Linear Contrasts

1
Linear Contrasts
2
Contrast Coefficients

• One for each group mean
• Sum to zero
• One set negative, one positive
• Groups A B C D E
• -3 -3 2 2 2 compares (AB) with (CDE)
• 0 0 -2 1 1 compares C with (DE)

3
Standard Contrast Coefficients

• n number of means in set
• Coefficients -1/n1 and 1/n2
• Sum 0
• Sum of absolute values 2
• -1/2 -1/2 1/3 1/3 1/3 codes (AB) vs. (CDE)
• 0 0 -1 1/2 1/2 codes C vs. (DE)

4
Standard Contrasts

• With equal sample sizes,

5
(AB) vs. (CD)

• Means are 2, 3, 7, 8, MSE .5, dfe 16
• F(1, 16) 125/.5 250, p ltlt .01

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Confidence Interval for

• 5 ? 2.12(.3162) 4.33, 5.67

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Standardized Contrast

• s pooled standard deviation, SQRT(MSE)
• That is a whopper effect

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Get Contrast F From SAS
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Approximate CI for Standardized Contrast

• Just take the unstandardized contrast and divide
  each end by s (.707).
• Runs from 6.12 to 8.02

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Exact CI for Standardized Contrast

• Conf_Interval-Contrast.sas
• Contrast t SQRT(250) 15.81, df 16
• Sample sizes 5, 5, 5, 5
• Coefficients -.5, -.5, .5, .5
• CI runs from 4.48 to 9.64

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Contrast ?2

• ?2 SScontrast / SStotal 125/138 .9058
• Partial ?2
• Conf-Interval-R2-Regr.sas gives a CI of .85, .96.

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CI for ?2

• For regular ?2 (not partial ?2), must adjust the
  F before using my SAS program.
• Add to the MSE all variance not included in the
  contrast.
• Our total SS was 138
• The contrast SS was 125
• So the adjusted SSE is 138-125 13.

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CI for ?2

• Add to the dferror all df not included the
  contrast.
• Our total df was 19
• The contrast df is 1
• So the adjusted dferror is 19-1 18.
• The adjusted contrast
• My SAS program gives CI .78, .94

14
Orthogonal Contrasts

• They are independent of one another.
• With k means, you can obtain k-1 orthogonal
  contrasts.
• For each pair of contrasts, the following must be
  true.
• See the handout if you have unequal sample sizes.

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Orthogonal Contrasts

• Example A B C D E 1/2 1/2 -1/3 -1/3
  -1/3 1 -1 0 0 0 0 0 1 -1/2 -1/2 0
  0 0 1 -1
• The sums of squares for these five contrast total
  to equal the treatment sum of squares from the
  ANOVA.