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KristanSchneider
Howoftenareyouhitbyaninfection?-Alikelihood
approachtodeterminethemultiplicityofinfection
ROeS2013,Sept.12,2013
Whatismalaria?
•Infectiousdiseasecausedbyparasites(genus
Pla
sm
od
ium;eukaryote)
•40%oftheworld’spopulationatmalariarisk
•Worldwide200-300millioninfections&1-3milliondeathsperyear
•Enormouseconomicdamageeveryyear
Transmissioncycle
Transmissioncycle
Transmissioncycle
Malariacontrol
•Goalofmalariacontrol:
◦reducedeceaseburden⇒drugtreatments
Malariacontrol
•Goalofmalariacontrol:
◦reducedeceaseburden⇒drugtreatments
◦reducetransmission⇒bednets,vectorcontrol,...
Malariacontrol
•Goalofmalariacontrol:
◦reducedeceaseburden⇒drugtreatments
◦reducetransmission⇒bednets,vectorcontrol,...
Q:Howtomeasureefficiencyofcontrolinterventions?
Malariacontrol
•Goalofmalariacontrol:
◦reducedeceaseburden⇒drugtreatments
◦reducetransmission⇒bednets,vectorcontrol,...
Q:Howtomeasureefficiencyofcontrolinterventions?
A:Multiplicityofinfection=metricfortransmissionintensity
Malariacontrol
•Goalofmalariacontrol:
◦reducedeceaseburden⇒drugtreatments
◦reducetransmission⇒bednets,vectorcontrol,...
Q:Howtomeasureefficiencyofcontrolinterventions?
A:Multiplicityofinfection=metricfortransmissionintensity
→co-infectionsmightincreasediseaseseverity
Malariacontrol
•Goalofmalariacontrol:
◦reducedeceaseburden⇒drugtreatments
◦reducetransmission⇒bednets,vectorcontrol,...
Q:Howtomeasureefficiencyofcontrolinterventions?
A:Multiplicityofinfection=metricfortransmissionintensity
→co-infectionsmightincreasediseaseseverity
→metrictransmissionintensity
Malariacontrol
•Goalofmalariacontrol:
◦reducedeceaseburden⇒drugtreatments
◦reducetransmission⇒bednets,vectorcontrol,...
Q:Howtomeasureefficiencyofcontrolinterventions?
A:Multiplicityofinfection=metricfortransmissionintensity
→co-infectionsmightincreasediseaseseverity
→metrictransmissionintensity
→impactstatisticsbasedongeneticdata
Malariacontrol
•Goalofmalariacontrol:
◦reducedeceaseburden⇒drugtreatments
◦reducetransmission⇒bednets,vectorcontrol,...
Q:Howtomeasureefficiencyofcontrolinterventions?
A:Multiplicityofinfection=metricfortransmissionintensity
→co-infectionsmightincreasediseaseseverity
→metrictransmissionintensity
→impactstatisticsbasedongeneticdata
Multiplicityofinfection-Whyisitimportant?
Multiplicityofinfection-Whyisitimportant?
Multiplicityofinfection-Whyisitimportant?
Multiplicityofinfection-Whyisitimportant?
Multiplicityofinfection-Whyisitimportant?
Multiplicityofinfection-Whyisitimportant?
Multiplicityofinfection-Whyisitimportant?
Multiplicityofinfection-Whyisitimportant?
Multiplicityofinfection-Whyisitimportant?
Multiplicityofinfection-Whyisitimportant?
•Largenumberofco-infections=hightransmission
Multiplicityofinfection-Whyisitimportant?
