In a study involving $n$ samples, if the likelihood ratio test statistic $\Lambda = -2 \ln \left( \frac{L(\theta_0)}{L(\hat{\theta})} \right)$ is used to test $H_0: \theta = \theta_0$, which of the following best describes the effect of increasing the dimensionality of the parameter space $p$ on the power of the test if the sample size $n$ remains constant?