Contrasts

1
Contrasts

• Harry R. Erwin, PhD
• School of Computing and Technology
• University of Sunderland

2
Resources

• Crawley, MJ (2005) Statistics An Introduction
  Using R. Wiley.
• Freund, RJ, and WJ Wilson (1998) Regression
  Analysis, Academic Press.
• Gentle, JE (2002) Elements of Computational
  Statistics. Springer.
• Gonick, L., and Woollcott Smith (1993) A Cartoon
  Guide to Statistics. HarperResource (for fun).

3
Introduction

• Contrasts are the basis of hypothesis testing and
  model simplification in ANOVA
• When you have more than two levels in a
  categorical variable, you need to know which
  levels are meaningful and which can be combined.
• Sometimes you know which ones to combine and
  sometimes not.
• First do the ANOVA to determine whether there are
  significant differences to be investigated.

4
Orthogonal Contrasts

• For a factor with k levels, there are only k-1
  orthogonal contrasts. (The missing one is used in
  the ANOVA testing to see if there is a
  significant difference.)
• The contrast coefficients form a vector in Rk.
• The last of the k coefficients is constrained by
  the previous k-1.
• Any two contrasts are orthogonal in Rk.

5
Example

• Suppose a factor has five levels a, b, c, d, and
  e.
• d is the control level.
• a and b are similar treatments, as are c and e.
• Orthogonal contrast coefficients would be
• (1, 1, 1, -4, 1)do the treatments contrast with
  the control?
• (1, 1, -1, 0, -1)do the different treatments
  contrast?
• (1, -1, 0, 0, 0)do the levels of the first
  treatment contrast?
• (0, 0, 1, 0, -1)do the levels of the second
  treatment contrast?

6
Model Reduction in ANOVA

• Basically how you reduce a model in ANOVA is by
  combining factor levels.
• Define your contrasts based on the science
• Treatment versus control
• Similar treatments versus other treatments.
• Treatment differences within similar treatments.
• You can also aggregate factor levels in steps.
• Book example.

7
Types of Contrasts

• Treatment contrasts (in R).
• Helmert contrasts (in S)
• Sum contrasts (not used).
• Book example

8
Aliasing

• No information to analyse.
• Intrinsic aliasing reflects the structure of the
  model. (E.g., the model has more parameters than
  data points.)
• Extrinsic aliasing reflects the nature of the
  data. (E.g., missing data.)
• Book example.

9
ANCOVA and Contrasts

• Book example.