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

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.

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.