# Different linear mixed model approaches for only estimating variance components once

I found these papers on linear mixed model with a single random intercept:

1.

https://www.genetics.org/content/177/1/577.long

Where they use an approach for implementing a linear mixed model, where they regress out the BLUP of the random effects (after having estimated the variance components under a null model) and then run a normal linear model on the residuals.

So they do this, for this linear mixed model ($$X$$ being the design matrix with also column of 1s for intercept:

$$y=X\beta+u+e$$

They run the model (null model) like this (meaning not including the genotype in their design matrix, only covariates):

$$y=X'\beta' + u +e$$

They can estimate the BLUP for $$u$$, from this null model:

$$\hat{u}=\sigma_e^2V^{-1}y$$

Where $$G$$ is the co-variance matrix of $$u$$ and $$V=ZGZ^T+R$$, where $$R$$ is the co-variance matrix of $$e$$. And where $$G=\sigma^2_uK$$ and $$R=\sigma_e^2I$$. Let us say that $$Z$$ is the identity matrix and that we therefore have a linear mixed model with only a random intercept.

They then run this model on the residuals of the null model:

$$\tilde{y}=y-(X'\beta' + \hat{u})$$
$$\tilde{y}=\alpha+g\beta+e'$$

(Where $$g$$ are the genotypes). So the regress out the random effects in the first step.

I have tried myself and this approach gives different results from this approach:

2.

Here they estimate $$\sigma^2_u$$ and $$\sigma_e^2$$ under the null model:

$$y=X'\beta' + u +e$$

And then use the same $$\sigma^2_u$$ and $$\sigma_e^2$$ values for each genetic variant, when using the full model, instead of re-estimating $$\sigma^2_u$$ and $$\sigma_e^2$$ each time.

$$y=X\beta+u+e$$

https://www.nature.com/articles/ng.548

So my issue is, why are these two models different?? They are both using the variance components from the null model?

Essentially can I regress out the random effects using the BLUP?

• Can you provide more information on the exact model specification of the two approaches? The second article is behind a paywall. Is it that both approaches seek to model the within-genetic variant associations? Jul 12, 2020 at 20:56
• From what I can see, the first method fits the full model on "markers showing test statistics greater than some predefined threshold are selected" (identified by the residual modelling), while in the second method they use either OLS for markers where $\sigma_A^2=0$ cannot be rejected (they assume no cluster variance) or the mixed effect model.
– user289381
Jul 12, 2020 at 22:34
• I have added some edits, to hopefully make stuff clearer.
– lo2
Jul 13, 2020 at 19:42
• What do you find different? $\beta$ coefficients? The first method does filtering of $g_i$, in the second method they don't filter: OLS for $\sigma_A^2=0$ or mixed effect model for the others.
– user289381
Jul 13, 2020 at 20:52
• Yeah different betas and P-values. What do you mean by filtering, could I ask you to elaborate on this? Am I fundamentally mistaken, in thinking that a random effects intercept can be modelled by regression out the BLUP of the random intercept??
– lo2
Jul 14, 2020 at 12:09