A contrast analysis approach to change

Abstract

This paper presents the foundations of contrast analysis as a method for examining change. Contrast analysis is a relatively high-powered, simple, and informative procedure for evaluating hypotheses about specific patterns of change. This paper reviews the general purpose and nature of contrast analysis, it discusses some of the advantages of contrast analysis as a method for examining change, it provides a conceptual overview of the relevant statistical procedures, it illustrates the approach by working through several examples, and it addresses important issues that should be considered when conducting and interpreting contrast analysis.

Keywords:

• contrast analysis
• analysis of variance
• effect size
• repeated measures
• longitudinal data analysis
• growth curve models
• change

Acknowledgements

This work was supported by a grant from the National Institute of Health (R01-MH63908). I thank Erika Carlson, Kris Gauthier, and Eric Stone for their comments on this manuscript.

Notes

1. Although the current paper presents examples of contrast analysis for a study with five time points, contrast analysis is applicable to studies with any number of time points. In fact, contrast analysis is the implicit basis of some familiar techniques that involve only two time points, such as repeated-measures t tests and ANOVA designs with a two-level repeated-measures factor. Such techniques can be seen as contrast procedures in which the two contrast weights are +1 and −1.

2. An alternative approach to obtaining SS_{CONTRAST-NORM} is by using “normalized” contrast weights in Step 1 of the contrast analysis procedure. A set of normalized contrast weights has a sums of squares value of 1.0. For example, the steady decline set of contrast weights in A1 have a SS value of 10:

• However, the steady decline hypothesis can be reflected in a set of normalized contrast weights (Set A5):

  Table

  Demonstrating that the SS of these values is 1.0:
  A set of weights can be normalized by dividing each weight by the SS value of the set, for example:
  If a set of normalized weights is used to conduct the contrast analysis, then the normalized SS is simply: