Anthony Greene1 Correlation The Association Between Variables.
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Transcript of Anthony Greene1 Correlation The Association Between Variables.
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Anthony Greene 1
Correlation
The Association Between Variables
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Anthony Greene 2
When to Use
t-test or ANOVA
When the independent variable is Categorical
0
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Control Treat A0
102030405060708090
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Control Treat A Treat B Treat C
MaleFemale
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Anthony Greene 3
When to UseCorrelation
& Regression
When the independent variable is Ratio or Interval
Tour De France Gatorade Consumption
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60 70 80 90 100 110 120Average Daytime Temperature
Lit
ers
of G
ator
ade
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Anthony Greene 4
ScatterPlots
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Anthony Greene 5
ScatterPlots
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Times and costs for five word-processing jobs
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Four data points
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Anthony Greene 8
Direct Relationship: Positive Slope
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Anthony Greene 9
Age and price data for a sample of 11 used cars
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Scatter diagram for the age and price data of used cars
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Anthony Greene 11
Inverse Relationship: Negative Slope
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Anthony Greene 12
Various degrees of linear correlation (Slide 1 of 3)
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Anthony Greene 13
Various degrees of linear correlation (Slide 3 of 3)
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Anthony Greene 14
Various degrees of linear correlation (Slide 2 of 3)
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Anthony Greene 15
Examples of positive and negative relationships
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The Simple Idea
• If the corresponding x and y z-scores are always in agreement, r will be high.
• If they are sometimes in agreement r will be moderate
• If they are generally different, r will be near zero• If they are in agreement, but in opposite
directions, r will be negative
n
zzr yx
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Anthony Greene 17
The Basic Idea n
zzr yx
Zx Zy Zx Zy Zx Zy
+1.2 +1.6 +1.2 -2.3 +1.2 -1.6
-1.1 -0.7 -1.1 -0.2 -1.1 +0.7
+0.8 +1.1 +0.8 +1.6 +0.8 -1.1
+3.2 +2.8 +3.2 -0.7 +3.2 -2.8
-2.7 -2.3 -2.7 +1.1 -2.7 +2.3
+0.1 -0.2 +0.1 +2.8 +0.1 +0.2
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Anthony Greene 18
We define SSx, SSp and SSy by
Notation Used in Regression and Correlation
nyySS
nyxxyS
nxxSS
MySS
MyMxS
MxSS
y
p
x
yy
yxp
xx
22
22
2
2
formula nalcomputatio Or the
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Anthony Greene 19
Obtaining the three sums of squares for the used car data using the computational formulas
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Anthony Greene 20
The linear correlation coefficient, r, of n data points is defined by
or by the computational formula
Linear Correlation Coefficient
yx
p
SSSS
Sr
n
zzr yx
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Anthony Greene 21
Linear Correlation Coefficient
n
zzr yx
22yx
yx
MyMx
MyMxr
n
My
n
Mxn
MyMxr
yx
yx
22
yx
p
SSSS
Sr
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Anthony Greene 22
Coefficient of Determination
The coefficient of determination, r2, is the proportion of variation in the observed values of the response variable that is explained by the regression:
The coefficient of the determination always lies between 0 and 1 and is a descriptive measure of of the utility of the regression equation for making predictions. Values of r2 near 0 indicate that the regression equation is not useful for making predictions, whereas values near 1 indicate that the regression equation is extremely useful for making predictions.
varianceexplained of proportion variancetotal
varianceexplained2 r
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Anthony Greene 23
t-Distribution for a Correlation Test
t r
1 r2
n 2
For samples of size n, the variable
has the t-distribution with df = n – 2 if the null hypothesis ρ = 0 ρ or rho is pronounced “row”
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Anthony Greene 24
The t-test for correlation (Slide 1 of 3)
With df = n-2 use table B.6
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Anthony Greene 25
The t-test for correlation (Slide 2 of 3)
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Anthony Greene 26
The t-test for correlation (Slide 3 of 3) Step 4 Compute the test statistic r. Table B.6 allows a
direct lookup. Alternatively, r has a t distribution and Table B.2 will yield an identical conclusion
Step 5 If the value of the test statistic falls in the rejection region, reject the null hypothesis.
Step 6 State the conclusion in words
21 2
nr
rt
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Anthony Greene 27
Criterion for deciding whether or not to reject the null hypothesis
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Anthony Greene 28
Correlation Matrix
a b c d
a 1.00
b 0.84 1.00
c 0.68 0.58 1.00
d 0.12 0.19 0.08 1.00
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Anthony Greene 29
Computer printouts for correlations