Comparison of Factor Analysis and Single Element Geochemical Predictions Using Linear Regression with Weighted Variance

0 Tetra Tech NUS, Inc.

Comparison of Factor Analysis and SingleElement Geochemical Predictions Using

Linear Regression with Weighted Variance

Russell Sloboda, Tetra Tech NUS

Poster Presentation for the 18th Annual

Association for Environmental Health and Sciences

West Coast Conference on Soils, Sediments, and Water

March 10 – 13, 2008, San Diego, California

Tetra Tech NUS, Inc.


I. ABSTRACT

•At a military base, metals concentrations were characterized inbackground soils using geochemical prediction methods appliedto a database representing several USDA soil types.

•Linear regression 95 percent Upper Prediction Limits (UPL) wereestimated for future comparisons of site data to background.

•Simple linear regressions were based on one predictor metal,such as iron, while factor analysis predicted soil metalconcentrations based on overall mineral patterns in a sample.

•Linear prediction equations were based on metals that exhibitfactor loadings onto the factor scores for a metal of interest.

•Factor analysis back-predictions subtracted the influence of themetal of interest and renormalized factor pattern coefficients.

•Accuracy of factor analysis predictive ability was assessed bystripping out the influence of a metal of interest and evaluatingthe residual errors of observed versus predicted values.


II. PROBLEM DEFINITION AND STUDY GOALS

•State Regulations for Arsenic Concentrations in Soil:

–Average < 7 mg/kg, <= 10% samples > 7 mg/kg, no samples > 15 mg/kg

•Within a military base, 1179 soil samples were analyzed for arsenic:

– Average = 10 mg/kg, 31% samples > 7 mg/kg arsenic, 19% > 15 mg/kg

•US Dept. of Agriculture (USDA) soil types found within base areas:

–Mansfield mucky silt loam (MA) –Merrimack sandy loam (MM)

–Newport silt loam (NE) –Pittstown silt loam (PM)

–Stissing silt loam (SE) –Beach soils (BA)

–Udorthents-Urban land complex (UD) = Soil disturbed by cutting/filling

•Background Sampling Goals to allow future comparisons to site data:

–Background database for 2 sample hypothesis tests & geochemical tests

–Assess soil type differences to see if can combine background soil types

–Geochemical prediction model applicability to disturbed soil or fill thatmay contain any combination of soil types in the background data

–Characterize all metals, natural or anthropogenic & unimpacted by IR sites


III. Box Plots of Background Soil Arsenic Data

• Interquartile rangevaries by soil type

•4 possible outliers

•All positive results

•Beaches (BASS):

–Lowest conc.

•MA, PM, & SE soil:

–conc.[SB] > [SS]

•NE soil type:

–conc.[SS] > [SB]

•MM soil type:

–conc.[SS] ~ [SB]


IV. Box Plots of Bedrock Arsenic Data

7.4

42.2

0

20

40

60

80

Phylite Conglomerate

Ars

en

ic,

mg

/kg

q1 (25%)

MIN

median

MAX

ND (o)

Hit (●)

outlier ?

q3 (75%)

Samples collected below the soil layers,up to 51 feet into bedrock.

Conglomerate:

Range = 0.2 to 27 mg/kgAverage = 9.6 mg/kg

2 out of 11 samples >15 mg/kg

RI Formation (Phylite):

Range = 1.3 to 79 mg/kgAverage = 38 mg/kg

14 out of 19 samples >15 mg/kg

Observations: ContributingSources of Arsenic in Bedrock


V. Approximate Arsenic Distributional Shape

Lognormal Q-Q Plot for ARSENIC

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2 3

Theoretical Quantiles

Ord

ere

dO

bs

erv

ati

on

s

Blue -- Subsurface Soil

Lavender - Surface Soil

Shapiro Francia Test:

Sample Statistic = 0.9924

Critical Value = 0.987

Data are lognormal


VI. Hypothesis tests show soil type differences

A statistical significance level (P value) of 0.025 is used for all tests. Overall decision isYES if any one of the Mann-Whitney/Gehan, Upper Ranks Test, or T-Test is YES,regardless of other test results. Overall decision is NO if at least one of Mann-Whitney/Gehan, Upper Ranks Test, or T-Test is NO, and none of the aforementionedtests are YES. Overall decision is YES/NO if Z/Fisher Test is YES/NO, respectively, andother tests are NA.


