multi-step polynomial regression method to model and forecast malaria incidence

8/14/2019 Multi-step polynomial regression method to model and forecast malaria incidence

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predicting Malaria incidencepredicting Malaria incidence

in Chennaiin Chennai

By

Chandrajit Chatterjee

M Sc (I), Statistics

University of Madras

, Chennai


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Introduction to malariaIntroduction to malaria

Malaria is a communicative disease, caused

through parasitic infection (mostly by the

parasite Plasmodiumfalciparum).

Causes 300-500 million cases of infection

around the world and kills 1.5 to 2.7 million

people each year.

Problems of drug resistance of the parasite and

no single, universally accepted control measure

around the world has aggravated the global

situation of malaria incidence.

The premier cause of concern is the high

incidence rates of disease in children below 5


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Four species of pathogens are found to be

causative of the major part of malaria around

namely Plasmodium falciparum, P. vivax, P.

malariae and P.ovale.

In India and other warmer parts of the world

the falciparumspecies is far more predominant

than its counterparts (death is caused due to

cerebral malaria and renal failure).

The parasite is transmitted from person to

person by mosquitoes of genusAnopheles.

Out of the 422 species of existingAnopheles

species worldwide only 40 are important from


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MALARIAMALARIA IN INDIAIN INDIA

Malaria is prevalent in all parts of the country except

in areas 5000 feet above sea-level.

In India, Malaria has been on a constant high from

1993 except the period between 1995 and 1999 (Fig. 2)

Fig 1: THE STATE OF MALARIA IN INDIA FROM 1961 TO

2006


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A complete orientation towards eradication needs

understanding the causative factors, their degree of influence

and disease transmission dynamics over a horizon-that is

where the need of a model arises.

New age modeling probably began with Harvey in 16th century

when he used quantitative reasoning in proving circular motion

of blood.

There are 2 broad methods in modeling one using abstract

differential equations (mathematical modeling) and the other

using data on previous incidences and related causes (databased or statistical modeling)

Our aim:

We aim at establishing a simple method that bypass the

Malaria controlneed of a modelMalaria controlneed of a model


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Malaria incidence data from the Corporation of Chennai.

(I) The first data set (large-over 30 time points) consists of the

monthly slide positivity rate of malaria (all types) in Chennai over 37

months from Jan 2002 to Jan 2005.

(II) The second data set (small-less than 30 time points) consists of

deaths due toplasmodium vivaxdistributed over the 10 zones of

Chennai city for 12 months of the year 2006 and the population ofthese months for all zones.

Climatic data was had for the same time points from the following

websites:

www.waterportal-india.orgwww.wunderground.com

www.imd.ernet.in

www.worldweather.com

The Data:The Data:


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Contd.

The population for Chennai city for the period between 2002

Jan and 2005 Jan was obtained from census data from 1901 to

2001 from the website www.gisd.tn.nic.in/census-paper1,census-paper2.

A third order polynomial was fitted to this data to impute

population at required time points as in fig 2.

Fig 2: IMPUTATION OF POPULATION(Dashed-observed; Straight line-predicted).


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The method in briefThe method in brief

Our methodology consists of the following steps:

Selection of variables

Identification of relationship between the modelvariables

Initial identification of the model

Model refinementPrediction and forecasting with the help of the modelequation

Testing the correctness of prediction with standardmethods

Tools of analysis: Microsoft excel,2000 & 2007 SPSS for windows 11.0 MATLAB 7.5.0.


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Variable selection and identification ofVariable selection and identification of

relationshiprelationship

The procedure begins with variable selection:

The first order models are considered first i.e. simple

linear regression between dependant variable and the

factors individually, purely based on the coefficient of

determination* we select the first variable.

In the next step we consider regression model of order

2 with the first variable already in model and based on

partial t-statistic**, we choose the second variable and go

on increasing the order with this procedure unless all

variables are exhausted or no other variable qualify the

criterion, whichever earlier.

* Coefficient of determinationa measure of goodness ofregression fit denoted as R-square** Partial t-statistic is a t test to determine the influence of a

particular variable in a multi variable model.


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Discussions - Analysis I-Slide Positivity Rates:Discussions - Analysis I-Slide Positivity Rates:

After variable selection we have:

Rainfall

Maximum temperature

Minimum humidity

Population.

