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

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?

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)

Nested Designs Typically, A is fixed and B is random

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

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

EMS for A

EMS for A

EMS for A

EMS for B(A)

EMS for B(A)

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

Example Tobacco Case Study SAS code Minitab Variance Components

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

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?

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

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

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