FOLLOW UP TESTS + LINEAR CONTRASTS

FOLLOW UP TESTS

A PRIOR (PLANNED):

• Before data collection

• Does not require significant F first

• More powerful

• Bonferroni t’

• Linear contrasts

POST HOC (EXPLORATORY):

• after examining data

• requires significant F first

• less powerful

• Scheffe test

• Pairwise contrasts (Tukey test)

LINEAR CONTRASTS

• Allows us to compare averages of group means 

• A way of organising multiple t-tests to evaluate non-pairwise comparisons  

ASSIGNING WEIGHTS

1. Choose sensible comparison (two chunks at a time) 

2. Groups coded with positive values will be compared against groups coded with negative values 

3. The sum of the weights for a comparison must equal zero 

4. If a group is not involved in a comparison, give the group a weight of zero 

5. The weights assigned to groups in one chunk needs to be equal to the number of groups in the other chunk.  

ORTHOGONALITY

• Ask independent questions of the data set  

  • Knowing the result of one contrast does not guess the results of another 

• Ensure minimal, non-redundant comparisons 

  • Prevents over-analysing data 

• Doing the minimum number of tests means we have greater power for those tests 

  • Fewer comparisons -> lower t' crit -> more likely to detect effect 

CHECKING ORTHOGANALITY

1. The number of contrasts should not exceed k – 1 

2. Each contrast in a set must be linear 𝛴a_j = 0 (sum of weights = 0) 

3. Contrasts must be independent of one another 𝛴a_jb_j = 0 (the sum of weights for each pair of contrasts must equal zero / dot product) 

LINEAR CONTRAST STEPS

1. Designate contrasts

2. Assign weights

3. Do orthoganality checks

4. Contrast 1 hypothesis

5. Contrast 1 L and aj²

6. Calculate contrast 1 t’

7. Find t’ critical

8. Contrast 1 interpretation

9. Contrast 2 hypothesis

10. Contrast 2 L and aj²

11. Calculate contrast 2 t’

12. Contrast 2 interpretation

13. Combined interpretations

BONFERRONI ADJUSTMENT

• The Bonferroni adjustment is a correction to the α for each 
  comparison, ensuring the family-wise error rate remains at a desired 
  level (typically, α_FW = .05) 

• Results in higher critical t 

CALCULATING BONFERRONI T ALPHA

Divide alpha by number of comparisons

SCHEFFE TEST

Process of calculating F statistic is the same, the critical value is the thing that changes.

Fcrit = (k-1) x Fcrit (omnibus)

• greater critical value

• harder to reject