View
227
Download
0
Category
Preview:
Citation preview
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
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
##
#
#
#
#
#
#
#
#
# #
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
# #
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
# #
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
##
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
##
##
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
##
#
#
##
#
#
##
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
# #
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
##
#
#
#
###
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
###
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
###
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
###
# #
#
#
#
#
##
# #
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
# #
#
#
#
#
#
#
#
##
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
##
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
###
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
$$$
$
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.
Recommended