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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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Statistical control The 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

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Randomized block design "Block what you can, randomize what you cannot.“ Blocking to remove effects of the most important nuisance variables Randomization to reduce contaminating effects of remaining nuisance variables Within each block elements must be homogenous with respect to response variable Heterogeneity among blocks Order blocks perpendicular to gradient of extraneous variable Treatments are randomized within each block

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Randomized block design Example from Netter & Wasserman (1974) Applied Linear Statistical Models Response: sales volume Explanatory: level of newspaper advertising Size of city correlated with response block according to population size perpendicular to gradient=from large to small randomize level of advertising within each block

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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

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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? GLM | Randomized block design BlocksGtypeWt 110.958 210.971 310.927 410.971 120.986 221.051 320.891 421.010 130.925 230.952 330.829 430.955

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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

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

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1. Construct Model Formal: Can we have an interaction term? Same situation as last lecture. If we were to include an interaction term then df res =0 MS res = SS res / df res = X

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1. Construct Model Formal: X

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2. Execute analysis lm1 <- lm(M~G+B, data=trib) BlocksGtypeWt 110.958 210.971 310.927 410.971 120.986 221.051 320.891 421.010 130.925 230.952 330.829 430.955

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2. Execute analysis GtypeBlock Effect FitsRes BlocksGtypeWtβ0β0 βGβG βBβB β 0 + β B + β G 110.9580.95220.004580.0040.9609-0.0029 210.9710.95220.004580.0390.9959-0.0249 310.9270.95220.00458-0.0700.88690.0401 410.9710.95220.004580.0270.9833-0.0123 120.9860.95220.032330.0040.9887-0.0027 221.0510.95220.032330.0391.02370.0273 320.8910.95220.03233-0.0700.9147-0.0237 421.0100.95220.032330.0271.0110-0.0010 130.9250.9522-0.036920.0040.91940.0056 230.9520.9522-0.036920.0390.9544-0.0024 330.8290.9522-0.03692-0.0700.8454-0.0164 430.9550.9522-0.036920.0270.94180.0133

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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?

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

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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

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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 ?

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4.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

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5.Decide on mode of inference. Is hypothesis testing appropriate? 6.State H A / H o 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

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6.State H A / H o pair, test statistic, distribution, tolerance for Type I error. Genotype Term: H A : E(M I ) ≠ E(M II ) ≠ E(M III )H A :Var(β G ) > 0 H 0 : E(M I ) = E(M II ) = E(M III ) H 0 :Var(β G ) = 0 Test Statistic Distribution of test statitstic Tolerance for Type I error

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7. ANOVA n = 12 SourcedfSSMSFp Blocks Genotype Res______ Total

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7. ANOVA n = 12 SourcedfSSMSFp Blocks30.0213 Genotype20.0097 Res__6__0.0048 Total110.0353

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7. ANOVA n = 12 SourcedfSSMSFp Blocks30.0213 Genotype20.00970.0048586.970.027 Res__6__0.00480.000697 Total110.0353

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7. ANOVA n = 12 SourcedfSSMSFp Blocks30.0213 Genotype20.00970.0048586.970.027 Res__6__0.00480.000697 Total110.0353 What would it look like had we not controlled for experiment? SourcedfSSMSFp Genotype20.00970.0048581.710.235 Res__9__0.02550.002841 Total110.0353

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7. ANOVA n = 12 SourcedfSSMSFp Blocks30.0213 Genotype20.00970.0048586.970.027 Res__6__0.00480.000697 Total110.0353 What would it look like had we not controlled for experiment? SourcedfSSMSFp Genotype20.00970.0048581.710.235 Res__9__0.02550.002841 Total110.0353

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7. ANOVA n = 12 SourcedfSSMSFp Blocks30.0213 Genotype20.00970.0048586.970.027 Res__6__0.00480.000697 Total110.0353 What would it look like had we not controlled for experiment? SourcedfSSMSFp Genotype20.00970.0048581.710.235 Res__9__0.02550.002841 Total110.0353

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7. ANOVA n = 12 SourcedfSSMSFp Blocks30.0213 Genotype20.00970.0048586.970.027 Res__6__0.00480.000697 Total110.0353 What would it look like had we not controlled for experiment? SourcedfSSMSFp Genotype20.00970.0048581.710.235 Res__9__0.02550.002841 Total110.0353

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7. ANOVA n = 12 SourcedfSSMSFp Blocks30.0213 Genotype20.00970.0048586.970.027 Res__6__0.00480.000697 Total110.0353 What would it look like had we not controlled for experiment? SourcedfSSMSFp Genotype20.00970.0048581.710.235 Res__9__0.02550.002841 Total110.0353

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7. ANOVA n = 12 SourcedfSSMSFp Blocks30.0213 Genotype20.00970.0048586.970.027 Res__6__0.00480.000697 Total110.0353 What would it look like had we not controlled for experiment? SourcedfSSMSFp Genotype20.00970.0048581.710.235 Res__9__0.02550.002841 Total110.0353

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7. ANOVA n = 12 SourcedfSSMSFp Blocks30.0213 Genotype20.00970.0048586.970.027 Res__6__0.00480.000697 Total110.0353 What would it look like had we not controlled for experiment? SourcedfSSMSFp Genotype20.00970.0048581.710.235 Res__9__0.02550.002841 Total110.0353 STATISTICAL CONTROL

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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) p ran = 0.0186 change in p: 0.027/0.186 = 1.45

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9.Declare decision about terms Only the fixed term was tested p=0.0186< α =0.05 Reject H 0 There is significant variation in mean dry weight among genotypes

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

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10.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 III Homozygote mutant Genotype I Heterozygote mutant Genotype II Wild

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Genotype III Homozygote mutant 10.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

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10.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 III Homozygote mutant Genotype I Heterozygote mutant Genotype II Wild

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