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Transcript of Where we are Where we are going
Where we are Where we are going
Statistical analyses by number of sample groups
1 - Single sample Z or t tests 2 – T tests (dependent, independent)
3 or more ?Analysis of variance
Between subjects (independent)Equal N (Easy)Unequal N (Harder – structural)
Within subjects (dependent – structural)Two-way (independent - structural)
Analysis of VarianceEqual N (Easy)
Used with 3 or more groupsExtends logic of independent groups t-testSome additional things to think about
ASSUMPTIONS of ANOVA:
equal variances (required for pooling)normality (required for test distribution)
The null hypothesis in ANOVA is always:
This implies that any combination of means are also equal
1 2 3 j
1 2 3( ) / 2
The alternative hypothesis in ANOVA is always- The population means are different (at least one
mean is different from another)
**** Null hypothesis is tested by comparing two estimates of the population variance (σ2 ):
(MSB) between-group estimate of (σ2 ) AFFECTED by whether the null is true
(MSW) within-group estimate of (σ2 )
UNAFFECTED by whether the null is true
ANOVA: Null Hypothesis Is TRUE
Score Distributions
Mean Distributions, N=9
Sample 3 ScoresSample 2 ScoresSample 1 Scores
Sample 3 MeanSample 2 MeanSample 1 Mean
ANOVA: Null Hypothesis Is False
Score Distributions
Mean Distributions (N=9)
-- when the null hypothesis is true (MSB)
F ratio = ------- = About 1 (MSw)
-- when the null hypothesis is not true (MSB)
F ratio = ------- = Much greater than 1 (MSw)
Let: K = # of groups (treatments)J = a particular group 1…KNJ = # of people in group J_Xj = Sample mean for group JNG = Total (grand) number of people_XG = The grand mean
I = Individual = Variance of the means
/ij Gx n
S2_X
Is an estimate of Variance of the meansS2_X
σ2_X
= σ2_X
σ2
--- Because the variance of the means is the n variance of the variable divided by sample size
S2_X (n) estimates the population variance (σ2 )
S2_X (n) = MSB IF groups are same size (n1=n2=n3…)
Between Groups variance
S2_X (n) estimates σ2 just as well as any random samples would
S2_X (n) = MSB will be higher than the populations variance because the means are farther away from each other than would be expected by chance
If Null is true, then these are just random samples
If Null is false, then these are not just random samples
Between Groups variance
S1
2 + S22 + S3
2 +… Sk2
= ----------------------- k
If Null is true, then these are just random samplesSo we can poll the variances just as we did with the independent samples t-test
Within Groups variance
S2pooled
S2pooledMSW =
MSB
F = ---------
MSW
• F Ratio
• Critical values of f depend on dfW and dfB
• Look up in table
Post-hoc tests
ANOVA tells us there is some difference, but it does not tell us which groups are different from each other
ANOVA is like a shotgun – firing many pellets at many different hypothesis like
u1=u2
u2=u3
(u1+u2)/2=u3
Post-hoc tests - Tukey
Tests all the pairwise comparisons – does not test complex hypotheses (such as (u1+u2)/2=u3)
u1=u2
u1=u3
u1=u4
u2=u3
u2=u4
u3=u4
Apriori tests
Also referred to as
“planned comparisons”
“planned contrast”
A rifle instead of a shotgun. Used to test a specific hypothesis that is a subset of all hypotheses. For example, with 3 groups – if you wanted to test if group 3 was different from the other two groups, then you would test the following:
(u1+u2)/2=u3
But WHAT IF the sample sizes are not the same?
Structural Model of ANOVA An alternative way of understanding ANOVA - used whenever Nj is not equal across groups All the basic logic stays the same, computationally however
GX = mean of all the scores For each score, deviation from grand mean is divided into two parts a) deviation of score from mean of its group b) deviation of group mean from grand mean
.
2( )ij GX X =
2( )j GX X + 2( )ij jX X
SST = SSB + SSW
DFB = 1K DFW = gN K
2 ( )
1j G
B B
X XS MS
K
2 ( )ij jW W
G
X XS MS
N K
F = MSB/MSW DFB = 1K DFW = gN K
22
STANDARD WAY OF SETTING UP ANOVA EFFECT SS DF MS F
BETWEEN 2( )j GX X 1K /B BSS DF /B WMS MS
WITHIN 2( )ij jX X gN K /W WSS DF
TOTAL 2( )ij GX X 1gN
2( )ij GX X =
2( )j GX X + 2( )ij jX X
• F Ratio
• Critical values of f depend on dfW and dfB
• Look up in table
Anova Effect size
dM = --------------
How far apart the means are divided by the standard deviation -
Similar to the effect size for independent t-test (mean difference/stdev)
Small Medium Large
.2 .5 .8
_ _Xmax - Xmin
Spooled =