I examine feeding behavior of an insect for several hours. One behavior is phloem ingestion. It has been postulated that the first instance of phloem ingestion is on average shorter than the second, and that is shorter than the third, and so forth. I take these measurements for several insects in several treatments. Given that the hypothesis that the duration of subsequent events is longer than prior events is true, and given that I cannot guarantee that all insects have the same number of phloem ingestion events, are any statistical methods valid?
Of course the interpretation of the mean changes to being the expected value at the midpoint of the number of instances of phloem ingestion. This applies to each insect. Since I cannot guarantee that all insects have the same number of events, I am less clear on the definition once I average these values over all insects because insects with more instances of phloem ingestion will invariably have a greater mean.
At the simplest level I just want to determine if there is a significant difference between treatments.
I should confess that I could use the literature and show that it is common practice to run any nonparametric test on these data. I am not yet comfortable with this practice.
The question is interesting.
What measure is actually relevant? The common practice to take the expected value at the midpoint might not be a sensible choice (I am not an entomologist!).
If I understood you correctly, a plot of the ingestion time against the instance number of an insect should give some increasing curve.It may also be useful to consider the times between events (if this is not constant).
The points can be smoothed by some regression model (linear or non-linear, one can even use splines).
It would now be interesting to know how this relationship (between time and instance) looks like. It is linear or is it some saturated course (i.e. does the time tend to a constant plateau value when the number of instances increases)? Or something else? Then: is there a reasonable model for this course?
If there is a plateu/saturation, then a resonable model could stating that the insect metabolism [or the insect-plant-system] reaches a steady state. In this case the "plateau ingestion time" would possibly be a sensible measure.
Also the slope or curvature of the curve (time vs. instance) could have a particular biological meaning and might be interesting to analyze.
If there is no model, then I would think that the ingestion time at some arbitrary (but constant) instance number should be taken. The course (time vs. instance) can still be fitted by some (regression)model, and this model can be used to get an estimate of the expected time at any instance. It is probably even possible to extrapolate the ingestion time to the "zero-th" event.
The data of all animals can be analyzed together in a mixed model approach. The fitted mixed model will then provide all estimates for between-group differences for all the parameters.
What are you wanting to compare?
From what you describe I think you want to determine if there is a significant effect size on feeding behaviour based on some treatments. Is it just total (or mean) time of feeding or interval between feeding, number of feeding events within a given time or volume of feeding?
Is it possible to fit trends to the increase in feeding time within each individual insect then test the coefficicents for those trends? This would measure the effect size of each treatment on feeding patterns in a more general way.
@ James The immediate goal is to show that there is a significant difference in the duration of phloem ingestion. A simple t-test would suffice if appropriate. I have 20 insects in each treatment. I am happy comparing either total duration of ingestion or mean duration of ingestion (mean for each insect). The experimental unit is the insect. However, the duration of phloem ingestion is not randomly distributed about the mean.
@ Jochen It is postulated that the relationship represents the adaptation of the insect to its new host. Thus, you are correct to assume that there would be a plateau at some point. However, the whole system changes over time. So we could wait 24 hours before recording, but then the plant has started to adapt to the relatively dark conditions of the laboratory. We could add lights, but the lights introduce electronic noise. We could add shielding, but the shielding reduces light levels.
Analysis of the slope is one possibility. While I get only a single number from each treatment, I could randomize the insects, calculate the difference in slopes: a randomization test approach. I could do a stratified randomization test: randomize within those insects with 3 events, and randomize within those insects with 4 events, ... and thereby correct for the influence of the number of observed events. Something to try at least.
Ah! Analysis of the duration of the nth event, rather than taking the average of all events. That would certainly avoid the problem. Great suggestion. Thank you Jochen. With 20 insects, some will only get to phloem ingestion once before the end of recording. Some will get there twice, and some will get there ten times. So there might be some issues with "missing data" as one increases the instance number.
I use a mixed model approach as implemented in Prooc GLIMMIX in SAS. GLIMMIX uses a restricted maximum likelihood approach. However, the current strategy is to average the duration for all instances rather than performing the analysis instance by instance.
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