1. Population Marginal Means Two factor model with replication Two factor model with replication

2. Population Marginal Means

3. The above expectation depends on the design The above expectation depends on the design Population marginal means depend only on the unknown parameters; it is these quantities that LSMEANS estimates Population marginal means depend only on the unknown parameters; it is these quantities that LSMEANS estimates

4. Population Marginal Means

5. Missing Cells Population Marginal Means Additive two factor model with replication Additive two factor model with replication Example from Searle Example from Searle  a=2, b=2, n 22 =0  Searle et al use unusual constraints—choice of constraints doesn’t affect estimators for either the observed or unobserved cell means

6. Missing Cells Population Marginal Means Table of expectations (note that      Table of expectations (note that          

7. Missing Cells Population Marginal Means Table of least squares estimates Table of least squares estimates

8. Population Marginal Means LSMEANS for the population marginal means: LSMEANS for the population marginal means:LSMEANS

9. Missing Cells Population Marginal Means Table of expectations for the interaction model Table of expectations for the interaction model

10. Missing Cells Estimability Table of least squares estimates for the interaction model Table of least squares estimates for the interaction model PMM(    PMM(    PMM(    are also non-estimable PMM(    PMM(    PMM(    are also non-estimable

11. Missing Cells Estimability Worksheet Example Worksheet Example Yandell notes that cell means in an additive model are always estimable if the design is connected Yandell notes that cell means in an additive model are always estimable if the design is connected –Connectedness is easy to verify in a two-way layout; difficult in other contexts.