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John Vokey
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Daniel,
Here is a simple example. Let us assume we have two factors, A with 2 levels and B with 3 levels, such that A crosses B, and there are 2 of observations per each of the 2 × 3 = 6 cells, for a total of 12 observations:

Table 1: Example table of vector codes using “effect coding” for a 2 × 3 factorial experiment, with two observations per cell. As the column sums and the sums of column products across sets are necessarily zero, the sets are necessarily orthogonal (and, hence, uncorrelated) with one another (it is in LaTeX, but you should get the point):

\begin{table}[t] \centering
\caption{Example table of vector codes using effect coding” for a $2 \times 3$ factorial experiment, with two observations per cell. As the column sums and the sums of column products across sets are necessarily zero, the sets are necessarily orthogonal (and, hence, uncorrelated) with one another.}
\label{tbl:effect}
\footnotesize

\begin{tabular}{@{\extracolsep{5pt}} c c c c c }
\\[-1.8ex]\hline
\hline \\[-1.8ex]
a & b1 & b2 & ab1 & ab2 \\
\hline \\[-1.8ex]
$1$ & $1$ & $0$ & $1$ & $0$ \\
$1$ & $1$ & $0$ & $1$ & $0$ \\
$1$ & $0$ & $1$ & $0$ & $1$ \\
$1$ & $0$ & $1$ & $0$ & $1$ \\
$1$ & $-1$ & $-1$ & $-1$ & $-1$ \\
$1$ & $-1$ & $-1$ & $-1$ & $-1$ \\
$-1$ & $1$ & $0$ & $-1$ & $0$ \\
$-1$ & $1$ & $0$ & $-1$ & $0$ \\
$-1$ & $0$ & $1$ & $0$ & $-1$ \\
$-1$ & $0$ & $1$ & $0$ & $-1$ \\
$-1$ & $-1$ & $-1$ & $1$ & $1$ \\
$-1$ & $-1$ & $-1$ & $1$ & $1$ \\
\hline \\[-1.8ex]
\normalsize
\end{tabular}
\end{table}

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