Hotelling test of linear hypotheses ($T_B^2=n(B\bar{X}{)}^T(BS{B}^T{)}^{-1}B\bar{X}$)

For a fixed rank-$m$ constraint matrix $B$, apply Hotelling's T-squared statistic to the transformed observations $B{X}_j$. Under $B\mu =0$, $(n-m)T_B^2/(m(n-1))\sim F_{m,n-m}$ for $n>m$. If the displayed contrasts are dependent, use a basis of their row space and replace $m$ by the rank.

At equally spaced distinct times, a mean profile is affine exactly when its consecutive second differences vanish. These form $p-2$ independent contrasts. At unequal times, compare adjacent slopes instead: $({\mu }_{i+1}-{\mu }_i)/(t_{i+1}-t_i)-({\mu }_i-{\mu }_{i-1})/(t_i-t_{i-1})$. A Hotelling test of linear hypotheses tests these restrictions jointly with unknown covariance.