Ch 13, ANOVA

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Ch 13, ANOVA

• If we have two normal populations, we can use a t
  test to see if the means are the same
• For gt2 populations, we have to do something
  different
• You cant subtract 3 avgs

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Ch 13, ANOVA

• In the Chi2 test, we used the sum of squared
  differences to see if several values (obs) were a
  given value (exp)
• In Analysis of Variance, we use the Sum of
  Squares (SS) for a similar reason

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Ch 13, ANOVA

• General terms
• TOTAL without regard for groups
• TREATMENT the different groups
• ERROR what happens within groups

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Ch 13, ANOVA

• In the t test, we found the pooled SD
• In ANOVA, we use S (ni-1)SDi2
• SSE
• Divide by S (ni-1) to get pooled variance
• S (ni-1) dfe (df for Error)

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Ch 13, ANOVA

• To see if all the avgs are the same, we compare
  them to the overall avg
• Overall avg Sniavgi / Sni
• S ni(avgi-overall)2 measures if the avgs are
  all equal to the overall avg
• SSTr

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Ch 13, ANOVA

• If we were to est the SD using all the data, we
  would use
• S (xi overall avg)2
• SSTotal
• And divide by N-1
• df Total

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Ch 13, ANOVA
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Ch 13, ANOVA

• The Mean Square is the SS divided by df

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Ch 13, ANOVA
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Ch 13, ANOVA

• FMStr/ MSE
• Measures how spread out the avgs are with
  respect to the variation within the groups
• Need to know df for numerator and denominator

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Ch 13, ANOVA

• PV is prob of F or greater

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Ch 13, ANOVA

• Excel example
• See Table13.3 on p 592
• Do Summary by Group in DDXL

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Ch 13, ANOVA
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Ch 13, ANOVA

• Overall avg is overall total/ N
• Use SUMPRODUCT to get overall total
• Put totals at TOP (so they will be in fixed
  locations)

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Ch 13, ANOVA
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Ch 13, ANOVA

• H0 all means are equal
• Ha not all means are equal
• (NOT all means different)
• If H0 true, then MSTr MSE, so F1
• If F large, then the avgs are relatively spread
  out and we reject H0

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Ch 13, ANOVA

• F distn
• Num, denominator df
• PV Prob(obs F or greater)

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Ch 13, ANOVA

• ?MSE RMSE Pooled SD for several groups
• Can use this to compare various means
• Again, use Sp ?(1/n11/n2), where SpRMSE

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Ch 13, ANOVA

• Multiple comparisons
• Suppose all means are actually equal and we do 20
  comparisons, each with sig level0.05
• Then we would expect one of the comparisons to be
  significant, even though H0 is true

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Ch 13, ANOVA

• Have to adjust significance level (or pvalue)
• Several ways to do it
• Most direct is Bonferroni method
• Have to adjust by a factor equal to the number of
  comparisons being made

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Ch 13, ANOVA

• If there are 4 groups, then there are 6 pairs to
  compare
• Must multiply each p-value by 6
• If we do 95 confidence intervals, we must solve
  for 0.025/6 instead of 0.025

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Ch 13, ANOVA

• Consider Table 13.3

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Ch 13, ANOVA
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Ch 13, ANOVA

• MSE5.14
• RMSE2.268
• Compare Midwest to NE
• For SD, use 2.268?1/61/5
• For df, use DFE

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Ch 13, ANOVA
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Ch 13, ANOVA

• But we need to multiply this by 2 (for 2 tailed)
• And by 6 because of 6 pairs of comparisons
• So PV is quite large
• No diff between NE and MidWest

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Ch 13, ANOVA

• Compare Midwest to South

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Ch 13, ANOVA
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Ch 13, ANOVA

• Again, multiply by 2 and by 6
• (Corrected) pvalue 0.0436
• So we can conclude that Midwest is diff from
  South

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Ch 13, ANOVA

• Note that A and B might not be different
• B and C might not be different
• But A and C are different
• Can get some complicated comparisons

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Ch 13, ANOVA

• We can also use the Bonferoni correction to do
  confidence intervals
• Usually, to do a 95 CI, we use 0.975 as the Left
  Prob
• With the Bonferoni correction, we use a higher
  value
• For a 95 CI, use 1-0.025/k
• Where k pairs

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Ch 13, ANOVA