Overdispersion study of poisson and zero-inflated poisson regression for some characteristics of the data on lamda, n, p

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1 Iteratioal Joural of Advaces i Itelliget Iformatics ISSN: Overdispersio study of poisso ad zero-iflated poisso regressio for some characteristics of the data o lamda,, p Lili Puspita Rahayu a,1,*, Kusma Sadik b,2, Idahwati b,3 a Miistry of Educatio ad Culture of The Republic of Idoesia, Idoesia b Departmet of Statistics, Bogor Agriculture Uiversity, Idoesia 1 lilipuspitarahayu@gmail.com *; 2 kusmasadik@gmail.com; 3 idah.stk@gmail.com * correspodig author ARTICLE INFO AB ST R ACT Article history: Received November 19, 2016 Revised November 25, 2016 Accepted November 25, 2016 Keywords: Overdispersio Poisso Zero iflated Poisso Regressio Simulatio data Poisso distributio is oe of discrete distributio that is ofte used i modelig of rare evets. The data obtaied i form of couts with oegative itegers. Oe of aalysis that is used i modelig cout data is Poisso regressio. Deviatio of assumptio that ofte occurs i the Poisso regressio is overdispersio. Cause of overdispersio is a excess zero probability o the respose variable. Solvig model that be used to overcome of overdispersio is zero-iflated Poisso (ZIP) regressio. The research aimed to develop a study of overdispersio for Poisso ad ZIP regressio o some characteristics of the data. Overdispersio o some characteristics of the data that were studied i this research are simulated by combiig the parameter of Poisso distributio (), zero probability (p), ad sample size () o the respose variable the comparig the Poisso ad ZIP regressio models. Overdispersio study o data simulatio showed that the larger,, ad p, the better is the model of ZIP tha Poisso regressio. The results of this simulatio are also stregtheed by the exploratio of Pearso residual i Poisso ad ZIP regressio Copyright 2016 Iteratioal Joural of Advaces i Itelliget Iformatics. All rights reserved. I. Itroductio Poisso distributio is oe of the discrete distributio that is ofte used for modelig o rare occasios. The data obtaied i the form of couts with o-egative itegers. Oe form of aalysis used to model cout data is Poisso regressio. Poisso regressio aalysis showed a relatioship betwee the explaatory variables with the respose variable that spread Poisso. Poisso regressio has equidispersio assumptios, a coditio i which the mea ad variace of the respose variable equal value. Deviatio assumptios that ofte evets o Poisso regressio is overdispersio/ uderdispersio. Overdispersio is the variace greater tha the mea, while the value uderdispersio is the variace smaller tha mea value o respose variable. Applicatio about uderdispersio o Poisso regressio is rare evetrig, it is because there is o low variace value of the respose variable o real data [1]. Problem ofte ecoutered i Poisso regressio was overdispersio. This coditio is caused by the explaatory variable that ca t be explaied i the model, so it is possible the high variability of the respose variable caused by other variables. Cause of overdispersio that ofte evet i Poisso regressio is zero probability that excess o the respose variable. Oe result is the stadard deviatio of parameter estimate to be uderestimate ad the sigificace of the explaatory variables to be overstate, resultig ivalid coclusios [2]. The hadlig model ca be used to overcome overdipersio due to zero probability excess o the respose variable Poisso regressio such as models of hurdle Poisso regressio, zero-iflated Poisso (ZIP) regressio, ad Semiparametric hurdle Poisso regressio [3]. Hadlig model that will be used i this paper is the model of ZIP regressio, because it is more coveiet tha models of hurdle Poisso regressio ad Semiparametric hurdle Poisso. Superiority of ZIP regressio is very easily applied to several fields [4] such as agriculture, aimal husbadry, biostatistics, ad idustry. DOI: W : E : ifo@ijai.org

