1. Biostatistics-Lecture 9 Experimental designs Ruibin Xi Peking University School of Mathematical Sciences

2. Two-way ANOVA A study investigated the effects of 4 treatments (A, B, C, D) on 3 toxic agents (I, II, III). 48 rats were randomly assigned to 12 factor level combinations.

3. Two-way ANOVA Y: the response variable Factor A with levels i=1 to a Factor B with levels j = 1 to b A particular combination of levels is called a treatment or a cell. There are treatments is the kth observation for treatment (i,j), k = 1 to n

4. Two-way ANOVA

7. The factor effect model

8. Two-way ANOVA The factor effect model: constraints

9. Interaction plots

10. Two-way ANOVA Estimates Sum of squares (SS)

11. Two-way ANOVA

12. Test for Factor A Test For Factor B

13. Two-way ANOVA Test for Interaction Effect

14. Two-way ANOVA One observation per cell (n=1) – Cannot estimate the interaction, have to assume no interaction

15. Randomized Complete Block Design Useful when the experiments are non- homogenous – Rats are bred from different labs – Patients belong to different age groups Randomized Block design can used to reduce the variance

16. Randomized Complete Block Design

17. A “block” consists of a complete replication of the set of treatments Block and treatments usually are assumed not having interactions Advantages: – Effective grouping can give substantially more precise results – Can accommodate any number of treatments and replications – Statistical analysis is relatively simple – If an entire block needs to be dropped, the analysis is not complicated thereby

18. Randomized Complete Block Design Disadvantages – The degree of freedom for experiment error are not as large as with a completely randomized design – More assumptions (no interaction between block and treatment, constant variance from block to block) – Blocking is an observational factor and not an experimental factor, cause-and-effect inferences cannot be made for the blocking variable and the response

19. Randomized Complete Block Design The model (similar to additive two-way ANOVA) block effect treatment effect

20. Random effect designs Fixed effect models – Levels of each factor are fixed – Interested in differences in response among those specific levels Random effect model – Random effect factor: factor levels are meant to be representative of a general population of possible levels If there are both fixed and random effects, call it mixed effect model

21. Random effect designs One way random model

22. Random effect designs

23. Random Effect designs

24. Hypothesis Testing statistic

25. Random effect designs Two random factors

26. Random effect designs There are five parameters in this model For balanced design

27. Random effect designs Hypothesis testing: main effects

28. Random effect designs Hypothesis testing: interaction effects

29. To be continued Mixed effect models Unbalanced two-way ANOVA