The optimum relations are then determined between

the dependant and the selected independent variables

from study of the following scatters.

(In the method trial and error in determining the

functional forms of individual relationships is the way we

chose).

The methodology of model refinement to attain the final

model was run then.


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FIG 3:SCATTER BETWEEN log(SPR) AND log(Max Temp) FIG 4: SCATTER BETWEEN log(SPR) ANDlog(Rainfall)

A 4th order polynomial explains the relationship A 3rd order polynomial explains

18% of variabilitybetween the variables above with an R square 49% between the two in

fig3.

FIG 5: SCATTER BETWEEN log(SPR) AND Population FIG 6: SCATTER BETWEEN log(SPR) AND MinTemp


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Model construction and refinementModel construction and refinement

We deviate here from the other similar works done in the

field, in that we consider a non-linear model as it is but

keeping methodology simple and not having to pre-define

the functional form of the model as it is determined in the

process.

In our multi step regression procedure in the modelrefinement we have considered each functional form of an

independent variable as a separate independent variable.

when the R2 is not a plausible option for comparison of

predictions made we have taken the adjusted R2

* as thecriterion of comparison.

We then construct and refine the model based on principle

ofprogressive improvement of residual sum of squares in

stepwise induction of variables ,as depicted in the nextflowchart .

* Adjusted R2 is a refinement over R2 in that it uses the unbiasedestimators for the sum of squares used in R2 calculated as 1-[(1-

R2)*(n-1)/(n-p-1)] where p is the number of independent variablesin model.


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NO

Is R2

increasing?

Model with the initialvariable (say Xi),with

highest R-square

Induce Xi2

YES

AcceptXi

2

NORemoveXi

2

Induce all higher orders ofXi (one by one)

as defined by its relation with dependant

variable, dependant on R2

Induce Xj

Is R2

increasing?

YES

AcceptXj

Remove

Xj

Induce all higher orders ofXj(one by one) as before,

dependant on R2

Are all

variablesexhausted?

Inducea newvariabl

e anditsfunction-forms

STOP inducing

YES

NO


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In this particular analysis the final model after refinement is:

NOTE: One has to get back SPR values by exponential calculation.

Results:

The model gave an R2 of 44.16% and an adjusted R2 of 42.51%

which says that the model will not vary much in its degree of

prediction when applied to a new data set of similar kind.

ANOVA indicates an F value of 2.720 (F(sig) =0.024) which ishighly significant at the 5% significance level indicating a good

explanative nature of the regression.

95% confidence intervals over all 36 time points contain thepredicted responses indicating significantly correct responseprediction (process slide 17).

The forecasted value of Jan 05 also falls in the 95% C.I. ofresponse indicating a fairly good degree of forecast by the model

log(SPR) = -26.492+5.343E-21*(POPULATION)3

+22.008*log(MAX TEMP)

-1.652*[ log(MAX TEMP)]4

+1.097E-02*(MIN TEMP)

-5.24E-07*[MIN TEMP]3

-5.03E-02* log(RAINFALL)

+ 9.128E-02*[ log(RAINFALL)]2

-2.45E-02*[ log(RAINFALL)]3

.


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FIG 7: PLOT OF FIT OF PREDICTED VALUES OF MODEL AGAINST OBSERVEDVALUES

Legends:

Dashed series with circular markers-predicted values

Straight line with square markers-observed values

Green circle-forecasted value

Light blue square-observed

The error bars are constructed according to 95% confidence intervals of


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Testing the model predictions:Let a general regression model be :

Yi=+jXij*j+i; i=1(1)m, j=1(1)n

n=number of independent variables

m=number of data points

S=iri2where ri are the residuals

Then by Gauss Markov GLM

est=(XTX)-1X

TY and

2est=S/(m-n)

Given independent variable values Xdj for a given time point d,

Then writing Z=(1, xd1, , xdn)

100(1- ) % C.I. for a predicted response for level is

[ZT t/2;(m-n)* est* sqrt[(1+ZT(XTX)-1Z)]

Where X =((Xij))j=1(1)n ; i=1(1)m-the matrix of independent variables


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Analysis II-Total P.V. deaths in Chennai cityAnalysis II-Total P.V. deaths in Chennai city

We had data on 10 months of total deaths due toPlasmodium Vivax, scaled with total population (henceour dependant variable) in Chennai and we wanted to see

the credibility of forecasts of the model for the next 2time points.

