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Lets say I have 10 different treatments and these 10 treatments can be considered combinations of 2 factors - as in a factorial design experiment. What do I miss if I treat this data as coming from a complete randomized design instead of a factorial design? What will be the consequences?

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The randomization in a completely randomized design refers to the fact that the experimental units are randomly assigned to treatments. Even though a factorial design is very structured, you can still assign the experimental units to the levels randomly. This prevents bias due to the differences in your experimental units from being confounded with effects you want to observe. So, in answer to your question, there would be no consequences.

If you can't completely randomize the allocation of units to treatments but you can randomize it to some degree then there are sometimes ways to handle it in the analysis (see split-plot designs for example). Otherwise you're in a bad spot, and reasonable people might hold that any inference from your experiment is invalid.

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  • $\begingroup$ Hm, isn't the main problem the reduction of information you get depending on which treatment design you adhere to? What about if experimental units were randomly assigned to the 10 treatments. The treatment design could then either be considered as 10 separate treatments or as a combination of 2 factors as in a factorial design. Then I could run a ANOVA with either the 10 separate treatments or I could run a ANOVA with the two factors and the interaction, that is to say, the factorial design would yield information regarding the factors and information instead of just the treatments? $\endgroup$ – Throwthatbear Feb 24 '14 at 9:57
  • $\begingroup$ You're referring to two different ANOVA models. You can fit the cell means model where each combination of treatments has its own mean, or you can fit a factor effects model with various levels of interactions. But you're not losing information: the cell means will be the sum of the mean, factor effects, and interactions at the various levels. $\endgroup$ – neverKnowsBest Feb 24 '14 at 14:17

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