# Restricted Maximum Likelihood (REML) Estimate of Variance Component.

#### Cynderella

##### New Member
Let, $$\mathbf y_i = \mathbf X_i\mathbf\beta + \mathbf Z_i\mathbf b_i+ \mathbf\epsilon_i,$$

where

$$\mathbf y_i\sim N(\mathbf X_i\mathbf\beta, \Sigma_i=\sigma^2\mathbf I_{n_i}+\mathbf Z_i \mathbf G\mathbf Z_i'),$$

$$\mathbf b_i\sim N(\mathbf 0, \mathbf G),$$

$$\mathbf\epsilon_i\sim N(\mathbf 0, \sigma^2\mathbf I_{n_i})$$

$$\mathbf y_i$$ is a $$n_i\times 1$$ vector of response for $$i^{th}$$ individual at $$1,2,\ldots, n_i$$ time points, $$\mathbf X_i$$ is a $$n_i\times p$$ matrix, $$\mathbf \beta$$ is a $$p\times 1$$ vector of fixed effect parameters, $$\mathbf Z_i$$ is a $$n_i\times q$$ matrix, $$\mathbf b_i$$ is a $$q\times 1$$ vector of random effects, $$\mathbf \epsilon_i$$ is a $$n_i\times 1$$ vector of within errors, $$\mathbf G$$ is a $$q\times q$$ covariance matrix of between-subject, $$\sigma^2$$ is a scalar.

Note that, $$\mathbf X_i$$, $$\mathbf Z_i$$, and $$\mathbf G$$ do NOT involve $$\sigma^2$$.

Now I have to find out the Restricted Maximum Likelihood (REML) Estimate of $$\sigma^2$$, that is,

$$\hat\sigma^2_R = \frac{1}{N_0-p}\sum_{i=1}^{N}(\mathbf y_i-\mathbf X_i\mathbf\beta)'(\mathbf I_{n_i}+\mathbf Z_i \mathbf G\mathbf Z_i')^{-1}(\mathbf y_i-\mathbf X_i\mathbf\beta),\ldots (1)$$

where $$N_0 = \sum_{i=1}^{N}n_i$$.

So first I wrote the Restricted Maximum Log-Likelihood :

$$l_R \propto -\frac{1}{2}\sum_{i=1}^{N}\log\det(\Sigma_i)-\frac{1}{2}\sum_{i=1}^{N}\log\det(\mathbf X_i'\Sigma_i^{-1}\mathbf X_i)-\frac{1}{2}\sum_{i=1}^{N}(\mathbf y_i-\mathbf X_i\mathbf\beta)'\Sigma_i^{-1}(\mathbf y_i-\mathbf X_i\mathbf\beta)$$

Then I have to differentiate log-likelihood, $$l_R$$, with respect to $$\sigma^2$$ and equate it to zero, i.e.,

$$-\frac{1}{2}\frac{\partial}{\partial\sigma^2}\{\sum_{i=1}^{N}\log\det(\Sigma_i)+\sum_{i=1}^{N}\log\det(\mathbf X_i'\Sigma_i^{-1}\mathbf X_i)$$ $$+\sum_{i=1}^{N}(\mathbf y_i-\mathbf X_i\mathbf\beta)'\Sigma_i^{-1}(\mathbf y_i-\mathbf X_i\mathbf\beta)\}|_{\sigma^2=\hat\sigma^2_R}=0$$

But basically I can't differentiate,

$$\frac{\partial}{\partial\sigma^2}\log\det(\Sigma_i)=\frac{\partial}{\partial\sigma^2}\log\det(\sigma^2\mathbf I_{n_i}+\mathbf Z_i \mathbf G\mathbf Z_i')$$

$$\frac{\partial}{\partial\sigma^2}\log\det(\mathbf X_i'\Sigma_i^{-1}\mathbf X_i)=\frac{\partial}{\partial\sigma^2}\log\det(\mathbf X_i'(\sigma^2\mathbf I_{n_i}+\mathbf Z_i \mathbf G\mathbf Z_i')^{-1}\mathbf X_i)$$ and

$$\frac{\partial}{\partial\sigma^2}\Sigma_i^{-1}= \frac{\partial}{\partial\sigma^2}(\sigma^2\mathbf I_{n_i}+\mathbf Z_i \mathbf G\mathbf Z_i')^{-1}$$.

How can I differentiate the above derivatives and get the REML estimate $$\hat\sigma^2_R$$ in equation $$(1)$$ ?

#### Cynderella

##### New Member
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.

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