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I'm using a mixed effects model (lmer) in R to model eye-tracking data using orthogonal polynomials (poly(time,3)) for time. The response variable is log(looks to target/looks to competitor). The fixed effects are condition, talker gender, accuracy, like so:

lmer(looks~(poly1+poly2+poly3)*condition*gender*accuracy+(1|stimulus)+(1|subject)

The polynomial time variables are of course, continuous (and centered), while the other factors are treatment coded so that each coefficient is compared to the baseline of the cells (where condition=0, gender=0, accuracy=0 (actually accuracy is recoded so that accurate trials are the baseline)). All of this is pretty much irrelevant because my question is more of a matter of mathematical interpretation.

I understand that the intercept reflects the estimate for the base level of each cell, but my question is-- what is the base level of the time polynomials? Mirman (2014) claims "For natural polynomials, the intercept term corresponds to the y-intercept [...] For orthogonal polynomials, the intercept term corresponds to the overall average [...] 'area under the curve'" (p. 48). While I don't claim to thoroughly understand mixed effects modeling (because matrix algebra!), it seems to me that the intercept should estimate a value for the zero points of the polynomials (so, at the midpoint of the linear time, at two points for the quadratic, and three points for the cubic, one of which is also the midpoint of the linear).

While it would be very convenient for me to pretend that this is the average, I can't make this claim while I am still confused about the math. Is there anyone who can explain to me whether my "5-point snapshot" hypothesis is correct, or if the intercept reflects the mean over time? I have no stake in which hypothesis is more accurate, but I don't want to misreport my results, and I would very much like to understand how mixed effects models work. Also, this is my first time posting a question here (though I've found many useful answers over the years), so please be kind even if you think I am completely stupid.

(Ref: Mirman, D. 2014. Growth Curve Analysis and Visualization Using R. CRC Press.)

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    $\begingroup$ Take a look here: danmirman.org/gca $\endgroup$ – Alex R. May 15 '17 at 18:36
  • $\begingroup$ Here, again, Dan Mirman makes the claim that "With orthogonal polynomials, the intercept term reflects the average overall curve height, rather than the height at the left edge of the time window, so if you are interested in differences at the very beginning of the time window, you may be better off sticking with natural polynomials." But I don't understand how using orthogonal polynomials changes the intercept to reflect the mean over the entire time window, whereas raw polynomials follow the expected pattern of producing an intercept that represents the pattern at time zero. $\endgroup$ – bsmith May 16 '17 at 14:07
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After some searching outside of my field, I came across Hedecker and Gibbons 2006 (http://rtksa.com/library1/wp-content/uploads/2015/11/523.pdf), who suggest that using a sufficiently large number of polynomials to model time will effectively yield an average over time for the intercept, but this is not the same thing as the mean over all the time values, but, as I suspected, the mean over the x-zero crossings (um, y-intercept, if you like), which will be evenly spaced for each orthogonal polynomial. If you have very many polynomial terms, this is likely very similar to a mean, but for the only three terms that I am interested in, it is too susceptible to random fluctuations, and will change based on which timepoint the zero-crossings happen to be at (such as I observe if I shift the center to the right or left by 20 ms). They suggest, in order to get a mean over time, to create an orthogonal constant (1*sqrt(n)) over all time, to replace the intercept term, which must be held to zero, so as to not have two factors predicting the same thing (and in fact, lmer will not run with both, but always drops one). So I have changed my analyses to reflect that, as such:

    lmer(looks~0+constant+(poly1+poly2+poly3)*condition*gender*accuracy+(1|stimulus)+(1|subject) 

and the result gives me essentially the same effects in the interactions, but additionally significant effects where they ought to have appeared for main effect coefficients, if they had represented a mean over time.

Does anyone know if this seems right, or am I just taking it upon myself to revise my field's understanding of mixed effects models using orthogonal polynomials? Any additional citions would be helpful, as I feel I am stepping out into uncharted territory, at least as far as my field has gotten.

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    $\begingroup$ This question does not seem to have anything to do with mixed models, right? If you kick out stimulus and subject random effects and use lm instead of lmer, your question about intercept would stay exactly the same. Mixed models might be just a complication that you are bringing in here. (I'm not saying that you should not be using mixed models! I am saying that they are irrelevant to the issue at hand.) $\endgroup$ – amoeba says Reinstate Monica Jun 16 '17 at 7:16
  • $\begingroup$ I agree with you. My main problem is that I can work through the math by hand for a linear regression model, but haven't taught myself enough to be able to do the same for a mixed effects model. So, while I have read that they can be interpreted in roughly the same way, I don't know quite enough to trust that. If I am going to interpret my results differently than has been done in my field, then I either need a mathematical proof, or an authority to cite. Luckily, I found a citation that verifies my hunch. But if reviewers press me, I could do a watered-down lm example. Thanks for the idea. $\endgroup$ – bsmith Jun 18 '17 at 22:41

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