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A repeated measures analysis (Bayesian or otherwise) with dependent measurements

Hi JASP experts,

This is a general question about assumptions that, I think, is applicable to both Bayesian and traditional repeated measures ANOVAs.

@Cherie and I have eye-movement data of participants searching through a set of books. I'll simplify the design a bit for the sake of the discussion, but we can provide the actual data if that's useful.

There are two book categories, A and B. For each trial we have quantified the gaze duration on each of the categories, giving two measures per trial. These measures are dependent, because if they look at A then they cannot look at B. In other words, high gaze durations for A are predictive (though not perfectly) of low gaze durations for B, and vice versa.

Then we have an experimental condition with two levels, X and Y. We're interested in whether this condition affects gaze duration, such that participants look more at A in condition X and more at B in condition Y.

An intuitive appealing way to analyze this is with a repeated measures, in which we treat book category as a factor, so we have a 2 (book category: A, B) × 2 (condition: X, Y) design with gaze duration as dependent measure. And then we'd be interested in the book category × condition interaction (not in the main effects of book category or condition).

Now here's where things get tricky.

  • I'm pretty sure that it's ok to look at the main effect of condition on gaze duration, because X and Y are independent.
  • I suspect that it's problematic to look at the main effect of book category on gaze duration, because A and B are not independent. But I'm not 100% sure about this.
  • And what about the book category × condition interaction. Is that valid? And if not, how would we ideally analyze a dataset like this?

I find it hard to wrap my head around this issue, so I really hope that someone can shed some light on this for us!



  • How dependent are A and B?

    If they are completely dependent (say 100% gaze = GazeA + GazeB), than no need to put both measurements into the model - the intercept will give an indication for both, the main effect for condition (X/Y) will actually be the interaction, with the conditional means in X and Y the effect of A vs. B.

    You can also convert your measurements to that this ^ is true:

    DV = GazeA / (GazeA + GazeB)

  • Hi MSB,

    Thanks for your reply. 😄

    The gaze durations on A and B are somewhat dependent, but not perfectly. (If they were, we could indeed recode it without losing data.) Basically, there are three possibilities:

    • People look at neither A nor B
    • People look only at A
    • People look only at B

    And the measures that we have are proportional gaze durations for A and B across a trial, which are generally values in the 0.1 to 0.3 range.

    So to restate the main question: Given this scenario, is it acceptable to treat this as a 2 (book category: A, B) × 2 (condition: X, Y) design with gaze duration as dependent measure?



  • Hmmm... Given your data and design, probably the most correct analysis would be a multinomial logistic regression...

    But let's stick to an ANOVA-like design.

    It seems %A and %B are dependent (negatively). You can deal with this dependance in two ways:

    1. Remove it (what I suggested above).
    2. Account for it.

    This would mean you use a liner-mixed model with a random intercept by trial (accounting for the differences between trials in %neither-A-nor-B), and a random slope for Book by trial (accounting for the negative dependence within each trial).

    In a lme4 type formula, your model would look like this:

    percent_looking ~ Book * Condition + 
                      (1 + Book | Trial) +
                      (1 + ... | Subject)

    ((...| Subject) indicating any within-subject effects)

  • Hi MSB,

    Thanks for this. That makes sense.

    Our design is actually a little more complicated than what I described here, in the sense that there are four book categories, and four conditions. Does that make any difference for your proposed approach?



  • I don't think this should matter.

    But, upon further reflection, the lme4 formula should be:

    percent_looking ~ Book * Condition + 
             (1 + Book | Trial:Subject) +
             (1 + Book + ... | Subject)

    to account for the fact the trials are nested in subjects (and a random book effect per subject)

    or, if you suspect there may be any random effect for trials across subjects:

    percent_looking ~ Book * Condition + 
             (1 + Book | Trial:Subject) + 
             (1 | Trial) + 
             (1 + Book + ... | Subject)

    Good luck!

  • > to account for the fact the trials are nested in subjects (and a random book effect per subject)

    Right, I was thinking about that too, but I was unsure how to indicate that.

    Thanks for all your help!

  • interesting problem!

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