Hypothesis formulation and testing

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• Lecture 9
• Hypothesis formulation and testing
• Contd..
• Experimental designs
• 1. Complete randomized design (CRD)
• 2. Randomized Complete Block Design (RCBD)
• 3. Latin Square Design (LSD)

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• Randomized Complete Block Design (RCBD)
• If the experimental units are not uniform,
  blocking is necessary to separate experimental
  error, therefore, block is considered as a random
  factor

Canal
Block 1 Block 2 Block 3
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• Randomized Complete Block Design (RCBD)
• Two factors (Trts Block) - Two-way ANOVA
• Most widely used
• Block can be
• Spatial pond, row, districts/village,
  households, etc.
• Time week, month or year
• Age different age group of animals
• Height tree trunk, shelf in the drier etc.
• Soil heterogeneity, slope, insect migration etc.
• Two null hypotheses (H0)
• no effects of block
• no effects of treatment

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• Comparisons of five means

Freq.
A
A
A
Block 2
Block 1
Block 3
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• Experimental designs
• 1. Randomized Complete Block Design (RCBD)
• all the experimental units within the block are
  considered uniform or identical
• treatment allocation in each block is completely
  random
• Blocks are considered as replications

Canal
Block 1 Block 2 Block 3
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• Two hypothesis are tested by comparing the
  variation, therefore, called as Two-way Analysis
  of Variance (ANOVA)
• between treatments with the variation among
  treatments and blocks
• Effect of block is separated
• If variation between treatments (Treatment
  effect) is higher than the variation within
  treatment (i.e. Random error), there is a
  significant difference
• Model

Yi ? Ti Bi Ri
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Separation of variation
If Ti gt Ri treatment effect is significant
after separation Of block effects
Yi Ri Bi Ti ?
Random errors
Block effect
Treatment effect
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• Also called as Two factor experiment/analysis
• Examples
• Fertilization trials in different types of soil
• - Organic, in-organic and combination
• - 0, 40 and 60 kg N/ha/week etc.
• Crop/vegetable/fruits varieties in different
  districts
• Animal breeds in different villages/families
• Drug efficacy on different age groups
• Tilapia seed production by week or month
• Weekly DO and Temp differences
• Fish growth trial in cages in ponds

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• Randomization and layout

• Determine the total number of experimental units
  (cages) t x b e.g. to test 6 feeds in 4
  ponds, you will need 24 cages (6/pond)
• Assign cage number in each pond (1 to 6)
• Assign all the treatments in each block (pond)
  randomly by using lottery or random table

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• Randomization

T6 T3 T5 T1 T2 T4
T2 T5 T4 T3 T6 T1
T2 T1 T6 T5 T3 T4
T3 T4 T2 T6 T5 T4
Pond 1
Pond 2
Pond 2
Pond 4
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Data analysis 1. Group the data by treatments
and calculate the treatment totals (T), block
totals (B) and grand total (G), the grand mean
and the coefficient of variation (c.v.) etc.2.
Using number of treatments (t) and the number of
blocks (b) 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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ANOVA table of a RCBD experiment
t number of treatments b number of
blocks/replicates per treatment
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• 4. Using Xi to represent the measurement of the
  ith plot, Ti as the total of the ith treatment,
  and n as the total number of experimental plots
  i.e. n (r) (t) , calculate the correction
  factor (CF) 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 MST/MSE and MSB/MSE)
• 7. Enter all the values computed in the ANOVA
  table

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Example Four different feeds were tested on 20
pigs. Following were the mean final weights (kg)
of 19 pigs (1 pig died). Here, H0 ?1 ?2 ?3
?4 and there is no effect of block
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• Step 1 Calculate Missing value
• Missing value estimation
• (t treatment total bblock total) Grand
  total
• (t-1)(b-1)
• (4401.45222) 1482.2/(34) 102.7
• Notes
• Missing values should not be more than 10 and
  it should be used only lost data due to
  unavoidable circumstances
• In case of more than 1 missing data, guess
  others and estimate first one. Then use it to
  estimate others/guessed ones iterate it until
  all the values become stable

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Example Four different feeds were tested on 20
pigs. Following were the mean final weights (kg)
of 19 pigs (1 pig died). Here, H0 ?1 ?2 ?3
?4 and there is no effect of block
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Step 1 Calculate sum of squares Correction
factor/Intercept ( C ) (Grand total)2 /n
(1584.9)2 /20 125,595 Total SS (60.8) 2
(57.0)2 -- (90.3)2 - C 4,934 Treatment SS
? (Treatment total)2/t C (303.1)2 /5
(346.5)2 /5 (504.1)2 /5 (431.2)2 /5
125,595 4,801 Block SS ? (Block total)2/n
C (320)2/4(311)2/43222/4(307)2/4(325)2/4 -
125,595 57 Error SSTotal SSSST-SSB
4,9344,801 5776
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Step 2 Prepare an ANOVA table
Note Numerator df 3, Denominator d.f. 12 for
treatment F, and 4 and 12 respectively for the
block F. Reject H0 for Treatment but accept H0
for block or Family had no effects on pig growth
but type of feed has effects
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Results from SPSS Analysis
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• Two-way (Multi-factor) ANOVA to analyze
• Check the block effect first,
• If there is an effect of block - treatment
  effects need to be compared within each block
  separately
• If there is no significant block effect data can
  be analyzed using one-way ANOVA and go for t-test
  and Multiple range tests to compare treatments
  (Tukeys test is widely accepted and most common)
• Note If ANOVA shows no significant difference
  then multiple range tests are not necessary

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• 3. Latin Square Design
• Two blocks (random factors) 1 treatment (fixed
  factor)

Canal (Block 1)
Road/ Shade/ Elevation etc. (Block 2)
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• Latin Square Design
• All the treatments should be allocated to each
  block of both the factors
• 3 null hypotheses (H0) there are no effects of
  block 1 and block 2 and the treatments
• For four treatments (T1, T2, T3 and T4)
• Randomization?

Canal (Block 1)
T3 T2 T4 T1
Road/ Shade/ Elevation etc. (Block 2)
T2 T4 T1 T3
T4 T1 T3 T2
T1 T3 T2 T4
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• Latin Square Design (Data table)

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• Three hypotheses are tested by comparing the
  variation, therefore, called as Multifactor ANOVA
• Effects of two way blocks are to be separated
• Effects of treatment
• Model

Yi ? Ti B1 B2 Ri
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ANOVA table of Latin Square Design
t number of treatments b number of
blocks/replicates per treatment
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• Calculation of SS and MSS, Missing values
• Calculation is same as in RCBD
• Keep for practical session

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• RCBD ANOVA vs Multiple range tests
• Most common
• Same number of treatments with the block
• Latin Square
• Less common in biological research
• Shouldnt loose single observation
• Need to have equal treatments with the block
• Very limited treatments can be tested (normally
  min- 4 and max - 8 only)

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Some useful websites related to
ANOVA http//www.physics.csbsju.edu/stats/anova.
html http//www.psychstat.smsu.edu/introbook/sbk2
7.htm

• Thank you!