Supplementary section 9.3: Are subjects in the same block really independent?

In Chapter 5 we examined in detail the importance of designing studies so that the replicates in the samples we wish to compare (and use in our statistical analysis) are independent of one another. In a randomized block design we will always be measuring multiple subjects within the same block; doesn’t that leave us with problems of pseudoreplication? Surely individuals within the same block are not independent of one another? This is an important issue, so let’s use an example to explore it further. Many recent studies using a variety of animals show that if you restrict their food intake this has the surprising effect of increasing their lifespan. Now suppose you want to test this using nematode worms. Your study will run like this. 100 worms will be placed individually in separate petri dishes with a supply of food. For 50 of the worms, the amount of food will be excess to requirements, whilst for the other 50 it will be restricted, and worms will be allocated at random to their treatments. The lifespan of each worm will be measured. So far, this is just a simple, one-factor fully randomized design. However the laboratory incubators that we keep our worms in can only hold a maximum of 60 petri dishes, so we are going to have to run our study using two incubators. Individual incubators will differ in lots of ways, and some of these differences may have an effect on worm lifespan. To avoid confounding any treatment effects with differences between incubators we sensibly decide to allocate 25 worms of each of the two treatments to both incubators, giving us a randomized block design with incubator as our blocking factor. Now let’s think about independence. In Chapter 5 we illustrated the idea of independence as the situation where two experimental subjects are expected to be more similar to one another than a random pair of subjects. Let’s assume that there are differences between incubators that do affect worm lifespan such that, all things being equal, worms in incubator A live slightly longer than those in incubator B. This means a random pair of worms both drawn from incubator A will be more similar to one another (on average) than a random pairs of worms drawn from across the whole sample, so by this definition worms in the same incubator are not independent. Does this mean that if we use our individual worm lifespans as independent measures to compare treatments we will be pseudoreplicating? The answer is no, and the reason is that when we analyse our randomized block design properly (with ‘incubator identity’ included as a factor), our treatments are being compared within incubators. So ad-lib fed worms in incubator A are being compared to restricted worms in incubator A, and not to restricted worms in incubator B. Similarly, ad-lib worms in incubator B are being compared to restricted worms in incubator B, and not to those in incubator A. Remember, this is exactly why the inclusion of the blocking factor in the design increases our power, the incubators are adding to the between-subject variation in lifespan among all the worms in the study, but by comparing worms within incubators this source of variation is removed, and the problems of non-independence are removed at the same time because worms within the same incubator are independent with respect to other worms in the same incubator (with which they are being compared) - just not to worms in other incubators (with which they aren’t being compared). Obviously, if we had made the mistake of putting all the ad-lib worms in incubator A and the restricted worms in incubator B we would be back to comparing treatments by comparing worms in different incubators, and all the problems of non-independence would return.

All of this makes an important assumption, that any confounding effect of the incubators on lifespan applies in the same way to worms in both treatment groups. If you are familiar with statistical jargon, you may recognize this as an assumption that there is no block-by-treatment interaction. That is, if food restriction increases longevity by a given amount, that amount (i.e. the treatment effect) will be constant across both incubators (i.e. will be independent of block); even if, on average (regardless of treatment), longevity differs between blocks. That is, in statistical jargon, there may be main effects of ‘treatment’ and ‘block’ but there is no interaction between the effects of these two factors. If this is not the case then the effect of the treatment measured in one incubator may not apply if it is measured in another incubator and the measures of individual worms in each incubator no longer provide us with independent measures of the general effects of the treatment. Each incubator then provides us with a single independent measure of general effects of the treatment. Going into much more detail on this issue is beyond the scope of this book. But it is important to keep in mind that the issue is really biological, and whether it is likely to be met for your study system is something you will need to think hard about, and be prepared to argue your case with a cynical Devil’s Advocate.