Experimental Design with Factorial Block Structure I'm designing a language-learning study with maybe 30 participants. My collaborators want each person to experience both of the methods we're using, so we are planning a crossover design. In addition to the "method" factor (and a random effect for each person's aptitude), we have two nuisance factors to block by. One is language: we'll use two different languages to reduce bleed-through (effects carrying from one round of treatment into the next in our crossover design). The other is order: half the people do method A followed by method B; half do the reverse.
We could have four types of people:


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*Swedish Method 1 followed by Norwegian Method 2

*Swedish Method 2 followed by Norwegian Method 1

*Norwegian Method 1 followed by Swedish Method 2

*Norwegian Method 2 followed by Swedish Method 1


Or we could have just two types:


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*Norwegian Method 1 followed by Swedish Method 2

*Norwegian Method 2 followed by Swedish Method 1


I have three related questions (sorry) (thanks). 


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*Can you help me conceptualize potential interactions here?

*Are either of these designs obviously bad for some reason? 

*If not, then which one is more efficient? 


In your answer, keep in mind that our goal is inference on which methods are "better," as measured by some yet-to-be-carefully-selected quantitative variables describing learning and learner preferences. I have looked through Gary Oehlert's experimental design textbook (pdf), especially ch. 13, and I read a few suggested questions here on CV. 
 A: For a simplified analysis, suppose that the data $Y$ come from this model. To define notation,


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*$X_S = 1$ for Swedish, 0 for Norwegian 

*$X_M = 1$ for Method 1, 0 for Method 2

*$X_1 = 1$ if measurement was taken first, 0 if second


and  $\beta_0$ is the average score for Method 2, Norwegian, second. Suppose average outcomes for the other conditions conform to
$$E[Y(X=x)] = \beta_0 + \beta_M x_M + \beta_S x_S + \beta_1 x_1.$$
If we use four types of people as in the question (one of each) then $X = [\ 1\ |\ x_M\ |\ x_S\ |\ x_1\ ]$ has rank 4, with 8 rows (2 per person) and 4 columns. 


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*Swedish Method 1 followed by Norwegian Method 2: 1111 and 1000 

*Swedish Method 2 followed by Norwegian Method 1: 1011 and 1100

*Norwegian Method 1 followed by Swedish Method 2: 1101 and 1010

*Norwegian Method 2 followed by Swedish Method 1: 1001 and 1110


We have 8 measurements and 4 coefficients to estimate (not counting variance parameters).
If we use only two types of people, language is aliased with time of measurement, so $X$ has rank 3. (The last two columns sum to the first column.) 


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*Norwegian Method 1 followed by Swedish Method 2: 1101 and 1010

*Norwegian Method 2 followed by Swedish Method 1: 1001 and 1110


We can recover the quantity of interest $\beta_M$ by estimating the model
$$E[Y(X=x)] = \beta_0 + \beta_M x_M + \beta_{N \& 1} x_{N \& 1}$$.
This has one more degree of freedom left over for the error, so all else being equal, it's a tiny bit more efficient.
All else may not be equal. Suppose the true data-generating mechanism takes the form 
$$E[Y(X=x)] = \beta_0 + \beta_M x_M + \beta_S x_S + \beta_1 x_1 + \gamma x_M x_S x_1,$$
so that people do slightly better when they learn Swedish first by method one. In an analysis neglecting interactions, this should hopefully show up as a slight increase in $\beta_M$; the improvement is proportional to $\gamma Pr(x_S x_1 = 1\ | \ x_M)$. The four-types-of-people design has this property, but the two-types doesn't: if Swedish is always second, we never even see this particular interaction.
