Spatial Data AnalysisSpatial Data Analysis

Iowa County Land Values (1926)Iowa County Land Values (1926)

Data DescriptionData Description

County Level Data (Circa 1926, n=99)County Level Data (Circa 1926, n=99) Latitude/Longitude Co-Ordinates of Latitude/Longitude Co-Ordinates of

County SeatCounty Seat Land Values per Acre (Federal/State)Land Values per Acre (Federal/State) Corn Yield per AcreCorn Yield per Acre Percent CornPercent Corn Percent Other GrainsPercent Other Grains Percent Un-plowable LandPercent Un-plowable Land

Map of Federal Land ValuesMap of Federal Land Values

Summary StatisticsSummary Statistics

Statistic Corn Yield/Acre Percent Corn Percent Grain Percent Un-plow Federal Value State Value

q1 36.5 27.5 17 12 97 88.5

min 30 13 10 8 66 49

median 39 33 22 17 118 108

max 46 48 33 44 173 161

q3 42 38 26 23.5 140 124

Mean 39.11 32.47 21.56 18.85 118.68 106.25

Std Dev 3.40 7.09 5.98 7.83 26.73 24.38

Box Plots of Iowa County Data

0

20

40

60

80

100

120

140

160

180

200

CornY/A PctCorn PctGrain PctUnplow LandFed LandState

Variable

q1

min

median

max

q3

Land Value (Federal)

0

5

10

15

20

25

60 75 90 105 120 135 150 165 180

Freq

uenc

y

Weight MatricesWeight Matrices

We consider 2 weight matrices:We consider 2 weight matrices:

Inverse distance:Inverse distance:

Queen’s Case:Queen’s Case:

Each is scaled to have rows sum to 1 with Each is scaled to have rows sum to 1 with WWiiii=0=0

22

1

))()(())()((

1

jlongilongjlatilatdW ijij

not if 0

touchcounties if 1ijW

Federal Land Values versus Average of Neighbors

60

70

80

90

100

110

120

130

140

150

160

170

50 60 70 80 90 100 110 120 130 140 150 160 170 180

Average of Neighbors

Fede

ral L

and

Val

ue

Test for AutocorrelationTest for Autocorrelation

Moran’s I statistic under Randomization:Moran’s I statistic under Randomization:

''

1 1

2'0 0

1

2 2 21 2 0 1 2 0

2 20

01

1 1'

11 '

1( )

1

3 3 3 ( 1) 2 6 1( )

( 1)( 2)( 3) ( 1)

where:

n n

ij i ji j

n

ii

n

ijj

W y y y y Y I J W I J Yn n e Wen n

I e Y YS S e ey y Y I J Y

n

E In

n n n S nS S k n n S nS SV I

n n n S n

S W

2 2'1 2

1 1 1 1

41

12

21

1

11 1

2

( )Test Statistic:

( )

n n n n

ij ji i ii i j i

n

ii

n

ii

obs

W S W W S W W

n y yk

n y y

I E Iz

V I

Moran’s I – Federal Land (Queen’s Moran’s I – Federal Land (Queen’s WW))

NN = 99 Counties = 99 Counties SS00 = 99 (Rows sum to 1) = 99 (Rows sum to 1)

SS11 = 34.82809 = 34.82809

SS22 = 400.6013 = 400.6013

kk = 2.6268 = 2.6268 e’We = e’We = 45395.696745395.6967 e’e = e’e = 70033.6566 70033.6566 II = 0.6482 = 0.6482 E(E(II) = -0.0102) = -0.0102 V(V(II) = 0.003373) = 0.003373 ZZobsobs = 11.34 = 11.34

Moran’s I – Federal Land (Inverse Moran’s I – Federal Land (Inverse Distance)Distance) NN = 99 Counties = 99 Counties SS00 = 99 (Rows sum to 1) = 99 (Rows sum to 1)

SS11 = 3.2919 = 3.2919

SS22 = 397.1385 = 397.1385

kk = 2.0925 = 2.0925 e’We = e’We = 9772.80233 9772.80233 e’e = e’e = 70033.656670033.6566 II = 0.1395 = 0.1395 E(E(II) = -0.0102) = -0.0102 V(V(II) = 0.00012972 ) = 0.00012972 ZZobsobs = 13.15 = 13.15

SemiVariogram EstimatesSemiVariogram Estimates

Counties assigned to 34 distance Counties assigned to 34 distance classes: <0.35,0.40 to 2.00 by 0.05classes: <0.35,0.40 to 2.00 by 0.05

^ 2

( )

41/2

( )

Classical (Matheron) Estimator:

1( ) ( ) ( )

2 | ( ) |

Robust (Cressie-Hawkins)Estimator:

1( ) ( )

| ( ) |( )

