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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 Lesson 1: General linear models and design of experiments
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:
Polynomial regression: Ex:: y = depth at disappearance x = nitrogen concentration of water
Multiple regression: Eks: y = depth at disappearance x1= Concentration of N x2 = Concentration of P
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)
Analysis of covariance (ANCOVA): Ex: y = depth at disappearance x1= Blue disc x2 = Green disc x3 = Concentration of N
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
What is not a general linear model? y = β0(1+β1x) y = β0+cos(β1+β2x)
Other topics covered by this course: • Multivariate analysis of variance (MANOVA) • Repeated measurements • Logistic regression
Experimental designs Examples
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
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
Note! Different persons
Treatments (p = 4) Blocks (b = 3) Randomized block design • All treatments are allocated to the same experimental units • Treatments are allocated at random
Blocks (b-1) Treatments (p-1)
Rows (a = 4) Columns (b = 4) Persons(b-1) Drugs (p-1) Sequence (a-1) Double block design (latin-square)
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)
Effect of A Effect of B Factorial designs No interaction between A and B
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
Effect of A Effect of B Interactions between A and B Factorial designs
Two-way factorial designwith interaction, but without replication
Two-way factorial designwithout replication Without replication it is necessary to assume no interaction between factors!
30 Three-way interactions Factor A Factor B Factor C 10 Main effects 31 Two-way interactions Three-way factorial design
Why should more than two levels of a factor be used in a factorial design?
High Medium Low Three-levelsfactor qualitative
Why should not many levels of each factor be used in a factorial design?
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