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Biostatistics-Lecture 9 Experimental designs

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

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Biostatistics-Lecture 9 Experimental designs

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  1. Biostatistics-Lecture 9Experimental 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

  5. Two-way ANOVA

  6. Two-way ANOVA

  7. Two-way ANOVA • 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. Two-way ANOVA • 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. Randomized Complete Block Design • 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. Random Effect designs • 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

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