Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X...
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![Page 1: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/1.jpg)
Review: The Logic Underlying ANOVA
• The possible pair-wise comparisons:
X11
X12
.
.
.X1n
X21
X22
.
.
.X2n
Sample 1 Sample 2
€
X 1
€
X 2means:
X31
X32
.
.
.X3n
Sample 3
€
X 3
![Page 2: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/2.jpg)
Review: The Logic Underlying ANOVA
• There are k samples with which to estimate population variance
X11
X12
.
.
.X1n
X21
X22
.
.
.X2n
Sample 1 Sample 2
€
X 1
€
X 2
X31
X32
.
.
.X3n
Sample 3
€
X 3€
ˆ σ 12 =
(X i − X 1)2∑
n −1
![Page 3: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/3.jpg)
Review: The Logic Underlying ANOVA
• There are k samples with which to estimate population variance
X11
X12
.
.
.X1n
X21
X22
.
.
.X2n
Sample 1 Sample 2
€
X 1
€
X 2
X31
X32
.
.
.X3n
Sample 3
€
X 3€
ˆ σ 22 =
(X i − X 2)2∑n −1
![Page 4: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/4.jpg)
Review: The Logic Underlying ANOVA
• There are k samples with which to estimate population variance
X11
X12
.
.
.X1n
X21
X22
.
.
.X2n
Sample 1 Sample 2
€
X 1
€
X 2
X31
X32
.
.
.X3n
Sample 3
€
X 3€
ˆ σ 32 =
(X i − X 3)2∑n −1
![Page 5: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/5.jpg)
Review: The Logic Underlying ANOVA
• The average of these variance estimates is called the “Mean Square Error” or “Mean Square Within”
€
MSerror =
ˆ σ j2
j=1
k
∑
k
![Page 6: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/6.jpg)
Review: The Logic Underlying ANOVA
• There are k means with which to estimate the population variance
X11
X12
.
.
.X1n
X21
X22
.
.
.X2n
Sample 1 Sample 2
€
X 1
€
X 2
X31
X32
.
.
.X3n
Sample 3
€
X 3€
ˆ σ 2 = n ˆ σ X 2 = n
(X j − X overall )2∑
k −1
![Page 7: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/7.jpg)
Review: The Logic Underlying ANOVA
• This estimate of population variance based on sample means is called Mean Square Effect or Mean Square Between
€
ˆ σ 2 = n ˆ σ X 2 = n
(X j − X overall )2∑
k −1
![Page 8: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/8.jpg)
The F Statistic
• MSerror is based on deviation scores within each sample but…
• MSeffect is based on deviations between samples
• MSeffect would overestimate the population variance when there is some effect of the treatment pushing the means of the different samples apart
![Page 9: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/9.jpg)
The F Statistic
• We compare MSeffect against MSerror by constructing a statistic called F
![Page 10: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/10.jpg)
The F Statistic
• F is the ratio of MSeffect to MSerror
€
Fk−1,k(n−1) =MSeffect
MSerror
![Page 11: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/11.jpg)
The F Statistic
• If the hull hypothesis:
is true then we would expect:
except for random sampling variation
€
μ1 = μ2 = μ3 = μ
€
X 1 = X 2 = X 3 = μ
![Page 12: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/12.jpg)
The F Statistic
• F is the ratio of MSeffect to MSerror
• If the null hypothesis is true then F should equal 1.0
€
Fk−1,k(n−1) =MSeffect
MSerror
![Page 13: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/13.jpg)
ANOVA is scalable
• You can create a single F for any number of samples
![Page 14: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/14.jpg)
ANOVA is scalable
• You can create a single F for any number of samples
• It is also possible to examine more than one independent variable using a multi-way ANOVA– Factors are the categories of independent
variables– Levels are the variables within each factor
![Page 15: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/15.jpg)
ANOVA is scalableA two-way ANOVA:
4 levels of factor 1
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
3 le
vels
of
fact
or 2
1 2 3 4
1
2
3
![Page 16: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/16.jpg)
Main Effects and Interactions
• There are two types of findings with multi-way ANOVA: Main Effects and Interactions– For example a main effect of Factor 1 indicates that the
means under the various levels of Factor 1 were different (at least one was different)
![Page 17: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/17.jpg)
Main Effects and Interactions4 levels of factor 1
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
3 le
vels
of
fact
or 2
1 2 3 4
1
2
3
€
X 1
![Page 18: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/18.jpg)
Main Effects and Interactions4 levels of factor 1
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
3 le
vels
of
fact
or 2
1 2 3 4
1
2
3
€
X 2
![Page 19: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/19.jpg)
Main Effects and Interactions4 levels of factor 1
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
3 le
vels
of
fact
or 2
1 2 3 4
1
2
3
€
X 3
![Page 20: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/20.jpg)
Main Effects and Interactions4 levels of factor 1
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
3 le
vels
of
fact
or 2
1 2 3 4
1
2
3
€
X 4
![Page 21: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/21.jpg)
Main Effects and Interactions
A main effect of Factor 1
Factor 11 2 3 4
Levels of Factor 2
123
depe
nden
t var
iabl
e
means of each sample
![Page 22: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/22.jpg)
Main Effects and Interactions
• There are two types of findings with multi-way ANOVA: Main Effects and Interactions– For example a main effect of Factor 1 indicates that the means
under the various levels of Factor 1 were different (at least one was different)
– A main effect of Factor 2 indicates that the means under the various levels of Factor 2 were different
![Page 23: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/23.jpg)
Main Effects and Interactions4 levels of factor 1
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