0.4942 0.457

( )

i jN h

i jN h

h Z s Z sN h

Z s Z sN h

h

N h

Classical EstimatorMidpnt Average

#N/A #N/A0.325 271.060.375 254.090.425 154.770.475 188.170.525 269.950.575 399.280.625 381.330.675 630.750.725 503.490.775 397.750.825 553.460.875 586.520.925 477.600.975 617.891.025 645.801.075 724.391.125 673.941.175 864.461.225 675.011.275 631.981.325 1136.081.375 677.001.425 750.941.475 1002.681.525 678.691.575 700.471.625 795.311.675 1147.861.725 847.101.775 745.271.825 784.441.875 832.841.925 751.591.975 858.93

Robust EstimatorMidpnt Estmate

#N/A #N/A0.325 205.350.375 268.260.425 181.780.475 173.960.525 227.890.575 377.860.625 392.650.675 445.470.725 507.550.775 411.320.825 661.640.875 719.740.925 501.090.975 683.251.025 654.171.075 782.741.125 879.511.175 913.761.225 902.221.275 672.221.325 1513.771.375 707.401.425 827.471.475 1324.551.525 745.001.575 755.381.625 918.011.675 1285.891.725 970.901.775 955.611.825 917.821.875 1009.001.925 824.861.975 942.62

Several Semivariogram ModelsSeveral Semivariogram Models

0 12

0

1

2

Exponential: 1 exp

where: 0 Nugget effect

Partial Sill (Vertical Extent)

3 Practical Range (Horizontal Extent, Reaching 95% of Max)

0

Spherical:

hh

h

3

0 1 232 2

0 1 2

2

0 1 22

0

30

2 2

3Gaussian: 1 exp

h

h hh

h

hh

Fitted Semivariograms – (R Fitted Semivariograms – (R

gstat)gstat)

2

2

Exponential: 1643.2 1 exp2.273

3Spherical: 869.5 0 1.98; 869.5 1.98

2 1.98 2 1.98

hh

h hh h h h

Regression ModelRegression Model

Response: FEDVAL = Federal land valueResponse: FEDVAL = Federal land value Predictors: Predictors: CORNYLD = Corn yield/acreCORNYLD = Corn yield/acre PCTCORN = Percent of land planted cornPCTCORN = Percent of land planted corn PCTGRAIN = Percent of land for other grainsPCTGRAIN = Percent of land for other grains PCTUNPLOW = Percent land un-plowablePCTUNPLOW = Percent land un-plowable

Regression OutputRegression OutputRegression StatisticsMultiple R 0.9050R Square 0.8190Adjusted R Square 0.8113Standard Error 11.6135Observations 99

ANOVAdf SS MS F P-Value

Regression 4 57355.66 14338.91 106.31 0.0000Residual 94 12678.00 134.87Total 98 70033.66

Coefficients Standard Error t Stat P-value Lower 95%Upper 95%Intercept -64.71 17.34 -3.73 0.0003 -99.14 -30.27CORNYLD 3.15 0.37 8.47 0.0000 2.41 3.89PCTCORN 1.82 0.31 5.88 0.0000 1.20 2.43PCTGRAIN 0.54 0.26 2.10 0.0382 0.03 1.05PCTUNPLOW -0.55 0.27 -2.05 0.0433 -1.09 -0.02

Federal land values are:• Positively associated with corn yield per acre• Positively associated with percent of land planted corn• Positively associated with percent of land planted other grains• Negatively associated with percent of land un-plowableNo evidence of autocorrelated residuals (see following slides)

Moran’s I – Residuals (Queen’s Moran’s I – Residuals (Queen’s WW))

NN = 99 Counties = 99 Counties SS00 = 99 (Rows sum to 1) = 99 (Rows sum to 1)

SS11 = 34.82809 = 34.82809

SS22 = 400.6013 = 400.6013

kk = 4.1487 = 4.1487 e’We = e’We = 998.2657998.2657 e’e = e’e = 12677.997 12677.997 II = 0.07874 = 0.07874 E(E(II) = -0.0102) = -0.0102 V(V(II) = 0.00330) = 0.00330 ZZobsobs = 1.548 = 1.548

Moran’s I – Residuals (Inverse Distance)Moran’s I – Residuals (Inverse Distance)

NN = 99 Counties = 99 Counties SS00 = 99 (Rows sum to 1) = 99 (Rows sum to 1)

SS11 = 3.2919 = 3.2919

SS22 = 397.1385 = 397.1385

kk = 4.1487 = 4.1487 e’We = e’We = 91.9083 91.9083 e’e = e’e = 12677.997 12677.997 II = 0.00725 = 0.00725 E(E(II) = -0.0102) = -0.0102 V(V(II) = 0.00012693 ) = 0.00012693 ZZobsobs = 1.549 = 1.549

Map of OLS ResidualsMap of OLS Residuals