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

2. Population Marginal Means Inference

4. Population Marginal Means How do we conduct hypothesis testing? How do we conduct hypothesis testing? Usual hypotheses: Usual hypotheses:

5. Population Marginal Means In the balanced case, testing In the balanced case, testing

6. Population Marginal Means For the example in the text (n 31 =5; n1. =18=n 2. ; n 3. =17), a test that the expectations of the sample marginal means are equal would test: For the example in the text (n 31 =5; n1. =18=n 2. ; n 3. =17), a test that the expectations of the sample marginal means are equal would test: The discrepancy is modest here, but can lead to unusual tests in other contexts. The discrepancy is modest here, but can lead to unusual tests in other contexts.

7. Population Marginal Means We will study inference for both the additive and interaction model We will study inference for both the additive and interaction model Tests for the additive model are actually harder to derive than for the interaction model Tests for the additive model are actually harder to derive than for the interaction model Sequential analyses (Type I and Type II analyses) can be misleading Sequential analyses (Type I and Type II analyses) can be misleading Type III analyses are appropriate for both Type III analyses are appropriate for both

8. Population Marginal Means— Additive Model The usual numerator for testing H oA, based on LS estimates of a model with A alone, will no longer be appropriate: The usual numerator for testing H oA, based on LS estimates of a model with A alone, will no longer be appropriate: This is actually the basis of the Type I hypothesis test statistic for A (if A is tested first), and tests This is actually the basis of the Type I hypothesis test statistic for A (if A is tested first), and tests

9. Population Marginal Means— Additive Model We base a test on: We base a test on: The direct minimization is difficult, but does test the correct hypothesis The direct minimization is difficult, but does test the correct hypothesis

10. Population Marginal Means— Additive Model Yandell uses reduction in SS formulae to derive the same tests, but these still require LS estimation Yandell uses reduction in SS formulae to derive the same tests, but these still require LS estimation SAS example SAS example

11. Population Marginal Means-- Interaction Model Under the interaction model, correct test statistics are actually easier to compute. Under the interaction model, correct test statistics are actually easier to compute. Appropriate estimates of PMMs would be: Appropriate estimates of PMMs would be:

12. Population Marginal Means-- Interaction Model Consider the partition: Consider the partition:

13. Population Marginal Means-- Interaction Model These deviations form the basis for the straightforward Type III SS seen in Table 10.3 These deviations form the basis for the straightforward Type III SS seen in Table 10.3 Type III SS test the usual PMM hypotheses, but they are not additive Type III SS test the usual PMM hypotheses, but they are not additive

14. Population Marginal Means-- Interaction Model Type I SS are additive, but tests on A, and then on B|A, are odd Type I SS are additive, but tests on A, and then on B|A, are odd We already discussed We already discussed H OB|A will test (for Type II SS as well): H OB|A will test (for Type II SS as well):

15. Population Marginal Means SAS Example SAS Example –n 11 =2, n 12 =3, n 13 =1, n 21 =2, n 22 =2, n 23 =2 –We can study contrasts that SAS uses to test Type I, Type II, and Type III hypotheses under both additive and interaction models.

16. Missing Cells Inference In the additive model, Type III hypotheses are reasonable provided the design is connected In the additive model, Type III hypotheses are reasonable provided the design is connected For the interaction model, there is little we can do For the interaction model, there is little we can do

17. Missing Cells Inference Type III SS often test uninteresting hypotheses Type III SS often test uninteresting hypotheses Type IV SS test accessible hypotheses, but may ignore much of the data in doing so Type IV SS test accessible hypotheses, but may ignore much of the data in doing so Worksheet example Worksheet example

18. Missing Cells Inference Yandell Yandell –Suggests Type IV-style contrasts for testing –Suggests analyzing non-empty cells as a one- way effects model