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Experimental design and statistical analyses of data

Experimental design and statistical analyses of data. Lesson 1: General linear models and design of experiments. Examples of G eneral L inear M odels (GLM). Slope. E x : Depth at which a white disc is no longer visible in a lake y = depth at disappearance

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Experimental design and statistical analyses of data

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

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

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

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

  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 ptreatments (e.g. drugs) are compared • Total number of experimental units (persons) is n • Treatment i is administrated to niunits • 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. Note! Different persons

  15. Treatments (p = 4) Blocks (b = 3) Randomized block design • All treatments are allocated to the same experimental units • Treatments are allocated at random

  16. Blocks (b-1) Treatments (p-1)

  17. Randomized block design

  18. Rows (a = 4) Columns (b = 4) Persons(b-1) Drugs (p-1) Sequence (a-1) Double block design (latin-square)

  19. Latin-square design

  20. 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)

  21. Effect of A Effect of B Factorial designs No interaction between A and B

  22. 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

  23. Factorial experiment with no interaction

  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 interaction

  29. Effect of A Effect of B Interactions between A and B Factorial designs

  30. Two-way factorial designwith interaction, but without replication

  31. Two-way factorial designwithout replication Without replication it is necessary to assume no interaction between factors!

  32. Two-way factorial designwith replications

  33. Two-way factorial designwith interaction (r = 2)

  34. 30 Three-way interactions Factor A Factor B Factor C 10 Main effects 31 Two-way interactions Three-way factorial design

  35. Three-way factorial design

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

  37. Two-levels of a factor

  38. High Medium Low Three-levelsfactor qualitative

  39. Three-levelsfactor quantitative

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

  41. 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

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