2 ISSN: Iteratioal Joural of Advaces i Itelliget Iformatics 141 I additio, estimate of the parameter o ZIP regressio model ca be iterpretatio easily, ad ca explai the reaso of the mea smaller i the respose variable. Research that has bee doe before, startig the ZIP regressio model developed as a solutio overdispersio hadlig of Poisso regressio models usig simulatio studies that Xie et. al. [5] usig a type II error i the simulatio with combiig the parameter of Poisso distributio, zero probability for sample size o the respose variable. Numa [6] developed a Wald test for compariso of Poisso ad ZIP regressio models. Wald test developmet performed simulatios with determiatio of zero probability o the respose variable based o the value parameter of the Poisso distributio. Developmet of the research that has bee doe previously, the researcher wated to study develops the overdispersio o some characteristics of the data. Overdispersio which will be examied i this study are simulated by combiig the value of the parameter of the Poisso distributio, zero probability, ad sample size o the respose variable. Furthermore, comparig Poisso ad ZIP regressio models based o exploratio of the respose variable, ad the evaluatio of the predictio models. Ay simulatio o characteristics data are expected to determie the cause of overdispersio o respose variable. II. Poissio Regressio Hardi ad Hilbe [7] stated that the Poisso regressio model provides a stadard framework for aalysis of data cout. Poisso regressio is a form of geeral liear model. Let y i, i=1,2,, represets cout of those rare occasios i the period with value of the parameter Poisso distributio lambda ( i). y i is a Poisso radom variable that spread by the mass fuctio probability of the followig f(y i ; i ) = e y i i i, y i = 0, 1, 2, y i! with assumig of Poisso regressio is E(Y i ) = Var(Y i ) = i If Poisso regressio is used for the coditio overdispersio, the the result is ot exact because the value of mea ad variace Poisso regressio cotai dispersio Var (Y) = τ E (Y) = τ with τ is ratio of dispersio. Whe there was overdispersio o Poisso regressio the value of τ is more tha oe ad costat. Dispersio is a measure of variat of a group of data to the mea data. Small dispersio values showed a homogeeous variety i the data, while the big dispersio values idicate heterogeeity i data. Method to estimate parameters of Poisso regressio coefficiets are maximum likelihood method. Suppose X is explaatory variable that size matrix x (p+1). Radom variables y i ad i is a row vector of X, will be liked with the log lik fuctio. l i i Xb i exp Xb The model i (5) is a Poisso regressio model with parameter b is the coefficiet estimate. III. Zero Iflated Poisso Regressio Jasakul ad Hide [1] state that if the Y i are idepedet radom variables that have a ZIP distributio, the the value of zero is assumed to arise from the same two steps. The first step

3 142 Iteratioal Joural of Advaces i Itelliget Iformatics ISSN: evets o probability that oly produces zero observatios deoted by pi. The secod step evets o the probability that result of data cout spread Poisso with parameter is deoted by (1- pi).i geeral, the zero value of the first step is called structural zeros, ad the zero value of the secod step is called samplig zeros. Variables Yi ~ ZIP i, pi have assumptios o the ZIP regressio 1 E Y p ad Var Y i i i i p 1 pi i 2 i i i variables Y i has overdispersio p i>0 if, the overdispersio will reduce to Poisso models whe p i=0. Value p i>0 explais that there is a icrease value of zero o the respose variable. Method to estimate parameters of ZIP regressio coefficiets is maximum likelihood method. Loglikelihood fuctio for observatios y 1,,y o ZIP regressio model is used to simplify the calculatio to get parameter estimate coefficiet. Maximizig of log-likelihood fuctio will give the same result as maximizig the likelihood fuctio. ZIP regressio models of divided ito two compoets, amely discrete data models for ad zero-iflatio models for p. If the explaatory variables used i l ad logit model o the same value, the the ZIP regressio model is l Xb p l Xg 1 p with X is the matrix of explaatory variables, while b ad g is the vector of parameter estimate of ZIP regressio coefficiets, each sized (q+1)x1 ad (r+1)x1 i equatio (8). Explaatory variables used i the model l be the same or differet from the explaatory variables used i the logit model. Maximum likelihood estimatio for b ad g are obtaied by usig the expectatio maximizatio (EM) algorithm which provides a simple way, so that it ca be applied to stadard software to match the geeral liear model. IV. Desig of Simulatio The data used i this study is the simulatio data. Simulatio data was geerated based o the characteristics of the data. Characteristics of the data i the form of lambda () startig from = 0.6, 0.8, 1, 6, 8, 10, ad 20, the zero probability (p) are p=0.1, 0.3, 0.5, ad 0.7, ad sample size () are =,, ad. The data geerated are useful to obtai parameter estimators of Poisso ad ZIP regressio. The coefficiet of regressio parameters were determied are 0=3, ad 1=0.01. Variables were determied to make the Poisso ad ZIP regressio models are explaatory variable (X), the respose variable (Y). Variable X cosists of variable X which is a ormal radom variable spreads (μ,1). Variable X is assumed as a fixed variable. Geeratig variables X ad Y o simulatio study carried out by stages, amely: 1. Geeratig variable Y based o the value of,, p have bee determied. 2. Geeratig variable X is with the first loop, are: Separate variables Y become Y zero ad Y ot zero. Trasform variable X with formula x i= (l (y i) - 0)/ 1, which is y i from variable Y ot zero.

