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Linear Contrasts

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Groups A B C D E -3 -3 2 2 2 compares (AB) with (CDE) 0 0 -2 1 1 compares C ... squares for these five contrast total to equal the treatment sum of squares from ... – PowerPoint PPT presentation

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Title: 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

6
Confidence Interval for
  • 5 ? 2.12(.3162) 4.33, 5.67

7
Standardized Contrast
  • s pooled standard deviation, SQRT(MSE)
  • That is a whopper effect

8
Get Contrast F From SAS
9
Approximate CI for Standardized Contrast
  • Just take the unstandardized contrast and divide
    each end by s (.707).
  • Runs from 6.12 to 8.02

10
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

11
Contrast ?2
  • ?2 SScontrast / SStotal 125/138 .9058
  • Partial ?2
  • Conf-Interval-R2-Regr.sas gives a CI of .85, .96.

12
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.

13
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.

15
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.
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