•Largenumberofco-infections=hightransmission
•Hightransmission=moregeneticvariation
→Multiplicityofinfection=keyquantityingeneticstudies
Approach
Approach
Approach
•Nbloodsamples
Approach
•Nbloodsamples
•nstrains(versionsofageneticmarkers)
Approach
•Nbloodsamples
•nstrains(versionsofageneticmarkers)
•n ii iobservednumbersampleswithconfigurationii i(0-1vector;lengthn)
Approach
•Nbloodsamples
•nstrains(versionsofageneticmarkers)
•n ii iobservednumbersampleswithconfigurationii i(0-1vector;lengthn)
•Qii iexpectedfrequencysampleswithconfigurationii i
Approach
•Nbloodsamples
•nstrains(versionsofageneticmarkers)
•n ii iobservednumbersampleswithconfigurationii i(0-1vector;lengthn)
•Qii iexpectedfrequencysampleswithconfigurationii i
Likelihood:
∏
ii i
Qn ii iii i
Log-likelihood:
L=
∑
ii i
n ii ilogQii i
HowtoderiveQii i?
•Assumptions:
HowtoderiveQii i?
•Assumptions:
◦pp p=(p1,...,pn)...frequencyvectorofstrain
HowtoderiveQii i?
•Assumptions:
◦pp p=(p1,...,pn)...frequencyvectorofstrain
◦Infectionsrare&independentevents
HowtoderiveQii i?
•Assumptions:
◦pp p=(p1,...,pn)...frequencyvectorofstrain
◦Infectionsrare&independentevents
•Implications:
◦Numberofinfectingstrains∼positivePoissondistribution(parameterλ)
HowtoderiveQii i?
•Assumptions:
◦pp p=(p1,...,pn)...frequencyvectorofstrain
◦Infectionsrare&independentevents
•Implications:
◦Numberofinfectingstrains∼positivePoissondistribution(parameterλ)
P(X=m)=
1eλ−1λm
m!
m>1
HowtoderiveQii i?
•Assumptions:
◦pp p=(p1,...,pn)...frequencyvectorofstrain
◦Infectionsrare&independentevents
•Implications:
◦Numberofinfectingstrains∼positivePoissondistribution(parameterλ)
P(X=m)=
1eλ−1λm
m!
m>1
◦InfectionconditionedonX=mstrainsmultinomiallydistributed∼Mult(m,pp p)
HowtoderiveQii i?
•Assumptions:
◦pp p=(p1,...,pn)...frequencyvectorofstrain
◦Infectionsrare&independentevents
•Implications:
◦Numberofinfectingstrains∼positivePoissondistribution(parameterλ)
P(X=m)=
1eλ−1λm
m!
m>1
◦InfectionconditionedonX=mstrainsmultinomiallydistributed∼Mult(m,pp p)
◦Quantityofinterest=
λ1−e−λ(meanofpositivePoissondistr.)
HowtoderiveQii i?
•Assumptions:
◦pp p=(p1,...,pn)...frequencyvectorofstrain
◦Infectionsrare&independentevents
•Implications:
◦Numberofinfectingstrains∼positivePoissondistribution(parameterλ)
P(X=m)=
1eλ−1λm
m!
m>1
◦InfectionconditionedonX=mstrainsmultinomiallydistributed∼Mult(m,pp p)
◦Quantityofinterest=
λ1−e−λ(meanofpositivePoissondistr.)
◦Likelihood:
L=L(λ,pp p)=−Nlog(eλ−1)+
n∑ k=1
Nklog(eλpk−1)
◦Nk...numberofsampleswithstraink
Results •Aims:
Results •Aims:
◦MLestimateofparametersθ̂θ θ=(λ̂,p̂p p)?
Results •Aims:
◦MLestimateofparametersθ̂θ θ=(λ̂,p̂p p)?
→multidimensionalNewtonmethod(nindependentparameters)
Results •Aims:
◦MLestimateofparametersθ̂θ θ=(λ̂,p̂p p)?
→multidimensionalNewtonmethod(nindependentparameters)
◦Existence,uniqueness,numericaleffort?
Results •Aims:
◦MLestimateofparametersθ̂θ θ=(λ̂,p̂p p)?
→multidimensionalNewtonmethod(nindependentparameters)
◦Existence,uniqueness,numericaleffort?
