## 8.3 Inference

### 8.3.1 Parameters $$\beta$$

#### 8.3.1.1 Wald test

We have

$\mathbf{\hat{\beta}(\theta) = \{X'V^{-1}(\theta) X\}^{-1}X'V^{-1}(\theta) Y} \\ var(\hat{\beta}(\theta)) = \mathbf{\{X'V^{-1}(\theta) X\}^{-1}}$

We can use $$\hat{\theta}$$ in place of $$\theta$$ to approximate Wald test

$H_0: \mathbf{A \beta =d}$

With

$W = \mathbf{(A\hat{\beta} - d)'[A(X'\hat{V}^{-1}X)^{-1}A']^{-1}(A\hat{\beta} - d)}$

where $$W \sim \chi^2_{rank(A)}$$ under $$H_0$$ is true. However, it does not take into account variability from using $$\hat{\theta}$$ in place of $$\theta$$, hence the standard errors are underestimated

#### 8.3.1.2 F-test

Alternatively, we can use the modified F-test, suppose we have $$var(\mathbf{Y}) = \sigma^2 \mathbf{V}(\theta)$$, then

$F^* = \frac{\mathbf{(A\hat{\beta} - d)'[A(X'\hat{V}^{-1}X)^{-1}A']^{-1}(A\hat{\beta} - d)}}{\hat{\sigma}^2 \text{rank}(A)}$

where $$F^* \sim f_{rank(A), den(df)}$$ under the null hypothesis. And den(df) needs to be approximated from the data by either:

• Satterthwaite method
• Kenward-Roger approximation

Under balanced cases, the Wald and F tests are similar. But for small sample sizes, they can differ in p-values. And both can be reduced to t-test for a single $$\beta$$

#### 8.3.1.3 Likelihood Ratio Test

$H_0: \beta \in \Theta_{\beta,0}$

where $$\Theta_{\beta, 0}$$ is a subspace of the parameter space, $$\Theta_{\beta}$$ of the fixed effects $$\beta$$ . Then

$-2\log \lambda_N = -2\log\{\frac{\hat{L}_{ML,0}}{\hat{L}_{ML}}\}$

where

• $$\hat{L}_{ML,0}$$ , $$\hat{L}_{ML}$$ are the maximized likelihood obtained from maximizing over $$\Theta_{\beta,0}$$ and $$\Theta_{\beta}$$
• $$-2 \log \lambda_N \dot{\sim} \chi^2_{df}$$ where df is the difference in the dimension (i.e., number of parameters) of $$\Theta_{\beta,0}$$ and $$\Theta_{\beta}$$

This method is not applicable for REML. But REML can still be used to test for covariance parameters between nested models.

### 8.3.2 Variance Components

• For ML and REML estimator, $$\hat{\theta} \sim N(\theta, I(\theta))$$ for large samples

• Wald test in variance components is analogous to the fixed effects case (see 8.3.1.1 )

• However, the normal approximation depends largely on the true value of $$\theta$$. It will fail if the true value of $$\theta$$ is close to the boundary of the parameter space $$\Theta_{\theta}$$ (i.e., $$\sigma^2 \approx 0$$)

• Typically works better for covariance parameter, than vairance prarmetesr.

• The likelihood ratio tests can also be used with ML or REML estimates. However, the same problem of parameters