ANOVA's Flashcards
One-Way ANOVA definition
One factor with 2 or more levels.
H0 and H1 of one-way ANOVA
H0: μ1 = μ2 = μ3.
H1 = not all μ’s are the same.
Assumptions of a One-way ANOVA
1) Population normal within each group.
2) Homogeneity of variance of population.
3) Independence of observations.
What are the 2 sources of variance in a One-Way ANOVA?
The variance explained by the model (MSM) & The variance within groups, or the residual variance (MSR)
MSm refers to…
Variance between groups that is due to the IV.
MSr refers to….
The random variation in the scores for the subjects within each group.
If there’s a large difference between groups, we have a ___ MSm and a ____ F ratio.
High, high.
F = (df___, df____)
dfm, dfr
With only two groups, either a t-test or an F-test can be used for testing for a significant difference between means. What equation represents this?
F = t**2
What are the two size measures that can be used?
eta squared (η2) & omega squared (ω2)
Which size measure (between η2 & ω2) is biased and which is unbiased?
eta squared (η2) is positively biased.
omega squared (ω2) is an unbiased estimator.
What’s a small, medium and large effect?
Small: 0.01
Medium: 0.06
Large: 0.14
What happens if assumptions are violated in a One-way ANOVA?
F ratio no longer follow F-distribution.
5 Assumptions in between-subjects ANOVA
1) Independence of observations (1 person doesn’t affect another person’s performance)
2-3) Identical distribution (within and between groups)
3) Homogeneity of variance (homoscedasticity)
4) Normal distribution
How can we violate the “independence of observations” assumption & what are the consequences of its violation?
1) Sampled from the same natural class (e.g friend)
2) Underestimation of true variability (MSr), Increase Type 1 error -> Sample size doesn’t solve problem
What is needed for the “identical distribution within groups” assumption? What are the consequences of its violation?
Needed: We don’t know more about 1 participant’s score than we do about others. No subtypes of population in the same group.
Consequences: Not accurate representation of population of interest - biased results. MSr inflated which reduces the power of the test.
How can we resolve the problem of “independence of observations & “identical distribution within groups”?
Random sampling
What is needed for the “identical distribution between groups” assumption?
Groups differ only by their means (same variance, skew…).
What is needed for the “homo of variance” assumption? How is it violated? What are the consequences of its violation?
Needed: Variance is the same for all groups.
Violated: groups defined by classification factor (e.g. nationality), problems with experimental manipulation.
Consequences: High type 1 error.
How can we alleviate some of the inflation of Type 1 error rate if homo of variance violated?
Higher sample size + equal groups
What are the consequences of violating the normal distribution assumption? (2)
F ratio is derived from the normality assumption.
Produce Type 1 error rate LOWER than nominal value -> Difficult to reject H0.
Hypotheses in a Two-Way ANOVA
1) Main effects (For factor A & B):
H0a : μa1 = μa2 = …
H1a : Not all μag are the same
2) Interaction effects:
H0axb : All μAgBj are equal.
H1axb : Not all AgBj are equal.
4 methods to assess normality
Skewness statistic, Kolmogorov-Smirnov test, Shapiro-Wilk test, normal quantile plot
Null hypothesis in Skewness statistic (assess normality)
H0: Skew is 0 in the population.
Which is the most powerful between the Kolmogorov-Smirnov & and Shapiro-Wilk test?
What are their H0?
Shapiro-Wilk test
H0: Distribution of the variable is normal.
Limitation of the normality tests
It is easy to find significant results (reject null hypothesis that data is normal) when sample size is large.
3 methods to assess homogeneity of variance
Fmax test of Hartley, Levene’s test, Brown and Forsythe test
(homo of variance) Which test is the more robust between Levene’s test & Brown and Forsythe test?
Brown-Forsythe test is slightly more robust than Levene’s test.
(homo of variance) What’s the problem with Levene’s test?
It is very easy to obtain significant results when the sample size is large.
(homo of variance) What is the H0 in Levene’s test & Brown and Forsythe test?
H0: The population variances are equal
If normality and homogeneity of variances are not met, consider _______ tests.
For example: _____
nonparametric, Kruskal-Wallis ANOVA
Prior requirements/assumptions in a Two-Way ANOVA (3)
- The population distribution of the DV is normal within each group
- The variance of the population distributions are equal for each group (homogeneity of variance assumption)
- Independence of observations
In a Two-Way ANOVA, SSM is partitioned. In which parts?
- SSA: Variation between means for Factor A
- SSB: Variation between means for Factor B
- SSA×B: Variation between cell means
Definition of SSr for one-way vs two-way ANOVA.
