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Thread: Variance of single element of random effects in linear mixed model

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    Variance of single element of random effects in linear mixed model




    Hi there,

    I have recently been working a bit with linear mixed model and in order to enhance my understanding of those, I would like to know the following.

    If we have a linear mixed model where the random effects have the following distribution:

    u ~ MVN(0,a*K)

    Where u is a nx1 vector and K is nxn matrix not being an identity matrix and a is some scaling factor denoting how much of the variance is described by the random effects.

    So my question is then what is var(u_i) (the i'th element of u)? And cannot this even be computed?
    If K were an identity matrix I guess it would be var(u_i)=a.

    These linear mixed models are for instance used in genetics research.

    If anything is unclear please ask

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    Re: Variance of single element of random effects in linear mixed model

    So is the shortage of answers due to the lack of validity of the answer or due to people not understanding the question?

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    Re: Variance of single element of random effects in linear mixed model

    Unless I'm missing something, then by definition, \text{var}(u_i) = aK_{ii}, where K_{ii} refers to the ith diagonal element of K.
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    Re: Variance of single element of random effects in linear mixed model

    Hi thanks for the answer!

    Does this also apply when K has non-zero non diagonal elements?
    I would think that they somehow contribute to the variance of the i'th element.

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    Re: Variance of single element of random effects in linear mixed model


    Yes Jake's answer is still correct. When you specify the covariance matrix the diagonal literally is "and the variance of this element is defined to be THIS". The off-diagonal elements tell you about the covariance which is important when you're looking at combinations of multiple variables but if all you want is the variance of a single element then all you have to do is grab the appropriate value from the covariance matrix.
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