SingleFactor Studies KNNL Chapter 16 SingleFactor Models Independent

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 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

Fitting ANOVA Models

Analysis of Variance

ANOVA Table

F-Test for H 0: m 1 =. . . = mr

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 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 = n. T!/(n 1!. . . nr!)

Power Approach to Sample Size Choice - Tables

Power Approach to Sample Size Choice – R Code

Power Approach to Finding “Best” Treatment

Effects of Model Departures • Non-normal Data – Generally not problematic in terms of the F-test, if data are not too far from normal, and reasonably large sample sizes • Unequal Error Variances – As long as sample sizes are approximately equal, generally not a problem in terms of F-test. • Non-independence of error terms – Can cause problems with tests. Should use Repeated Measures ANOVA if same subject receives each treatment

Tests for Constant Variance H 0: s 12=. . . =st 2

Bartlett’s Test General Test that can be used in many settings with groups • H 0: s 12 = … = st 2 (homogeneous variances) • Ha: Population Variances are not all equal • MSE ≡ Pooled Variance

Remedial Measures • Normally distributed, Unequal variances – Use Weighted Least Squares with weights: wij = 1/si 2 § Welch’s Test • Non-normal data (with possibly unequal variances) – Variance Stabilizing and Box-Cox Transformations – – Variance proportional to mean: Y’=sqrt(Y) Standard Deviation proportional to mean: Y’=log(Y) Standard Deviation proportional to mean 2: Y’=1/Y Response is a (binomial) proportion: Y’=2 arcsin(sqrt(Y)) • Non-parametric tests – F-test based on ranks and Kruskal-Wallis Test

Welch’s Test – Unequal Variances

Nonparametric Tests – Non-Normal Data