Why do contrasts have to be orthogonal

If the control belongs to a different level of A, then the rows of the contrast coefficients can be rearranged accordingly without losing orthogonality. A four-level factor A can have the following alternative sets of three orthogonal contrasts B to D in any permutation of coefficient rows for each set, and analysed in GLM by requesting the fixed contrast terms with sequential SS :. Contrast set 2. Contrast set 3. A five-level factor A can have the following alternative sets of four orthogonal contrasts B to E in any permutation of coefficient rows for each set, and analysed in GLM by requesting the fixed contrast terms with sequential SS :.

Contrast set 1. Contrast set 4. Analysis of contrasts on a factor A does not require a significant A effect. If it is significant, however, at least one of the orthogonal sets will contain at least one significant contrast. For a priori planned orthogonal contrasts, the conceptual unit for error rate is conventionally taken to be the individual contrast rather than the family of contrasts in the full set , just as it is taken to be the individual term in multi-factorial ANOVA partitioned into treatment effects and interactions rather than the full experiment.

The family-wise Type-I error must apply, however, if contrasts are used for post hoc comparisons to locate the biggest differences amongst levels of a treatment. An example of a non-practical contrast is the normalized maximal contrast estimate. Thus, is maximal with Y in L". Supposing the above considerations the following expression may be written:.

In order to maximize , the following conditions are imposed:. Adding 11 in , thus. If the tested hypothesis in terms of comparisons is not rejected, the resulted implication is in the correspondent comparison in populational terms that it is not significantly different from zero.

Moreover, and. Thus, is defined as the contrast between treatment means obtained from data with unequal number of replications. Winer and Kirk showed that the sum of squares due to Y h , for the case of unequal number of replications, is given by the following expression:.

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Design and analysis of experiments. One can actually show that the formula to determine whether contrasts are centered i. As we have discussed above, when the intercept is included in the hypothesis matrix, the weights for this intercept term should sum to one, as this yields a column of ones for the intercept term in the contrast matrix.

We can see that considering the intercept makes a difference for the treatment contrast. We take a look at the resulting contrast matrix:. This shows a contrast matrix that we know from the treatment contrast. The intercept is coded as a column of 1s.

And each of the comparisons is coded as a 1 in the condition which is compared to the baseline, and a 0 in other conditions. The point is here that this gives us the contrast matrix that is expected and known for the treatment contrast. Interestingly, the resulting contrast matrix now looks very different from the contrast matrix that we know from the treatment contrast.

Indeed, this contrast also estimates a reasonable set of quantities. It again estimates whether the condition mean m1 differs from the baseline and whether m2 differs from baseline. The intercept, however, now estimates the average dependent variable across all three conditions i. This can be seen by explicitly adding a comparison of the average of all three conditions to The resulting contrast matrix is now the same as when the intercept was ignored, which confirms that these both estimate the same comparison.

Dobson, Annette J, and Adrian Barnett. An Introduction to Generalized Linear Models. CRC press. Schad, Daniel J. All contrasts discussed here are centered except for the treatment contrast, in which the contrast coefficients for each contrast do not sum to zero: colSums contr.