Repeated Measures ANOVA vs multiple simple ANOVAs

Question

I have calculated plant growth rates at 2, 4, and 6 weeks of growth and would like to answer the question, "at each of these time points, do the soil treatments affect plant growth rates?". I have been advised that I need to run a related measure ANOVA as I have sampled the same individuals multiple times, however, I am not convinced that my data will meet the assumptions or that this is the correct choice of test.

The plant growth rate for each time was calculated by taking the change in plant growth (from the start to end of that fortnight) and dividing by 14 days. So the week 2 growth rate was calculated by taking each individuals day 0 height from its day 14 height and then dividing by 14 days. The same was done for week 4 heights but using day 14 to day 28. And so on.

My issues are:

• I do not have the same number of observations per treatment (some treatments resulted in higher germination success than others), nor are they consistent through time (some plants died throughout the course of measurements). As such, it appears that my I may be violating the need of a balanced design (or is this simply a missing values issue? I can't get my r {anova_test} function to run using my data for this reason)
• I am not so interested in if plant growth rates change between times (this is to be expected), but more so if treatments have an affect at each time (three seperate questions; that is to say, at week 2, was there a sig difference?; and week 4, was there a sig difference?...), hence why I am more inclined to do multiple simple ANOVA instead (one for each time). Would this be suitable, or is it still improper to do so given the same individual was measured multiple times for the same variable (despite one one measurement per individual being used per ANOVA)?

Answer 1

It is very difficult to understand plant growth conditional on the plant not dying. I would opt for an unconditional analysis of an ordinal outcome Y where the lowest level of Y codes for death and all other levels are actual plant heights. That way you don't need to use a hard-to-interpret "missing" data imputation. Consider a longitudinal ordinal model as discussed here - either a first-order Markov model or using random effects.

It is generally not a good idea to model differences, because in doing so one must have perfectly transformed Y before the subtraction. It's usually better to model trajectories, then after fitting the model to compute any contrast of interest. But the simplest thing to assess, from the fitted longitudinal model, is the plant height at the final time point, adjusted for the baseline height. This captures growth.

Answer 2

As Frank Harrell says in his answer (+1), "It is very difficult to understand plant growth conditional on the plant not dying." That makes it hard to address your interest in whether treatments have different effects on "plant growth" at different times.

Would you, for example, be interested in finding rapid growth of a single plant even if all of the others treated the same way failed to germinate or died in the interim? Or do you care more about something related to net biomass production, like the sum of living plant heights? Think carefully about the specific question you are trying to answer.

You certainly don't want to do "multiple simple ANOVAs" in the way you propose. For one, they would ignore the deaths. For another, you are much better off modeling the actual plant heights as a function of time and treatment first and examining differences ("contrasts of interest," as Frank Harrell put it) later. And, as you note in the question, multiple ANOVAs would ignore the correlations within individual plants that should be taken into account.

Standard repeated-measures ANOVA won't work either, so you need some form of regression model that takes those correlations and the deaths into account.

Frank Harrell's suggestion for an ordinal outcome model with the lowest outcome level representing death is an interesting way to approach the problem of modeling both growth and death together, which I hadn't previously considered. His course notes and book provide more details on both longitudinal data analysis and ordinal outcome models.

There are other ways to model longitudinal data and events (deaths) jointly, with this paper about the R JM package providing some useful literature references and a way to proceed. But any approach will still come up against what you mean by "treatment effects on plant growth" when the treatments might also affect death.