VII. Arsenic Elemental Correlations: Surface Soil


VIII. Arsenic Elemental Correlations: Subsurface Soil


IX. Scatter Plot: Arsenic (Untransformed) vs Iron

0

12

24

36

48

60

72

0 10000 20000 30000 40000 50000 60000

Iron, mg/kg

Ars

en

ic,m

g/k

g

BASS MASB MASS MMSB MMSS NESBNESS PMSB PMSS SESB SESD SESS


X. Scatter Plot: Arsenic (0.67 Power) vs Iron

0

2

4

6

8

10

12

14

16

18

0 10000 20000 30000 40000 50000 60000Iron, mg/kg

Ars

en

ic0

.67

Po

we

r

BASS MASB MASS MMSB MMSS NESBNESS PMSB PMSS SESB SESD SESS


XI. Linear Regression with Weighted Residuals

•Why weight the residuals in geochemical regression?

–Residuals (Y-observed minus Y-predicted) increase with X

–Wedge-shaped scatter plot

•What is weighted Least-Squares Regression Analysis?

–Modification of ordinary least-squares that accommodatesnonconstant variance: As X increases, so does spread in observedY values

•Mathematics: Instead of minimizing sum of squares of thedeviations of the predicted Y values from the line, minimize thesum of the square of deviations multiplied by a weighting factorfor each point, Wj.

•Goals for prediction limits so that percent coverage is correct:

–Weighted residuals have constant variance with increasing X

–Weighted residuals are normally distributed (probability plot)

–The number of outliers is roughly 5 percent and similar by soil type


XII. Weighted Regression Prediction Formula


XIIIa. (Arsenic)0.67 Regressed on Iron: Surface Soil

0

2

4

6

8

10

12

14

16

18

0 10000 20000 30000 40000 50000

FE

AS

^0.6

7

All Data

BASS

MASS

MMSS

NESS

PMSS

SESS

AS^0.67=(2.34E-4)xFE+-0.63 R^2=0.81 Std.Error Y-est.=1.06

Weighted 1/SQRT(MAX(x-Xmin,4273.5)*MAX(y-Ymin,1.0))


XIIIb. (Arsenic)0.67 Regressed on Iron: Subsurf. Soil

0

2

4

6

8

10

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14

16

18

0 10000 20000 30000 40000 50000

FE

AS

^0

.67

All Data

MASB

MMSB

NESB

PMSB

SESB

AS^0.67=(2.34E-4)xFE+-0.63 R^2=0.81 Std.Error Y-est.=1.06



XIVa. (Arsenic)0.67 Regressed on As Predicted by FA

0

2

4

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10

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18

-5 0 5 10 15 20 25 30 35 40 45

AS predict (from Factor Anal.)

AS

^0.6

7

All Data

BASS

MASS

MMSS

NESS

PMSS

SESS

AS^0.67=0.30xAS predict (from Factor Anal.)+1.25 R^2=0.84 Std.Error Y-est.=0.96



XIVb. (Arsenic)0.67 Regressed on As Predicted by FA

0

2

4

6

8

10

12

14

16

18

-5 0 5 10 15 20 25 30 35 40 45

AS predict (from Factor Anal.)

AS

^0

.67

All Data

MASB

MMSB

NESB

PMSB

SESB

AS^0.67=0.30xAS predict (from Factor Anal.)+1.25 R^2=0.84 Std.Error Y-est.=0.96



XV. Arsenic Factor Pattern Matrix & Contributions


XVI. Factor Matrix Manipulations to Predict Metals


XVII. Prediction Accuracy: Factor Analysis vs. 1 Metal

• Regressions apply to majority of base – 70% of base consists of UD,unknown combination of soils disturbed by cutting or filling

• Two useful regressions for arsenic – one based on iron, the other based onfactor analysis (linear combination of all metals)

• Site-related samples can be plotted to see if arsenic <95% prediction limits

• Uncertainty and accuracy of regressions are listed (next slide):–Low regression residual errors were attained after back-transforming data into

original units (arsenic mg/kg)

–Even coverage across regression domain

–Good regression statistics: standard error of the Y-estimate and r2

• Geochemical regressions were developed for a total of 12 metals:–Single-metal predictions were compared to factor analysis for 11 metals

–4 Different Factor Analyses used different SS/SB data sets, transformations,and numbers of factors

–All factor analyses used Varimax rotation (other rotations had inferior results)


XIXa. All Metals: Regression Accuracy & Coverage


XIXb. All Metals: Regression Accuracy & Coverage


XIXc. All Metals: Regression Accuracy & Coverage