1. Experimental design and statistical analyses of data
Lesson 1:
General linear models and design of experiments

2. Examples of General Linear Models (GLM)

3. Simple linear regression:
Ex:
Depth at which a white disc is no longer visible in a lake
y = depth at disappearance
x = nitrogen concentration of water
Dependent
variable β0
Intercept β1
Slope
The residual ε expresses
the deviation between the model
and the actual observation
Independent
variable

4. Polynomial regression:
Ex::
y = depth at disappearance
x = nitrogen concentration of water

5. Multiple regression: Eks: y = depth at disappearance
Eks:
y = depth at disappearance
x1 = Concentration of N
x2 = Concentration of P

6. Analysis of variance (ANOVA)
Ex:
y = depth at disappearance
x1 = Blue disc
x2 = Green disc
x1= 0; x2 = 0
x1= 0; x2= 1
x1= 1; x2= 0

7. Analysis of covariance (ANCOVA):
Ex:
y = depth at disappearance
x1 = Blue disc
x2 = Green disc
x3 = Concentration of N

8. Nested analysis of variance:
Ex:
y = depth at disappearance
αi = effect of the ith lake
β(i)j = effect of the jth measurement in the ith lake

9. What is not a general linear model?
y = β0(1+β1x)
y = β0+cos(β1+β2x)

10. Other topics covered by this course:
Multivariate analysis of variance (MANOVA)
Repeated measurements
Logistic regression

11. Experimental designs
Examples

12. Randomised design Effects of p treatments (e.g. drugs) are compared
Total number of experimental units (persons) is n
Treatment i is administrated to ni units
Allocation of treatments among units is random

13. Example of randomized design
4 drugs (called A, B, C, and D) are tested (i.e. p = 4)
12 persons are available (i.e. n = 12)
Each treatment is given to 3 persons (i.e. ni = 3 for i = 1,2,..,p) (i.e. design is balanced)
Persons are allocated randomly among treatments

14. Drugs A B C D Total y1A y2A y3A y1B y2B y3B y1C y2C y3C y1D y2D y3D
Note!
Different persons

15. Source
Degrees of freedom
Estimate of
Treatments ( )
Residuals 1
p - 1 = 3
n-p = 8
Total
n = 12

16. Randomized block design
All treatments are allocated to the same experimental units
Treatments are allocated at random B C A D
Treatments (p = 4)
Blocks (b = 3)

17. Treatments
Persons A B C D
Average 1 2 3
Blocks (b-1)
Treatments (p-1)

18. Randomized block design
Source
Degrees of freedom
Estimate of
Blocks (persons)
Treatments ( drugs )
Residuals 1
b - 1 = 2
p-1 = 3
n-[(b-1)+(p-1)+1] = 6
Total
n = 12

19. Double block design (latin-square)
Person
Sequence 1 2 3 4 B D A C
Rows (a = 4)
Columns (b = 4)
Sequence (a-1)
Persons (b-1)
Drugs (p-1)

20. Latin-square design Source Degrees of freedom Estimate of
Rows (sequences)
Blocks (persons)
Treatments ( drugs )
Residuals 1
a-1 = 3
b - 1 = 3
p-1 = 3
n-[3(p-1)+1] = 6
Total
n = p2 = 16

21. Factorial designs
Are used when the combined effects of two or more factors are investigated concurrently.
As an example, assume that factor A is a drug and factor B is the way the drug is administrated
Factor A occurs in three different levels (called drug A1, A2 and A3)
Factor B occurs in four different levels (called B1, B2, B3 and B4)

22. Factorial designs Factor B Factor A B1 B2 B3 B4 Average A1 y11 y12 y13
Effect of A
Effect of B
No interaction between A and B

23. Factorial experiment with no interaction
Survival time at 15oC and 50% RH: 17 days
Survival time at 25oC and 50% RH: 8 days
Survival time at 15oC and 80% RH: 19 days
What is the expected survival time at 25oC and 80% RH?
An increase in temperature from 15oC to 25oC at 50% RH decreases survival time by 9 days
An increase in RH from 50% to 80% at 15oC increases survival time by 2 days
An increase in temperature from 15oC to 25oC and an increase in RH from 50% to 80% is expected to change survival time by –9+2 = -7 days

24. Factorial experiment with no interaction

25. Factorial experiment with no interaction

26. Factorial experiment with no interaction

27. Factorial experiment with no interaction

28. Factorial experiment with no interaction

29. Factorial experiment with interaction

30. Factorial designs Factor B Factor A B1 B2 B3 B4 Average A1 y11 y12 y13
Effect of A
Effect of B
Interactions between A and B

31. Two-way factorial design with interaction, but without replication
Source
Degrees of freedom
Estimate of
Factor A (drug)
Factor B (administration)
Interactions between A and B
Residuals 1
a-1 = 2
b - 1 = 3
(a-1)(b-1) = 6
n- ab = 0
Total
n = ab = 12

32. Two-way factorial design without replication
Source
Degrees of freedom
Estimate of
Factor A (drug)
Factor B (administration)
Residuals 1
a-1 = 2
b - 1 = 3
n- a-b+1 = 6
Total
n = ab = 12
Without replication it is necessary to assume no interaction between factors!

33. Two-way factorial design with replications
Source
Degrees of freedom
Estimate of
Factor A (drug)
Factor B (administration)
Interactions between A and B
Residuals 1
a-1
b - 1
(a-1)(b-1)
ab( r-1)
Total
n = rab

34. Two-way factorial design with interaction (r = 2)
Source
Degrees of freedom
Estimate of
Factor A (drug)
Factor B (administration)
Interactions between A and B
Residuals 1
a-1 = 2
b – 1 = 3
(a-1)(b-1) = 6
ab( r-1) = 12
Total
n = rab = 24

35. Three-way factorial design
Factor B
Factor C
Factor A
10 Main effects
31 Two-way interactions
30 Three-way interactions

36. Three-way factorial design
Source
Degrees of freedom
Estimate of
Factor A
Factor B
Factor C
Interactions between A and B
Interactions between A and C
Interactions between B and C
Interactions between A, B and C
Residuals 1
a-1 = 2
b – 1 = 5
c-1 = 3
(a-1)(b-1) = 10
(a-1)(c-1) = 6
(b-1)(c-1) = 15
(a-1)(b-1)(c-1) = 30
abc( r-1) = 0
Total
n = rabc = 72

37. Why should more than two levels of a factor be used in a factorial design?

38. Two-levels of a factor

39. Three-levels factor qualitative
Low
Medium
High

40. Three-levels factor quantitative

41. Why should not many levels of each factor be used in a factorial design?

42. Because each level of each factor increases the number of experimental units to be used
For example, a five factor experiment with four levels per factor yields 45 = 1024 different combinations
If not all combinations are applied in an experiment, the design is partially factorial