Example 22. Hierarchical Model: Purity Data

A company buys raw material from three suppliers and wishes to check if the purity of the material from each supplier is the same. Four batches of raw material are available from each supplier, and on each batch, three determinations of purity are to be done. The responses are coded (Response = Purity - 93) and given in the following table.

Source: Montgomery, D.C. (2001). The Design and Analysis of Experiments. John Wiley & Sons: New York.

Table 22: Purity Data

	`Suppliers`
	`S1`				`S2`				`S3`
`Batches`	`B1`	`B2`	`B3`	`B4`	`B1`	`B2`	`B3`	`B4`	`B1`	`B2`	`B3`	`B4`
	`1`	`-2`	`-2`	`1`	`1`	`0`	`-1`	`0`	`2`	`-2`	`1`	`3`
	`-1`	`-3`	`0`	`4`	`-2`	`4`	`0`	`3`	`4`	`0`	`-1`	`2`
	`0`	`-4`	`1`	`0`	`-3`	`2`	`-2`	`2`	`0`	`2`	`2`	`1`

Questions:

We will discuss whether the variation that is believed to exist in the raw material can be solely attributed to differences in the suppliers.
Treating both suppliers and batches as fixed effects, perform an ANOVA and report findings.
Suppose that suppliers are fixed effects and batches denote random effects. How is the analysis carried out?
Discuss why this is a nested two-way model. Contrast a nested model with a 2-way crossed model.

Keywords: Nested classification, fixed effects, random effects (variance components), expected mean squares, F-test

Numerical Examples for use with
A First Course in Linear Model Theory by Ravishanker and Dey
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