• 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

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R2=0.79

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R2=0.79

R2=0.10

Theta-TVDI ScatterplotsGlyndon Field Sampled Soil Moisture

versus TVDI from a 3x3 kernel

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rePond Branch Field Sampled Soil Moisture

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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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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

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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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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