# Are crossed factorial design and full factorial design the same concept?

About crossed factorial design, two factors A and B are crossed, if every level of A can occur in every level of B.

A full factorial/crossed experiment is an experiment whose experimental units take on all possible combinations of these levels across all such factors.

If I am correct about the above, are crossed factorial design and full factorial design the same concept?

• How do Graeco-Latin squares meet your definitions? – Henry Jun 25 '13 at 6:52
• "Crossed" means that the factors are formally independent, not nested. "Full factorial" = "all main effects + all interaction effects" between factors (plus covariate main effects, if any) – ttnphns Jun 25 '13 at 7:24
• You might also have a "fractional factorial design", for instance a $2^{3-1}$ design, meaning three variables but only four runs so inteactions cannot be estimated. Such a design is also "crossed" but it is not "full factorial". – kjetil b halvorsen Jun 25 '13 at 14:52
• @kjetilbhalvorsen; Thanks! why isn't " crossed design = full factorial design", given that "two factors A and B are crossed, if every level of A can occur in every level of B"? – Tim Jun 25 '13 at 19:43

(Crossed) factorial design is not always equal to Full factorial design.

In this context, the word "crossed" is a synonym of "factorial" (see here). Both indicates the presence of interactions between factors.

A factorial design can be either:

• Full (all possible interactions) or,
• Fractional (without all interactions, but some).

Therefore, a "factorial design" does not necessarily implies it is "fully crossed".

In the OP's case, where:

two factors A and B are crossed, if every level of A can occur in every level of B

It will be a full factorial design.

• Thanks! "two factors A and B are crossed, if every level of A can occur in every level of B." So isn't a crossed design full factorial? – Tim Jun 25 '13 at 16:47
• Yes, it is but you did not write the word "full" on question. And there is no need to use "crossed" and "factorial" together, because they are synonyms. – Andre Silva Jun 25 '13 at 16:49
• I meant why isn't " crossed design = full factorial design", given that "two factors A and B are crossed, if every level of A can occur in every level of B"? – Tim Jun 25 '13 at 16:53
• Then, What is the difference between a crossed factorial design and a full factorial design? Why are not igual designs? I don't understand. From a crossed factorial design is it possible to get the analysis of all interactions of the factors A, B, AB, AA, BB? Thanks. – cabita Jul 6 '14 at 14:47
• @cabita, the terminology crossed factorial design is redundant, because crossed and factorial means the same thing, i.e., an experiment has factors w/ usually 2 levels/factor. E.g., factor A = I (levels: with irrigation, without irrigation), factor B = F (with fertilization, without fert.). A full factorial design has all possible combinations between factors and levels (in this case $2^2$ --> I+F, no_I+no_F, I+no_F, no_I+F). Yet, a factorial design can be fractional, where there are interactions, but not all possible ones. – Andre Silva Jul 6 '14 at 15:15