Part IVThe General Linear Model

Multiple Explanatory Variables

Chapter 13.4 Fixed * Random Effects

Randomized block

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

FieldNot possible at large scalesExpensiveCan generate artifacts study with well

designed statistical control can be more informative

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 variables

Within 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

Randomized block design

Example from Netter & Wasserman (1974) Applied Linear Statistical Models

Response: sales volumeExplanatory: level of newspaper advertising

Size of city correlated with response

block according to population sizeperpendicular to gradient=from large to

smallrandomize level of advertising within each block

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 settingobserver, time of processing, machine, tank, batch, measuring instrument, etc

• 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

Blocks Gtype Wt

1 1 0.958

2 1 0.971

3 1 0.927

4 1 0.971

1 2 0.986

2 2 1.051

3 2 0.891

4 2 1.010

1 3 0.925

2 3 0.952

3 3 0.829

4 3 0.955

1. Construct ModelResponse 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

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

1. Construct ModelFormal:

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 =

X

1. Construct ModelFormal:

X

2. Execute analysis

lm1 <- lm(M~G+B, data=trib)

Blocks Gtype Wt

1 1 0.958

2 1 0.971

3 1 0.927

4 1 0.971

1 2 0.986

2 2 1.051

3 2 0.891

4 2 1.010

1 3 0.925

2 3 0.952

3 3 0.829

4 3 0.955

2. Execute analysis

Gtype Block

Effect Effect Fits Res

Blocks Gtype Wt β0 βG βB β0+ βB+ βG

1 1 0.958 0.9522 0.00458 0.004 0.9609 -0.0029

2 1 0.971 0.9522 0.00458 0.039 0.9959 -0.0249

3 1 0.927 0.9522 0.00458 -0.070 0.8869 0.0401

4 1 0.971 0.9522 0.00458 0.027 0.9833 -0.0123

1 2 0.986 0.9522 0.03233 0.004 0.9887 -0.0027

2 2 1.051 0.9522 0.03233 0.039 1.0237 0.0273

3 2 0.891 0.9522 0.03233 -0.070 0.9147 -0.0237

4 2 1.010 0.9522 0.03233 0.027 1.0110 -0.0010

1 3 0.925 0.9522-

0.03692 0.004 0.9194 0.0056

2 3 0.952 0.9522-

0.03692 0.039 0.9544 -0.0024

3 3 0.829 0.9522-

0.03692 -0.070 0.8454 -0.0164

4 3 0.955 0.9522-

0.03692 0.027 0.9418 0.0133

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?

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

?

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

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

?

4. State the population and whether the sample is representative.

Genotype fixed effectsWe 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

5. Decide on mode of inference. Is hypothesis testing appropriate?

6. 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

6. 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

7. ANOVA n = 12

Source df SS MS F p

Blocks

Genotype

Res______

______

Total

7. ANOVA n = 12

Source df SS MS F p

Blocks 30.0213

Genotype

20.0097

Res __6__0.0048

Total 110.0353

-1

7. ANOVA n = 12

Source df SS MS F p

Blocks 30.0213

Genotype

20.0097

0.004858

6.97 0.027

Res __6__0.0048

0.000697

Total 110.0353

-1

7. ANOVA n = 12Source df SS MS F p

Blocks 30.0213

Genotype

20.0097

0.004858

6.97 0.027

Res __6__0.0048

0.000697

Total 110.0353

What would it look like had we not controlled for experiment?

Source df SS MS F p

Genotype

20.0097

0.004858

1.71 0.235

Res __9__0.0255

0.002841

Total 110.0353

7. ANOVA n = 12Source df SS MS F p

Blocks 30.0213

Genotype

20.0097

0.004858

6.97 0.027

Res __6__0.0048

0.000697

Total 110.0353

What would it look like had we not controlled for experiment?

Source df SS MS F p

Genotype

20.0097

0.004858

1.71 0.235

Res __9__0.0255

0.002841

Total 110.0353

7. ANOVA n = 12Source df SS MS F p

Blocks 30.0213

Genotype

20.0097

0.004858

6.97 0.027

Res __6__0.0048

0.000697

Total 110.0353

What would it look like had we not controlled for experiment?

Source df SS MS F p

Genotype

20.0097

0.004858

1.71 0.235

Res __9__0.0255

0.002841

Total 110.0353

7. ANOVA n = 12Source df SS MS F p

Blocks 30.0213

Genotype

20.0097

0.004858

6.97 0.027

Res __6__0.0048

0.000697

Total 110.0353

What would it look like had we not controlled for experiment?

Source df SS MS F p

Genotype

20.0097

0.004858

1.71 0.235

Res __9__0.0255

0.002841

Total 110.0353

7. ANOVA n = 12Source df SS MS F p

Blocks 30.0213

Genotype

20.0097

0.004858

6.97 0.027

Res __6__0.0048

0.000697

Total 110.0353

What would it look like had we not controlled for experiment?

Source df SS MS F p

Genotype

20.0097

0.004858

1.71 0.235

Res __9__0.0255

0.002841

Total 110.0353

7. ANOVA n = 12Source df SS MS F p

Blocks 30.0213

Genotype

20.0097

0.004858

6.97 0.027

Res __6__0.0048

0.000697

Total 110.0353

What would it look like had we not controlled for experiment?

Source df SS MS F p

Genotype

20.0097

0.004858

1.71 0.235

Res __9__0.0255

0.002841

Total 110.0353

7. ANOVA n = 12Source df SS MS F p

Blocks 30.0213

Genotype

20.0097

0.004858

6.97 0.027

Res __6__0.0048

0.000697

Total 110.0353

What would it look like had we not controlled for experiment?

Source df SS MS F p

Genotype

20.0097

0.004858

1.71 0.235

Res __9__0.0255

0.002841

Total 110.0353

STATISTICAL CONTROL

8. Decide whether to recompute p-valueResiduals

homogenousnot independentdeviated slightly from normality

n=12

p=0.027 needs to change 2-fold to change decision

Randomization (100 000 runs)pran= 0.0186change in p: 0.027/0.186 = 1.45

9. Declare decision about terms

Only the fixed term was tested p=0.0186< α =0.05

Reject H0 There is significant variation in mean dry weight among genotypes

10.Report and interpret parameters of biological interest

Means per genotype with 95% CI, not controlled for among experiments variation

Genotype IIIHomozygote mutant

Genotype IHeterozygote mutant

Genotype IIWild

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 IIIHomozygote mutant

Genotype IHeterozygote mutant

Genotype IIWild

Genotype IIIHomozygote mutant

10.Report and interpret parameters of biological interest

Means per genotype with 95% CI, controlled for among experiments variation

Genotype IHeterozygote mutant

Genotype IIWild

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 IIIHomozygote mutant

Genotype IHeterozygote mutant

Genotype IIWild