4 ISSN: Iteratioal Joural of Advaces i Itelliget Iformatics 143 Iitialize the result of trasformatio variable X as X ot zero. 3. Geeratig variable X o variable Y ot zero ad Y zero with the secod loop are If the variable Y is zero, the the variable x i is gotte by samplig with replacemet o variable X ot zero. If the variable Y is ot zero, the the variables x i is geerated from Normal distributio with mea of trasformatio result from 2(ii) ad variace is 1 with sample size =1. Simulatio data o the variables X ad Y geerated by the software program R ver ad will be repeated r= replicatios. There are 84 simulatio coditios used i this study. The accuracy of parameter estimators i the Poisso ad ZIP regressio models ca be see from the relative absolute bias (RAB) ad relative root mea square error (RRMSE) [8]. Furthermore, the accuracy of the estimate y i Poisso ad ZIP regressio models ca be see from Pearso residual (PR) ad sum of absolute Pearso residual (SAPR) [9]. The equatio of value RAB, RRMSE, PR, ad SAPR are defied respectively i (9), (10), (11), ad (12). - RAB = i i=1 x% (9) θ )2 RRMSE = ( i i=1 % (10) PR i = j=1 (y ij y ij ) Var(Y) the SAPR = i=1 PR i (12) where r is the umber of simulatio replicatios, i is the parameter estimators of Poisso ad ZIP regressio i th, is the actual parameter. The, y ij is the respose variable o repeat to-i ad observatios to- j, ad y ij is the estimate y of the Poisso ad ZIP regressio model o repeat to-i ad observatios to- j, ad Var (Y) is the estimate variace of Poisso ad ZIP regressio. The, the smaller values of RAB, RRMSE, ad SAPR o regressio model ca be said to be gettig better. (11) V. Results Simulatio study cosisted of 84 cases of simulatio which is a characteristic of data from the combiatio of,, ad p. Simulatios were performed to evaluate the results of estimatig the parameters of the Poisso ad ZIP regressio usig percetage of ARB, RRMSE, ad average of SAPR. The value obtaied from the simulatio was repeated times. The evaluatio results of the simulatio data to be clarified with the results of exploratio ad testig of the variable Y. A. Exploratio ad testig at the variable Y Characteristics of data simulatio agaits,, ad p is tested idicates that the rise of the value of p effect o. Value of 0.6 with p that is tested 0.3, the the variables Y produces rage p of 0.3 to 0.5. This is because the that has small value still has p from Poisso distributio relatively large. This statemet ca be explaied by the cumulative Poisso distributio table. Value is tested to 0.6, 0.8, 1, 6, ad 8 still have zero probability of a Poisso distributio, while for the other is tested, amely 10 ad 20 already have ot zero probability of a Poisso distributio. Exploratio variable Y to,, ad p is tested idicatio excess zero probability, so that the ecessary tests o variables Y. The test is i the form of scores test ad chi-square test were able to geeralize the geeral coclusios o the results of exploratio at variables Y. The coditio of excess zero probability at the variable Y due to overdispersio. Fly ad Fracis [10] state that whe the score test results value of zero excess o a variable, the chaces are the variables do ot spread Poisso distributio, but has ZIP distributio. The score test results with α of 0.05 o simulatio data at variable Y agaist a combiatio of,, p are show i Table 1. Scores test idicate that the larger,, ad p is tested, the the larger percetage of excess zero at the variable Y.

5 144 Iteratioal Joural of Advaces i Itelliget Iformatics ISSN: Table 1. Percetage of score test o combiatio of,, p (%) Furthermore, the results of chi-square test with α of 0.05 for the Poisso ad ZIP distributio agaist the combiatio of,, p are show i Table 2. The chi-square test for Poisso distributio shows that the larger,, ad p is tested, the percetage Poisso distributio will be smaller at the variable Y. The results of the score test ad the chi-square test for Poisso distributio is iversely proportioal to the larger,, ad p is tested. Chi-square test for ZIP distributio idicates that ZIP regressio able to overcome overdispersio due to excess p at the variable Y. This coditio is idicated by the larger value of, the the percetage Poisso distributio reaches 0%, while the percetage of ZIP distributio i the rage of 60% to 80%. Table 2. Percetage of chi-square test o combiatio of,, p (%) Poisso ZIP Poisso ZIP Poisso ZIP Poisso ZIP

