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Causal Analysis TechniquesLecture 1: One-Way Between-Subjects Analysis of Variance
(ANOVA)
Lecture objectives
- In which situation ANOVA is applicable
- What hypotheses can be tested with ANOVA
- The key logic behind ANOVA
- How to calculate the deviations from the mean(s)
Logic of ANOVA
Substantive hypothesis: A person’s degree of organizational commitment (Y) depends on the
team in which the person works (X).
Team in which someone works (X) Organizational Commitment (Y)
- Question: if the hypothesis is correct, what would you expect to find with regard to
differences in average commitment between the teams?
- Imagine that we have collected data of measurements of organizational commitment
for 3 teams.
- 2 scenarios with regard to the data.
In which of the data scenarios would you be more inclined to conclude that there is a
connection between the team in which someone works and organizational commitment?
,Key idea of ANOVA: when there are 2 or more groups, can we make a statement about
possible – significant – differences between the means scores of the groups?
Intermezzo: What could we do if there were only 2 groups? → Answer: t-test
Fundamental principle of ANOVA
ANOVA analyses the ratio of the two components of total variance in data: between-group
variance and within-group variance.
,Fundamental principle of ANOVA
ANOVA analyses ratio in which between-group variance measures systematic differences
between groups and all other variables that influence Y, either systematically or randomly
(‘residual variance’ or ‘error’)
And
Within-group variance measures influence of all other variables that influence Y either
systematically or randomly (‘residual variance’ or ‘error’)
Important to realize
1. Any differences within a group cannot be due to differences between the groups
because everyone in a particular group has the same group score; so, within-group
differences must be due to systematic unmeasured factors (e.g. individual
differences) or random measurement error.
2. Any observed differences between groups are probably not only pure between-group
differences, but also differences due to systematic unmeasured factors or random
measurement error.
Compare…
Between-group variability (= systematic group effect + error)
To
Within-group variability (= error)
… to learn about the size of the systematic group effect
Example from biology
Statistical null hypothesis of one-way between-subjects ANOVA:
Mean scores of k population populations corresponding to the groups in the study are all
equal to each other:
, When do we reject H0? → When at least one mean is significantly different from the other
means.
Intermezzo
Why prefer One-Way Between-Subjects ANOVA instead of separate t-tests for means
(Warner I, p. 390).
In our example with 3 teams, we could also conduct 3 separate t-tests for means:
- Problem of this approach: the larger the number of test that is applied to a dataset,
the larger the chance of rejecting the null hypothesis while it is correct (Type I error).
- Why? Follows from logic of hypothesis testing: we reject the null hypothesis if a result
is exceptional, but the more tests we conduct, the easier it is to find an exceptional
result
- One will easier make the mistake of concluding that there is an effect, while there is
not
- This is called: ‘inflated risk of Type I error’ (Warner I, p. 390)
Formula for calculation of chance of 1 or more Type I errors in a series of C tests with
significance level 𝛼:
Therefore, with 3 separate tests with 𝛼 = 0.05 the chance of unjustified rejection of the null
hypothesis is:
Solution: One-Way ANOVA → one single omnibus test for the null hypothesis that the
means of K populations are equal, with chance of Type I error = 0.05.
Calculations: F-statistic
If we want to test the statistical null hypothesis
with an ANOVA, the F-distribution is used.
In order to determine if a specific sample result is exceptional (‘significant’) under the
assumption that the statistical null hypothesis is correct, the test-statistic F has to be
calculated.
Calculations: Deviations
Strategy: Partition of scores into components
- Component of score that is associated with ‘group’
- Component of score that is not associated with ‘group’
How can you do this? Calculate deviation scores
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