Randomized Block Design

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Randomized Block Design
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Overview

• To frame our discussion, consider

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Outline
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Nuisance Factors

• A nuisance factor is any possible source of
  variability other than the conditions you wish to
  compare.
• We can handle such nuisance factors, or potential
  sources of variation which is not important,
  using block design.

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Block Design

• Group experimental units into groups (blocks).
  Randomly assign treatments to blocks.
• Blocks organize units with the features you are
  not interested in, but which may influence the
  response variable.
• Each level of the factor must be applied to
  blocks.

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Rationale
Small differences within the blocks.
Large differences between the blocks.
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Example
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One Factor Model

• yijmajeij
• yij is the value of the response variable of the
  i-th observation for the j-th alternative
  (level).
• m is the mean response.
• aj is the effect of alternative j.
• eij is the error.

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Block Model

• yijmbiajeij
• yij is the value of the response variable of the
  i-th observation for the j-th alternative
  (level).
• m is the mean response.
• bi is the blocking variable effect.
• aj is the effect of alternative j.
• eij is the error.

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Alternatively
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(No Transcript)
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Assumptions

• No interaction between factor and blocking
  variable.
• The error distribution is normal.
• The error variance in the blocks or alternatives
  must be equal.

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Calculations

• SSE sum of the square of the residuals
• a is number of levels of the factor.
• b is number of blocks.

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Significance Test

• MSASSA/(a-1)
• MSBSSB/(b-1)
• MSESSE/(a-1)(b-1)
• F0MSA/MSE
• F0 gtFa,(a-1),(a-1)(b-1)

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Example

• A consumer product-testing organization wished to
  compare the annual power consumption for five
  different brands of dehumidifier. Because power
  consumption depends on the prevailing humidity
  level, it was decided to monitor each brand at
  four different levels ranging from moderate to
  heavy humidity. The data provides the power
  consumption in kilowatt-hours (kWh).

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Example Data
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Example Data
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More on Blocking

• Blocking entails potential gain and loss of
  precision.
• The effect of blocking is to filter out variation
  which may overwhelm the one-factor model.
• The one-factor test uses I(J-1) df where the
  two-factor test uses (I-1)(J-1) df. Fewer
  degrees of freedom is generally a loss of power
  in a test.