4' Factorial experiments

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• Lecture 10
• 4. Factorial experiments

• In a real biological system, organisms are
  exposed to many factors simultaneously
• Response to one factor may vary with the response
  to the levels other factors
• Single-factor experiment is the simplest method
  of study but this cannot handle and explain
  complicated biological system where there are
  interactions among various factors

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• Factorial Design Multifactor ANOVA
• Two or more than two fixed factors (treatment
  factors) are tested at a time
• Factors have graded levels
• Simplest design is 2 x 2 (two factors with 2
  levels)
• Test of main effects (e.g. effects of Factor A
  and Factor B separately
• Test of interactions (e.g. A x B or N x P)
• H0 three null hypotheses (2 x 2 factorial
  design)
• There is no effect of factor A
• There is no effect of factor B
• There is no interaction effect of A and B

3

• Hypothesis formulation and testing
• Contd..
• Simplest Factorial Design
• 2 x 2 factorial

4

• Hypothesis formulation and testing
• Contd..
• Simplest Factorial Design
• 2 x 3 factorial

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• Factorial Design 3 factors
• 2 x 2 x 2 or 23 (all the three factors have 3
  levels)
• 4 x 3 x 2 (all the three factors have different
  levels)
• Examples of factorial designs

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• Layout and randomization
• 2 x 2 Factorial in CRD

T2 T3 T1 T2
T4 T1 T2 T3
T3 T4 T1 T4
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• Layout and randomization
• 2 x 2 Factorial in RCBD

Canal or road
Block 1
T2 T3 T4 T1
Block 2
T4 T1 T3 T2
Block 3
T3 T1 T2 T4
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• Factorial designs
• Null hypotheses (H0)
• There is no effects of N
• There is no effects of P
• There is no interaction of N and P
• There is no effect of block

CRD
RCBD
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• ANOVA Models

2-Factors Yi ? A B (AB) Ri
3-factors Yi ? A B C AB AC BC
ABC Ri
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• ANOVA Models

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Separationof variation
Random errors
If A or B gt R effect of A or B is significant
after separation of block effects
AB effect
Factor A Factor B
Block effect
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What is an interaction?Positive interaction
means both the factors add to the
production/growth or the dependent variable
2 kg P
Positive interaction
Production or growth
0 kg P
0 kg N 4 kg N
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What is an interaction?Interaction can be
negative also
Negative interaction
2 kg P
Production or growth
0 kg P
0 kg N 4 kg N
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What is an interaction?Interaction in 3 x 2
factorial experiment
2 kg P
Positive interaction
Production or growth
0 kg P
0 kg N 2kg N 4 kg N
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• Some examples
• Fertilization trials N P levels (ive
  interaction?)
• Irrigation and fertilization levels (ive?)
• Feeding and fertilization levels (ive?)
• Vitamin E Lipid (ive?)
• Vitamins and minerals (ive or ive?)
• Temperature and salinity (-ive?)
• Temperature and light (?)
• Temperature and pH (?)

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• Randomization and layout

• Determine the total number of experimental units
  (cages, plots, animals etc.)
• Assign the unit numbers
• Follow the procedures in either CRD or RCBD for
  randomization depending upon your experimental
  facility

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Example of 3 x 5 factorial design
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(Block1)
(Block2)
(Block3)
(Block4)
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Table 1
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Data analysis 1. Group the data by each factor
and calculate the factor totals means, and grand
total (G), the grand mean and the coefficient of
variation (c.v.) etc.2. Determine the degree of
freedom (d.f.) for each source of variation3.
Construct an outline/table (next slide) of the
analysis of variance
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• 4. Calculate the correction factor (C) and the
  various sums of square (SS)
• 5. Calculate the mean square (MS) for each source
  of variation by dividing SS by their
  corresponding d.f.
• 6. Calculate the F- values (R.A. Fisher) for
  testing significance of the treatment difference
  (F MSA/MSE and MSB/MSE)
• 7. Enter all the F- values computed in the ANOVA
  table and find out the P-values

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Two-factor ANOVA table
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• ANOVA Models

Notes If there is no interaction, AB MS and the
Error MS will be the same.
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Three-factor ANOVA table
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• Confusion with the Exp. design

Note It looks like 3 x 2 factorial but not
because the feed types are different in each
district
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• Re-arranged to simple 3 x 4 factorial

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• Nesting in factorial experiments

You can further split
x 3 rep per sample x twice a year
Note needs advanced knowledge for analysis
(contact if interested)
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• Other designs (just names)
• Split-plot design
• Split-split plot design
• Strip-split plot design
• Lattice design
• Fractional factorial design

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• Multivariate analysis MANOVA
• In reality, two or more than two variables are
  affected by the treatments simultaneously
• These variables can be associated each other
  therefore, it is necessary to measure so that it
  makes easier to explain/interpret the results
• For example, feeding can have effects on
• Body weight (DWG)
• Body length
• Survival ()
• Fish yield (Net fish yield)
• Body compositions (CP, Lipids, vitamins, minerals
  etc.)
• etc. etc.

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• Multivariate Analysis of Variance MANOVA
• The analysis method is same as in ANOVA, only
  difference is that we take all the variables at a
  time
• Choose multivariate instead of Univariate
  function in SPSS
• You can select many variables at a time
• There will be a separate F-value (and P-value)
  for each factor which determines the effects
• Practice Lab session!

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• Analysis of Covariance ANCOVA
• There can be still some variables which can not
  be fully controlled in experimental field but
  they can be measured/collected e.g. temp, DO, pH
  etc.
• ANCOVA reduces the variability among experimental
  units by adjusting their values
• Is used to adjust the values distracted by
  natural calamities or unavoidable circumstances
  e.g. damage of plants by insects
• These might affect main variables e.g. pond
  history, initial weight of fish/animals,
  temperature, DO, pH, etc. which could be used as
  covariates and their effects could be
  separated/assessed at the same time

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• Analysis of Covariance ANCOVA
• Select these variables and move them right under
  Covariates in SPSS program (both in Univariate
  and multivariate models)
• There will be a separate F-value (and P-value)
  for covariates as well which determines their
  effects on dependent variables

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Form more information about ANOVA/ANCOVA http//w
ww.physics.csbsju.edu/stats/anova.html http//www
.psychstat.smsu.edu/introbook/sbk27.htm http//ww
w.statsoft.com/textbook/stathome.html

• See you in the lab after 15 min!
• Thank you!