Balanced Incomplete Block Design BIBD

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Balanced Incomplete Block Design (BIBD)

• Notation
• a-number of treatments
• m-number of treatments/block
• r-number of treatment reps
• b-number of blocks
• Narbm
• r(m-1)l(a-1)

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BIBD

• Example
• Block
• 1 2 3 4
• A A A B
• B B C C
• C D D D
• Verify that l2

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BIBD

• One way to generate BIBDs (assuming a and m have
  been selected) is to choose
• This worked well in the previous example
• It can be inefficient for large a or ma/2

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Least Squares estimator of a

• These are the adjusted means that one obtains
  from LSMEANS in SAS

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Least Squares estimator of a

• The solution for ai

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Least Squares estimator of a

• These estimators are called intrablock estimators
  because they can be expressed as contrasts of
  within block observations.

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Properties of LS estimators
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Properties of LS estimators

• The variance estimate is not the square of the
  standard error for the LSMEANS that one obtains
  in SAS (its not corrected for the overall mean).

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Example

• Region II Science Fair Winner
• Treatment is organic chemical repellent
• Block is Day
• Response is average number of flies on underside
  of lid (smaller is better)
• Originally presented as a Latin Square
  (additional block was Fly Group)

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Example
  AAscorbic Acid BCitric Acid CControl
DHesperidin ELemon Juice FBlack Pepper
GPiperine
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Interblock Estimators

• When Block is a random effect, alternative
  estimators of treatment effects can be found

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Interblock Estimators
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Interblock Estimators

• Some properties

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Interblock Estimators

• For the random block model,
• Source df SS EMS
• Trt a-1 SSTrt
• Blockadj b-1 SSBlock(adj)
• Error N-a-b1 s2
• Total N-1

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Interblock Estimators
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BIBD Estimators

• Suppose we want to minimize the variance of
  estimators of the form
• The best weighted estimator is

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Power Analysis

• The noncentrality parameters look similar to all
  other one factor designs we have so far seen