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Statistics > ANOVA's > Flashcards

ANOVA's Flashcards

1
Q

One-Way ANOVA definition

A

One factor with 2 or more levels.

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2
Q

H0 and H1 of one-way ANOVA

A

H0: μ1 = μ2 = μ3.
H1 = not all μ’s are the same.

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3
Q

Assumptions of a One-way ANOVA

A

1) Population normal within each group.
2) Homogeneity of variance of population.
3) Independence of observations.

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4
Q

What are the 2 sources of variance in a One-Way ANOVA?

A

The variance explained by the model (MSM) & The variance within groups, or the residual variance (MSR)

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5
Q

MSm refers to…

A

Variance between groups that is due to the IV.

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6
Q

MSr refers to….

A

The random variation in the scores for the subjects within each group.

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7
Q

If there’s a large difference between groups, we have a ___ MSm and a ____ F ratio.

A

High, high.

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8
Q

F = (df___, df____)

A

dfm, dfr

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9
Q

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?

A

F = t**2

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10
Q

What are the two size measures that can be used?

A

eta squared (η2) & omega squared (ω2)

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11
Q

Which size measure (between η2 & ω2) is biased and which is unbiased?

A

eta squared (η2) is positively biased.
omega squared (ω2) is an unbiased estimator.

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12
Q

What’s a small, medium and large effect?

A

Small: 0.01
Medium: 0.06
Large: 0.14

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13
Q

What happens if assumptions are violated in a One-way ANOVA?

A

F ratio no longer follow F-distribution.

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14
Q

5 Assumptions in between-subjects ANOVA

A

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

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15
Q

How can we violate the “independence of observations” assumption & what are the consequences of its violation?

A

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

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16
Q

What is needed for the “identical distribution within groups” assumption? What are the consequences of its violation?

A

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.

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17
Q

How can we resolve the problem of “independence of observations & “identical distribution within groups”?

A

Random sampling

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18
Q

What is needed for the “identical distribution between groups” assumption?

A

Groups differ only by their means (same variance, skew…).

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19
Q

What is needed for the “homo of variance” assumption? How is it violated? What are the consequences of its violation?

A

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.

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20
Q

How can we alleviate some of the inflation of Type 1 error rate if homo of variance violated?

A

Higher sample size + equal groups

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21
Q

What are the consequences of violating the normal distribution assumption? (2)

A

F ratio is derived from the normality assumption.
Produce Type 1 error rate LOWER than nominal value -> Difficult to reject H0.

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22
Q

Hypotheses in a Two-Way ANOVA

A

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.

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23
Q

4 methods to assess normality

A

Skewness statistic, Kolmogorov-Smirnov test, Shapiro-Wilk test, normal quantile plot

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24
Q

Null hypothesis in Skewness statistic (assess normality)

A

H0: Skew is 0 in the population.

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25
Q

Which is the most powerful between the Kolmogorov-Smirnov & and Shapiro-Wilk test?

What are their H0?

A

Shapiro-Wilk test

H0: Distribution of the variable is normal.

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26
Q

Limitation of the normality tests

A

It is easy to find significant results (reject null hypothesis that data is normal) when sample size is large.

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27
Q

3 methods to assess homogeneity of variance

A

Fmax test of Hartley, Levene’s test, Brown and Forsythe test

28
Q

(homo of variance) Which test is the more robust between Levene’s test & Brown and Forsythe test?

A

Brown-Forsythe test is slightly more robust than Levene’s test.

29
Q

(homo of variance) What’s the problem with Levene’s test?

A

It is very easy to obtain significant results when the sample size is large.

30
Q

(homo of variance) What is the H0 in Levene’s test & Brown and Forsythe test?

A

H0: The population variances are equal

31
Q

If normality and homogeneity of variances are not met, consider _______ tests.

For example: _____

A

nonparametric, Kruskal-Wallis ANOVA

32
Q

Prior requirements/assumptions in a Two-Way ANOVA (3)

A
  • 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
33
Q

In a Two-Way ANOVA, SSM is partitioned. In which parts?

A
  • SSA: Variation between means for Factor A
  • SSB: Variation between means for Factor B
  • SSA×B: Variation between cell means
34
Q

Definition of SSr for one-way vs two-way ANOVA.

