Multiplicative-Binomial Distribution: Some Results on Characterization, Inference and Random Data Generation

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1 Joural of Statistical Theory ad Applicatios, Vol. 1, No. 1 (May 013), Multiplicative-Biomial Distributio: Some Results o Characterizatio, Iferece ad Radom Data Geeratio Elsayed A.H. Elamir Departmet of Mathematics ad Statistics, Faculty of Commerce, Beha Uiversity, Egypt & Maagemet ad Marketig Departmet, College of Busiess, Uiversity of Bahrai, P.O. Box 3038, Kigdom of Bahrai shabib40@gmail.com Multiplicative-biomial distributio is oe of the distributios that allows for over-dispersio, ad uderdispersio relative to the stadard biomial distributio. It will be show that the multiplicative-biomial distributio ca be a very useful model for these situatios. Moreover, the cofidece iterval for the parameters of the multiplicative-biomial distributio is ivestigated by the profile likelihood methods. The first four momets ad simulatio procedures for geeratig data from the multiplicative-biomial distributio usig R-software are give. By usig four applicatios to simulated ad real data it is show that the multiplicative-biomial distributio is the same as or outperformig the stadard biomial ad beta-biomial distributios. Keywords: biomial distributio; likelihood ratio test; over-dispersio; profile likelihood; zero-iflated data. 1. Itroductio The stadard biomial distributio provides a stadard framework for the aalysis of cout data uder the assumptios of idepedet trials ad fixed trial probability. I recet years there has bee cosiderable iterest i models for cout data that allows for over-dispersio, uder-dispersio ad excess zeros relative to the stadard biomial distributio. Applicatio areas are diverse; see, Lambert (199), Bowma ad George (1995), Kibria (006), Baik ad Kibria (009), Kolev ad Paiva (008) Siegel ad Castella (1988) ad Yu ad Zelterma (00, 008). The Loviso s multiplicative-biomial distributio (LMBD) is itroduced by Loviso (1998) as a geeralizatio of the stadard biomial distributio i the case of depedece amog the trials that allows for over-dispersio ad uder-dispersio ad icludes the stadard biomial distributio as a special case. I this paper, it will be argued that the LMBD could be used to model data with over-dispersio ad uder-dispersio relative to the stadard biomial distributio. Moreover, appropriateess of the stadard biomial distributio ad cofidece itervals for the LMBD parameters are ivestigated by the likelihood ratio test ad the profile likelihood method, respectively. The Published by Atlatis Press 9

2 E.A.H. Elamir first four momets ad simulatio procedures for geeratig radom variates from LMBD usig R- software are studied. Four applicatios to simulated ad real data ad comparisos with biomial ad beta-biomial distributios are give. I Sectio the LMBD is reviewed ad its momets are obtaied. The Likelihood ratio test ad profile likelihood for the LMBD are studied i Sectio 3. The procedures for geeratig radom variates from LMBD are give i Sectio 4. Four applicatios to simulated ad real data ad compariso with biomial ad beta-biomial distributios are preseted i Sectio 5. Sectio 6 is devoted to the coclusio.. Loviso s multiplicative-biomial distributio Cosider the radom vector Z =[Z 1,...,Z ], Z i beig a biary respose which measures whether some evet of iterest is preset, 1 or abset, 0 for a sample of uits ad let Y = i=1 Z i deote the sample frequecy of successes. To accommodate for the possible depedece amog trials ad uder the assumptio that the uits are exchageable, Loviso (1998) proposed a alterative form of the multiplicative-biomial distributio itroduced by Altham (1978) based o origial Cox s represetatio (197) as P(Y = y)= ( ) ψ y (1 ψ) y ω y( y) y (, y = 0,1,...,, (.1) )ψ t t (1 ψ) t ω t( t) t=0 where 0 < ψ < 1adω > 0 are the parameters; for more details about this distributio; see, Loviso (1998). This distributio provides a wider rage of distributios tha is provided by the stadard biomial distributio ad icludes the stadard biomial distributio as a special case at ω = 1. Figures 1 ad show the distributio of Y for = 10, ψ = 0.5, 0.95, ad differet values of ω. For small values of ψ ad ω < 1 the distributio gives higher probability for zero tha the stadard biomial distributio. Also, for large values of ψ ad ω > 1 the distributio gives higher probability for values at..1. Momets of LMBD Let (i) = ( 1) ( i + 1), (0) = 1, π i = ψ i κ i(ψ,ω) κ (ψ,ω), ad κ a (ψ,ω)= a x=0 ( ) a ψ x (1 ψ) ( a x) ω ( a x)(x+a) x Published by Atlatis Press 93

