GEOG 371 Study Guide - Final Guide: Chi-Squared Test, Soil Type, Contingency Table
5 May 2018
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Department
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Professor
GEOG 371 Final Exam Review
Week 12
4/9 – Spatial Association
Spatial Association
• Relationship between two or more spatial variables
• Map comparison and overlay
• Is variable X associated with variable Y?
• How can we quantitatively analyze the relationships b/w maps?
Maps as Outcomes of Spatial Stochastic Processes
• How would the maps look if there was no association between them?
• Presence of X at a location is not associated with presence of Y
• Null hypothesis = presence of x at location not associated w/ y presence
Analyzing Spatial Association Between Two Variables
Variable 1
Variable
2
Nominal
Ordinal
Interval
Nominal
Chi Square
Mann-
Whitney, etc.
T-test
ANOVA
Ordinal
Rank
correlation
Rank
correlation
Interval
Correlation
Chi Square Test
• Two sample – analyze association between two nominal variables based on counts/frequencies
• Compare two maps, each containing nominal data
• Based on contingency tables
• Ex – x: soil type, y: dominant flower species
o X: climate type, y: terrain type
o X: industry type, y: voting outcome
• Ex – is crop type associated with soil type?
o Data: Counts of grid points with crop type i and soil type j
o Oij = # of grid points with crop type i and soil type j
o Null Hypothesis (Ho): Soil type is not related to crop type\
▪ No relationship or no difference
o Alternative Hypothesis (Ha): Soil type is related to crop type
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GEOG 371 Final Exam Review
▪ Is the actual relationship
o Data in contingency table: Observed frequencies (Oij)
▪ Oij = frequency of values in category i, j
Soil Type A
Soil Type B
Soil Type C
Total
Corn
15
30
25
70
Soybeans
10
15
5
30
Total
25
45
30
100
o What would the contingency table look like if Crop type and Soil type were statistically
independent (Ho)?
▪ Multiplication rule for independent events:
)()()( jiji SoilPCropPSoilCropP
▪ Expected frequency under the Ho:
▪
n
CR
Eji
ij
• Ri = Row i Total
• Cj = Column j Total
• n = number of observations (table total)
▪ Expected contingency table under Ho:
Soil
Type A
Soil
Type B
Soil
Type C
Total
Corn
17.5
31.5
21
70
Soybeans
7.5
13.5
9
30
Total
25
45
30
100
• P (Corn ∩ A) = P (Corn) x P(A) = 0.7 x 0.25 = 0.125
o P (Corn) = 70/100 = 0.7
o P (Soil A) = 25/100 = 0.25
o 0.175 x N = 0.175 x 100 = 17.5
•
n
CR
Eji
ij
o Ecorn (A) = (70x25)/100 = 17.5
o Ecorn (B) = (70x45)/100 = 31.5
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GEOG 371 Final Exam Review
o Ecorn (C) = (70x30)/100 = 21
• Chi Square Test Statistic:
o
r
i
c
jij
ijij
E
EO
1 1
2
2)(
▪ O = observed
▪ E = expected
▪ r = # rows
▪ c = # columns
▪ Degrees of Freedom (df) = (r-1)x(c-1)
o Cell calculation – go thru each cell and calculate expected and observed
▪
9
)59(
5.13
)5.1315(
5.7
)5.710(
21
)2125(
5.31
)5.3130(
5.17
)5.1715(
222
222
2
▪ X2 = 3.96
▪ Df = 2
▪ P value (online calculator or excel (CHIDIST)) = 0.14
• P > 0.05 → do NOT reject Ho
• Crop type and soil are NOT related
• Assumptions of chi square test
o Epeted feueies ≥5
o Nominal data
o Although degrees of freedom only depend on number of categories, the Chi square test
statistic value increases with the number of observations in each category (sample size).
Its easie to ejet Ho when you have a large sample size
4/9 – Correlation Analysis
Correlation Analysis
• Association between two interval scale variables
o Ex – crime rate and poverty
• Enables us to quantify special association b/w 2 variables
• 1st step – display data in scatterplot
Coelatio Coeffiiet Peasos
• R =
(-)
(+)
(+)
(-)
• Covariance tells you directionality
o If = 0 → no correlation
yx
xy
SS
S
yx
xy
StdDevStdDev
Covariance
*
=
=
yx
n
i
ii
SS
nYYXX
1
}/))({(
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Document Summary
Is the actual relationship: data in contingency table: observed frequencies (oij, oij = frequency of values in category i, j. Soil type a soil type b soil type c total. 100: what would the contingency table look like if crop type and soil type were statistically independent (ho), multiplication rule for independent events, expected frequency under the ho: Cr i n j: ri = row i total, cj = column j total, n = number of observations (table total, expected contingency table under ho: 100: p (corn a) = p (corn) x p(a) = 0. 7 x 0. 25 = 0. 125, p (corn) = 70/100 = 0. 7, p (soil a) = 25/100 = 0. 25, 0. 175 x n = 0. 175 x 100 = 17. 5. Cr i n j: ecorn (a) = (70x25)/100 = 17. 5, ecorn (b) = (70x45)/100 = 31. 5. Geog 371 final exam review: ecorn (c) = (70x30)/100 = 21, chi square test statistic:
