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Single-Factor Studies

Single-Factor Studies. KNNL – Chapter 16. Single-Factor Models. Independent Variable can be qualitative or quantitative If Quantitative, we typically assume a linear, polynomial, or no “structural” relation If Qualitative, we typically have no “structural” relation

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Single-Factor Studies

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  1. Single-Factor Studies KNNL – Chapter 16

  2. Single-Factor Models • Independent Variable can be qualitative or quantitative • If Quantitative, we typically assume a linear, polynomial, or no “structural” relation • If Qualitative, we typically have no “structural” relation • Balanced designs have equal numbers of replicates at each level of the independent variable • When no structure is assumed, we refer to models as “Analysis of Variance” models, and use indicator variables for treatments in regression model

  3. Single-Factor ANOVA Model • Model Assumptions for Model Testing • All probability distributions are normal • All probability distributions have equal variance • Responses are random samples from their probability distributions, and are independent • Analysis Procedure • Test for differences among factor level means • Follow-up (post-hoc) comparisons among pairs or groups of factor level means

  4. Cell Means Model

  5. Cell Means Model – Regression Form

  6. Model Interpretations • Factor Level Means • Observational Studies – The mi represent the population means among units from the populations of factor levels • Experimental Studies - The mi represent the means of the various factor levels, had they been assigned to a population of experimental units • Fixed and Random Factors • Fixed Factors – All levels of interest are observed in study • Random Factors – Factor levels included in study represent a sample from a population of factor levels

  7. Fitting ANOVA Models

  8. Analysis of Variance

  9. ANOVA Table

  10. F-Test for H0: m1= ... = mr

  11. General Linear Test of Equal Means

  12. Factor Effects Model

  13. Regression Approach – Factor Effects Model

  14. Factor Effects Model with Weighted Mean

  15. Regression for Cell Means Model

  16. Randomization (aka Permutation) Tests • Treats the units in the study as a finite population of units, each with a fixed error term eij • When the randomization procedure assigns the unit to treatment i, we observe Yij=m. + ti+ eij • When there are no treatment effects (all ti = 0), Yij=m. + eij • We can compute a test statistic, such as F* under all (or in practice, many) potential treatment arrangements of the observed units (responses) • The p-value is measured as proportion of observed test statistics as or more extreme than original. • Total number of potential permutations = nT!/(n1!...nr!)

  17. Power Approach to Sample Size Choice - Tables

  18. Power Approach to Sample Size Choice – R Code

  19. Power Approach to Finding “Best” Treatment

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