3 Multiplicative-Biomial Distributio Fig. 1. LMB distributio for selected values of ω = 1, 0.9, 0.7, ad 0.56 ψ = 0.5 ad = 10. The factorial momets for LMBD ca be obtaied as ( ) ψ y (1 ψ) y ω y( y) y μ (i) = E(Y (Y i + 1)) = y (y i + 1) ( = (i) π i. (.) i=0 )ψ t t (1 ψ) t ω t( t) For example, μ () = E(Y (Y 1)) = i=0 The first four o-cetral momets are ( 1) y(y 1) t=0 y(y 1) t=0 μ 1 = (1) π 1 = E(Y )=μ, ( ) ψ y (1 ψ) y ω y( y) y ( = () π )ψ t t (1 ψ) t ω t( t) Published by Atlatis Press 94

4 E.A.H. Elamir Fig.. LMB distributio for selected values of ω = 0.85, 1, 1.5, ad, ad ψ = 0.95 ad = 10. μ = (1) π 1 + () π, μ 3 = (1) π () π + (3) π 3, ad μ 4 = (1) π () π + 6 (3) π 3 + (4) π 4, The cetral momets are μ = (1) π 1 + () π (1) π 1 = V (Y )=σ, μ 3 = [ (1) π () π + (3) π 3 ] 3(1) π 1 [ (1) π 1 + () π ] + 3 (1) π 3 1, ad μ 4 = [ (1) π () π + 6 (3) π 3 + (4) π 4 ] 4(1) π 1 [ (1) π () π + (3) π 3 ] + 6 (1) π 1 [ (1)π 1 + () π ] 3 4 (1) π4 1 Published by Atlatis Press 95

5 Multiplicative-Biomial Distributio Fig. 3. Skewess γ of LMBD with selected values of ω = 0.5, 0.8, 1, ad 1.5 over the rage ψ =(0, 1) ad = 10. The skewess measure is The kurtosis measure is γ = μ 3 (σ ) 3/. β = μ 4 μ. Figures 3 ad 4 show the skewess ad kurtosis measures for some selected values of ψ ad ω at = 10. From the expected value the relatioships amog, p, ψ,adω are p = P(Z = 1)=ψ κ 1(ψ,ω) κ (ψ,ω), ad ψ = p κ (ψ,ω) κ 1 (ψ,ω). Thus p ca be cosidered as a probability of success ad ψ as a probability of success weighted by the itra-uits associatio measure ω goverig the depedece amog the biary resposes of Published by Atlatis Press 96

6 E.A.H. Elamir Fig. 4. Kurtosis β of LMBD with selected values of ω = 1, 0.8, ad 1.5 over the rage ψ =(0, 1) ad = 10. the uits. If ω = 1thep = ψ represets the probability of success of the uits whe they are idepedet. If ω 1 the biary uits are ot idepedet ad therefore p ψ. 3. Likelihood iferece o the parameters of the LMBD Whe the LMBD fits the data, the profile likelihood ad the likelihood ratio ca be used to test the appropriateess of the stadard biomial distributio ad the profile likelihood ca be used for estimatig the cofidece itervals for the LMBD parameters Cofidece itervals based o the profile likelihood Whe the aalyst is iterested i oe of the parameters, the profile likelihood is a useful geeral tool obtaied by maximizig out the remaiig parameters for each fixed value of the parameter of iterest. It provides a graphical summary of the most plausible values of the parameters of iterest ad ca provide cofidece itervals that are more accurate tha the stadard itervals based o ormal theory; see, Pawita (001), Severii (000) ad Vezo ad Moolgavkar (1988). From Published by Atlatis Press 97

