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

20. 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

21. Tests for Constant Variance H0:s12=...=st2

22. Bartlett’s Test
General Test that can be used in many settings with groups
H0: s12 = … = st2 (homogeneous variances)
Ha: Population Variances are not all equal
MSE ≡ Pooled Variance

23. Remedial Measures
Normally distributed, Unequal variances – Use Weighted Least Squares with weights: wij = 1/si2
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 mean2: Y’=1/Y
Response is a (binomial) proportion: Y’=2arcsin(sqrt(Y))
Non-parametric tests – F-test based on ranks and Kruskal-Wallis Test

24. Welch’s Test – Unequal Variances

25. Nonparametric Tests – Non-Normal Data