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Publication date: 04/21/2023

Statistical Details for Variance Components

The exact model type that you choose in the Variability Chart platform depends on how the data were collected. For example, if the operators are measuring the same parts, you have a crossed design. If they measuring different parts, you have a nested design. To illustrate, in a model where B is nested within A, multiple measurements are nested within both B and A, and there are nanbnw measurements, the following statements hold:

na random effects are due to A

nanb random effects due to each nb B levels within A

nanbnw random effects due to each nw levels within B within A:

Equation shown here.

The Zs are the random effects for each level of the classification. Each Z is assumed to have a mean of zero and to be independent from all other random terms. The variance of the response y is the sum of the variances due to each z component:

Equation shown here.

Table 6.3 shows the supported models and what the effects in the model would be.

Table 6.3 Models Supported by the Variability Charts Platform

Model

Factors

Effects in the Model

Main Effects

1

2

unlimited

A

A, B

and so on, for more factors

Crossed

1

2

3

4

unlimited

A

A, B, A*B

A, B, A*B, C, A*C, B*C, A*B*C

A, B, A*B, C, A*C, B*C, A*B*C, D, A*D, B*D, A*B*D, C*D, A*C*D, B*C*D, A*B*C*D,

and so on, for more factors

Nested

1

2

3

4

unlimited

A

A, B(A)

A, B(A), C(A,B)

A, B(A), C(A,B), D(A,B,C)

and so on, for more factors

Crossed then Nested

3

A, B, A*B, C(A,B)

Nested then Crossed

3

A, B(A), C, A*C, C*B(A)

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