1. Nested Designs
Ex 1--Compare 4 states’ low-level radioactive hospital waste
A: State
B: Hospital
Rep: Daily waste

2. Nested Designs Ex 2--Compare tobacco yield/acre in 5 counties
A: County
B: Farm
Rep: Field Yield/acre
Other Ex--Tick study, Classroom studies
Are A and B crossed in these studies?

3. Nested Designs
We say B is nested in A when levels of B are unique (or intrinsic) to a given level of A
Nested designs can also be thought of as incomplete factorial designs (with most treatment combinations impossible)

4. Nested Designs
Typically, A is fixed and B is random

5. Nested Designs We can decompose the sum of squares:
SSTO=SSA+SSB(A)+SSE
Note that SSB(A)=SSB+SSAB

6. Expected Mean Squares
All other assumptions being satisfied for an F-test, we can compute Expected Mean Squares (EMS) to construct appropriate F-tests for factor effects

7. EMS for A

8. EMS for A

9. EMS for A

10. EMS for B(A)

11. EMS for B(A)

12. ANOVA table
Source df SS EMS A a-1 SSA B(A) a(b-1) SSB(A) Error ab(n-1) SSE s2 Total abn-1 SSTO

13. Example
Tobacco Case Study
SAS code
Minitab
Variance Components

14. Cost Analysis
Supposed we want to minimize the variance of our treatment means (for a balanced design)
This will depend on the cost of sampling another level of the nested factor vs the cost of adding a replication

15. Cost analysis
The total number of units to be sampled is fixed at bn. If cost isn’t a factor, how should these units be allocated?

16. Cost Analysis Assume we have a fixed project cost C
D1=“dollars” per nested factor level
D2=“dollars” per rep
Total cost C=b D1+bn D2

17. Cost Analysis Subject to the cost constraint, we minimize
The answer turns out to be

18. Power Analysis
Note that increasing n does not really improve power for testing treatment effects