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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!

Sebastiaan

• 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,

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?

Cheers!

Sebastiaan

• 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?

Cheers!

Sebastiaan

• 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.