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Transcript of Relationship between two variables e.g, as education , what does income do? Scatterplot Bivariate...
• Relationship between two variables
• e.g, as education , what does income do?
• Scatterplot
Bivariate Methods
Correlation
Linear Correlation
Source: Earickson, RJ, and Harlin, JM. 1994. Geographic Measurement and Quantitative Analysis. USA: Macmillan College Publishing Co., p. 209.
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TMIT
het
a
Wet – May 29/30 Avg. – June 26/28 Dry – August 22
Pond B
ranch - P
G 11.25m
DE
MG
lyndon – LID
AR
0.5m
DE
M 11x11
R2=0.71
R2=0.29
R2=0.79
R2=0.24
R2=0.79
R2=0.10
Theta-TVDI ScatterplotsGlyndon Field Sampled Soil Moisture
versus TVDI from a 3x3 kernel
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TVDI (3x3 kernel)
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rePond Branch Field Sampled Soil Moisture
versus TVDI from a 3x3 kernel
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TVDI (3x3 kernel)
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Oxford Tobacco Research Station Field Sampled Soil Moisture versus TVDI from a 3x3 kernel
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TVDI (3x3 kernel)
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API-TVDI Scatterplot
Covariance: Interpreting Scatterplots
• General sense (direction and strength)
• Subjective judgment
• More objective approach
• Extent to which variables Y and X vary together
• Covariance
Covariance Formulae
Cov [X, Y] = (xi - x)(yi - y)i=1
i=n1
n - 1
Covariance Example
Glyndon Field Sampled Soil Moisture versus TVDI from a 3x3 kernel
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TVDI (3x3 kernel)
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TVDISoil
Moisture
0.274 0.4140.542 0.3590.419 0.3960.286 0.4580.374 0.3500.489 0.3570.623 0.2550.506 0.1890.768 0.1710.725 0.119
Covariance Example
TVDI (x)Soil
Moisture (y)
(x - xbar) (y - ybar)(x - xbar) * (y - ybar)
0.274 0.414 -0.227 0.107 -0.0243050.542 0.359 0.042 0.052 0.00216240.419 0.396 -0.082 0.090 -0.0073230.286 0.458 -0.215 0.151 -0.0324240.374 0.350 -0.127 0.044 -0.0055530.489 0.357 -0.011 0.050 -0.0005660.623 0.255 0.122 -0.052 -0.0063740.506 0.189 0.005 -0.118 -0.0006180.768 0.171 0.267 -0.136 -0.0362820.725 0.119 0.225 -0.188 -0.042289
Mean 0.501 0.307 -0.15357-0.017063
SumCovariance
1
2 3
45
How Does Covariance Work?
• X and Y are positively related
• xi > x yi > y
• xi < x yi < y
• X and Y are negatively related
• xi > x yi < y
• xi < x yi > y
__ __
__ __
__ __
__ __
Interpreting Covariances
• Direction & magnitude
• Cov[X,Y] > 0 positive
• Cov[X, Y] < 0 negative
• abs(Cov[X, Y]) ↑ strength ↑
• Magnitude ~ units
Covariance Correlation
• Magnitude ~ units
• Multiple pairs of variables not comparable
• Standardized covariance
• Compare one such measure to another
Pearson’s product-moment correlation coefficient
Cov [X, Y]
sXsY
r =
r (xi - x)(yi - y)i=1
i=n
(n - 1) sXsY
=
ZxZyr i=1
i=n
(n - 1)=
Pearson’s Correlation Coefficient
• r [–1, +1]
• abs(r) ↑ strength ↑
• r cannot be interpreted proportionally
• ranges for interpreting r values 0 - 0.2 Negligible
0.2 - 0.4 Weak
0.4 - 0.6 Moderate
0.6 - 0.8 Strong
0.8 - 1.0 Very strong
Example
• X = TVDI, Y = Soil Moisture
• Cov[X, Y] = -0.017063
• SX = 0.170, SY = 0.115
• r ?
Glyndon Field Sampled Soil Moisture versus TVDI from a 3x3 kernel
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TVDI (3x3 kernel)
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Pearson’s r - Assumptions
1. interval or ratio
2. Selected randomly
3. Linear
4. Joint bivariate normal distribution
Interpreting Correlation Coefficients
• Correlation is not the same as causation!
• Correlation suggests an association between
variables
1. Both X and Y are influenced by Z
Interpreting Correlation Coefficients
2. Causative chain (i.e. X A B Y)
e.g. rainfall soil moisture ground water runoff
3. Mutual relationship
e.g., income & social status
4. Spurious relationship
e.g., Temperature (different units)
5. A true causal relationship (X Y)
Interpreting Correlation Coefficients
6. A result of chance
e.g., your annual income vs. annual population of the world
Interpreting Correlation Coefficients
7. Outliers
(Source: Fang et al., 2001, Science, p1723a)
Interpreting Correlation Coefficients
• Lack of independence
– Social data
– Geographic data
– Spatial autocorrelation
A Significance Test for r
• An estimator
r
= 0 ?
• t-test
A Significance Test for r
ttest = r
SEr
=r
1 - r2
n - 2
=r n - 2
1 - r2
df = n - 2
A Significance Test for r
H0: = 0
HA: 0
ttest = r n - 2
1 - r2