Cumulative Geographic Cumulative Geographic Residual TestResidual Test

Example: Example:

Taiwan Petrochemical StudyTaiwan Petrochemical Study

Andrea CookAndrea Cook

OutlineOutline

1.1. Motivation Motivation Petrochemical exposure in relation to childhood Petrochemical exposure in relation to childhood

brain and leukemia cancersbrain and leukemia cancers

2.2. Cumulative Geographic ResidualsCumulative Geographic Residuals UnconditionalUnconditional ConditionalConditional

3.3. ApplicationApplication Childhood Leukemia Childhood Leukemia Childhood Brain CancerChildhood Brain Cancer

Taiwan Petrochemical StudyTaiwan Petrochemical Study

Matched Case-Control StudyMatched Case-Control Study• 3 controls per case3 controls per case• Matched on Age and GenderMatched on Age and Gender• Resided in one of 26 of the overall 38 Resided in one of 26 of the overall 38

administrative districts of Kaohsiung administrative districts of Kaohsiung County, TaiwanCounty, Taiwan

• Controls selected using national Controls selected using national identity numbers (not dependent on identity numbers (not dependent on location). location).

Study PopulationStudy Population

Due to dropout approximately 50% 3 to 1 matching, Due to dropout approximately 50% 3 to 1 matching, 40% 2 to 1 matching, and 10% 1 to 1 matching.40% 2 to 1 matching, and 10% 1 to 1 matching.

LeukemiaLeukemia Brain CancerBrain Cancer

CasesCases 121121 111111

ControlsControls 287287 259259

Map of KaohsiungMap of Kaohsiung

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$$$

$

Nantze

Jenwu

Linyuan

Tsoying

# Study Participants$ Petro Plants

Cumulative ResidualsCumulative Residuals

Unconditional (Independence)Unconditional (Independence)• Model definition using logistic regressionModel definition using logistic regression• Extension to Cluster DetectionExtension to Cluster Detection

Conditional (Matched Design)Conditional (Matched Design)• Model definition using conditional logistic Model definition using conditional logistic

regressionregression• Extension to Cluster DetectionExtension to Cluster Detection

Logistic ModelLogistic ModelAssume the logistic model where,Assume the logistic model where,

and the link function,and the link function,

ii Y1i

Yiii )p1(p)p|Y(L

. )p(logit)p(g ii iβX

Residual FormulationResidual Formulation

Then define a residual as,Then define a residual as,

Assuming the model is correctly specified Assuming the model is correctly specified would imply there is no pattern in residuals.would imply there is no pattern in residuals.

=> Use Residuals to test for misspecification.=> Use Residuals to test for misspecification.

)ˆexp(1

)ˆexp(Ye ii

i

i

Xβ

Xβ

Cumulative Residuals for Model Checking; Lin, Wei, Ying 2002

Hypothesis TestHypothesis Test

Hypothesis of interest,Hypothesis of interest,

Geographic Location, (rGeographic Location, (r ii, t, tii ) )

Independent Independent of Outcome, Yof Outcome, Yii|X|Xii

Cumulative Geographic Residual Cumulative Geographic Residual

Moving Block Process is PatternlessMoving Block Process is Patternless

Unconditional Cluster DetectionUnconditional Cluster DetectionDefine the Cumulative Geographic Residual Moving Block Process as,Define the Cumulative Geographic Residual Moving Block Process as,

n

1ii2i221i112121loc ext)bx(,xr)bx(I

n

1),bb|x,x(W

Asymptotic DistributionAsymptotic DistributionHowever, the distribution of,However, the distribution of,

is hard to define analytically, but we have found another is hard to define analytically, but we have found another distribution that is asymptotically equivalent,distribution that is asymptotically equivalent,

which consists of a fixed component of data and random which consists of a fixed component of data and random variables variables

)1,0(~,...,G1

iid

nG

),bb|x,x(W 2121loc

),bb|x,x(W 2121loc

Significance TestSignificance TestTesting the NULLTesting the NULL

• Simulate N realizations ofSimulate N realizations of

by repeatedly simulating , while fixing the data at their by repeatedly simulating , while fixing the data at their observed values.observed values.

