Estimating an interaction effect when the data is collected not in a usual way I am having a bit hard time analyzing the data I have.
59 participants all read two stories – one in Format A and one in Format B. In addition, participants each read one story about Birds and one story about Fish. Participants read TWO of the following four possible combinations, but only read ONE of each format and ONE of each topic:
Format A Bird or
Format A Fish
Format B Bird or
Format B Fish
We are interested in the main effect of format, the main effect of topic, and the interaction between them.
Each participant has scores of format A and B. But again, if a given participant read fish in format A then the topic should be bird in format B, and vice versa (bird in format A then fish in format B).
If we had the score for format and the score for topic, we would have analyzed it by using ANOVA or multilevel modeling, but the data structure is a bit strange to us so not sure how to analyze this data now to estimate the main effects and the interaction effect.
We would appreciate any advice.
Thank you.
 A: Your data (in long format) will look like
id    method     story     score
1       A         fish      .
1       B         bird      .
2       A         bird      .
2       B         fish      . 
3   ...
3   ... 

Assuming this structure, a mixed model can be written in R, lme4 syntax) as
lme4::lmer( score ~ method*story + (1 | id), data=your_data_frame )

A test with simulated dummy data (no effects, apart from the individual random effect) is
set.seed(7*11*13)#My public seed
your_data_frame <- data.frame( id = rep(1:50, each=2), 
                                method=rep(LETTERS[1:2], 50), 
                                story=rep(c("fish", "bird", "bird", "fish"), 25), 
                                score=rep(rnorm(50, 0, 1), each=2)+
                                      rnorm(100, 0, 2) )

 mod0 <- lme4::lmer(score  ~ factor(method)*factor(story) + (1 | id), data=your_data_frame)
> summary(mod0)
Linear mixed model fit by REML ['lmerMod']
Formula: score ~ factor(method) * factor(story) + (1 | id)
   Data: your_data_frame

REML criterion at convergence: 430.9

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-1.93603 -0.73508  0.07433  0.67022  2.52069 

Random effects:
 Groups   Name        Variance Std.Dev.
 id       (Intercept) 0.5919   0.7693  
 Residual             4.0048   2.0012  
Number of obs: 100, groups:  id, 50

Fixed effects:
                                  Estimate Std. Error t value
(Intercept)                       -0.16342    0.42880  -0.381
factor(method)B                    0.08652    0.60641   0.143
factor(story)fish                  0.08722    0.60641   0.144
factor(method)B:factor(story)fish  0.38252    0.91114   0.420

Correlation of Fixed Effects:
            (Intr) fct()B fctr()
fctr(mthd)B -0.707              
fctr(stry)f -0.707  0.564       
fctr(m)B:()  0.531 -0.751 -0.751  

there is no estimation problems.
