Two vectors are orthogonal if the sum of the products of their corresponding elements is 0. For example, consider the following vectors a and b:
You can multiply the corresponding elements of the vectors to show the following result:
a*b = 2(–4) + 3(1) + 5(1) + 0(4) = –8 + 3 + 5 + 0 = 0
This shows that the two vectors are orthogonal.
The concept of orthogonality is important in Design of Experiments because it says something about independence. Experimental analysis of an orthogonal design is usually straightforward because you can estimate each main effect and interaction independently. If your design is not orthogonal, either by plan or by accidental loss of data, your interpretation might not be as straightforward.
A | B | C |
---|---|---|
1 | –1 | –1 |
1 | –1 | 1 |
–1 | –1 | 1 |
–1 | 1 | –1 |
–1 | 1 | 1 |
–1 | –1 | –1 |
1 | 1 | 1 |
1 | 1 | –1 |
So in a sense, factor A is estimated independently from B and C and vice versa.
The estimates for the effects and coefficients will remained unchanged when you remove interactions from the model. The other output will change as the experimental error (MSE) is adjusted accordingly with more degrees of freedom.
In conclusion, a designed experiment is orthogonal if the effects of any factor balance out (sum to zero) across the effects of the other factors. Orthogonality guarantees that the effect of one factor or interaction can be estimated separately from the effect of any other factor or interaction in the model.
When analyzing factorial designs, if the design is displayed in uncoded units in the worksheet, first choose , select Coded units, and click OK.
You can also do this for Response Surface, Taguchi, and Mixture designs. To store a design matrix in Taguchi, you must be fitting a linear model.
When analyzing a factorial design, the design matrix will store the terms in uncoded units if the worksheet is in uncoded units. will perform the analysis in coded units. When analyzing a response surface design, the design matrix will store the terms in coded or uncoded units depending on the units in which you choose to analyze the data.