# Testing the random slope with correlated random effects

I have a mixed/random effects model $$\mathbf{y}_i=\mathbf{X}_i\boldsymbol\beta+\mathbf{Z}_i\mathbf b_i+\boldsymbol\epsilon_i,$$ where random effects $\mathbf b_i$ has variance-covariance matrix $$D=\text{cov}(\mathbf b_i)=\left(\begin{array}{cc} \sigma_{1}^{2} & \sigma_{12}\\ \sigma_{12} & \sigma_{2}^{2} \end{array}\right).$$

I would like to test $H_0:\sigma_2^2=0$. I know that under the null $\sigma_{12}=0$. But I am not willing to restrict that $\sigma_{12}=0$ under the alternative, i.e., $\sigma_{12}$ and $\sigma^2_1$ are unspecified.

I know there are some tests developed assuming independent random effects. For example, the RLRsim package can do the simulation-based exact (restricted) likelihood ratio test for testing the presence of a single variance component. Besides, according to @StasK, "the bootstrap on the boundary of the parameter space has its own set of issues and problems leading to inconsistency."

So I wonder if there is any potential solutions or references for the individual variance component test without assuming independent random effects. It would be perfect if it is readily available or can be easily implemented. Thanks!

In Zhang and Lin (2008), Case 2 on Page 23, for likelihood ratio test (LRT), the asymptotic null distribution is a 50:50 mixture of $\chi^2_1$ and $\chi^2_2$. This may be not the most powerful test, but the loss of power as compared to the score test may not be much. Table 1 therein shows the power comparison of LRT and score test for a single variance component.
Alternatively, confint.merMod() in the lme4 package computes confidence intervals for variance components numerically using likelihood profile (and LRT) or bootstrap. So it can be used for testing potentially.