In One-way ANOVA, SSR is the sum of the squared difference between a group mean and group observations, across all k groups.
In Two-way ANOVA, SSR is the sum of squared differences between a cell mean and cell observations, across all (a × b) cells
Hypotheses in one-way repeated measures ANOVA
H0 : μ1 =μ2···=μk
H0 : Not all means are equal.
What are all the SS’s in a One-Way repeated measures ANOVA? Explain how it is partitioned.
SS(A), SS(S), SS(AxS)
-> SS(W) = SS(A) + SS(S)
Explain what is SS(W) in a One-Way repeated measures ANOVA
Within-participant variation.
Explain what is SS(A) in a One-Way repeated measures ANOVA.
Variation between group means (IV). (columns).
Explain what is SS(S) in a One-Way repeated measures ANOVA.
Variation between row means (subject means). Not interesting (simply tells us that participants are different). Between-participant effect.
Explain what is SS(AxS) in a One-Way repeated measures ANOVA.
Variation between cell means (all cell means in the table). Also called SS error.
Assumptions in one-way repeated measures ANOVA
Normality within each level of the factor, Homogeneity of variance, Homogeneity of covariance
What is compound symmetry in a one-way repeated measures ANOVA?
Homogeneity of variance & Homogeneity of covariance
Homogeneity of covariance
The population covariance between any pair of repeated measurements is equal (homogenous covariance).
Sphericity
The variance of differences of a pair of observations is the same across all pairs.
Compound symmetry can be replaced by the assumption of _______.
Sphericity.
What is the test that checks the assumption of sphericity?
Mauchly’s test
H0 in Mauchly’s test
H0: Variances of differences between conditions are equal
In Mauchly’s test, if sphericity holds, epsilon ____. If violated, epsilon = _____.
Holds: ε = 1.
Violated: ε < 1.
What happens to the df’s when sphericity is violated?
Reduces dfA and dfSxA and thus gives a larger critical value for F.
What are the 2 steps to deal with violation of sphericity?
(1) measuring the degree of violation of sphericity ε.
(2) using the critical value equal to the value of the F distribution that corresponds to εdf (the adjustment is made for both df numerator and df denominator)
In One-way repeated measures ANOVA, we use a ___ omega squared.
Partial.
What is the formula of F in one-way repeated measures ANOVA?
F = MS(A) / MS(SxA)
Partial Omega squared (ω2) in a one-way repeated measures ANOVA excludes ______.
The variability due to differences between subjects: MS(S).
Three possible ways to calculate ε (correction)
Greenhouse-Geisser approach, the Huynh-Feldt approach, and the Lower bound ε can attain which is ε = 1/(a − 1)
(sphericity) Which test for ε is the most conservative?
Greenhouse-Geisser
How the Greenhouse-Geisser & Huynh-Feldt approach help in the correction of sphericity?
Reduce df + conservative critical value
What is ε (epsilon) & its formula ?
Extent to which sphericity was violated.
ε = 1 / (a - 1)
What are the sources of variation in a two-factor mixed design? (5)
1) Main effect of the between-subjects factor A
2) Subject variation at levels of the between-subjects factor S/A
3) Main effect of the within-subjects factor B
4) Interaction between the between-subjects factor and the within-subjects factor AxB
5) Interaction between the within-subjects factor and subjects nested in levels of the between-subjects factor BxS/A
What is S/A in a two-factor mixed design?
Subject variation at levels of the between-subjects factor
What is BxS/A in a two-factor mixed design?
Interaction between the within-subjects factor and subjects nested in levels of the between-subjects factor
(mixed anova) What is the SSresidual of MS(A), MS(B) and MS(AxB)?
MS(A) = S/A
MS(B) and MS(AxB) = BxS/A
(two-factor mixed design) When the main effect of factor A is significant, what does it mean?
Not all group means are equal.
Assumptions in a two-factor mixed design
Between subject:
- Normal distribution
- Homo of variance (at each combination of A and B)
- Independence of observations
Within-subject:
- Sphericity
In a two-factor mixed ANOVA what are the consequences of the violation of sphericity?
No consequences for the between-subject factor.
Type 1 error rate increases for the within-subject effects.
In a two-factor mixed ANOVA what are the consequences of the violation of homo of variance?
Consequences depend on the equality of within-group sample sizes.
- Type I error rate > α when smaller groups have higher variability
Type I error rate < α when smaller groups have lower variability (lower power to detect effects).
R2 is ______ η2
the same as
