& 0.314 & 0.412 & 0.457 & 100 & 0.005 & 0.005 & 0.006 & 0.005\
200 & 0.212 & 0.286 & 0.383 & 0.407 & 200 & 0.005 & 0.005 & 0.006 & 0.006\

![ Box-plots of simulated rejection probabilities $(\bar{p}_r)_{r=1}^{100}$ (gray crosses) of the $F$-test with $n=50$, $p=5$ and $d=10,50,100,1000$, for different design distributions. Every $r\in\{1,\dots, 100\}$ corresponds to a different $R_r$ applied to the standardized design $\tilde{z}$. []{data-label="fig:Boxplots1"}](t3t5Unif.pdf){width="\textwidth"}

![ Box-plots of simulated rejection probabilities $(\bar{p}_r)_{r=1}^{100}$ (gray crosses) of the $F$-test with $n=50$, $p=5$ and $d=10,50,100,1000$, for Bernoulli$\{-1,1\}$ and exponential design distributions and a benchmark panel of Binomial samples with different success probabilities. []{data-label="fig:Boxplots2"}](BernExpBench.pdf){width="\textwidth"}

In Table \[tab:NullMod\], the effect of the orthogonal matrix $R$ on the actual significance level of the $F$-test was compressed into one summary statistic, namely the mean absolute deviation from the nominal significance level. To get a more comprehensive picture, Figures \[fig:Boxplots1\] and \[fig:Boxplots2\] show plots of the sample $(\bar{p}_r)_{r=1}^{100}$ (gray crosses) and superimposed box-plots for different design distributions. Due to limited space we present only the results for sub-models of dimension $p=5$. In view of Theorem \[t1\], we expect that the size $\mathbb U$, i.e., the family of matrices $R$ for which and 