•Results:
◦θ̂θ θ=(λ̂,p̂p p)existsandisunique
Results •Aims:
◦MLestimateofparametersθ̂θ θ=(λ̂,p̂p p)?
→multidimensionalNewtonmethod(nindependentparameters)
◦Existence,uniqueness,numericaleffort?
•Results:
◦θ̂θ θ=(λ̂,p̂p p)existsandisunique
◦Eθ̂θ θNk=Nkexpected=observednumberofsamplescontainingstraink
Results •Aims:
◦MLestimateofparametersθ̂θ θ=(λ̂,p̂p p)?
→multidimensionalNewtonmethod(nindependentparameters)
◦Existence,uniqueness,numericaleffort?
•Results:
◦θ̂θ θ=(λ̂,p̂p p)existsandisunique
◦Eθ̂θ θNk=Nkexpected=observednumberofsamplescontainingstraink
◦λ̂foundbyiterating
λt+1=λt−
λt+
n∑ k=1
log(
1−NkN(1−e−λt))
1−
n∑ k=1
Nk
Neλt−Nk(e−λt−1)
,
convergesfromallstartingvaluesλ0>λ̂
Results •Aims:
◦MLestimateofparametersθ̂θ θ=(λ̂,p̂p p)?
→multidimensionalNewtonmethod(nindependentparameters)
◦Existence,uniqueness,numericaleffort?
•Results:
◦θ̂θ θ=(λ̂,p̂p p)existsandisunique
◦Eθ̂θ θNk=Nkexpected=observednumberofsamplescontainingstraink
◦λ̂foundbyiterating
λt+1=λt−
λt+
n∑ k=1
log(
1−NkN(1−e−λt))
1−
n∑ k=1
Nk
Neλt−Nk(e−λt−1)
,
convergesfromallstartingvaluesλ0>λ̂
◦p̂k=−1 λ̂log(
1−NkN(1−e−λ̂))
.
Results
Results
•Cameroon
Results
•Cameroon
•331bloodsample
Results
•Cameroon
•331bloodsample
Results
•Cameroon
•331bloodsample
Results
•Cameroon
•331bloodsample
Results
•Cameroon
•331bloodsample
Results
•Cameroon
•331bloodsample
•8microsatellitemarkers
Results
•Cameroon
•331bloodsample
•8microsatellitemarkers
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1.
1.1
1.2
1.3
1.4
1.5
microsatellitelocus
ΛeΛ
eΛ-1
Results •Aims:
◦Confidenceintervalls
Results •Aims:
◦Confidenceintervalls
◦Testingtheestimates
Results •Aims:
◦Confidenceintervalls
◦Testingtheestimates
•Approach:ProfileLikelihood
◦λfixedL(λ,pp p)→max?
Results •Aims:
◦Confidenceintervalls
◦Testingtheestimates
•Approach:ProfileLikelihood
◦λfixedL(λ,pp p)→max?
◦2(maxλ,pp pL(λ,pp p)
︸︷︷
︸max.LH
−max pp pL(λ0,pp p)
︸︷︷
︸profileLH
)∼χ2 1
Results •Aims:
◦Confidenceintervalls
◦Testingtheestimates
•Approach:ProfileLikelihood
◦λfixedL(λ,pp p)→max?
◦2(maxλ,pp pL(λ,pp p)
︸︷︷
︸max.LH
−max pp pL(λ0,pp p)
︸︷︷
︸profileLH
)∼χ2 1
◦Findλ,λwith:max pp pL(λ,pp p)=max pp pL(λ,pp p)=maxλ,pp pL(λ,pp p)−χ2 1(1−α)/2=ℓ∗
Results •Aims:
◦Confidenceintervalls
◦Testingtheestimates
•Approach:ProfileLikelihood
◦λfixedL(λ,pp p)→max?
