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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 MeansInference • Two factor model with replication
Population Marginal Means • How do we conduct hypothesis testing? • Usual hypotheses:
Population Marginal Means • In the balanced case, testing
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.
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
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
Population Marginal Means—Additive Model • We base a test on: • The direct minimization is difficult, but does test the correct hypothesis
Population Marginal Means—Additive Model • Yandell uses reduction in SS formulae to derive the same tests, but these still require LS estimation • SAS example
Population Marginal Means--Interaction Model • Under the interaction model, correct test statistics are actually easier to compute. • Appropriate estimates of PMMs would be:
Population Marginal Means--Interaction Model • Consider the partition:
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
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):
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.
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
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
Missing CellsInference • Yandell • Suggests Type IV-style contrasts for testing • Suggests analyzing non-empty cells as a one-way effects model