We demonstrate here that our method can work equallywell for smaller data sets for which normality may not bereadily assumed (as in the earlier case of the data set

with 36 time points, which had good chances of tendingtowards normality).

In this analysis with the same process, variable

selection yielded:

Minimum Temperature

Maximum Temperature

Maximum Humidity

The initial relationships were studied through scattersand then the model refined b the multi ste rocedure


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Fig 8: A 6th order polynomial explains 85.8% between Fig 9: A 4th order polynomial explains30.6%

P.V. Deaths (scaled) and Min Temp. between scaled P.V. deaths and MaxTemp

Fig 10: A 3rd order polynomial tells 61.8% of the relationship


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The final model equation was:

NOTE: After the predictions for scaled p.v. deaths are obtained from

the model one needs to multiply the values by population total toobtain total p.v. deaths.

Results:

The model gave a coefficient of determination of 82.565% and anadjusted R2 of 60.77% which as before implies the goodness of model in

application to a new data set.

ANOVA indicates an F value of 3.78 of regression against a critical

value of 0.11 at 5% level thereby signaling a good regressive nature of

the model.

95% C.I. over the 10 time points all include the predicted P.V. deaths

(except 1 value) and the observed values as well lie in the C.I. thereby

validating the confidence intervals at level 0.05. The 95% C.I. contains both the forecasted values of November and

(Scaled)PV deaths=2.309E-04+8.313E-06* (min temp)

+7.375E-13*(min temp)6

-8.72E-07*(max hum)

-5.81E-11*(max hum)3

-2.98E-10*(max temp)4


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FIG 11:PLOT OF FIT OF PREDICTED VALUES OF MODEL AGAINST OBSERVED

VALUES.

Legends:Dashed series with circular markers-predicted valuesStraight line with square markers-observed valuesGreen circle-forecasted valueBlue square-observed values in forecasting horizonThe error bars are constructed according to 95% confidence intervals of

predicted values.


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A special study of individual zonesA special study of individual zones

In order to analyze the zones separately we needed to construct

the models for each zone separately. However that is cumbersome

and may be redundant.

We hence did a Factor Analysis with the ten zones of data of

deaths (scaled by population) due to P.V. and the climatic factors

together, with the extraction method as Principal Component

Analysis (the data set was normalized before the analysis to reduce

the large analytical data set of intercorrelated variables into a

smaller and interpretable set of factors).

Subsequently we did a hierarchial Cluster Analysis to reconfirm

results from FA.

The clustering of the variables obtained from here, were used to

frame models with our methodology and the predictions were

tested through the same method as before to yield fits as shown

later.

The Dendrogram in cluster analysis looks as below from which two


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Where

HT - highest temperature,

TL - lowest temperature,

LH - lowest humidity,

HH -highest humidity,

TR- total rainfall,

Z1-Z10 - scaled values of P.V. deaths over the 10 zones of Chennai city.


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CLUSTER 1-TOT DEATHS OF ZONES 2,3,5,6,7,8-FITTED

series with circular markers-predictedt line with square markers-observedars constructed on the basis of 95% C.I. of predicted values


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CLUSTER 2-DEATHS OF ZONE 1-FITTEDCLUSTER 2-DEATHS OF ZONE 1-FITTED

Dashed series with circular markers-predicted

Straight line with square markers-observed

Error bars constructed on the basis of 95% C.I.of predicted values

CONCLUSIONCONCLUSION ::


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CONCLUSIONCONCLUSION ::

The method yielded good fits to data sets, both large and

small with equally apt predictions and for two different

formats of incidence of the disease-one with SPR values and

the other with deaths.

The response predictions of our model for majority of

cases, lie in the 95% C.I. of the response predictions,

keeping in mind, the tests that we have applied for

prediction response has its basis in the most general format

of linear models-the Gauss Markov models.

The method may well be applicable to still larger data sets

with several other variables as the socio-economic factors,

geographical influences, etc. and one may follow the method

to establish a regression model for future predictions of

malaria incidence as we have also shown-using methods like

FA and cluster analysis and then adopting our method yields

predictions of equal caliber as before.


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multi-step polynomial regression method to model and forecast malaria incidence

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