STAT 652

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STAT 652
Lecture 17
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The Random Efeect Models
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Examples

• A pharmaceutical company would like to examine
  the potency of a liquid medication mixed in large
  vats.
• To do this, a random sample of five vats from a
  months production was obtained, and four
  separate samples were selected from each vat.

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• Maternal ability of mice-response is weight of 10
  day old litters. The data consists of 6 litters
  from each of 4 dams, all of same breed.
• Here we wont be interested in a particular dam
  but more interested in the variation s²(A) from
  dam to dam and the variation in litters from the
  same dam s²(e).

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Comparison of two Models
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Estimating variance components
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Possible Problem
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Yield Data

• Manufacturer will often receive numerous batches
  of raw materials for use in making chemical
  compounds.
• The manufacturer wishes to obtain a high yield
  from each batch of raw material.
• To determine the variability in yield due the
  different batches, three sample determinations of
  yield are made for each of five batches of raw
  material.

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Analysis

• MSTrVar (among batches)
• MSEVar (within batches)
• FMSTr/MSE20.5 (same as fixed effect) compare
  with F(a,4,10) so reject H0 that s²(Batch)0.
• Also 11.71/13.51.867 of the variation is due to
  batches.

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Two factor Models

• Both factors Fixed Factorial Experiment
• Both Factor random Random Effect
• One fixed, one random Mixed effect
• The E(MS) and corresponding F tests will depend
  on the nature of the factor.

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Two factor Random effects Model
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Consequence of Model
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Variance Components

• s²MSE
• s²(AB)MSAB-MSE/n
• s²(A)MSA-MSE-n s²(AB)/bn

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Test For Interaction
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Test for factor A
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Test for Factor B
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Estimation of Variance Components
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Strength Data

• The factors that affect the breaking strength of
  synthetic fiber are being studies.
• Four machines are chosen at random from a large
  population of machines, and three operators are
  chosen random from a large population of
  operators.
• Two replications of a factorial experiment are
  run using fiber from the same production batch.
• The response is the breaking strength of the
  fiber.

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Analysis

• Test for Interaction F9.54/3.332.86.
  F(6,12)3. so we fail to reject H_0 of no
  interaction.
• Now s²(AB)0 so need to perform a pooled test.
• MS(pooled)SSABSSE/61297.25/185.4
• F(Machine)10.5/5.4 compare with F(3,6) so not
  significant
• F(Operator)63.8/5.4 is significant.
• Half of the total variation is due to the
  operators.

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Mixed Effect Model
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Also the interactions are independent to main
effect and error
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Consequence of the Models
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Expected Mean Squares
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Test for Interaction
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Test for Fixed Factor A
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Test for Random Factor B
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Estimation of Variance Components
ns²(AB)
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Comparison of three models
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Gasket data

• A company wishes to investigate whether the type
  of gasket material affects the amount of
  production of gaskets.
• Three materials, cork, rubber and plastic are to
  be studied.
• Two machines were randomly selected from a large
  number of machines that are used to produce the
  gasket.
• The response is the number of gaskets (in
  thousand) produced.