In this mixed linear model(LMM) y is our response variable, XB is our fixed effects Matrix and Coefficient, Wu is our random effects matrix and coefficient and E is our error term with variance y for our model which is where our question comes in. The variance is defined as our standardized genetic matrix times the variance of the random effects plus the variance of errors times the identity matrix. Why is there no variance for the fixed effects involved in the variance for the model?
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$\begingroup$ I am really looking for at least if and why there is no variance in the fixed effects. $\endgroup$– DaveRowanCommented Jun 23, 2015 at 14:30
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2 Answers
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Note that y in your equation is a vector of observations (say N in number), and the value for var(y) is then an N by N matrix, not the scalar value you are thinking about for the variance among the N values of the variable y. For each of those N observations, there is a predicted value given by the corresponding entry in the vector X$\beta$. The variance formula you display is for the (co)variance matrix around those predicted values, not the scalar variance of the N original observations.
This might be considered a slightly different way of thinking than in traditional analysis of variance, where you start with the scalar variance among the N observations and then partition it into the variance associated with the fixed effects, with the random effects, and with the error term. Think of the formula for variance you display as derived by a multi-dimensional generalization of the additivity of independent variances, with X$\beta$ in this case being a vector of constants and thus contributing no variance to the matrix result.
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Fixed effects are fixed -- they are not random quantities, so there is no variability.
answered Jun 23, 2015 at 14:47