The motivation for this post was to create a pipeline for generating publication-ready plots entirely within ggplot and avoid post-generation touch-ups in Illustrator or Inkscape. These scripts are a start. The ideal modification would be turning the chunks into functions with personalized detail so that a research team could quickly and efficiently generate multiple plots. I might try to turn the scripts into a very-general-but-not-ready-for-r-package function for my students.
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A factorial experiment is one in which there are two or more factor variables (categorical \(X\)) that are crossed, resulting in a group for each combination of the levels of each factor. Factorial experiments are used to estimate the interaction effect between factors. Two factors interact when the effect of one factor depends on the level of the other factors. Interactions are ubiquitous, although sometimes they are small enough to ignore with little to no loss of understanding.
motivating source: Integration of two herbivore-induced plant volatiles results in synergistic effects on plant defense and resistance
What is synergism or antagonism? (this post is a follow up to What is an interaction?)
In the experiment for Figure 1 of the motivating source article, the researchers were explicitly interested in measuring any synergistic effects of hac and indole on the response. What is a synergistic effect? If hac and indole act independently, then the response should be additive – the HAC+Indole effect should simply be the sum of the independent HAC and Indole effects.
Some background (due to Sewall Wright’s method of path analysis) Given a generating model:
\[\begin{equation} y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 \end{equation}\] where \(x_3 = x_1 x_2\); that is, it is an interaction variable.
The total effect of \(x_1\) on \(y\) is \(\beta_1 + \frac{\mathrm{COV}(x_1, x_2)}{\mathrm{VAR}(x_1)} \beta_2 + \frac{\mathrm{COV}(x_1, x_3)}{\mathrm{VAR}(x_1)} \beta_3\).
If \(x_3\) (the interaction) is missing, its component on the total efffect is added to the coefficient of \(x_1\).
I was googling around and somehow landed on a page that stated “When effect coding is used, statistical power is the same for all regression coefficients of the same size, whether they correspond to main effects or interactions, and irrespective of the order of the interaction”. Really? How could this be? The p-value for an interaction effect is the same regardless of dummy or effects coding, and, with dummy coding (R’s default), the power of the interaction effect is less than that of the coefficients for the main factors when they have the same magnitude, so my intuition said this statement must be wrong.
The post motivated by a tweetorial from Darren Dahly
In an experiment, do we adjust for covariates that differ between treatment levels measured pre-experiment (“imbalance” in random assignment), where a difference is inferred from a t-test with p < 0.05? Or do we adjust for all covariates, regardless of differences pre-test? Or do we adjust only for covariates that have sustantial correlation with the outcome? Or do we not adjust at all?
