Summary Notes Contrasts | Statistics Educational Science | KU Leuven | 2025/26

HS 3: CONTRAST

HC4: OPDELING VD LES

1. Research questions
2. Contrasts: notation, types, and sampling distribution
3. One planned contrast
4. Multiple planned contrasts
6. Pairwise planned contrasts - Tukey-Kramer is not evaluated in this course
5. Post hoc contrasts - Not evaluated in this course

PLANNED RESEARCH QUESTION USING ONE-WAY ANOVA.

BY PLANNED RESEARCH QUESTIONS
- We meant research questions that are determined before data collection or data analysis

- Goal is to test whether 2/more groups have different population means: one-way ANOVA

AFTER CONDUCTING A ONE-WAY ANOVA: what we often want?
- Identify which of the (4 eye-color) groups differ
- Estimate how much they differ
- Test differences in specific, theory-driven ways

- → Solution: Use Contrasts: they let you test targeted, interpretable comparisons that match the research questions

CONTRASTS WHY?
- Beperkingen vd F-test
o Zegt alleen  er is ergens een verschil tussen de groepen (of niet)
o Maar zegt NIET
 Welke groepen verschillen?
 Hoe groot is dat verschil?

o Oplossing: contrasts.

CONTRASTS/VERGELIJKING WHAT?
- = Een gerichte vergelijking/verschil tussen gemiddelden (eventueel combinaties van gemiddelden)
- Inhoud
o Welk verschil
o Hoe groot is verschil
o Hoe zeker ben je over verschil

- Formeel:
o Contrast = lineaire (= rechtlijnig) combinatie van groepsgemiddelde
o c 1 y1 + c2 y 2+ ⋯+ c a y a
a
- Waarbij: ∑ c j=0
j=1
 Die c j zijn contrast-coëfficiënten  via die coëfficiënten druk je je wetenschappelijke vraag uit
 Coëfficiënten moeten dus opgeteld = 0

EXAMPLE CONTRASTS
- After conducting a one-way ANOVA, we may want to test the following research questions:

o Is there a difference in the mean brand attitude between
 Participants who saw an advertisement where the model’s eye color was visible (Blue, Brown, Green)
VS
 Those who saw an advertisement where the eye color was not distinguishable (Down)?

o Is there a difference in mean brand attitude between
 The group that saw the advertisement featuring a model with green eyes
 And the group that saw the advertisement featuring a model with blue eyes?

,
, NOTATION

μ j is population mean of group j (with j = 1, . . . , a where a is the number of groups)

A contrast is a linear combination of population means:
- Waarbij contrast-coëfficienten optellen tot 0
o = Subject to the key constrain (=regel/voorwaarde):


INTERPRETATION:
- The values C j are contrast-coefficients (known, chosen by the researcher)
o Deze drukken je hypothese uit
o Bepalen welke groepen je vergelijkt

- A contrast represents a specific difference among the population means
- It allows us to translate a research question into a precise, testable comparison across the a groups
 NIET data kiezen het contrast
 WEL onderzoeker kiest het contrast

HYPOTHESES
- In a contrast analysis, the H0 is given by:


- H1 can be
o One-sided (i.e., H1 : γ > 0 or H1 : γ < 0)
OR
o Two-sided (i.e., H1 : γ̸ ≠ 0)

In practice, a contrast is estimated using the sample means:

CONTRASTS TYPES

PLANNED VS POST-HOC
- Planned contrasts (a priori):
o Specified before collecting/analyzing the data
o Based on theory

- Post Hoc Contrasts (a posteriori)
o Decided after inspecting the data
o Are exploratory contrasts:
 Test the comparisons that become interesting after seeing the results or based on data patterns

o Risicovoller
 Je gebruikt data 2X  zowel om idee te krijgen + idee te testen onterecht “echte” patronen zien

PAIRWISE VS COMPLEX
- Pairwise contrast:
o = Compares 2 groups only.
o Coefficients focus on one mean VS another mean

o Example: Is there a difference in mean brand attitude between
 The group that saw the advertisement featuring a model with green eyes
 And the group that saw the advertisement featuring a model with blue eyes?
 Contrast:


 What are the contrast coefficients?:


 Thus:

o !!! coefficienten zijn 1 en - 1