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