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Part IV The General Linear Model Multiple Explanatory Variables Chapter 13.4 Fixed * Random Effects Randomized block

Part IV The General Linear Model Multiple Explanatory Variables Chapter 13.4 Fixed * Random Effects Randomized block.

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Part IV The General Linear Model Multiple Explanatory Variables Chapter 13.4 Fixed * Random Effects Randomized block

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  1. Part IVThe General Linear ModelMultiple Explanatory VariablesChapter 13.4 Fixed * Random EffectsRandomized block

  2. Statistical controlThe effect of one variable (random: subject in last lecture) are removed to arrive at a better test for the variable of interest (fixed: drug in last lecture)It is used when manipulative control is not possible Field Not possible at large scales Expensive Can generate artifacts  study with well designed statistical control can be more informative

  3. Randomized block design"Block what you can, randomize what you cannot.“Blocking to remove effects of the most important nuisance variablesRandomization to reduce contaminating effects of remaining nuisance variablesWithin each block elements must be homogenous with respect to response variableHeterogeneity among blocksOrder blocks perpendicular to gradient of extraneous variableTreatments are randomized within each block

  4. Randomized block designExample from Netter & Wasserman (1974) Applied Linear Statistical ModelsResponse: sales volumeExplanatory: level of newspaper advertisingSize of city correlated with response block according to population size perpendicular to gradient=from large to small randomize level of advertising within each block

  5. Randomized block design Some blocking criteria: Characteristics associated with the unit • For persons: gender, age, etc • For geographic areas: population size, average income, etc • Characteristics associated with the experimental setting observer, time of processing, machine, tank, batch, measuring instrument, etc

  6. GLM | Randomized block design • Data from Sokal & Rohlf 1995. Dry weights of 3 genotypes (wild, hetrozygote mutant, homozygote mutnt) of flour beetle Tribolium in 4 experiments • Does weight vary among genotypes?

  7. 1. Construct Model Response variable: M = beetle mass Explanatory variables: 1. Genotype Fixed effect 2. Experiment Random effect Experiments were laborious and carried out several months apart Each experiment is a block

  8. 1. Construct Model Verbal: Does weight vary among genotypes, after controlling for differences among genotypes? Graphical:

  9. 1. Construct Model X Can we have an interaction term? Same situation as last lecture. If we were to include an interaction term then dfres=0  MSres = SSres / dfres =  Formal:

  10. 1. Construct Model X Formal:

  11. 2. Execute analysis lm1 <- lm(M~G+B, data=trib)

  12. 2. Execute analysis

  13. 3. Evaluate model a. Straight line • Straight line model ok? b. Need to revise model? • Errors homogeneous? c. Assumptions for computing p-values • Errors normal? • Errors independent?

  14. 3. Evaluate model a. Straight line • Straight line model ok? b. Need to revise model? • Errors homogeneous? c. Assumptions for computing p-values • Errors normal? • Errors independent? NA ?

  15. 3. Evaluate model a. Straight line • Straight line model ok? b. Need to revise model? • Errors homogeneous? c. Assumptions for computing p-values • Errors normal? • Errors independent? NA ? X

  16. 3. Evaluate model a. Straight line • Straight line model ok? b. Need to revise model? • Errors homogeneous? c. Assumptions for computing p-values • Errors normal? • Errors independent? NA ? X

  17. State the population and whether the sample is representative. Genotype  fixed effects We will infer only to those genotypes Experiment, i.e. time and other condition  random effects All possible measurements that could have been made on Tribolium, given the mode of collection

  18. Decide on mode of inference. Is hypothesis testing appropriate? • State HA / Ho pair, test statistic, distribution, tolerance for Type I error. Interaction Term: Removed by experimental design (genotypes weighed in random order) Block Term experiment: We are interested in this effect. Only included in model to remove the variance from the error term

  19. State HA / Ho pair, test statistic, distribution, tolerance for Type I error. Genotype Term: HA: E(MI) ≠ E(MII) ≠ E(MIII) HA:Var(βG) > 0 H0: E(MI) = E(MII) = E(MIII) H0:Var(βG) = 0 Test Statistic Distribution of test statitstic Tolerance for Type I error

  20. 7. ANOVA n = 12

  21. 7. ANOVA n = 12 -1

  22. 7. ANOVA n = 12 -1

  23. 7. ANOVA n = 12 What would it look like had we not controlled for experiment?

  24. 7. ANOVA n = 12 What would it look like had we not controlled for experiment?

  25. 7. ANOVA n = 12 What would it look like had we not controlled for experiment?

  26. 7. ANOVA n = 12 What would it look like had we not controlled for experiment?

  27. 7. ANOVA n = 12 What would it look like had we not controlled for experiment?

  28. 7. ANOVA n = 12 What would it look like had we not controlled for experiment?

  29. 7. ANOVA n = 12 What would it look like had we not controlled for experiment? STATISTICAL CONTROL

  30. 8. Decide whether to recompute p-value Residuals homogenous not independent deviated slightly from normality n=12 p=0.027  needs to change 2-fold to change decision Randomization (100 000 runs) pran= 0.0186 change in p: 0.027/0.186 = 1.45

  31. Declare decision about termsOnly the fixed term was tested p=0.0186< α =0.05 Reject H0 There is significant variation in mean dry weight among genotypes

  32. Report and interpret parameters of biological interest Means per genotype with 95% CI, not controlled for among experiments variation Genotype I Heterozygote mutant Genotype II Wild Genotype III Homozygote mutant

  33. Report and interpret parameters of biological interest Means per genotype with 95% CI, not controlled for among experiments variation library(effects) effect("G", lm2,se=TRUE, confidence.level=.95) Genotype I Heterozygote mutant Genotype II Wild Genotype III Homozygote mutant

  34. Report and interpret parameters of biological interest Means per genotype with 95% CI, controlled for among experiments variation Genotype I Heterozygote mutant Genotype II Wild Genotype III Homozygote mutant

  35. Report and interpret parameters of biological interest Means per genotype with 95% CI, controlled for among experiments variation effect("G", lm1,se=TRUE, confidence.level=.95) Genotype I Heterozygote mutant Genotype II Wild Genotype III Homozygote mutant

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