Paired or unpaired t test in two samples of horses and their wound size

Question

There is one population of horses. Assume two groups are randomly selected. Every member of each group has the exact same wound given to them.

You want to determine whether applying manuka honey to the wounds affects their size after a period of time. So one group has the wound left untreated, and the other treated with manuka honey.

The question is whether the two groups are independent in the sense that the wound size in the untreated group is independent or dependent on the wound size in the other group.

This will determine whether a paired or unpaired t test will be used.

It seems to me that the wound size from an untreated horse will be independent of the wound size of the treated horse. Yes both horses have a natural level of healing but this would be independent between the horses (they are not cloned pairs), and as they may have independent genetics and immune systems, possibly their environments will vary too, etc.

What I think is that the wound size of the untreated horse could vary in size, but this would not affect the wound size of a treated wound, and vice versa, and hence independent variables. The horses or wounds in the two different groups are not paired.

Answer 1

Since the described study does not entail repeated measures, where each horse is wounded twice, and treated once with honey the measurements of wound size over time are not paired.

Since each horse in the manuka honey treatment group does not have a specific horse matching on characteristics pertinent to wound healing (and perhaps to honey-reactivity) in the untreated group, the measurements of wounds size over time are not matched. (Aside: matching does not have to be strictly 1:1, but that's the conceptually simplest form of individual matching, and you aren't describing any other kind either.)

Since your measurements are neither paired nor matched, a paired t test is inappropriate, since it would bias standard errors and p values to be small. An unpaired t test would be more appropriate.

Since you are randomizing horses to two groups, the version of the unpaired t test assuming equal variances may be reasonable, particularly if your sample size is large enough.

Also I am tickled that your question and my answer discusses collections of horses in a fashion where "herd" would be an inappropriate collective noun. Tee hee.

Answer 2

Alexis is correct that the design that you describe is unpaired. However, you might like to consider a paired design that would probably be more powerful. It is worth noting that paired designs are often called "within subject" designs. If you give each horse two wounds and treat only one of them then the treated and untreated woulds would be paired. Including the pairing into the analysis is very useful because it often decreases the overall variability.