orthogonal

Examples of orthogonal in the following topics:

• Experimental Design

  • Orthogonality: Orthogonality concerns the forms of comparison (contrasts) that can be legitimately and efficiently carried out.
  • Contrasts can be represented by vectors and sets of orthogonal contrasts are uncorrelated and independently distributed if the data are normal.
  • Because of this independence, each orthogonal treatment provides different information to the others.
  • If there are $T$ treatments and $T-1$ orthogonal contrasts, all the information that can be captured from the experiment is obtainable from the set of contrasts.
  • Outline the methodology for designing experiments in terms of comparison, randomization, replication, blocking, orthogonality, and factorial experiments

• Specific Comparisons (Independent Groups)

  • Independent comparisons are often called orthogonal comparisons.
  • There is a simple test to determine whether two comparisons are orthogonal: If the sum of the products of the coefficients is 0, then the comparisons are orthogonal.
  • Therefore, the two comparisons are orthogonal.
  • Table 7 shows two comparisons that are not orthogonal.

• Specific Comparisons (Correlated Observations)

  • Although mathematically possible, orthogonal comparisons with correlated observations are very rare.

• Two-Way ANOVA

  • Each level of one factor is tested in combination with each level of the other(s), so the design is orthogonal.

• Tests Supplementing ANOVA

  • Important issues concerning multiple comparisons and orthogonal comparisons are discussed in the Specific Comparisons section in the Testing Means chapter.
  • Naturally, the same consideration regarding multiple comparisons and orthogonal comparisons that apply to other comparisons among means also apply to comparisons involving components of interactions.

• Polynomial Regression

  • Although the correlation can be reduced by using orthogonal polynomials, it is generally more informative to consider the fitted regression function as a whole.

• Exercises

  • True/false: The contrasts (-3, 1 1 1) and (0, 0 , -1, 1) are orthogonal.

• Glossary

  • When comparisons among means provide completely independent information, the comparisons are called "orthogonal. " If an experiment with four groups were conducted, then a comparison of Groups 1 and 2 would be orthogonal to a comparison of Groups 3 and 4 since there is nothing in the comparison of Groups 1 and 2 that provides information about the comparison of Groups 3 and 4.