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X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
3 le
vels
of
fact
or 2
1 2 3 4
1
2
3€
X 1
![Page 24: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/24.jpg)
Main Effects and Interactions4 levels of factor 1
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
3 le
vels
of
fact
or 2
1 2 3 4
1
2
3
€
X 2
![Page 25: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/25.jpg)
Main Effects and Interactions4 levels of factor 1
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
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Xn
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X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
X1
X2
Xn
3 le
vels
of
fact
or 2
1 2 3 4
1
2
3
€
X 3
![Page 26: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/26.jpg)
Main Effects and Interactions
A main effect of Factor 2
Factor 11 2 3 4
Levels of Factor 2
123
depe
nden
t var
iabl
e
![Page 27: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/27.jpg)
Main Effects and Interactions
• There are two types of findings with multi-way ANOVA: Main Effects and Interactions– For example a main effect of Factor 1 means that the means under
the various levels of Factor 1 were different (at least one was different)
– A main effect of Factor 2 means that the means under the various levels of Factor 2 were different
– An interaction means that there was an effect of one factor but the effect is different for different levels of the other factor
![Page 28: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/28.jpg)
Main Effects and Interactions
An Interaction
Factor 11 2 3 4
Levels of Factor 2
123
depe
nden
t var
iabl
e
![Page 29: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/29.jpg)
Correlation
• We often measure two or more different parameters of a single object
![Page 30: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/30.jpg)
Correlation
• This creates two or more sets of measurements
![Page 31: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/31.jpg)
Correlation
• These sets of measurements can be related to each other– Large values in one set correspond to
large values in the other set– Small values in one set correspond to
small values in the other set
![Page 32: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/32.jpg)
Correlation
• examples:– height and weight– smoking and lung cancer– SES and longevity
![Page 33: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/33.jpg)
Correlation
• We call the relationship between two sets of numbers the correlation
![Page 34: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/34.jpg)
Correlation
• Measure heights and weights of 6 people
Person Height Weight
a 5’4 120
b 5’10 140
c 5’2 100
d 5’1 110
e 5’6 140
f 5’8 150
![Page 35: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/35.jpg)
Correlation
• Height vs. Weight
5’ 5’2 5’4 5’6 5’8 5’10
100 110 120 130 140 150Weight
Height
![Page 36: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/36.jpg)
Correlation
• Height vs. Weight
5’ 5’2 5’4 5’6 5’8 5’10
100 110 120 130 140 150
a
a
Weight
Height
![Page 37: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/37.jpg)
Correlation
• Height vs. Weight
5’ 5’2 5’4 5’6 5’8 5’10
100 110 120 130 140 150
a
a
b
b
Weight
Height
![Page 38: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/38.jpg)
Correlation
• Height vs. Weight
5’ 5’2 5’4 5’6 5’8 5’10
100 110 120 130 140 150
a
a
b
b, e
c
c
d
d
e f
f
Weight
Height
![Page 39: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/39.jpg)
Correlation
• Notice that small values on one scale pair up with small values on the other
5’ 5’2 5’4 5’6 5’8 5’10
100 110 120 130 140 150
a
a
b
b, e
c
c
d
d
e f
f
Weight
Height
![Page 40: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/40.jpg)
Correlation
• Scatter Plot shows the relationship on a single graph
• Like two number lines perpendicular to each other
5’ 5’2 5’4 5’6 5’8 5’10
100 110 120 130 140 150
a
a
b
b, e
c
c
d
d
e f
f
Think of this as the y-axis
Think of this as the x-axis
![Page 41: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/41.jpg)
Correlation
• Scatter Plot shows the relationship on a single graph
5’ 5’2 5’4 5’6 5’8 5’10
a bcd e f
100
110
120
130
140
150
ab,
ec
df
Wei
ght
Height
*
*
*
*
*
*
![Page 42: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/42.jpg)
Correlation
• The relationship here is like a straight line
• We call this linear correlation
*
*
*
*
*
*
![Page 43: Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3.](https://reader030.fdocuments.in/reader030/viewer/2022033106/56649d615503460f94a4338b/html5/thumbnails/43.jpg)
Various Kinds of Linear Correlation
• Strong Positive
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Various Kinds of Linear Correlation
• Weak Positive
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Various Kinds of Linear Correlation
• Strong Negative
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Various Kinds of Linear Correlation
• No (or very weak) Correlation
• y values are random with respect to x values
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Various Kinds of Linear Correlation
• No Linear Correlation
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Correlation Enables Prediction
• Strong correlations mean that we can predict a y value given an x value…this is called regression
• Accuracy of our prediction depends on strength of the correlation
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Spurious Correlation
• Sometimes two measures (called variables) both correlate with some other unknown variable (sometimes called a lurking variable) and consequently correlate with each other
• This does not mean that they are causally related!
• e.g. use of cigarette lighters positively correlated with incidence of lung cancer
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Next Time: measuring correlations