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Analysis of Variance. Schaum’s Outline Probability and Statistics Chapter 9 Examples by Steve Brochu Mark Thomas. Outline Chapter 9. t test versus F test Analysis of variance Test differences of means across groups Variation within groups Variation between groups

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schaum s outline probability and statistics chapter 9 examples by steve brochu mark thomas

Analysis of Variance

Schaum’s Outline

Probability and Statistics

Chapter 9

Examples by Steve Brochu

Mark Thomas

outline chapter 9
Outline Chapter 9
  • t test versus F test
  • Analysis of variance
  • Test differences of means across groups
  • Variation within groups
  • Variation between groups
  • Consider (Variation between)/ (Variation within)
  • Explanatory Power of Regression
  • (Variation explained/Variation unexplained)
analysis of variance f test
Analysis of Variance – F test
  • t tests
  • inferences on one parameter
  • unknown variances, small sample
  • F tests
  • Analysis of variance
  • difference of means
  • often groups > 2
  • across models
    • Do variables in regression model explain y
    • Which model is better
analysis of variance f test1
Analysis of Variance – F test
  • Uranium Mines
  • j different sized mines
  • do costs differ for the j different sized mines
  • (j = 1,. . .,a a=3)
  • 1 = small
  • 2 = medium
  • 3 = large
  • sample 15 mines, 5 (k) in each category
  • sample k mines in each category k = 1,5
  • Cost per ton
  • 44 =  + ejk ei ~ N(0, 2)
analysis of variance uranium mine cost
Analysis of Variance Uranium Mine Cost

Xjk =j + ejk

Ho: 1 = 2= 3

H1: not all equal

variation within groups
Variation Within Groups
  • Vw = jk(Xjk- Xj.)2
  • tons produced
  • j Cost Per Ton (k) xj.= 5k=1xkj
  • 25000 100 110 120 130 140 120
  • 50000 100 105 110 115 120 110
  • 100000 95 98 100 102 105 100
  • Within group variation
  • (100-120)2 ( ) 0 100 400
  • (100-110)2 25 0 25 100
  • ( 95- 100)2 4 0 4 25 Vw =1308
distribution of variation within groups
Distribution of Variation Within Groups
  • (Xjk- j)2/2 ~ 21
  • jk(Xjk- j)2~ 2T = 2ab
  • Vw = jk(Xjk- Xj.)2/ 2= 2T-a = 2ab-a
variation between groups
Variation Between Groups
  • Vb = jk (Xj.- X)2 =bj (Xj.- X)2
  • tons produced
  • j Cost Per Ton (k) Xj.
  • 25000 100 110 120 130 140 120
  • 50000 100 105 110 115 120 110
  • 100000 95 98 100 102 105 100 x 110
  • 5(100-110)2 + (110-110)2 + ( )2 = 1000
distribution of variation between groups
Distribution of Variation Between Groups
  • Total Variation
  • V = jk(Xjk- X)2 = jk(Xjk- Xj.)2 +jk(Xj. - X)2
  • Vw + Vb
  • V = jk(Xjk- X)2= jk(Xjk- Xj.)2+jk(Xj. - X)2
  • 2 2 2
  • If all s the same then
  • T-1 = T-a + ?
  • ? ~ T-1 - T-a = T-1-(T-a) = a-1
  • Vb/2~ a-1
aov hypothesis tests
AOV Hypothesis Tests
  • 2df1
  • df1 ~ Fdf1,df2
  • 2df2
  • df2
  • Under null hypothesis
  • Ho: 1 = 2= 3
  • H1: not all equal
  • Vb/(a-1) = ŝb2= 500/(3-1) = 4.587
  • Vw/(ab-a) ŝw2 1308/(15-3)
  • Critical F2, 12 = 3.89
aov with unequal number of observations
AOV with Unequal Number of Observations
  • Vb = jk (Xj.- X)2 =jnj (Xj.- X)2
  • Vw = jk(Xjk- Xj.)2
  • Fa-1,T-a = vb/a-1
  • vw/T-a
fit of whole regression
Fit of Whole Regression
  • y = 1 + 2x2 +3x3 + . . . kxk+ e
  • R2 = 1 – Sêi2 /Sy\'i2
  • Sy\'i2 =S( ŷ -x)2 + Sêi2
  • Ho:2 = 3 = . . . = k = 0
  • H1: 2, 3,, .. k not all equal to zero
  • Total SS = Explained SS + Error SS
  • Under null hypothesis
  • Total SS/2 ~ T-1
fit of whole regression1
Fit of Whole Regression
  • Explained SS = Total SS - Error SS
  • 22 2
  • Under null ~ T-1- ~ T-K
  • Explained SS ~T-1-(T-K) =K-1
  • 2
  • Under null
  • Explained SS/2
  • K-1 ~ FK-1, T-K
  • Error SS/ 2
  • T-K
fit of whole regression2
Fit of Whole Regression
  • Under null
  • Explained SS/2
  • K-1 ~ FK-1, T-K
  • Error SS/ 2
  • T-K
analysis of variance summary
Analysis of Variance – Summary

9-155

  • Differences between t and F testing
  • Analysis of Variance (ANOVA)
    • Tests for equivalence of multiple means (μ1 = μ2 …)
    • Utilizes identity that: Total SS = Explained SS + Error SS
    • Compares variation between groups to variation within groups using F test
    • Test statistic is:
  • Need Modification if unequal observations each group
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