Statistical Analysis

1
Statistical Analysis Design in Research
Structure in theExperimental MaterialPGRM 10
2
Blocking the idea

• Detecting differences between treatments depends
  on the background noise (BN)
• BN is
• caused by inherent differences between the
  experimental units
• measured by the residual (error) mean square RMS
  (alternatively! MSE)
• Comparing treatments on similar units would
  reduce background noise
• With blocks of units of differing contributing
  characteristics we measures the variation due to
  blocks and reduce residual variation

3
Blocking the benefit

• Reducing background noise
• Gives more precise estimates
• Allows a reduction in replication, without loss
  of power(the probability of detecting an effect
  of a specified size)
• Reduces cost!

4
Blocking and experimental material

• Examples
• A field with fertility increasing from top to
  bottomWith 3 treatments group plots into BLOCKS
  of 3, starting at top and continuing to
  bottom.Randomise treatments within each block

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Block Design
What is the experimental unit?
How many replicates per treatment?
What is the block?
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Example

• 2 drugs (A, B) to control blood pressure
• 100 subjects randomly assign 50 each to A and B
  
• Valid - but is it efficient?
• If subjects are heterogenous - likely to be a
  large variation (?2) in the responses within each
  group.
• Design may not be very efficient.

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Factors affecting BP variation
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Blocking and experimental material

• 100 subjects are selected to compare new drug to
  control BP with a Control

Block into pairs by age weight (believed to
affect BP) In each pair one is selected at
random to receive the new drug, the other
receives Control
Alternatively see next slide
9
Groups (Blocks)
10
Groups (Blocks)
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Blocking and experimental material

• Examples
• A field with fertility increasing from top to
  bottomWith 3 treatments group plots into BLOCKS
  of 3, starting at top and continuing to
  bottom.Randomise treatments within each block
• 100 subjects are selected to compare new drug to
  control BP with a ControlBlock into pairs by age
  weight (believed to affect BP)In each pair one
  is selected at random to receive the new drug,
  the other receives Control
• 3 products to be compared in 15 supermarketsAll
  3 compared in each supermarket, regarded as BLOCKS

12
Blocking and experimental material

• Examples (contd)
• A crop experiment will take 5 days to
  harvest.The material is blocked into 5 sets of
  plots, and treatments assigned at random within
  each setA BLOCK of plots is harvested each
  dayHere day effects, such as rain etc will be
  allowed for in the ANOVA table, not clouding the
  estimation of treatment effects, and reducing
  residual variation.

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Blocking factors in your work area?
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Reasons to BLOCK

1. Reduce BN (as above)
2. Material is naturally blocked (eg identical
   twins)so using this a part of the design may
   reduce BN
3. To protect against factors that may influence the
   experimental outcomes, and so cloud comparison of
   treatments
4. To assess block variation itselfeg day to day
   variation large may indicate a process that is
   not well controlled.

15
Typical Randomised Block Design (RBD) Layout
4 treatments T1 T4 ? BLOCKS of size 4
Example of random allocation within blocks
Block
1 T3 T1 T2 T4
2 T2 T3 T1 T4
3 T1 T2 T3 T4
4 T2 T4 T1 T3
5 T4 T2 T3 T1
6 T3 T1 T4 T2
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ANOVA table
each treatment occurs once in each blockt
treatmentsb blockstb experimental units
Source DF SS MS F Pr gt F
Treatments t 1 TSS TMS TMS/RMS Small?
Blocks b 1 BSS BMS BMS/RMS Small?
Residual (t-1)(b-1) RSS RMS
Total tb - 1
MS SS/DF
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ExamplePGRM pg 10-2

• Compare effect of washing solution used in
  retarding bacterial growth in food processing
  containers.
• Only 3 trials can be run each day, and
  temperature is not controlled so day to day
  variability is expected.
• BLOCKS day
• Treatments 2, 4, 6 of active ingredient
• Randomisation 3 containers randomly allocated to
  3 treatments on each of 4 days.
• Response bacterial count on each container each
  day (low score cleaner)

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Example (contd)
Day Solution() Count
1 2 13
1 4 10
1 6 5
2 2 18
2 4 20
2 6 6
3 2 18
3 4 17
3 6 7
4 2 30
4 4 31
4 6 10
Day,Solution(),Count 1,2,13 1,4,10 1,6,5 2,2,18 2
,4,20 ...
csv
Excel

• Note
• Response values in a single column
• Extra column to identify
• BLOCK (day)
• TREATMENT (solution)

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SAS GLM code
proc glm data randb class solution
day model score solution day lsmeans
solution lsmeans day estimate 2-6 solution
1 0 -1 estimate linear ok? solution 1
-2 1 quit
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GLM OUTPUT ANOVA
425.17 322.92 748.09 So the Model SS has been
partitioned into TREATMENT (solution) and BLOCK
(Day)
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GLM OUTPUT means
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ANOVA table
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More Blocking Latin square designs
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Latin Square design blocking by 2 Sources of
variation

• Variation in milk yield among cows is large (CV
  25)
• Variation in Yield across lactation is large
• Use different treatments in sequence on each cow
• Need to allow for a standardisation period (1-2)
  weeks between treatments

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Data
Columns for period,cow and treatment codes
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SAS GLM code
proc glm data latinsq class period cow
treat model yield period cow treat lsmeans
treat lsmeans period lsmeans cow estimate
1v2 treat 1 -1 0 0 Run
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Results
Cow and Period removed much variation
Means
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Conclusions on Latin square design

• CV greatly reduced to 6 - When the effect of
  period is allowed for, repeated measurements
  within a cow are not very variable.
• Periods and cows are nuisance variables.
  Sometimes the row and column variables are of
  interest in themselves and so design is very
  efficient information on 3 factors. (e.g.
  treatments, machines, operators).
• Useful for screening but questionable whether
  short term results would apply for the long term.