What's the difference between a randomized block design and two factor design?

Question

What's the difference between a randomized block design and a two factor design? They both use two-way ANOVA, and your blocks can be your factor.

Answer 1

In both cases, you have two categorical variables and numerical response variable but in a randomised block design the second variable is a nuisance variable, while in the two factor factorial design the second variable is also of interest and you would like to understand the interaction. I think this is the main difference.

It's a bit confusing because ANOVA is really a family of methods and two way ANOVA can refer to two distinct but related models. I'll try to illustrate this with examples and a bit of maths.

Suppose you were studying 3 different types of barley, this is your treatment variable $\alpha_i$. You want to determine their typical yield in terms of tons per hectare, this your response variable $x_{ij}$. Yields wills vary based on local conditions so you pick 10 different fields and split each one into thirds randomly assigning assigning one wheat variety to each 1/3 of a field. The fields are your blocks $\beta_j$. There a still a bunch of things that you haven't controlled for like rainfall, soil type, pests, sunlight hours, and the calibration of your scales these are described by the measurement error $\epsilon_{ij}$. In this situation, you have a randomised block design. The model describing the yields is given by: $$ x_{ij} = \mu + \alpha_i + \beta_j + \epsilon_{ij} $$ where $i$ records the barley variety and $j$ records which field you are. This is the additive model.

Suppose again you are studying those 3 barley varieties ($\alpha_i$), but you are interested in the effects of soil salinity which can be low, medium or high. Soil salinity is another treatment variable $\beta_j$. Your response is again $x_{ij}$. Because you want to understand how yield is effected by the barley type and the soil salinity, for each different barley type you grow a sample with each different level of salinity. As you say, you can think of salinity level as being a block with respect to barley type and visa versa. This is a two factor factorial design. The model describing the yields is given by $$ x_{ij} = \mu + \alpha_i + \beta_j + \alpha_i \beta_j + \epsilon_{ij} $$ where notice you have a term describing the interaction $\alpha_i \beta_j$ effect. If you were to just use the randomised block design, the interaction term $\alpha_i \beta_j$ would but lumped in with the error term $\epsilon_{ij}$. This is the interaction model.

This distinction between the additive and interaction models then carries over into the details of how you conduct the two way ANOVA test.

Answer 2

I agree with MachineEpsilon's answer but will clarify two issues. First, there is a design difference between the models even if the two-way ANOVA is estimated in the same way. With the randomized-block design, randomization to conditions on the factor occurs within levels of the blocking variable. That is, the sample is stratified into the blocks and then randomized within each block to conditions of the factor. In a two-way factorial design, the sample is simply randomized into the cells of the factorial design. Second, there are situations where you might be interested in the interaction between the factor and block in a block randomized design. This would assess whether the effect of the factor (e.g., treatment effect) differs across blocks (e.g., person's with different characteristics).