Download Many Possible Explanations Exist

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 Variance
Equal N (Easy)
Used with 3 or more groups
Extends logic of independent groups t-test
Some additional things to think about
The alternative hypothesis in ANOVA is always
- The population means are different (at least one
mean is different from another)
ASSUMPTIONS of ANOVA:
equal variances (required for pooling)
normality (required for test distribution)
The null hypothesis in ANOVA is always:
1  2  3   j
This implies that any combination of means are also equal
(1  2 ) / 2  3
**** 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
Sample 3 Scores
Sample 2 Scores
Sample 1 Scores
Mean Distributions, N=9
Sample 3 Mean
Sample 2 Mean
Sample 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
J
NJ
_
Xj
NG
_
XG
I
S2_
X
= # of groups (treatments)
= a particular group 1…K
= # of people in group J
= Sample mean for group J
= Total (grand) number of people
= The grand mean
x
= Individual
= Variance of the means
ij
/ nG
Between Groups variance
S2_
Is an estimate of
X
σ2_
X
σ2_
X
Variance of the means
σ2
= --- Because the variance of the means is the
n variance of the variable divided by sample
size
S2_ (n) estimates the population variance (σ2 )
X
S2_ (n) = MSB
X
IF groups are same size (n1=n2=n3…)
Between Groups variance
If Null is true, then these are just random samples
S2_ (n) estimates σ2 just as well as any random samples would
X
If Null is false, then these are not just random samples
S2_ (n) = MSB will be higher than the populations
X
variance because the means are farther away
from each other than would be expected by
chance
Within Groups variance
If Null is true, then these are just random samples
So we can poll the variances just as we did with the
independent samples t-test
2 + S 2 + S 2 +… S 2
S
1
2
3
k
2
S pooled = ----------------------k
MSW = S2pooled
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
XG
= 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
(
X

X
)
 ij G
SST
=
2
(
X

X
)
 j G
=
SSB
DFB =
.
S  MS B 
2
B
F = MSB/MSW
 (X j  XG )
K 1
DFB =
+
2
(
X

X
)
 ij j
+
K 1
DFW =
SSW
Ng  K
2
SW2  MSW 
K 1
2
(
X

X
)
 ij j
NG  K
DFW = N g  K
STANDARD WAY OF SETTING UP ANOVA
EFFECT
SS
DF
MS
BETWEEN
2
(
X

X
)
 j G
K 1
WITHIN
2
(
X

X
)
 ij j
N g  K SSW / DFW
TOTAL
2
(
X

X
)
 ij G
N g 1
2
(
X

X
)
 ij G
=
2
(
X

X
)
 j G
F
SS B / DFB MS B / MSW
+
2
(
X

X
)
 ij j
• F Ratio
• Critical values of f depend on dfW and dfB
• Look up in table
Anova Effect size
_
_
Xmax - Xmin
dM = -------------Spooled =
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