◦2(maxλ,pp pL(λ,pp p)
︸︷︷
︸max.LH
−max pp pL(λ0,pp p)
︸︷︷
︸profileLH
)∼χ2 1
◦Findλ,λwith:max pp pL(λ,pp p)=max pp pL(λ,pp p)=maxλ,pp pL(λ,pp p)−χ2 1(1−α)/2=ℓ∗
◦Trick:maximizeconditionedonL(λ,pp p)=ℓ∗
→Lagrangemultiplies&(n+1)-dimensionalNewtonmethod
Results
Results:
◦1−αconfidenceintervalsexist&uniquelydefined
Results
Results:
◦1−αconfidenceintervalsexist&uniquelydefined
◦Boundsfoundbyiterating2-dimrecursion
Results
Results:
◦1−αconfidenceintervalsexist&uniquelydefined
◦Boundsfoundbyiterating2-dimrecursion
◦Approachconverges(locally)quadratically
Results
Results:
◦1−αconfidenceintervalsexist&uniquelydefined
◦Boundsfoundbyiterating2-dimrecursion
◦Approachconverges(locally)quadratically
◦TestforH0:λ=λ0vs.HA:λ6=λ0
rejectH0ifλ06∈[λ,λ]
p-value:
χ2 1
(
2(maxλ,pp pL(λ,pp p)−max pp pL(λ0,pp p)))
Results
Results:
◦1−αconfidenceintervalsexist&uniquelydefined
◦Boundsfoundbyiterating2-dimrecursion
◦Approachconverges(locally)quadratically
◦TestforH0:λ=λ0vs.HA:λ6=λ0
rejectH0ifλ06∈[λ,λ]
p-value:
χ2 1
(
2(maxλ,pp pL(λ,pp p)−max pp pL(λ0,pp p)))
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1.
1.1
1.2
1.3
1.4
1.5
microsatellitelocus
ΛeΛ
eΛ-1
Results
Results:
◦1−αconfidenceintervalsexist&uniquelydefined
◦Boundsfoundbyiterating2-dimrecursion
◦Approachconverges(locally)quadratically
◦TestforH0:λ=λ0vs.HA:λ6=λ0
rejectH0ifλ06∈[λ,λ]
p-value:
χ2 1
(
2(maxλ,pp pL(λ,pp p)−max pp pL(λ0,pp p)))
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1.
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microsatellitelocus
ΛeΛ
eΛ-1
302kb313kb 319kb379kb335kb363kb 383kb429kb
302kb
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319kb
379kb
335kb
363kb
383kb
429kb
Results (θ̂θ θ−θθ θ0)∼N(00 0,I−1N(θθ θ0))
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1.
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microsatellitelocus
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302kb
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379kb
335kb
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383kb
429kb
302kb313kb 319kb379kb335kb363kb 383kb429kb
302kb
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Results
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microsatellitelocus
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0.91.
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microsatellitelocus
Venezuela
Kenya
Results
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microsatellitelocus
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K6
0.91.
1.1
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1.9
microsatellitelocus
Venezuela
Kenya
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Cameroon
Venezuela
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mean±sd
Conclusions
•MLestimatefornumberofco-infections
(keyparameterinmalaria)
Conclusions
•MLestimatefornumberofco-infections
(keyparameterinmalaria)
•ML-approachworkswell
Conclusions
•MLestimatefornumberofco-infections
(keyparameterinmalaria)
•ML-approachworkswell
•Robustnessstudy→futurework
Conclusions
•MLestimatefornumberofco-infections
(keyparameterinmalaria)
•ML-approachworkswell
•Robustnessstudy→futurework
•Includingseveralmarkersatthesametime
Conclusions
•MLestimatefornumberofco-infections
(keyparameterinmalaria)
•ML-approachworkswell
•Robustnessstudy→futurework
•Includingseveralmarkersatthesametime
McCollumetal,2012
MalariaJ
Schneider&Kim,2010,
Theo.Pop.Biol.
Conclusions
•MLestimatefornumberofco-infections
(keyparameterinmalaria)
•ML-approachworkswell
•Robustnessstudy→futurework
•Includingseveralmarkersatthesametime
McCollumetal,2012
MalariaJ
Schneider&Kim,2010,
Theo.Pop.Biol.
