1. Lecture 4 zToday: More Sections 2.1-2.3 zPlease read these sections. You are responsible for all material in these sections…even those not discussed in class zSuggested Problems…not to be handed in: derive results a-c on page 35 zAssignment #1 (due in 1 week): yChapter 1: 11, 13, 16, 18(i) yChapter 2: 8

2. Blocking zPaired comparisons (Section 2.1) is a special case of a Randomized Complete Block (RCB) design zMore generally: yHave k treatments yhave b blocks yeach of the k treatments is applied (in random order) to each block

3. Blocking zUnits within a block are more homogeneous than units between blocks zCan remove variability due to blocks (e.g., boy to boy variability) from the comparison of treatments

4. Model

5. Estimating Model Parameters

6. ANOVA Table

7. Hypothesis Tests

8. Multiple Comparisons

9. Example: Penicillin experiment (source: Box, Hunter, Hunter, p. 209) zObjective: Compare four processes for making penicillin zThe raw material used in the process is thought to vary substantially from batch to batch zExperiment Design: yUse five separately produced batches of raw material yDivide each batch into four sub-batches yRandomly assign each process to one sub-batch. yRandomize the production order within each batch yMeasure the yield (%) zThis is a RCB design with b = a =

10. Data: Penicillin Example

11. Yield versus Process (grouped by blocks)

12. Observations: zSome consistent differences among batches: generally, B1 high, B5 low zNo apparent consistent differences among processes

13. ANOVA – Randomized Block Design

14. Conclusions zF-value for Processes is not significant at zF-value for Batches (P =.04) is significant at … indicates some differences among batches of raw material zWe suspected batch differences; that’s why the design was done this way. This result is no surprise or of particular interest, in this case. zWhich would you use?

15. Randomized Block Design -- Summary zObjective: yCompare several treatments for a factor yeliminate source of variability from comparison of treatments ybroaden conclusions zExperimental Method: ycreate b blocks each with a experimental units yin each block, randomly assign each treatment to one experimental unit zAnalysis: yANOVA: Blocks, Treatments, Error are sources of variation

16. Why Bother? zCan remove variability due to blocks (e.g., boy to boy variability) from the comparison of treatments zRemoving source of variability often increases power to detect treatment differences zMake comparisons on more homogeneous units

17. Examples of Blocking Variables zBlocks are units that can be sub-divided into sub-units yTime: ySpace yPeople: yBatches: