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Statistical Analysis

Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 15 Analysis of Data from Fractional Factorials and Other Unbalanced Experiments; Population Marginal Means. Issues with Unbalanced Data. Are the usual analysis of variance methods appropriate?

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Statistical Analysis

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  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. 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 Evaluating Factor Effects with Unbalanced Data Only fit and analyze subset hierarchical models

  4. Reduction in Error Sums of Squares R(M1 | M2) = SSE2 - SSE1 df = n2 - n1 Testing Effects Evaluating Factor Effects with Unbalanced Data Error Sums of Squares Models 1 & 2 are hierarchical Model 2 has a subset of model 1 terms SSE2 SSE1 n2 > n1

  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(MA) = TSS - SSE1 = SSA R(MAB|MA) R(MABC|MAB) R(MABCD|MABC) R(MABCDE|MABCD) R(MABCDEF|MABCDE) R(MABCDEFG|MABCDEF) Each effect adjusted for previous effects

  6. SAS GLM Type III Sums of Squares • 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 • 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 variablesexceptthe main effect or interaction to be tested • 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 caseif 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

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