Experimental design and statistical analyses of data

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Experimental design and statistical analyses of
data

• Lesson 1
• General linear models and design of experiments

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Examples of General Linear Models (GLM)
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Simple linear regression  
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Polynomial regression  
Ex   y depth at disappearance x
nitrogen concentration of water
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Multiple regression  
Eks   y depth at disappearance x1
Concentration of N x2 Concentration of P
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Analysis of variance (ANOVA)
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Analysis of covariance (ANCOVA)
Ex   y depth at disappearance x1 Blue
disc x2 Green disc x3 Concentration of N
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Nested analysis of variance
Ex   y depth at disappearance ai effect of
the ith lake ß(i)j effect of the jth
measurement in the ith lake
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What is not a general linear model?

• y ß0(1ß1x)
• y ß0cos(ß1ß2x)

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Other topics covered by this course

• Multivariate analysis of variance (MANOVA)
• Repeated measurements
• Logistic regression

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Experimental designs

• Examples

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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

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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

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Drugs Drugs Drugs Drugs Drugs
A B C D Total
y1A y2A y3A y1B y2B y3B y1C y2C y3C y1D y2D y3D

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Source Degrees of freedom
Estimate of Treatments ( ) Residuals 1 p - 1 3 n-p 8
Total n 12
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Randomized block design

• All treatments are allocated to the same
  experimental units
• Treatments are allocated at random

B C B
A B D
D A A
C D C
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Treatments Treatments Treatments Treatments Treatments Treatments Treatments
Persons A B C D Average
Persons 1
Persons 2
Persons 3
Average
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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
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Double block design (latin-square)
Person Person Person Person Person
Sequence 1 2 3 4
Sequence 1 B D A C
Sequence 2 A C D B
Sequence 3 C A B D
Sequence 4 D B C A
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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
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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)

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Factorial designs
Factor B Factor B Factor B Factor B Factor B Factor B
Factor A B1 B2 B3 B4 Average
Factor A A1 y11 y12 y13 y14
Factor A A2 y21 y22 y23 y24
Factor A A3 y31 y32 y33 y34
Average
No interaction between A and B
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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 92 -7 days

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Factorial experiment with no interaction
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Factorial experiment with no interaction
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Factorial experiment with no interaction
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Factorial experiment with no interaction
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Factorial experiment with no interaction
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Factorial experiment with interaction
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Factorial designs
Factor B Factor B Factor B Factor B Factor B Factor B
Factor A B1 B2 B3 B4 Average
Factor A A1 y11 y12 y13 y14
Factor A A2 y21 y22 y23 y24
Factor A A3 y31 y32 y33 y34
Average
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Two-way factorial designwith 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
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Two-way factorial designwithout replication
Source Degrees of freedom
Estimate of Factor A (drug) Factor B (administration) Residuals 1 a-1 2 b - 1 3 n- a-b1 6
Total n ab 12
Without replication it is necessary to assume no
interaction between factors!
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Two-way factorial designwith 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
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Two-way factorial designwith 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
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Three-way factorial design
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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
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Why should more than two levels of a factor be
used in a factorial design?
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Two-levels of a factor
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Three-levelsfactor qualitative
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Three-levelsfactor quantitative
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Why should not many levels of each factor be used
in a factorial design?
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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