Outline

1
Outline

• Comparing Group Means
• Data arrangement
• Linear Models and Factor
• Analysis of Variance (ANOVA)
• Partitioning Variance
• F-test (Computation)
• T-test and ANOVA
• Conclusion

2
Comparing group means

• Compares the means of independent variables x1
  and x2 (independent sample t-test)
• Compares two group means of a variable x
  examines how group makes difference in x.
• Data arrangements SPSS requires the second
  arrangement for independent sample t-test

3
Data Arrangement
x group
10 12 9 15 13 6 8 3 0 2 0 0 0 0 0 1 1 1 1 1
x1 x2
10 12 9 15 13 6 8 3 0 2
4
Linear Model and Factor

• Examines how group makes difference in the
  variable x.
• X µ gi ei
• µ is the overall mean, gi is group is mean
  difference from the overall mean
• G is called factor (categorical independent
  variable) that makes difference in the left-hand
  side variable (dependent variable) x.

5
Comparing More Variables

• What if you need to compare x1, x2, and x3? Or
  compares means of three groups?
• Compares x1 and x2, x1 and x3, and x2 and x3?
• How about comparing 4 and 5 groups?
• Any way to make comparison easy?
• ANOVA is the answer

6
T-test and ANOVA 1

• T-test directly compares means of two variables
  (groups)
• ANOVA (Analysis of Variance) partitions overall
  variance and examines the impact of factors on
  mean difference
• So, t-test is a special case of ANOVA that
  considers only TWO GROUPS

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Data Arrangement xij

• An observation of i group and jth data point xij

x group
x11 x12 x13 x14 x15 x21 x22 x23 x24 x25 0 0 0 0 0 1 1 1 1 1
x1 x2
x11 x12 x13 x14 x15 x21 x22 x23 x24 x25
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Partitioning of Variance 1

• Overall mean x group i mean xi
• Sum of squares between group (treatment)
• Sum of squares within group
• SST SSM (between) SSE (within)

9
Partitioning of Variance 2 µ7.7
Treatment x (x-xbar)2 xibar (xibar-xbar)2 (x-xibar)2
1 10 5.138 11.8 16.538 3.24
1 12 18.204 11.8 16.538 0.04
1 9 1.604 11.8 16.538 7.84
1 15 52.804 11.8 16.538 10.24
1 13 27.738 11.8 16.538 1.44
2 6 3.004 3.8 15.471 4.84
2 8 0.071 3.8 15.471 17.64
2 3 22.404 3.8 15.471 0.64
2 0 59.804 3.8 15.471 14.44
2 2 32.871 3.8 15.471 3.24
3 5 7.471 7.6 0.018 6.76
3 9 1.604 7.6 0.018 1.96
3 12 18.204 7.6 0.018 19.36
3 8 0.071 7.6 0.018 0.16
3 4 13.938 7.6 0.018 12.96
Total   264.933   160.133 104.800
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ANOVA F test 1

• H0 all group have the same mean
• Ratio of MSM to MSE
• Degrees of freedom 1 is t-1 (t is the number of
  groups)
• Degrees of freedom 2 is N-t (N is the number of
  overall observation)

Sources Sum of Squares DF Mean Squares F
Model (treatment) SSM t-1 MSMSSM/(t-1) MSM/MSE
Error (residual) SSE N-t MSESSE/(N-t)
Total SST N-1
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ANOVA F test 2

• T3 (three groups) N15 (5 in each group)
• SST264.9160.1104.8
• SSM160.1, MSM80.1160/(3-1)
• SSE104.8, MSE8.7104.8/(15-3)
• F9.280.1/8.7 (CV5.10 p.639)
• df123-1, df21215-3

Sources Sum of Squares DF Mean Squares F
Model (treatment) 160.1 2 80.1 9.2
Error (residual) 104.8 12 8.7
Total 264.9 14
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ANOVA F-test 3

• If F score is larger than the critical value or
  the p-value is smaller than the significance
  level, reject the null hypothesis
• Rejection of H0 is interpreted as there is at
  least one group that has a mean different from
  other group means.
• Rejection of H0 does not, however, say which
  groups have different means.

13
ANOVA Short Cuts

• Compute sum of observations (overall and
  individual groups) or overall mean x
• Compute variances of groups (si)2

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Independent sample t-test

• X1bar26,800, s1600, n110
• X2bar25,400, s2450, n28
• Since 5.47gt2.58 and p-value lt.01, reject the H0
  at the .01 level.

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ANOVA for two group means

• sum1268,00026,80010, s1600, n110
• sum2203,20024,4008, s2450, n28
• Overall sum471,200268,000203,200, N18
• SSTSSMSSE13,368,6118,711,1114,657,500
• FMSM/MSE29.9254 sqrt(29.9254)5.4704t

16
T-test versus ANOVA

• T-test examines mean difference using t
  distribution (mean difference/standard deviation)
• ANOVA examines the mean difference by
  partitioning variance components of between and
  within group (F-test)
• T-test is a special case of ANOVA
• T score is the square root of F score when df1 is
  1 (comparing two groups)

17
Summary of Comparing Means