6 ISSN: Iteratioal Joural of Advaces i Itelliget Iformatics 145 B. The testig overdipserio o Poisso ad ZIP regressio Results of exploratio ad testig of the simulatio variables Y is a step that must be checked before performig Poisso regressio aalysis. Whe the variable Y has the characteristics of the data that led to overdispersio due to excessi zero value, the ZIP regressio became oe of the completio of the Poisso regressio. Overdipersio coditios o ay combiatio of,, ad p are tested i Poisso ad ZIP regressio ca be traced from the dispersio ratio (τ) ad the Pearso chi-square test at 5% sigificace level. Ratio of τ shows the value of a statistical result Pearso chi-square test of the degrees of freedom (-k). The value of degrees of freedom Poisso ad ZIP regressio is differet, because the Poisso regressio usig k=2, are the parameter estimate b 0 ad b 1. ZIP regressio usig k=4 is based o discrete models for ad zero-iflatio modesl for p are g 0 ad g 1, the l 0 ad l 1. The average ratio of τ obtaied from times of repetitio of the combiatio of,, ad p is tested i Table 3. The overdispersio idicated by the ratio τ is greater tha oe. The ratio τ o Poisso regressio will be compared with ZIP regressio. The ratio τ most at risk are at p=0.7 i Poisso regressio with the value is the larger. The ratio τ of the Poisso regressio showed that the larger ad p, the the ratio τ more tha oe i every is tested. Poisso regressio suffered overdispersio by the larger, p, ad is tested. The ratio τ of ZIP regressio has a value of less tha oe i every, p, ad is tested, so ZIP regressio able to overcome overdispersio caused excess zero probability at the variable Y. Table 3. Dispersio ratio o Poisso ad ZIP regressio Poisso ZIP Poisso ZIP Poisso ZIP Poisso ZIP * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Overdispersio The percetage of overdispersio have the similar result with the ratio of τ. Pearso chi-square test that sigificat shows overdispersio. Value of overdispersio cotaied i Table 4 show that i every, p, ad is tested to get overdispersio value reachig 0% for ZIP regressio. These results idicate that ZIP regressio is better to hadle overdispersio caused excess zero probability o the variable Y. The percetage of overdispersio o Poisso regressio showed that the larger ad p, the the greater overdispersio i each is tested that idicated by the value reached %. These results are cosistet with the score test ad the chi-square test that idicates the larger,, ad p is tested, the the greater zero probability that appears ad the less spread Poisso at the variable Y.

7 146 Iteratioal Joural of Advaces i Itelliget Iformatics ISSN: Table 4. Percetage of overdispersio o Poisso ad ZIP regressio Poisso ZIP Poisso ZIP Poisso ZIP Poisso ZIP * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0.0 * Overdipersio C. Evaluatio of the estimatio o Poisso ad ZIP regressio model The goodess of fit o Poisso ad ZIP regressio model showed that the larger,, ad p is tested, the ZIP regressio is better tha Poisso regressio. Furthermore, the compariso of Poisso ad ZIP regressio models is take based o the evaluatio parameter estimate ad y. The average of ARB for each combiatio of,, ad p are tested o estimate θ 1 i Table 5. The average of ARB o ZIP regressio i Table 5 produces a value is smaller as elargemet,, ad p is tested agaist estimate θ 1. The value of ARB o Poisso regressio has a miimum value that is cotaied i = 6. The average of ARB o estimate θ 1 idicates that Poisso regressio is better tha ZIP regressio o the value of are 0.6, 0.8, ad 1 i each of p ad is tested. Value of are 6, 8, 10, ad 20 i each of the p ad is tested showed that ZIP regressio is better tha Poisso regressio. Table 5. Average of ARB agaits estimate θ 1 o Poisso ad ZIP regressio Poisso ZIP Poisso ZIP Poisso ZIP Poisso ZIP * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Average of ARB is smaller The result of RRMSE is similar with the ARB o estimate θ 1 i Table 6. MSE cotais two compoets, amely variace estimate (accuracy) ad the bias (accuracy) [11]. The estimatio with