• Calculate P-valueCalculate P-value

)t,r(|Y:H iiio iX

)b,b|,(W 21loc

)b,b|,(W),...,b,b|,(W 21loc,N21loc,1

)G,...,G( n1

)b,b|x,x(Wsup)b,b(S and )b,b|x,x(Wsup)b,b(S

whereN

)b,b(S)b,b(SI

value-P

2121locx,x

21loc2121locx,x

21loc

N

1j21loc,j21loc

2121

Conditional Logistic ModelConditional Logistic ModelType of Matching: 1 case to MType of Matching: 1 case to Ms s controlscontrols

Data Structure:Data Structure:

Assume that conditional on , an unobserved stratum-specific intercept, Assume that conditional on , an unobserved stratum-specific intercept, and given the logit link, implies,and given the logit link, implies,

The conditional likelihood, conditioning on is,The conditional likelihood, conditioning on is,

.)exp(

)exp()s|Y(E 1M

1j

isis s

is

is

βX

βX

.)exp(

)exp()(L

1 s

is

s

N

1s

1M

1i

Y

1M

1j j

s

is

βX

βXβ

0Y,...,0Y,1Y s)1M(s2s1 s

s

1YY s)1M(s1 s

Conditional ResidualConditional Residual

Then define a residual as,Then define a residual as,

=> Use these correlated Residuals to test for => Use these correlated Residuals to test for patterns based on location.patterns based on location.

1M

1j js

sisis s )ˆexp(

)ˆexp(Ye

Xβ

Xβ i

Conditional Cumulative ResidualConditional Cumulative ResidualHowever, the distribution of,However, the distribution of,

is hard to define analytically, but we have found another is hard to define analytically, but we have found another distribution that is asymptotically equivalent,distribution that is asymptotically equivalent,

which consists of a fixed component of data and random which consists of a fixed component of data and random variables variables

)1,0(~G,...,Giid

N1 1

),bb|x,x(W 2121loc

),bb|x,x(W 2121loc

Significance TestSignificance TestTesting the NULLTesting the NULL

Simulate N realizations ofSimulate N realizations of

by repeatedly simulating , while fixing the data at their by repeatedly simulating , while fixing the data at their observed values.observed values.

Calculate P-valueCalculate P-value

)t,r(|Y:H iiio iX

)b,b|,(W 21loc

)b,b|,(W),...,b,b|,(W 21loc,N21loc,1

),...,(11 NGG

)b,b|x,x(Wsup)b,b(S and )b,b|x,x(Wsup)b,b(S

whereN

)b,b(S)b,b(SI

value-P

2121locx,x

21loc2121locx,x

21loc

N

1j21loc,j21loc

2121

ApplicationApplication

Study: Study:

Kaohsiung, Taiwan Matched Case-Control Kaohsiung, Taiwan Matched Case-Control StudyStudy

Method: Method:

Conditional Cumulative Geographic Conditional Cumulative Geographic Residual Test (Normal and Mixed Residual Test (Normal and Mixed Discrete)Discrete)

ResultsResults

Odds Ratio (p-values)Odds Ratio (p-values)

Marginally Significant Clustering for both outcomes Marginally Significant Clustering for both outcomes without adjusting for smoking history.without adjusting for smoking history.

Unadjusted Adjusted Unadjusted AdjustedDiscrete 2.10 (0.055) 2.19 (0.143) 1.97 (0.058) 2.08 (0.104)

Normal 2.10 (0.050) 2.19 (0.122) 1.97 (0.052) 2.08 (0.104)

Leukemia Brain Cancer

Childhood LeukemiaChildhood Leukemia

165000 170000 175000 180000 185000 190000

24

90

00

02

50

00

00

25

10

00

02

52

00

00

25

30

00

02

54

00

00

X1

X2

Cu

mu

lativ

e R

esi

du

als

Unadjusted

P-Values:Discrete = 0.055 Normal = 0.050

(a)

165000 170000 175000 180000 185000 190000

24

90

00

02

50

00

00

25

10

00

02

52

00

00

25

30

00

02

54

00

00

X1

X2

Adjusted

(b)

P-Values:Discrete = 0.143 Normal = 0.122

CasesControlsPlants

Childhood Brain CancerChildhood Brain Cancer

165000 170000 175000 180000 185000 190000

24

90

00

02

50

00

00

25

10

00

02

52

00

00

25

30

00

02

54

00

00

X1

X2

P-Values:Discrete = 0.052 Normal = 0.058

(a)

Cu

mu

lativ

e R

esi

du

als

Unadjusted

165000 170000 175000 180000 185000 190000

24

90

00

02

50

00

00

25

10

00

02

52

00

00

25

30

00

02

54

00

00

X1

X2

Adjusted

P-Values:Discrete = 0.104 Normal = 0.104

(b)CasesControlsPlants

DiscussionDiscussion

Cumulative Geographic ResidualsCumulative Geographic Residuals• Unconditional and Conditional Methods for Binary Unconditional and Conditional Methods for Binary

OutcomesOutcomes• Can find multiple significant hotspots holding type I Can find multiple significant hotspots holding type I

error at appropriate levels.error at appropriate levels.• Not computer intensive compared to other cluster Not computer intensive compared to other cluster

detection methodsdetection methods

Taiwan StudyTaiwan Study• Found a possible relationship between Childhood Found a possible relationship between Childhood

Leukemia and Petrochemical Exposure, but not with Leukemia and Petrochemical Exposure, but not with the outcome Childhood Brain Cancer.the outcome Childhood Brain Cancer.