ANOVA II

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ANOVA II

• Engineering Experimental Design
• Valerie L. Young

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CRD vs. RBD

• CRD completely randomized design
• Independent groups of data
• RBD randomized block design
• Data in different groups is linked by some other
  factor
• Like data for a paired t-test
• Remember, Excels single factor ANOVA is only
  used to analyze a one-factor CRD experiment

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An RBD Design for the Chicken Feed Analysis
Problem

• As a chicken nutritionist, you need reliable
  measures of lasalocid sodium levels in chicken
  feed.
• You send 10 samples of different feeds to three
  independent labs, and get back the results at
  left.
• Do the labs agree with each other?

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What Does Blocking Do?

• The blocking factor is the brand of feed
• Note that we expect analyses of Feed 1 by all
  three labs to agree (null hypothesis), but we do
  not expect any relationship between results for
  Feed 1 and results for any other feed.
• Blocking prevents the variation among feeds from
  obscuring the variation among laboratories.

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ANOVA Models Source of Variability

• CRD
• Total Variability (variability among
  treatments) (variability within treatments)
• Variability within treatments is assumed to
  represent random experimental error
• SST SSTr SSE
• F (mean variability among treatments) / (mean
  variability within treatments)
• F MSTr / MSE
• RBD
• Adds effect of block to model
• SST SSTr SSB SSE
• Calculate an F for the treatment (MSTr/MSE) and
  an F for the block (MSB/MSE)

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ANOVA for RBD Total Variability

• Calculate the mean of ALL responses
• We have measurements of 10 feeds from 3 labs, so
  thats the mean of 30 measurements
• Calculate the difference between each response
  and the grand mean
• Square the differences and add them up
• Thats SST

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Calculating SST
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ANOVA for RBD Effect of Treatment

• For each treatment, calculate the difference
  between the mean response and the grand mean
• Square the differences and add them up
• Multiply by the number of blocks (and by the
  number of replicates per block if you have
  replicates)
• Thats SSTr
• Divide by ( treatments 1) to get MSTr

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Calculating MSTr
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ANOVA for RBD Effect of Blocking

• For each block, calculate the difference between
  the mean response and the grand mean
• Square the differences and add them up
• Multiply by the number of treatments (and by the
  number of replicates per treatment if you have
  replicates)
• Thats SSB
• Divide by ( blocks 1) to get MSB

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Calculating MSB
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ANOVA for RBD Effect of Random Error

• Method 1 Assume that random error is any
  variability that is NOT from the treatment or the
  blocking factor
• SSE SST SSTr SSB
• MSE SSE / (( treatments 1)( blocks 1)

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ANOVA for RBD Effect of Random Error

• Method 2 Assume that variability apparently due
  to interaction is really due to random error
• ?ij yij (mean of ith row) (mean of jth
  column) (grand mean)
• MSE SSE / (( treatments 1)( blocks 1)

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ANOVA for RBD Calculation of F

• F (treatment) MSTr / MSE
• F (block) MSB / MSE
• But what do the Fs mean?

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The Meaning of F

• Null hypotheses
• H0,Tr The treatment has no effect.
• H0,B The blocking factor has no effect.
• Significance level
• Lets use 0.05
• F critical values (from Table B7)
• F-criticalTr F(0.052,18) 3.55
• F-criticalB F(0.059,18) 2.46

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The Meaning of F

• F(treatment) 9.5027 gt 3.55
• At least one labs analysis differs significantly
  from at least one other labs analysis
• F(block) 13.80927 gt 2.46
• At least one type of feed has significantly
  different composition from at least one other
  type of feed
• Blocking is important in this experimental design
• Without blocking, the difference among
  laboratories would be obscured by the difference
  among feeds

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ANOVA on RBD with Excel

• Excel can handle an RBD with one treatment factor
  and one blocking factor
• Tools gt Data Analysis gt ANOVATwo factor