= Drift rejection by contrast orthogonality
\b[Drift rejection by contrast orthogonality] removes specified additive time trends from a load comparison by choosing contrast weights orthogonal to those trends. In an <aerodynamic interference> experiment, it prevents an assumed form of balance-zero drift from being confused with nonadditivity between component configurations.
For observations at times $t_r$, write
$$
\boldsymbol d=\boldsymbol\mu+B\boldsymbol\gamma+\boldsymbol\epsilon,\qquad
\widehat\Delta=\boldsymbol w^T\boldsymbol d.
$$
The columns of $B$ specify nuisance trends and $\boldsymbol\gamma$ their unknown amplitudes. The drift contribution vanishes for every amplitude exactly when
$$
B^T\boldsymbol w=0.
$$
For <polynomial instrumental drift> through degree $p$, the columns are $1,t,\ldots,t^p$, so the conditions are $\sum_r w_rt_r^j=0$ for $j=0,\ldots,p$. This <orthogonality> is a statement about actual observation times, not merely the number of repetitions.
An original eight-run construction uses equally spaced times $t=0,1,\ldots,7$ in a fixed time unit and the configuration order
$$
(00,10,01,11,10,00,11,01).
$$
For the <factorial contrast for aerodynamic interference>, choose
$$
\boldsymbol w=\tfrac12(1,-1,-1,1,-1,1,1,-1)^T.
$$
Each configuration occurs twice and its total weight is the coefficient required for $\Delta=D_{11}-D_{10}-D_{01}+D_{00}$. Direct summation gives
$$
\boldsymbol w^T\boldsymbol1=\boldsymbol w^T\boldsymbol t
=\boldsymbol w^T\boldsymbol t^{\circ2}=0,
\qquad \boldsymbol w^T\boldsymbol t^{\circ3}=-24.
$$
The schedule rejects constant, linear and quadratic additive drift, but a cubic term $ct^3$ leaves bias $-24c$. If equal run spacing changes, the moments must be recomputed. If independent residual errors have common variance $\sigma^2$, the contrast variance is $\sigma^2\boldsymbol w^T\boldsymbol w=2\sigma^2$; correlated errors instead require the full <covariance matrix>.
Rejection must preserve the physical signal. If a desired signal column lies in the nuisance column space, every vector rejecting that space also rejects the signal. <Orthogonal projection> exposes this identifiability limit; collecting more readings in the same confounded arrangement need not fix it. Gain drift, configuration-dependent tare and abrupt offsets are also outside the stated additive polynomial model.
<Kenneth E. Ward>'s original attached- and isolated-strut measurements motivate separating combined load from component drag.<Interference Effects and Drag of Struts on a Monoplane Wing by Kenneth E. Ward>[Report abstract and Discussion of Results, interference versus isolated strut drag]{ref} The eight-run schedule here is a constructed design, not a reconstruction of that historical experiment.
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