# Cross over design 2x2 with one baseline

I have an experiment that consist in two stages and 24 subjects between three groups.

On the first stage, subjects were managed for them to answer in a certain way given the group (growth curve model).

On the second stage, once our subjects had already learnt I want to test a different factor (Context) between groups. For that, I have a cross over design 2x2. The subjects where assigned to sequences of "treatment" (Context A and B) with 2 periods.

Since the first stage is a forerunner, I would like to add the last session of the first stage as a baseline. However I am having trouble with understanding how to do it.

My approach was to code the baseline as a level of Context, and at the Period factor set 0 as baseline.

The data is like this:

A7 is the baseline measurement.

I run the next model that gave me a really nice adjust $$R^2=0.88$$

M<-lmer(Response~Period+Context*Group+Secuence
+(1|Subject:Secuence), data=Prueba)


I´m new in this kind of designs, so I not sure if my model is valid.

I know that in the way I define my model I have problems with interactions since Period 0 just have "Context"=baseline, and in Periods 1 and 2 just have levels of Context A and B. But I´m just interested in making planned comparisons.

Is it correct to add baseline in this way?

• The way you use the term "baseline" is a little confusing to me. Can you explicitly write out your research questions? That would help me understand whether the model you are running is answering your questions. Jan 11, 2020 at 19:36
• My research cuestions are: Is there a difference between the responses at Context A, and Context B in each group ? And, Is there a global effect of the Context factor? As baseline, I mean a reference point of the responses before testing the factor of Context. I thought that I could use the last session of the first stage as reference of what the subjects learned in the trainning stage. Am I misunderstanding the term baseline? Jan 11, 2020 at 23:39

Some thoughts based on your research questions...

Is there a difference between the responses at Context A, and Context B in each group?

The lmer model you posted addresses this through the Context*Group interaction - you get a main effect of context for those in the reference group (note that the reference group should have Group==0), a main effect for Group (at Context==0), and then the interaction - how the effect of Group varies by Context for the non-zero categories of Group and Context.

Is there a global effect of the Context factor?

Here you are asking whether Context has a main effect - are different levels of Context associated with more or less of the outcome. To estimate this, I would do the following after running your lmer model:

require(multcomp) #for testing main effect of a coefficient, or linear combinations
summary(glht(M,  linfct = "Context = 0")) #test of Context main effect


Regarding your full lmer model, why does Sequence appear in both the fixed and random parts of the model? For the random part of the model, you specify (1|Subject:Secuence). This means you are specifying a random intercept that is the interaction of Subject and Sequence. That seems strange to me. I would think you want Subject to have it's own intercept (i.e., (1|Subject)) and then perhaps the interaction of subject and sequence (i.e., + (1|Subject:Sequence)). See here.

However I do not understand how Sequence plays into your design, so I cannot say for sure. You do say,

On the first stage, subjects were managed for them to answer in a certain way given the group (growth curve model).

This suggests that you have a growth component, and if so, then I would expect to see Sequence in both the fixed and random parts of the model. But not in the way you have it. Instead, it would be represented as a slope allowed to vary across groups (i.e., (Sequence|Subject)).

As baseline, I mean a reference point of the responses before testing the factor of Context. I thought that I could use the last session of the first stage as reference of what the subjects learned in the trainning stage.

Currently your baseline measurement is part of your dependent variable. If sequence is coded such that the last session from the first stage == 0, then it is merely the starting point for your growth curve. That is fine. Your current lmer syntax does not estimate a growth model, however.

Often people use a baseline measure as a control variable. For example, in a two time point design where you have "pre" and "post" scores. Since you have multiple measures of the same thing for each individual, however, you could not model the baseline as a covariate.

• Thank you very much for your response and your follow-up. I'm afraid I didn't have enough clarity in describing my experiment. I modeled the first stage separately, in the way you pointed out, taking the time component as fixed and random (random slope). My questions are just for the second stage, where I have 2x2 cross over design. I have only found models where the baseline is added as a covariate. I tried that but the fit was not good, so I thought the model I wrote earlier could be an option. Jan 13, 2020 at 2:11
• By sequence I mean the order in which the subjects were assigned to both treatments (in this case Context). The subjects were placed in context A, and then in context B, in sequences AB or BA. I see your comment about the random part of the model. I got wrong the specification. Only are needed random intercepts (1|Subject). And thanks a lot for the code. It was very helpful. Jan 13, 2020 at 2:18