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Population Marginal Means Inference

Population Marginal Means Inference. Two factor model with replication. Population Marginal Means Inference. Population Marginal Means Inference. Population Marginal Means. How do we conduct hypothesis testing? Usual hypotheses:. Population Marginal Means. In the balanced case, testing .

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Population Marginal Means Inference

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  1. Population Marginal MeansInference • Two factor model with replication

  2. Population Marginal MeansInference

  3. Population Marginal MeansInference

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

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

  6. Population Marginal Means • For the example in the text (n31=5; n1.=18=n2.; n3.=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.

  7. Population Marginal Means • 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 • Sequential analyses (Type I and Type II analyses) can be misleading • Type III analyses are appropriate for both

  8. Population Marginal Means—Additive Model • The usual numerator for testing HoA, 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

  9. Population Marginal Means—Additive Model • We base a test on: • 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 • SAS example

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

  12. Population Marginal Means--Interaction Model • 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 • 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 • We already discussed • HOB|A will test (for Type II SS as well):

  15. Population Marginal Means • SAS Example • n11=2, n12=3, n13=1, n21=2, n22=2, n23=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 CellsInference • In the additive model, Type III hypotheses are reasonable provided the design is connected • For the interaction model, there is little we can do

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

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

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