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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 means inference
Population Marginal MeansInference
  • Two factor model with replication
population marginal means
Population Marginal Means
  • How do we conduct hypothesis testing?
  • Usual hypotheses:
population marginal means1
Population Marginal Means
  • In the balanced case, testing
population marginal means2
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 means3
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
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 model1
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 model2
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
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 model2
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 model3
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 means4
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 cells inference
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 cells inference1
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 cells inference2
Missing CellsInference
  • Yandell
    • Suggests Type IV-style contrasts for testing
    • Suggests analyzing non-empty cells as a one-way effects model