When comparing the growth of plants under 2 treatments, how to deal with dead individuals?

Question

I have carried a study of plant growth under 2 different treatments, each with 100 sample and measured the stem height after 3 months.

I then want to carry out a t-test to compare if there is any significant difference of plant growth (ie stem height) between the 2 treatments. But But a few plants died after 3 months and no stem height can be measured. 3 died in treatment A (out of 100 samples) and 8 died in treatment B (out of 100 samples), should I discard these data? Or should I put zero value of stem height for those dead plant when doing the y test?

Answer 1

This is a well-known application of principal stratification in the potential outcomes approach to causal inference.

You can get started with this article: Causal Inference Through Potential Outcomes and Principal Stratification: Application to Studies with “Censoring” Due to Death by Donald Rubin.

One of the main conceptual difficulties in a study like yours is that it is possible that one of your treatments causes plants to die faster than the other treatment. This seems plausible in your application since 3% died in treatment A and 8% died in treatment B. In particular, you would like to know/infer whether a plant that died in treatment arm B would also have died if it had been in the treatment arm A (and vice versa).

If you can infer whether the unit would live/die in each arm, then you can make progress on estimating causal effects for units that would survive under both treatments, die under both treatments, or survive in one but die in the other. The paper cited above provides a roadmap for doing that.

Answer 2

You could use a joint model, as a function of treatment, of the probability of survival (height > 0) and the probability distribution of plant heights given survival. This page and this page discuss general issues of modeling multiple 0 outcome values along with positive values. Both point to a paper by Min and Agresti, "Modeling Nonnegative Data with Clumping at Zero: A Survey," JIRSS 1: 7-33, 2002. Section 2.2 of the paper shows, in a situation like yours, that such a joint model is equivalent to separate maximum-likelihood modeling of the probability of height > 0 and the probability distribution of height given that height > 0.

Here, modeling height > 0 could be a binomial (e.g., logistic) regression and the other model could be a t-test restricted to the surviving plants.

In your data, survival fractions under the two treatments (97 versus 92 surviving out of 100 each), aren't distinguishable by the usual standard of $p< 0.05$. The logistic regression p value is 0.14; other binomial-family link functions and the R prop.test() and fisher.test() provide similar p values.

How you use the results of that model would then depend on your answer to the question that @whuber posed in a comment: what is your study question? Is it just the height of plants that lived, or is plant height a proxy for something like crop yield?

If it's mainly the height of living plants, then you might use the lack of "statistically distinguishable" survival rates to argue that limiting analysis to living plants is adequate.

If you are using plant height as a proxy for something like crop yield, then you probably should include the survival fractions in some way in your analysis. Including height values of 0 for the plants that died in a t-test might not meet the usual requirement for normally distributed errors around the mean values (more precisely, for normal distributions of the mean-value estimates), however, as the plants that died will have extreme deviations from the corresponding treatment means.

For evaluating height as a proxy for crop yield, there are several levels of complexity in how you might then choose instead to combine the two parts of the joint model. Simplest would be to scale the living-plant heights by the corresponding survival fractions to illustrate the basic issue and (perhaps) document that the survival differences aren't of practical importance if the height difference among the living is very large. That, however, would ignore the uncertainty in the survival-fraction estimates. You could instead sample repeatedly from the distributions of treatment-specific survival-fraction estimates and of heights given survival to get distributions of average heights over all plants, by including 0 height values for those that didn't survive.

This is also a good opportunity to examine alternate approaches, like Bayesian models or bootstrapping, that might be particularly useful in more complicated scenarios.