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Transform each observation to $Y_j=A^T{X}_j$. If $A^T$ has full row rank $m$, the transformed sample is multivariate normal with mean $A^T\mu$ and positive-definite covariance matrix $A^T\Sigma A$. Its sample mean is $A^T\bar{X}$ and its unbiased sample covariance matrix is $A^{T}SA$. The Hotelling test of linear hypotheses therefore usesHere $n>m$ is required. Reject for a large scaled statistic using the relevant upper F-distribution quantile. If the rows of $A^T$ are dependent, choose a basis for their row space, use that basis as the contrast matrix, and replace $m$ by its rank $r$. The null hypothesis is unchanged, but redundant contrasts must not be treated as independent dimensions. Rank zero gives no restrictions and hence no nontrivial test.

At equal spacing $h$, the adjacent increments of an affine mean profile are all $\beta h$. Their differences are therefore zero. Define the $(p-2)\times p$ matrix $B$ whose row associated with the interior time $i$ has coefficients $1,-2,1$ in columns $i-1,i,i+1$. ThenConversely these equations make all adjacent first differences equal, so ${\mu }_i$ is affine in $i$, equivalently affine in the equally spaced times. Thus the restriction is exactly $B\mu =0$, not just a necessary condition. Its null space has basis $1$ and $t=(t_1,\dots ,t_p{)}^T$, so $B$ has rank $r=p-2$.

For $p\ge 3$ and $n>r$, the second-difference test of a linear mean profile is the Hotelling test of linear hypotheses withReject the affine-profile hypothesis when the scaled statistic exceeds its level-$\eta$ upper F-distribution critical value. This removes the nuisance intercept and slope without estimating them separately.

For unequal ordered distinct times, put $h_i=t_{i+1}-t_i>0$ and replace each second-difference row by the adjacent-slope contrastIts three coefficients are $1/h_{i-1}$, $-(1/h_{i-1}+1/h_i)$ and $1/h_i$. Vanishing means all adjacent slopes agree, again giving null space $\mathrm{span}(1,t)$ and rank $p-2$. Use this new $B$ in the same statistic and null distribution. For $p\le 2$, every mean profile at distinct times is affine, so there is no lack-of-linearity restriction to test.

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.