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I have the following data in the repeated measure :

Participant   Stimulus    Response
         #1          A          30
         #1          A          25
                  ...
         #1          B          10
         #1          B          13
                  ...
         #2          A          35
         #2          A          30
                  ...
         #2          B          55
         #2          B          65
                  ...

I would like to do linear regression of Response on Stimulus.

But the thing is, since if I ignore the effect of Participant, this can clear confound the result.

For some reasons I would like to exclude the use of mixed effects model, and I would like to figure out a way to evade the hazard of confounding.


Try

First Approach

I considered the following linear regression model,

Response ~ Stimulus 

and this gave me

Variable    Coefficient            p.value
       B             +1    not significant

Second Approach

When I added the participant as a factor to the model, i.e.

Response ~ Stimulus + factor(Participant)

and I got

Variable    Coefficient            p.value
       B             -1                ***
      #2             -2                ***
      #3             +3                ***
      #4             -4                ***
                    ...

Question

But the results confuse me. So I would like to ask the following :

  1. Which approach is better, in terms of robustness to confounding?

  2. How should I interpret the result of the second approach? Should I interpret B's coefficient "-1" as...

    • the effect of Stimulus B on expected Response with respect to Stimulus A as a reference group, when the effects of the variable Participant is statistically controlled?

    • the effect of Stimulus B on expected Response with respect to Stimulus A and Participant #1 as a reference group, when the effects of the variable Participant is statistically controlled?

  3. How should I interpret the coefficients of each factor of Participant? e.g. should I interpret #2's coefficient "-2" as...

    • the effect of Participant #2 on expected Response with respect to Stimulus A and Participant #1 as a reference group?
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    $\begingroup$ Why not use GEE to fit your model (as a more suitable alternative to mixed effects modelling, though it's not clear why you'd want to avoid mixed effects modelling in the first place)? With multiple observations per participant, the concern is that ignoring the correlation of Response values collected from the same participant in your first approach will lead to violations of model assumptions and render your inferences invalid. You can plot the residuals from your linear model fitted via the first approach against the participant id and see what I mean! $\endgroup$ Commented Nov 30, 2018 at 16:33
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    $\begingroup$ So, the first approach doesn't really make sense because it ignores the within-participant correlation of the Response values. The second approach would only make sense if the participants included in this study were the only ones you were interested in and you had no interest in generalizing the findings of your study to a larger set of participants of which the ones in your study are representative. If that is not the case and your participants are a random selection from a larger set of participants, the second approach won't make sense. $\endgroup$ Commented Nov 30, 2018 at 16:42

1 Answer 1

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  1. Given that your observations are clustered according to individual participants, which can be considered random samples from a population distribution, the best and most robust approach is provided by mixed-effects models. Your first approach neglect the fact that the effect of stimulus may differs across participants, and your second approach treat participants as fixed-effects and therefore inferences based on it apply, strictly speaking, only to the sample and not to the population. If you wanted to fit a mixed-effect model, using the lme4 library the model could be formulated as Response ~ Stimulus + (Stimulus | Participant))
  2. The coefficient for B in the second model codes for the difference in the response that should be expected when the stimulus is B, relative to when the stimulus is A. Note that although this model control for differences across participants in the response (pooled across the two stimuli A and B), it does not take into account subject-specific differences in the effect of the stimulus.
  3. Yes, that would be the difference between participant #2 relative to participant #1, conditioned on the stimulus A, assuming standard treatment contrasts (i.e. factor levels are coded so that all coefficients represent the difference relative to a baseline or reference level).
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