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Single-Factor Studies

This chapter provides a comprehensive overview of Single-Factor Models, covering independent variables, including both qualitative and quantitative types. It discusses the assumptions underlying ANOVA models and the significance of balanced designs. The analysis procedure focuses on testing differences among factor level means and includes follow-up post-hoc comparisons. Key concepts like fixed and random factors, regression models, and randomization tests are explored in detail, emphasizing how to analyze experimental data effectively using statistical principles and software tools.

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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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