Contrast Analysis

202307141112
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Tags: Statistics Follow-up tests

Substantively motivated comparisons are known as planned comparisons, or more commonly contrasts. A pair of contrasts is statistically independent, or orthogonal, if the sum of the products of coefficients equals zero. This indicates the various contrasts describe non-overlapping parts of variation in the dependent variable.

When studying contrasts, first look at the interaction effect. Then look for the main effects.

Since planned comparisons are theoretically motivated and can be considered to correspond to separate ‘families’ of hypotheses, there is no need for multiple testing correction.

Example

We have four groups in a 2x2 design, with difficulty and priming as factors. We are interested in a main effect of difficulty, priming, and a possible interaction effect.

If Groups 1 and 2 had in common Easy difficulty, then we can average those two together and compare them against Groups 3 and 4 to test a possible Difficulty-effect.

$H_0:\frac{{\mu }_1+{\mu }_2}{2}=\frac{{\mu }_3+{\mu }_4}{2}$ or $H_0:0.5{\mu }_1+0.5{\mu }_2-0.5{\mu }_3-0.5{\mu }_4=0$.

The coefficients in the denominator can be multiplied with any constant without altering the substance of the hypothesis. Traditionally, these contrast coefficients are expressed as integers. We end up with the following weights: $(1,1,-1,-1)$.

If we have two contrasts for difficulty $(1,1,-1,-1)$ and priming $(1,-1,1,-1)$ we can multiply them together to get the interaction contrast $(1,-1,-1,1)$.

• Null hypothesis for interaction contrast: $H_0:{\mu }_1-{\mu }_2={\mu }_3-{\mu }_4$.
  • Differences between priming effect under difficult conditions and under easy conditions → if the priming effect is difference for different groups of the Difficulty factor, this indicates an interaction effect.

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