Part IVThe General Linear Model.

Multiple Explanatory Variables

Chapter 13.3 Fixed *Random Effects

Paired t-test

Overview of GLM

GLM

Regression

ANOVA

ANCOVA

One-Way ANOVA

Two-Way ANOVA

Simple regression Multiple regression

Two categories (t-test)

Multiple categories - Fixed (e.g., treatment, age)

- Random (e.g., subjects, litters)

2 fixed factors 1 fixed & 1 random

(e.g., Paired t-test)

Multi-Way ANOVA

GLM: Paired t-test

Two factors (2 explanatory variables on a nominal

scale)

One fixed (2 categories)

The other random (many categories)

+Fixed factor

Random factor

Remove var. among units → sensitive test

• Sleep data example, used by W. Gosset (1908) in the paper that introduced the t-test

• Are the effects of 2 sleep inducing drugs: Hyoscyamine (Drug A) and L Hyoscine (Drug B), controlled for among subject variation, different?

GLM | Paired t-test

Subject DrugA Drug B

1 0.7 1.9

2 -1.6 0.8

3 -0.2 1.1

4 -1.2 0.1

5 -0.1 -0.1

6 3.4 4.4

7 3.7 5.5

8 0.8 1.6

9 0.0 4.6

10 2.0 3.4Data are means

1. Construct ModelResponse variable: T=hours of extra sleep – ratio scale

Explanatory variables:

1. Drug. XD = Drug A, Drug B. Nominal scale

Fixed effect

2. Subject. XS = [1,2,…,10]. Nominal scale

Random effect

Mean value for each subject varies randomly and is not under the control of the investigator

1. Construct ModelVerbal: Hours of extra sleep depends on drug.Graphical:

1. Construct ModelVerbal: Hours of extra sleep depends on drug.Graphical:

1. Construct ModelVerbal: Hours of extra sleep depends on drug.Graphical:

1. Construct ModelFormal:

Can we have an interaction term?

Let’s look at the df

dfDrug =

dfSubject =

dfDrug*Subject =

Dfresidual =

1. Construct ModelVerbal: Hours of extra sleep depends on drug.Graphical:

1. Construct ModelFormal:Revised Model:

2. Execute analysis

lm1 <- lm(T~XS+XD, data=sleep)XS T XD

1 0.7 A

2 -1.6 A

3 -0.2 A

4 -1.2 A

5 -0.1 A

6 3.4 A

7 3.7 A

8 0.8 A

9 0.0 A

10 2.0 A

1 1.9 B

2 0.8 B

3 1.1 B

4 0.1 B

5 -0.1 B

6 4.4 B

7 5.5 B

8 1.6 B

9 4.6 B

10 3.4 B

R: multiple ways to model random effectsInstead of lm:

lmer{lme4}lme{nlme}use aov() , specifying

Error(subject)

2. Execute analysis

1. Compute

2. Compute mean per drug mean (TD=A)= 0.75 hs

3. Compute drug effect

4. Compute mean per subject mean(TS=1)= 1.3 hs

5. Compute subject effect

6. Compute fits

7. Compute residuals residuals = T - fits

hs54.1ˆ0

hshsAD 79.0)54.175.0(ˆ

hshsS 24.0)54.13.1(ˆ1

SDfits ˆˆˆ0

hshshs SAD 24.0ˆ;79.0ˆ;54.1ˆ10

2. Execute analysis

XS T XD β0 βD βS fits res

1 0.7 A1.54

-0.79

-0.24

0.51-

0.19

2 -1.6 A1.54

-0.79

-1.94

-1.19

0.41

3 -0.2 A1.54

-0.79

-1.09

-0.34

-0.14

4 -1.2 A1.54

-0.79

-2.09

-1.34

-0.14

5 -0.1 A1.54

-0.79

-1.64

-0.89

-0.79

1 1.9 B1.54

0.79-

0.242.09 0.19

2 0.8 B1.54

0.79-

1.940.39

-0.41

3 1.1 B1.54

0.79-

1.091.24 0.14

4 0.1 B1.54

0.79-

2.090.24 0.14

5 -0.1 B1.54

0.79-

1.640.69 0.79

3. Evaluate model

□ Straight line model ok?

□ Errors homogeneous?

□ Errors normal?

□ Errors independent?

3. Evaluate model

□ Straight line model ok?

□ Errors homogeneous?

□ Errors normal?

□ Errors independent?

NA

3. Evaluate model

□ Straight line model ok?

□ Errors homogeneous?

□ Errors normal?

□ Errors independent?

NA

3. Evaluate model

□ Straight line model ok?

□ Errors homogeneous?

□ Errors normal?

□ Errors independent?

NA

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

Drugs set by experimental design fixed effectsWe will infer only to those drugs

Subjects, chosen at random. Hopefully from a larger population random effects

Population of all possible measurements of hours of extra sleep, given the mode of collection

Infer to a population of subjects with characteristics similar to those in the study

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

6. State HA / Ho pair, test statistic, distribution, tolerance for Type I error.

– Assume no interaction, i.e. effect of drug is consistent across subjects

– Focus on drug effect

6. State HA / Ho pair, test statistic, distribution, tolerance for Type I error.

Test Statistic

Distribution of test statitstic

Tolerance for Type I error

7. ANOVA n = 20

Source df SS MS F p

Subject

Drug

Res______

______

Total

7. ANOVA n = 20

Source df SS MS F p

Subject 958.078

Drug 112.482

Res___9__

_6.808

Total 19 77.37

7. ANOVA n = 20

Source df SS MS F p

Subject 958.078

6.453

Drug 112.482

12.48

Res___9__

_6.808

0.7564

Total 19 77.37

7. ANOVA n = 20

Source df SS MS F p

Subject 958.078

6.453

Drug 112.482

12.48 16.5 0.0028

Res___9__

_6.808

0.7564

Total 19 77.37

7. ANOVA n = 20Source df SS MS F p

Subject 958.078

6.453

Drug 112.482

12.48 16.5 0.0028

Res___9__

_6.808

0.7564

Total 19 77.37

Source df SS MS F p

Drug 1 12.48 12.48 3.4626 0.079

Res__18__

64.886

3.6048

Total 19 77.37

BUT we did this before Ch 10.2 2 sample t-test

STATISTICAL CONTROL

r2 = 0.91

r2 = 0.16

8. Decide whether to recompute p-value

Slight deviation from normalityn<30, p=0.0028 not near α no need to recompute

9. Declare decision about terms

Only the fixed term was tested p=0.0028 < α =0.05Reject H0 extra sleep depends on drug administered

We did a 2 way ANOVA, also known as a paired t-test. 1 random factor1 fixed factor with 2 levels

9. Declare decision about terms

A B A-B fits res

0.7 1.9 1.2 1.58 -0.38

-1.6 0.8 2.4 1.58 0.82

-0.2 1.1 1.3 1.58 -0.28

-1.2 0.1 1.3 1.58 -0.28

-0.1 -0.1 0.0 1.58 -1.58

3.4 4.4 1.0 1.58 -0.58

3.7 5.5 1.8 1.58 0.22

0.8 1.6 0.8 1.58 -0.78

0.0 4.6 4.6 1.58 3.02

2.0 3.4 1.4 1.58 -0.18

Paired t-test:

1. Calculate difference within each random category

2. Test if the mean diff differs from zero

p=0.0028

10.Report and interpret parameters of biological interest

Means per drug, not controlled for among subject variation

SE LCL (5%) UCL(95%)

mean(TA)=0.75 hs

0.5657 -0.53 hs 2.03 hs

mean(TB)=2.33 hs

0.6332 0.89 hs 3.76 hs

Confidence limits for the average difference, controlled for among subject variation

SE LCL (5%) UCL(95%)

mean(TB-TA)=1.58 hs

0.388 0.7 hs 2.46 hs

Quizz 7

Good luck!

Clock