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 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

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

Cell Means Model

Cell Means Model – Regression Form

Model Interpretations Factor Level Means  Observational Studies – The  i represent the population means among units from the populations of factor levels  Experimental Studies - The  i 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

Fitting ANOVA Models

Analysis of Variance

ANOVA Table

F-Test for H 0 :    r

General Linear Test of Equal Means

Factor Effects Model

Regression Approach – Factor Effects Model

Factor Effects Model with Weighted Mean

Regression for Cell Means Model

Randomization (aka Permutation) Tests Treats the units in the study as a finite population of units, each with a fixed error term  ij When the randomization procedure assigns the unit to treatment i, we observe Y ij =  i +  ij When there are no treatment effects (all  i = 0), Y ij =  ij 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 = n T !/(n 1 !...n r !)

Power Approach to Sample Size Choice - Tables

Power Approach to Sample Size Choice – R Code

Power Approach to Finding “Best” Treatment