1. Stat 470-5 Today: General Linear Model Assignment 1:

2. General Linear Model ANOVA model can be viewed as a special case of the general linear model or regression model Suppose have response, y, which is thought to be related to p predictors (sometimes called explanatory variables or regressors) Predictors: x 1, x 2,…,x p Model:

3. Example: Rainfall (Exercise 2.16) In winter, a plastic rain gauge cannot be used to collect precipitation because it will freeze and crack. Instead, metal cans are used to collect snowfall and the snow is allowed to melt indoors. The water is then poured into a plastic rain gauge and a measurement recorded. An estimate of snowfall is obtained by multiplying this measurement by 0.44. One observer questions this and decides to collect data to test the validity of this approach For each rainfall in a summer, she measures: (i) rainfall using a plastic rain gauge, (ii) using a metal can What is the current model being used?

4. Example: Rainfall (Exercise 2.16)

5. Seems to be a linear relationship Will use regression to establish linear relationship between x and y What should the slope be?

6. Example: Rainfall (Exercise 2.16)

10. Comments General linear model may have many predictors Is suitable for many situations Easily done in all stats packages

11. Designs So Far… Have considered 1-factor designs: –Paired comparisons (paired t-test) –Completely randomized design (ANOVA) Frequently have more than one factor We will learn to design and analyze such experiments

12. Example: Penicillin Experiment Objective: Compare four processes for making penicillin The raw material used in the process is thought to vary substantially from batch to batch Experiment Design: –Use five separately produced batches of raw material –Divide each batch into four sub-batches –Randomly assign each process to one sub-batch. –Randomize the production order within each batch –Measure the yield (%)

13. Blocking Paired comparisons (Section 2.1) is a special case of a Randomized Complete Block (RCB) design More generally: –Have k treatments –have b blocks –each of the k treatments is applied (in random order) to each block

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

15. Model i=1, 2, …, b; j=1, 2, …,k;

16. ANOVA Table

17. Hypothesis Tests

18. Multiple Comparisons

19. Example: Penicillin Experiment Objective: Compare four processes for making penicillin The raw material used in the process is thought to vary substantially from batch to batch Experiment Design: –Use five separately produced batches of raw material –Divide each batch into four sub-batches –Randomly assign each process to one sub-batch. –Randomize the production order within each batch –Measure the yield (%) This is a RCB design with b = k =

20. Data: Penicillin Example

21. Yield versus Process (grouped by blocks)

22. Observations: Some consistent differences among batches: generally, B1 high, B5 low No apparent consistent differences among processes

23. ANOVA – Randomized Block Design

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

25. Diagnostic Checking Residual plots -- penicillin experiment –To check Normality assumption: plot all residuals: dot chart, histogram, Normal prob. plot –To check assumption of equal variances: dot plot of residuals by Treatment dot plot of residuals by Block –Other possible checks: plot residuals vs. testing order plot residuals vs. other potential sources of variability –e.g., vs. technician, or machine, etc.

26. Randomized Block Design -- Summary Objective: –Compare several treatments for a factor –eliminate source of variability from comparison of treatments –broaden conclusions Experimental Method: –create b blocks each with a experimental units –in each block, randomly assign each treatment to one experimental unit Analysis: –ANOVA: Blocks, Treatments, Error are sources of variation

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

28. Examples of Blocking Variables Blocks are units that can be sub-divided into sub-units –Time: –Space –People: –Batches: