Week 2 - ANOVA I Flashcards
What is the difference between treatment variance and error variance ? (x2)
Treatment is systematic variance due to our IV levels
Error is unsystematic - random, or due to unmeasured influence
What are the sources of variance in a one-way ANOVA? (x2, and x2)
Between groups variance: systematic variance due to membership in different groups/treatments
o Distribution of group means around the grand mean
Within-groups variance: error variance (random chance, unmeasured influences)
o Distribution of individual DV scores around the group mean
How are the sources of variance in a 1-way ANOVA related to error variance and treatment variance ? (x2)
Between-groups is the systematic/IV variance
Within-groups is the error/unmeasured influence variance
In a 1-way ANOVA, what is SStreatment? (equation x1, words x2)
n∑(X bar j – X bar dot)2
People per group x sum of squared differences between group means and grand mean
Estimate of between groups variability
In a 1-way ANOVA, what is SSerror? (equation x1, words x 2)
∑(X ij – X bar j)2
Sum of squared differences between individual scores and group mean
Estimate of within groups variability
In a 1-way ANOVA, what is MStreatment? (words x1, equation x 2)
Index of variability among treatment means
SStreat/dftreat or SSj/dfj
In a 1-way ANOVA, what is MSerror? (words x4, equation x1)
Index of variability among participants within a cell,
i.e. pooled within-cell variance
Average of s2 from each sample,
*Good estimate of σe2 (population error variance)
SSerror/dfError
How do you calculate the F-ratio in 1-way ANOVA? (equation x1)
F = MStreat/MSerror
Describe the structural (linear) model of 1-way ANOVA (x6)
Xij = μ. + τj + eij
o For i cases and j treatments
Xij, any DV/individual’s score at a given level of j is a combination of:
o μ. - the grand mean
o τj - the effect of the j-th treatment (μj - μ.)
o eij - error for i person in j-th treatment
How is variance partitioned in a 2-way ANOVA? (x6)
Total = Between-groups (treatment) variance, which = (Variance due to Factor A, plus Variance due to Factor B, plus Variance due to A x B), plus Within-groups (error) variance
For a 2-way ANOVA, explain the SS for each main effect (x4)
Sum the squares of
(Marginal means for each level of the factor minus the
Grand mean), and times the lot by
[n x levels of other factor (ie, the # of observations behind each factor marginal mean)]
For a 2-way ANOVA, explain the SS for the interaction (x4, or x2)
Sum the squares of
Grand mean minus each cell mean,
(Adjusting for the 2 marginal means),
Times the lot by n (# of observations behind each cell mean)
Simple: Between-groups variance minus Factor A variance minus Factor B variance
*SStreat - SSa - SSb
For a 2-way ANOVA, explain the SSerror (x4)
Same as in 1-way -
∑(X ij – X bar j)2
Sum of squared differences between individual scores and group mean
Estimate of within groups variability
For 2-way ANOVA, how do you calculate df for: total effects? (x2)
df total = N - 1
Where N is the total number of observations
For 2-way ANOVA, how do you calculate df for: effect of each factor? (x3)
df factor = # of levels of the factor – 1
o dfB = b – 1
o dfA = a – 1
For 2-way ANOVA, how do you calculate df for: effect of the interaction? (x2)
df interaction = product of df for factors in the interaction
o dfBA = (b – 1) x (a –1)
For 2-way ANOVA, how do you calculate df for: error? (x4)
dferror = total # of observations – # of treatments
o = N – ba
• = df for each cell x # of cells
• = (n – 1) ba
What does the null hypothesis represent in tests of main effects in 2-way ANOVA? (equation x1, words x1)
H0: μ1 = μ2 = μ3
No differences among means across levels of the factor
What does the null hypothesis represent in tests of the interaction in 2-way ANOVA? (equation x2, words x3)
H0: μ11 - μ21 = μ12 - μ22 = μ13 - μ23
(where μ11 = mean of the following group/1st level of Factor A and 1st level of Factor B)
H0: μ1k - μ.. = μ2k - μ..
So it’s comparing simple effects,
And saying that all those cell-mean differences are the same
What are the three omnibus test in a 2-way design?
Which we test by…? (x2)
Main effect of A
Main effect of B
Interaction AB
Calculating F = MStreat / MSerror for each
(for interactions, MStreat is the effect of factor AB)
Describe the structural model of 2-way factorial ANOVA (x8)
Xijk = μ. + αj + βk + αβjk + eijk
For i cases, factor A with j treatments, factor B with k treatments, and the AxB interaction with jk treatments:
Xijk, any DV/individual’s score is combo of:
μ. - the grand mean
α j - alpha-j, the effect of the j-th treatment of factor A (μAj - μ.)
βk - beta-k, the effect of the k-th treatment of factor B (μBk - μ.)
αβjk - the effect of the interaction/differences in factor A treatments at different levels of factor B treatments (μ. - μAj - μBk + μjk)
eijk - error for i person in j-th and k-th treatments
What are the main assumptions of ANOVA? (x7)
That treatment populations are normally distributed, and
Have the same variance (homogenous)
That samples are independent,
Randomly selected (no systematic allocation),
Have equal n, and
At least 2 observations
That DV data is measured on a continuous scale
What are the numerator/denominator MS terms for each F-ratio in the omnibus tests in 2-way ANOVA? (x3, x2)
Factor A or B main effect:
MStreat for effect the factor, over MSerror (the remainder once factor and interaction SS subtracted from total)
For interaction:
MS of effect of the interaction, over MSerror
In a 2-way factorial ANOVA, what does a significant F test for Factor A tell us? (X1)
That there is a meaningful difference in DV scores for the factor across its different levels
In a 2-way factorial ANOVA, was does a significant F test of the AxB interaction tell us? (X1)
That there is a meaningful difference among the simple effects -
The (cell mean) differences are different
Conceptually, what is an ANOVA test doing? (x2)
Looking for an effect as the difference among means
Which means used is dependent on which effect we’re looking at (main/interaction)
What does mu-j represent in ANOVA? (x2)
Population mean of group j
The mean of any particular level of j
What does mu. represent in ANOVA? (x1)
Grand mean
What is the null hypothesis in a 1-way ANOVA? (equation and words)
H0: μj = μ.
No differences between any treatment means
What is the alternative hypothesis in a 1-way ANOVA? (equation and words)
H1: μj ≠ μ.
There is a difference between treatment means
What are three general indicators for interpreting F-values? (x2, x2, x1)
If MStreat is a good estimate of error variance,
o F = MStreat/MSerror = 1
If MStreat is more than just error variance,
o F = MStreat/MSerror > 1
Larger values of F indicate that H0 is probably wrong
What are the 4 conceptual steps to calculating ANOVA?
Estimate of between-groups variability
Estimate of within-groups variability
Weight each variability estimate by # of observations used to generate the estimate (“degrees of freedom”)
Compare ratio: between over within
What is meant by ‘expected mean squares error’? (ANOVA) (x3)
Expected value of a statistic is defined as the ‘long-range average’ of a sampling statistic
So:
E(MSerror) - σe2
• i.e., long term average of variances within each sample (s2) would be the population variance σe2
What is meant by ‘expected mean squares treatment’ (ANOVA)? (x5)
Expected value of a statistic is defined as the ‘long-range average’ of a sampling statistic
So:
(Sigma tells us we’re talking about population variance, tau = treatment effect variance)
E(MStreat) - σe2 + n (στ)2
• Where (στ)2 is long term average of variance between sample means, and n is number of observations in each group
• i.e., long term average of variances within each sample PLUS any variance between each sample
If group means don’t vary then what would our expected mean squares be (ANOVA) (x3)
(n σ τ)2 = 0, and so then
E(MStreat) = σe2 + 0 = σe2
= E(MSerror) = σe2
How can you double-check that you have calculated the df correctly in a 2-way ANOVA? (x2)
Total equals between plus within
Between equals total of A, B and AxB
What is the alternative hypothesis for testing interactions in 2-way ANOVA? (equation x1, words x1)
H1: μ1k - μ.. ≠ μ2k - μ..
That at least one of the simple effects of B within A1 is different from that same comparison for A2
What is meant by an ‘additive effect’ in factorial ANOVA outcomes? (x1)
As distinct from? (x1)
To take one group, and add a constant to get the pattern for the other group
Interactive effect - non-parallel lines
