ANOVA Lecture 13 revision Flashcards
Statistical inference
From sample statistics, draw conclusions about population parameters
The critical value of the test statistic is determined by sample statistics
α = p(Type I error) = p(reject H0 | H0 true)
Type 1 vs Type 2 error
Retain H0 when Ho is True
Correct 1 – α
Retain H0 when Ho is false
Type II error
β
Reject H0 when ho is true
Type I Error
α
Reject H0 when ho is false
Correct (Power)
1 – β
Between Subjects ANOVA
Each participant only experiences one condition (i.e., different groups of people in each condition)
Interest is in the effect parameters (αj) because they represent systematic between-group differences.
How much the group mean deviates from the grand mean.
One family in One Way ANOVA across the entire experiment
null for between s anova
H0: 𝜇1=𝜇2=⋯=𝜇𝐽 H0: 𝛼1=𝛼2=⋯=𝛼𝐽=0
If H0 is true, Yij = μ + εij (because all αj = 0)
If H0 is false, Yij = μ + αj + εij
Are all the alpha js equal to zero? Are all the group means exactly the same? Contrasts let us know
Experiments with two between-subjects IVs
Factorial design – every level of IVA appears in combination with every level of IVB
A (row variable) has J levels
B (column variable) has K levels
J x K = number of conditions in the experiment / cells in the design / independent groups of participants
E.g., 4 x 5 = 20 cells, or combinations of conditions
Advantages of multifactor experiments:
More power (possibly), can account for more of our IV. •observed F will be possibly larger, Can assess interaction effects – only possible when there are combinations of variables
how many families for A two-way ANOVA
A three-way ANOVA
A four-way ANOVA
EER ≠ FER for multifactor experiments
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A two-way ANOVA (2 IVs in 1 experiment) will have 3 families: A, B and AB
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A three-way ANOVA (3 IVs in 1 experiment) will have 7 families: A, B, C, AB, AC, BC and ABC
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A four-way ANOVA (4 IVs in 1 experiment) will have 15 families
The two-way between-subjects ANOVA model:
Main effect of A, df = J-1, Inference about αj
Main effect of B df K-1, Inference about βk
Interaction effect AxB, df = (J-1)(K-1), Inference about αβjk (J-1)(K-1) Inference about αβjk Error df N-JK Total df N-1
Contrasts
1df test, asking focussed questions, allows directional conclusions
Linear combination of weighted means, whose coefficients sum to 0
Three types of contrasts:
Mean difference contrasts • Trend contrasts • Interaction contrasts
Mean difference contrasts
Is the average of ‘this lot’ the same/different from the average of ‘that lot’ in terms of some DV of interest?
standard form vs integor
psy hat (contrast estimate) is immediately interpretable when using standard form coefficients, while • Hand-calculations are often easier when using integer form coefficients
how do we know psy is signifiant?
NHST for contrasts
For between-subjects contrasts
df effect is always 1.
ms contrast is SS cont/1 = SSc, so F = SS cont/MSE
Critical F for contrasts
For any contrast, observed F = F(1, v2) three methods
what crit F for Planned, orthogonal
DER, good power, no further control over FER or EER, not great flexibility, c.v. F.05;1,ν2
what crit F for Planned contrasts?
Bonferroni, Power - Less than DER when k > 1; more than Scheffé when k ≤ (J-1) Yes controls FER/EER ≤ α Flexibility - Reasonable, but planned c.v. Fα/k;1,ν2
what crit F for Planned or post hoc
Scheffé
Power - Least, most of the time, but not always, controls Yes FER/EER ≤ α
f;exibility - Unlimited,
c.v. ν1F.05;ν1,ν2
Orthogonality
A pair of contrasts is orthogonal if: Σcj1cj2 = 0
orthogonal contrasts will account for …
which error rate?
SS for a set of (J-1) orthogonal contrasts will account for all of SSB without overlap
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Current practice allows use of DER when testing planned and orthogonal contrasts (if the researcher is not concerned about the inflating EER)
Trend contrasts
(orthogonal polynomials)
Special type of contrast asking questions about quantitative aspects of the IV
Interpretation not in terms of mean differences, but in terms of trends: How does the pattern of DV means change as levels of the quantitative IV increase/decrease?
Contrasts for two-way designs
Main effect contrast A – focussed questions about the αj
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Main effect contrast B – focussed questions about the βk
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Interaction contrast AB – focussed questions about the αβjk
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Does this difference = that difference?
FER for two-way designs
If contrasts are evaluated a family-wise error rate (FER), critical values must be worked out for each family
v1 = df(effect) in ANOVA
v2 = df(error)
k = number of questions about that family
DER Bonferroni Scheffé c.v. F.05;1,ν2 Fα/k;1,ν2 ν1F.05;ν1,ν2
Sources of variability
Between-subjects -> condition =group
Total variability SS(Total) is Between groups SS(Condition)
and Within groups SS(Error)
Within-subjects -> condition = repeated measures
Total variability SS(Total) is Within group, partitioned into SS(Subjects), SS(Condition) and SS(Error) ( Subject x Treatment Interaction)
5 points for Within-subjects (repeated measures) variables
- Difference scores = contrast scores
- Partitioning within-subjects variability
- Different ways of testing planned contrasts for repeated measures variables
- Normalising coefficients (a question of scaling)
- Pooled vs. separate error terms for within-subjects contrasts
when might we not be able to provide a direction of effect for a 1 df question?
cross over
