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Thread: Mixed models: Calculating ICC for model with a random intercept and a random slope

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    Mixed models: Calculating ICC for model with a random intercept and a random slope




    Hello All,

    I have a question about how to calculate an intraclass correlation coefficient in a mixed model that has both a random intercept and a random slope. I have a very nice paper by Judith Singer that describes how to do this for a random intercept model. The paper can be accessed at:

    http://www.gse.harvard.edu/~faculty/...oc%20Mixed.pdf

    At the top of page 330, there is a formula for calculating the ICC for a random intercept model. There's also a section on individual growth models, that starts on page 340. The growth models have a random intercept and a random slope. Is there some way to extend the formula on page 330, so I can calculate the ICC for growth models like that described on pages 340 and 341? If so, how is this done?

    Thanks,

    Paul

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    Re: Mixed models: Calculating ICC for model with a random intercept and a random slop

    Hi Paul. It's a good question. I thumbed through Snijders & Bosker (2012) and couldn't find an answer. So I did a small amount of Googling and the only clue I found (in the minute or so that I spent looking online) was the page here (LINK) which had the following to say:
    For a random intercept model, the intraclass correlation was identical to the variance partitioning coefficient, and it was quite simple to calculate. For a random slopes model, the intraclass correlation is not equal to the variance partitioning coefficient because the intraclass correlation will depend on the value of x1 for each of the two elements in question. The variance partitioning coefficient just depended on one value of x1 but if two different people each have a different value of x1, both those values are going to go into the formula for the intraclass correlation. The exact expression for the intraclass correlation is quite complicated; we're not going to give it here because the important thing is simply to note that the intraclass correlation will depend on the two values of x1 as well as σ2u1, σ2u0 and σu01.
    If you find a better answer be sure to update us in this thread!
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    Re: Mixed models: Calculating ICC for model with a random intercept and a random slop

    From Kreft & De Leeuw's Introducing Multilevel Modeling, p. 63:

    "The concept of intra-class correlation is based on a model with a random intercept only. No unique intra-class correlation can be calculated when a random slope is present in the model. The value of the between variance in models with a random slope and a random intercept is a combination of slope and intercept variance (and covariance). We know from the discussin of th basic RC model that the variance of the slope (and, as a consequence, the value of the covariance) is realted to the value of the explanatory variable x. Thus the intra-class correlation between individuals will be different, in models with random slopes, for individuals with different x-values. As a result the intra-class correlation is no longer uniquely defined".
    so some people say it can't be done. other people say it can be done but it's complicated. i guess this is one of those moments where you go back to the literature and cite your favourite author in case he or she has come up with something...
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    Re: Mixed models: Calculating ICC for model with a random intercept and a random slop


    Hi "Spunky" and Jake,

    Thanks for your replies. So it looks like this can't be done. Or it is difficult to do correctly and will be seen as controversial by some. So I think I'll leave off trying to do this.

    Appreciate your help.

    Paul

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