates's classic oat-yield experiment (data available as Oats in the nlme package in R). This experiment has six blocks (factor Block); each block is subdivided into 3 plots and randomly assigned to the three levels of factor Variety; and each plot is divided into four subplots and randomly assigned to the four levels of factor nitro. I have questions about two ways that these data could be analyzed.

Homogeneous mixed model

I think this is the easy question... In this model, we assume that the Block effects are iid $$N(0,\sigma_B^2)$$, the whole-plot effects (identify Block:Variety) are iid $$N(0,\sigma_P^2)$$, and the residual (subplot) effects are iid $$N(0,\sigma_E^2)$$. An additive model with fixed effects for Variety and nitro fits pretty well.

What I surmise is that we can estimate Cohen's $$d$$ values for either Variety comparisons or nitro comparisons as the observed comparison, divided by an estimate of the total SD, $$\sigma_T = \sqrt{\sigma_B^2 + \sigma_P^2 + \sigma_E^2}$$. Is that correct? Or do people use some other $$\sigma$$ as the reference (other than perhaps modeling blocks and/or plots as fixed effects)?

Multivariate model

Another possibility is to model the four observations on each plot (corresponding to the 4 levels of nitro) as a multivariate response variable. In that case, the Block effects, if modeled as random, have a multivariate distribution, and so do the errors. The plot effects are subsumed in these multivariate distributions. We can form meaningful comparisons of marginal means for Variety and nitro, as well as for combinations thereof, inasmuch as this model implicitly contains interaction effects.

But my question is, what, if anything, is a sensible reference value $$\sigma$$ for defining Cohen's $$d$$-style effect sizes? Each level of nitro has its own error variance. I can see that for certain interaction comparisons (comparing Variety with nitro held fixed at one level), this is clear-cut; but it is not at all clear to me what, if anything, defines a Cohen's $$d$$ for comparing levels of nitro, either marginally or at a fixed Variety. Can anybody enlighten me?

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