Which of the following statements is/are true ? (I) A two-way classified data will be orthogonal when the all frequencies are proportional. (II) In a given set of \(n\) values \(y_1, y_2, \ldots, y_n\), the maximum number of mutually orthogonal contrasts among them is \(n - 1\).
Both (I) and (II)
(I) Two-way data with cell frequencies \(n_{ij}\) are orthogonal when \(n_{ij} = \frac{n_{i.}\, n_{.j}}{n_{..}}\) for all \(i, j\), i.e. all frequencies are proportional; then the row and column sums of squares separate additively. True.
(II) A contrast \(\sum_{i=1}^{n} c_i y_i\) has \(\sum_{i=1}^{n} c_i = 0\), so its coefficient vector lies in the \((n - 1)\)-dimensional subspace orthogonal to \(\mathbf{1}\).
Mutually orthogonal contrasts have mutually orthogonal (hence linearly independent) coefficient vectors, so there are at most \(n - 1\) of them; Helmert contrasts attain \(n - 1\). True.
Hence, Both (I) and (II).
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