A

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

35
Q

Hypotheses in one-way repeated measures ANOVA

A

H0 : μ1 =μ2···=μk
H0 : Not all means are equal.

36
Q

What are all the SS’s in a One-Way repeated measures ANOVA? Explain how it is partitioned.

A

SS(A), SS(S), SS(AxS)

-> SS(W) = SS(A) + SS(S)

37
Q

Explain what is SS(W) in a One-Way repeated measures ANOVA

A

Within-participant variation.

38
Q

Explain what is SS(A) in a One-Way repeated measures ANOVA.

A

Variation between group means (IV). (columns).

39
Q

Explain what is SS(S) in a One-Way repeated measures ANOVA.

A

Variation between row means (subject means). Not interesting (simply tells us that participants are different). Between-participant effect.

40
Q

Explain what is SS(AxS) in a One-Way repeated measures ANOVA.

A

Variation between cell means (all cell means in the table). Also called SS error.

41
Q

Assumptions in one-way repeated measures ANOVA

A

Normality within each level of the factor, Homogeneity of variance, Homogeneity of covariance

42
Q

What is compound symmetry in a one-way repeated measures ANOVA?

A

Homogeneity of variance & Homogeneity of covariance

43
Q

Homogeneity of covariance

A

The population covariance between any pair of repeated measurements is equal (homogenous covariance).

44
Q

Sphericity

A

The variance of differences of a pair of observations is the same across all pairs.

45
Q

Compound symmetry can be replaced by the assumption of _______.

A

Sphericity.

46
Q

What is the test that checks the assumption of sphericity?

A

Mauchly’s test

47
Q

H0 in Mauchly’s test

A

H0: Variances of differences between conditions are equal

48
Q

In Mauchly’s test, if sphericity holds, epsilon ____. If violated, epsilon = _____.

A

Holds: ε = 1.
Violated: ε < 1.

49
Q

What happens to the df’s when sphericity is violated?

A

Reduces dfA and dfSxA and thus gives a larger critical value for F.

50
Q

What are the 2 steps to deal with violation of sphericity?

A

(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)

51
Q

In One-way repeated measures ANOVA, we use a ___ omega squared.

A

Partial.

52
Q

What is the formula of F in one-way repeated measures ANOVA?

A

F = MS(A) / MS(SxA)

53
Q

Partial Omega squared (ω2) in a one-way repeated measures ANOVA excludes ______.

A

The variability due to differences between subjects: MS(S).

54
Q

Three possible ways to calculate ε (correction)

A

Greenhouse-Geisser approach, the Huynh-Feldt approach, and the Lower bound ε can attain which is ε = 1/(a − 1)

55
Q

(sphericity) Which test for ε is the most conservative?

A

Greenhouse-Geisser

56
Q

How the Greenhouse-Geisser & Huynh-Feldt approach help in the correction of sphericity?

A

Reduce df + conservative critical value

57
Q

What is ε (epsilon) & its formula ?

A

Extent to which sphericity was violated.
ε = 1 / (a - 1)

58
Q

What are the sources of variation in a two-factor mixed design? (5)

A

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

59
Q

What is S/A in a two-factor mixed design?

A

Subject variation at levels of the between-subjects factor

60
Q

What is BxS/A in a two-factor mixed design?

A

Interaction between the within-subjects factor and subjects nested in levels of the between-subjects factor

61
Q

(mixed anova) What is the SSresidual of MS(A), MS(B) and MS(AxB)?

A

MS(A) = S/A
MS(B) and MS(AxB) = BxS/A

62
Q

(two-factor mixed design) When the main effect of factor A is significant, what does it mean?

A

Not all group means are equal.

63
Q

Assumptions in a two-factor mixed design

A

Between subject:
- Normal distribution
- Homo of variance (at each combination of A and B)
- Independence of observations

Within-subject:
- Sphericity

64
Q

In a two-factor mixed ANOVA what are the consequences of the violation of sphericity?

A

No consequences for the between-subject factor.
Type 1 error rate increases for the within-subject effects.

65
Q

In a two-factor mixed ANOVA what are the consequences of the violation of homo of variance?

A

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).

66
Q

R2 is ______ η2

A

the same as

Statistics (3 decks)

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