1. Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 15 Analysis of Data from Fractional Factorials and Other Unbalanced Experiments; Population Marginal Means

2. Issues with Unbalanced Data Are the usual analysis of variance methods appropriate? How does imbalance affect statistical tests? What are correct test procedures? How does one compare mean estimates for factor-level combinations?

3. Evaluating Factor Effects with Unbalanced Data Hierarchical Models If an interaction effect is included, all lower-order interaction effects and main effects of the factors in the interaction are also included If powers or products of covariates are included, all lower-order powers and products of the covariates are also included Subset Hierarchical Models Two models Each model contains only hierarchical terms One model has a subset of terms of the other model Only fit and analyze subset hierarchical models

4. Evaluating Factor Effects with Unbalanced Data Error Sums of Squares Models 1 & 2 are hierarchical Model 2 has a subset of model 1 terms SS E2 SS E1 Reduction in Error Sums of Squares R(M 1 | M 2 ) = SS E2 - SS E1 df = 2 - 1 Testing Effects 2 > 1

5. SAS GLM Type I Sums of Squares: Reduction in Error Sums of Squares Source A B C D E F G Sum of Squares R(M A ) = TSS - SS E1 = SS A R(M AB |M A ) R(M ABC |M AB ) R(M ABCD |M ABC ) R(M ABCDE |M ABCD ) R(M ABCDEF |M ABCDE ) R(M ABCDEFG |M ABCDEF ) Each effect adjusted for previous effects Each effect adjusted for previous effects

6. SAS GLM Type III Sums of Squares Indicator variables for each main effect Indicator variables for each main effect k - 1 indicator variables for a k level factor Usually are not orthogonal if the design is unbalanced Interactions are products of main effect indicator variables Interactions are products of main effect indicator variables Usually are not orthogonal if the design is unbalanced Fit a model with all main effect and all interaction indicator variables Fit a model with all main effect and all interaction indicator variables Fit a model with all main effect and all interaction indicator variablesthe main effect or interaction to be tested Fit a model with all main effect and all interaction indicator variables except the main effect or interaction to be tested Calculate the difference in residual sums of squares Calculate the difference in residual sums of squares Differences are necessarily positive Main effect and interaction sums of squares calculated with higher-order interactions present Primary rationale: population marginal means (see below) Hypotheses to be tested are the same as in the balanced case Hypotheses to be tested are the same as in the balanced case if no missing data

7. Choosing Between Type I and Type III Sums of Squares Ordinarily use Type III sums of squares for unbalanced data Complete or fractional factorials with no missing factor-level combinations Tests are the same as those for balanced data Type I sums of squares should be used only when the order of the factors in the model statement is meaningful There is no well accepted estimation method for experiments with missing factor-level combinations, apart from designed fractional factorials