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Table 5 Discrete probability distribution used to model Colombia's malaria cases

From: The role of ENSO in understanding changes in Colombia's annual malaria burden by region, 1960–2006

Poisson Model

(PM)

Negative Binomial Regression Model (NBRM)

Link Function

E(Y t /X t ) = λ t = exp(X t β) > 0

E(Y t /X t ) = λ t = exp(X t β) > 0

Log

V(Y t /X t ) = λ t = exp(X t β) > 0

V(Y t /X t ) = λ t + α (λ t )2-k; k = 0,1

Log

E(Y t /X t ): Expected value of number of malaria cases in year t given the information of X t .

V(Y t /X t ): Variance of the number of malaria cases in year t given the information of X t .

β: unknown set of parameters

α: dispersion parameter [α > 0 over-dispersion; α < 0 under-dispersion]

Y t : dependent variable: Number of Malaria cases per year (Total, R1,..., R5)

X t : set of independent or explanatory variables.

Y t : {Mal_Tot, Mal_R1,..., Mal_R5}: set of dependent variables.

X t : {Base Line Trend, ENSO Measure} = {BLT, ENSO}

BLT: {Intercept, Trend1, Trend2, Vextre}: some or all of them

ENSO: {ENSO_Avg, ENSO_Dom}: one of them

Probability Distribution (PD)

Poisson PD: P [ Y t = y t ] = ( λ t ) y t exp ( λ t ) y t ! MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaa1aamWaaeaacqWGzbqwdaWgaaWcbaGaemiDaqhabeaakiabg2da9iabdMha5naaBaaaleaacqWG0baDaeqaaaGccaGLBbGaayzxaaGaeyypa0tcfa4aaSaaaeaadaqadaqaaiabeU7aSnaaBaaabaGaemiDaqhabeaaaiaawIcacaGLPaaadaahaaqabeaacqWG5bqEdaWgaaqaaiabdsha0bqabaaaaiGbcwgaLjabcIha4jabcchaWjabcIcaOiabeU7aSnaaBaaabaGaemiDaqhabeaacqGGPaqkaeaacqWG5bqEdaWgaaqaaiabdsha0bqabaGaeiyiaecaaaaa@4C9F@

Negative Binomial PD: P [ Y t = y t ] = Γ ( y t + 1 α ) Γ ( y t + 1 ) 1 Γ ( 1 + α ) ( α λ t ) y t ( 1 + α λ t ) y t + 1 α MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@68EC@

  1. The expected value and variance of the Poisson Regression Model and the Negative Binomial Regression Model are shown at top, their distributions are defined at bottom, and symbols are defined in middle.