# What is a block in experimental design?

I have two questions about the notion of block in experimental design : (1) What is the difference between a block and a factor ? (2) I tried to read some books but something is not clear: it seems that the authors always assume that there is no interaction between the "block factor' and other factors. Is it right, and if it is, why ?

1. The block is a factor. The main aim of blocking is to reduce the unexplained variation $(SS_{Residual})$ of a design -compared to non-blocked design-. We are not interested in the block effect per se , rather we block when we suspect the the background "noise" would counfound the effect of the actual factor.
We group experimental units into "homogeneous" blocks where all levels of the main factor are equally represented. The analysis of variance of a Randomized Control Block design splits the residual term of an equivalent single factor Complete Randomized design in block and residual components. We should note, however, that the latter component has fewer degrees of freedom than in single factor CR designs, leading to higher estimates for $MS_{Residual} = {SS_{Residual}}/{d.f.}$.
The decision to block or not to block should be made when we reckon that the decrease in the residuals will more than compensate for the decrease in d.f.

2. Usually an additive model is fitted to RCB design data, in which the response variable is an additive combination of the factor and the block effects and it is assumed that no interaction exists between the two. I think this is accounted for by the fact that RCB does not enable us to tell apart the interaction BxF from the within Block variability and the variability within experimental units. The bottom line is that we have to assume no interaction since we can't measure it. We can test whether it is present either visually or with Tukey's test, though.

A good resource on experimental design is this.

• (+1) Another good read is Montgomery's Design and Analysis of Experiments. – chl Jan 9 '12 at 14:58
• Thank you @chl. Montgomery's was on my shopping list but I chosed not to buy it since it was more geared towards engineering than ecology. I have noticed a new edition is due to be published on April, 2012, will you update your R companion to it? – Charlie Jan 9 '12 at 17:49
• Thanks everybody. I rather have a mathematical mind, then I have difficulties to read books such as the one of Montgomery in which there is too much text and not enough mathematics – Stéphane Laurent Jan 9 '12 at 18:38
• @Charlie Yup, that's a project that dates back to 2006 when the Doe CRAN Task view didn't exist at all. I will continue to work on the 6th version with the hope of finishing it this year (but I say that each new year, so...). Besides the 'biased' field of application, I still think the text remains excellent for psychologists and biologists. – chl Jan 9 '12 at 20:04
• @Stéphane I can suggest to have a look at Plane Answers to Complex Questions, by Christensen: less DoE, more math, and a nice intro to Linear Models. – chl Jan 9 '12 at 20:09

Here is a concise answer. A lot of details and examples might be found in most documents treating the design of experiments; especially in agronomy.

Often, the researcher is not interested in the block effect per se, but he only wants to account for the variability in response between blocks. So, I use to view the block as a factor with a particular role. Of note, the block effect is typically considered as a random effect. Finally, if you expect the 'treatment effect' to differ from block to block, then interactions should be considered.

Here's a paraphrase of my favorite explanation, from my former teacher Freedom King.

You are studying how bread dough and baking temperature affect the tastiness of bread. You have a rating scale for tastiness. And let's say you're purchasing packaged bread dough from some food company rather than mixing it yourself. Each baked loaf of bread is an experimental unit.

Let's say that you have 2 doughs and 8 temperatures, you can fit 4 loaves of bread in the oven at once and you want to run $n=160$ loaves.

In a completely randomized $2\times2$ factorial layout (no blocks), you would completely randomly decide the order in which the breads are baked. For each loaf, you would preheat the oven, open a package of bread dough, and bake it. This would involve running the oven 160 times, once for each loaf of bread.

Alternatively, you could treat oven run as a blocking factor. In this case, you would run the oven 40 times, which might make data collection faster. Each oven run would have four loaves, but not necessarily two of each dough type. (The exact proportion would be chosen randomly.) You would have 5 oven runs for each temperature; this could help you to account for variability among same-temperature oven runs.

Even fancier, you could block by dough as well as oven run. In this design, you would have exactly two of each type of dough in each of the oven runs.

When I have time to think it through, I'll update this further with the appropriate fancy names for those experiment designs.

• BTW, it is a $2 \times 8$ factorial treatment structure, not a $2 \times 2$. – Dennis Jul 11 '14 at 16:24
• Is this an example of incomplete blocking? – SmallChess Sep 18 '16 at 4:22

Experimental designs are a combination of three structures:

1. The treatment structure: How are treatments formed from factors of interest?
2. The design structure: How are experimental units grouped and assigned to treatments?
3. The response structure: How are observations taken?

Blocks are "factors" that belong to the design structure (to distinguish, it's not a bad idea to call them "blocking factors" vs "treatment factors"). They are good examples of nuisance parameters: model parameters you have to have and whose presence you must account for, but whose values are not particularly interesting. Please note that this has nothing to do with the nature of a factor -- blocking factors may be fixed or random, just as treatment factors may be fixed or random.