8 ISSN: Iteratioal Joural of Advaces i Itelliget Iformatics 147 good character of MSE is that ca cotrols the variace ad bias. The big value of RRMSE show the big variace estimate, so the risk to the estimatio results, the accuracy estimatio is lower. Table 6. Average of RRMSE agaits estimate θ 1 o Poisso ad ZIP regressio Poisso ZIP Poisso ZIP Poisso ZIP Poisso ZIP * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Average of RRMSE is smaller The average of SAPR o the estimate y is gotte by Poisso ad ZIP regressio models. The estimate y i ZIP regressio model usig two models are discrete model for ad zero-iflatio model for p. I Table 7 shows that the larger,, ad p is tested, the the larger the value of SAPR. ZIP Regressio have the average of SAPR that is smaller tha Poisso regressio i every,, ad p is tested. Table 7. Average of SAPR agaits estimate y o Poisso ad ZIP regressio Poisso ZIP Poisso ZIP Poisso ZIP Poisso ZIP * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Average of SAPR is smaller

9 148 Iteratioal Joural of Advaces i Itelliget Iformatics ISSN: A good estimator should have a small bias ad variace. Poisso ad ZIP regressio compared based o the ability to cotrol bias ad variace estimate to get the high accuracy estimatio. Bias ad variace of Poisso ad ZIP regressio parameter estimate show o average of ARB ad RRMSE. The, to determie the predictability of Poisso ad ZIP regressio models show o average of SAPR o the value estimate y. Value of ARB ad RRMSE o estimate θ 1 i Poisso ad ZIP regressio idicates that the larger,, ad p is tested, the ZIP regressio is better tha Poisso regressio. The average of SAPR estimate y from the Poisso ad ZIP regressio model showed that ZIP regressio is better tha Poisso regressio i every,, ad p is tested. ZIP regressio is better tha Poisso regressio i solvig the overdispersio due to the may of excess zero value at the variable Y. The estimatio evaluatio results of Poisso ad ZIP regressio model i accordace with the results of the testig overdispersio, exploratio ad testig of the variable Y. VI. Coclusio Overdispersio study of the simulatio data from combiatio of,, p that is tested idicates that the greater,, ad p, the the scores test get excess the zero probability are gettig bigger ad chisquared test get percetage of Poisso distributio is gettig smaller. The compariso showed that ZIP regressio is better tha Poisso regressio based o the percetage of overdispersio ad ratio of dispersio, the value of ARB ad RRMSE o estimate θ 1, ad the average of SAPR o estimate y, alog with the growig,, ad p that is tested. VII. Ope Problem The desig of simulatio performed i this study started from geerate variable Y ad the variable X is based o the determiatio of parameters, so there is the relatioship betwee the two variables. I a subsequet study to do differet phases of desig simulatio ad there is a residual effect. Refereces [1] N. Jasakul ad J. P. Hide, Score tests for zero-iflated Poisso models, Comput. Stat. Data Aal., vol. 40, o. 1, pp , [2] N. Ismail ad A. A. Jemai, Hadlig overdispersio with egative biomial ad geeralized Poisso regressio models, i Casualty Actuarial Society Forum, 2007, pp [3] M. Ridout, C. G. B. Demétrio, ad J. Hide, Models for cout data with may zeros, i Proceedigs of the XIXth iteratioal biometric coferece, 1998, vol. 19, pp [4] D. Lambert, Zero-iflated Poisso regressio, with a applicatio to defects i maufacturig, Techometrics, vol. 34, o. 1, pp. 1 14, [5] M. Xie, B. He, ad T. N. Goh, Zero-iflated Poisso model i statistical process cotrol, Comput. Stat. Data Aal., vol. 38, o. 2, pp , [6] S. Numa, Aalysis of extra zero couts usig zero-iflated Poisso models, Price of Sogkla Uiversity, [7] J. W. Hardi, J. M. Hilbe, ad J. Hilbe, Geeralized liear models ad extesios. Stata press, [8] R. Savic ad M. Lavielle, Performace i populatio models for cout data, part II: a ew SAEM algorithm, J. Pharmacokiet. Pharmacody., vol. 36, o. 4, pp , [9] A. C. Camero ad P. K. Trivedi, Regressio aalysis of cout data, vol. 53. Cambridge uiversity press, [10] M. Fly ad L. A. Fracis, More flexible GLMs zero-iflated models ad hybrid models, Casualty Actuar. Soc, vol. 2009, pp , [11] G. Casella ad R. L. Berger, Statistical iferece, vol. 2. Duxbury Pacific Grove, CA, 2002.