My personal rule of thumb regarding where a factor belongs in an experimental design is this: If I want to estimate the parameters associated with the factor and compare them either within the factor or other factor parameters, then it belongs to the treatment structure. If I don't care about the values of the associated parameters and don't care to compare them, the factor belongs to the design structure.

Thus, in the bread example elsewhere in this thread, I have to worry about run-to-run differences. But I don't care to compare Run 1 vs Run 24. Oven run belongs to the design structure. I do want to compare the two dough recipes: recipe belongs to the treatment structure. I care about oven temperature: that belongs to the treatment structure, too. Let's build an Experimental Design.

The Design Structure has one factor (oven run, Run), and the Treatment Structure two factors (Recipe and Temperature). Because every run has to be a single (nominal) temperature, Temperature and Run must occur at the same level of the experimental design. However, there is space for 4 loaves in each Run. Obviously, we can choose to bake 1, 2, 3 or 4 loaves per run.

If we bake one loaf per run, and randomize the order of Recipe presentation we get a Completely Randomized Design (CRD) Structure. If we bake two loaves, one of each Recipe per Run, we have a Randomized Complete Block Design (RCB) Structure. Please note that it is important that each Recipe occur within each Run. Without that balance, Recipe comparisons will be contaminated by Run differences. Remember: the goal of blocking is to get rid of Run differences. If we bake three loaves per Run, we would probably be crazy: 3 is not a factor of 160, so we will have one or two different-sized blocks. The other reasonable possibility is four loaves per Run. In this case we would bake two loaves of each recipe in each Run. Again, this is a RCB Structure. We can estimate the within-run variability using differences between the two loaves of each Recipe in each Run.

If we choose one of the RCB Design Structures, Temperature effects are completely randomized at the Run level. Recipe is nested within temperature and has a different error structure than temperature, because each dough appears within each run. The contrasts looking at recipe and recipe by dough non-additivity (interaction) do not have run-to-run variability in them. Technically, this is called variously a split-plot design structure or a repeated-measures design structure.

Which would the investigator use? Probably the RCB with four loaves: 40 runs vs 80 vs 160 carries a lot of weight. However, this can be modified -- if the concern is home ovens rather than industrial production, there may well be reason to use the CRD if it is believed that home bakers rarely bake multiple loaves.

• I do not follow your analysis of the bread experiment, perhaps because several different designs of that experiment were mentioned and you do not specify which one(s) you are referring to. That makes most of your comments confusing rather than illuminating. If you could clear this up I believe your answer would stand out. – whuber Jul 11 '14 at 13:00
• The importance of #2 deserves to be brought out. Analysis can be carried out based on the random assignment of experimental treatments: blocks represent restrictions on that random assignment. – Scortchi - Reinstate Monica Jul 11 '14 at 13:28
• @whuber That's because I wasn't analyzing it, I was designing an experiment from those parameters de novo. Clarified in the edit. – Dennis Jul 11 '14 at 14:46

I think most of the time it’s just a matter of convention, likely proper to each field. I think that in medical context, in a two factors anova one of the factors is almost always called "treatment" and the other "block".

Typically, as ocram says, the block effect will be a random effect, but I don’t think this is systematic. Let says you want to assess the effectiveness of different medical treatments:

• First design: each patient takes only one treatment, and the efficiency is measured on an appropriate scale. You suspect that the sex of the patient is of interest: you will have a "block" of male and a block of female patients. In this case, the block is a factor with a fixed effect.

• Second design: each patients tries all treatments at different moments. As there is some variability between patients, you consider each patient as a "block". You are interested in the existence of such a variability in the population, but not in its value in these particular patients. In this case, the block is a factor with a random effect.

Well, I only teach this stuff, trying to stick with the conventions of the domain (in France) as I got them from textbooks, but I never participated to a clinical trial (and don’t want to)... so this is just my two cents...!

• It seems to me that blocking denotes a more general strategy to reduce experimental errors (by accounting for the within-block homogeneity), one manifestation of this approach being the use of repeated measurements. I mean, blocking as seen in RBD is a way to combine randomization and control for potential confounder(s) at the level of statistical units. Blocking is also used when we cannot perform a complete replicate of a ($2^k$ or other) factorial design in a single block. – chl Jan 9 '12 at 13:05
• @chl I think you’re right! I was just giving some very basic examples... – Elvis Jan 9 '12 at 15:48
• They are good (one example with a yield/crop experiment from agronomy would have make it even clearer); I was just pointing out that "blocking" extends beyond the "factor" concept and the fixed vs. random distinction. – chl Jan 9 '12 at 20:00
• @chl, if you have a good (and simple) reference on this kind of stuff I’d be glad to read it (and you should post as an answer)... (not so simple references accepted as well!) – Elvis Jan 9 '12 at 20:05
• You mean example(aov) or the R agricolae package? :-) – chl Jan 9 '12 at 20:11