7 Multiplicative-Biomial Distributio Habib (010) the logarithm of the likelihood of the multiplicative-biomial ca be writte as l(ψ,ω)=logr! + f y log P(y;ψ,ω) log f y! Note that R is the umber of sets of trials, f y the umber of sets givig y success, ad P(y;ψ,ω) is the LMBD. Give the joit likelihood L(ψ,ω) the profile likelihood of ω ca be writte as l p (ω)=maxl(ψ,ω), (3.1) ψ where maximizatio of ψ is performed at a fixed value of ω. To obtai the cofidece iterval for ω, the profile likelihood is assumed to be a icreasig fuctio to the left of its maximum ad the followig steps are used: 1. Compute the ML estimates ( ψ, ω) ad the correspodig log-likelihood value,. Compute a reasoable lower ad upper boud ω 1 ad ω, respectively (e.g., ω 5 SE( ω) ad) ω + 5 SE( ω), 3. Defie a grid of values ragig from ω 1 to ω. 4. For each grid value ω i compute the profile log-likelihood value l p (ω i ) by maximizig the l(ψ,ω i ) over ψ-values, 5. Cosider the smallest value of ω i as the lower boud (ω L ) for the (1 α)% CI such that l p (ω i ) l p ( ω) χ 1,1 α = l( ψ, ω) χ 1,1 α. 6. Cosider the largest value of ω i as the upper boud (ω U ) for the (1 α)% CI such that l p (ω i ) l p ( ω) χ 1,1 α = l( ψ, ω) χ 1,1 α. If ecessary, refie or exted the grid of values aroud ω L ad ω U to obtai greater accuracy. Similarly i which ψ is cosidered as the parameter of iterest ad ω is the uisace parameter, the profile likelihood of ψ ca be writte as l p (ψ)=maxl(ψ,ω). (3.) ω To obtai the cofidece iterval for ψ the followig steps ca be used: 1. Compute the ML estimates ( ψ, ω) ad the correspodig log-likelihood value,. Compute a reasoable lower ad upper boud ψ 1 ad ψ respectively, (e.g., ψ 5 SE( ψ) ad) ψ + 5 SE( ψ), 3. Defie a grid of values ragig from ψ 1 to ψ. 4. For each grid value ψ i compute the profile log-likelihood value l p (ψ i ) by maximizig the l(ψ i,ω) over ω-values, 5. Cosider the smallest value of ψ i as the lower boud (ψ L ) for the (1 α)% CI such that l p (ψ i ) l p ( ψ) χ 1,1 α = l( ψ, ω) χ 1,1 α, 6. Cosider the largest value of ψ i as the upper boud (ψ U ) for the (1 α)% CI such that l p (ψ L ) l p ( ψ) χ 1,1 α = l( ψ, ω) χ 1,1 α. If ecessary, refie or exted the grid of values aroud ψ L ad ψ U to obtai greater accuracy. Published by Atlatis Press 98

8 E.A.H. Elamir 3.. Likelihood ratio test for appropriateess of the stadard biomial versus LMBD Cosider a LMB distributio with likelihood fuctio L(ψ,ω) ad ( ψ, ω) the maximum likelihood (ML) estimates. The likelihood ratio test statistic (Λ) of the hypothesis H 0 : ω = 1agaistH 1 : ω 1 is Λ = [ll( ψ, ω) ll( ψ,1)] = [l( ψ, ω) l( ψ,1)] (3.3) ψ is the ML estimate of the reduced model. From (3.1) the model uder the alterative hypothesis is l( ψ, ω)=logr! + The model uder the ull hypothesis is Hece, Therefore, l( ψ,1)=logr! + l( ψ, ω) l( ψ,1)= Λ = f y logp(y; ψ, ω) f y logp(y; ψ,1) f y logp(y; ψ, ω) log f y!. log f y!. f y [ logp(y; ψ, ω) logp(y; ψ,1) ]. f y logp(y; ψ,1). Uder the ull hypothesis, the test statistic Λ approximately follows a chi-square distributio with oe degree of freedom; see, Severii (000). 4. Geeratig radom data from LMBD usig R The probability mass fuctio for the LMB distributio is ( ) ψ y (1 ψ) y ω y( y) y P(y;ψ,ω)= (, y = 0,...,. )ψ t t (1 ψ) t ω t( t) t=0 Geeratig radom data from this distributio for give, ψ, ad ω ca easily be accomplished i R by usig the probability mass fuctio ad the sample commad, which has the followig geeral sytax. >\#x:a vector of elemets from which to choose, size: \# of item to choose >sample(x, size, replace = FALSE, prob = NULL) Example 4.1. This example is to draw ad sample data from LMB distributio with = 4, ψ = 0.0, ad ω = From the probability mass fuctio of LMBD we obtai the probability distributio i Table 1. The followig commad is used. Published by Atlatis Press 99

9 Multiplicative-Biomial Distributio Table 1. Exact ad simulated mea ad variace from LMBD with = 4, ψ = 0.0, ad ω = y P(y) Simulated Size Exact Mea Variace Skewess >x=sample(c(0,1,,3,4),size,replace=true,prob= c( , , , , )) The size ca be replaced by the desired value of ad ad therefore the variable x cotais radom data draw from the LMBD with = 4, ψ = 0.0, ad ω = The mea, the variace ad skewess of simulated data are compared with exact distributio i Table 1. A program i R-software to geerate radom data from LMBD (lmbd.sim) is give as follows: >MBD.dist=fuctio(,y,p,w) \{\#prob. of LMB distributio \#:\#of trials, y:radom variable, p:psi value, w:omega value\# t=0:; t1=choose(,y)*p^y*(1-p)^(-y) *w^(y*(-y)); t=choose(,t)*p^t*(1-p)^(-t) *w^(t*(-t)); t1/sum(t) \} >lmbd.sim=fuctio(r,,p,w)\{\#simulatio data from LMBD \#r:\# of observatios, :size, p:psi value, w:omega value yr=0:; pr=mbd.dist(,yr,p,w); sample(yr,r,replace=true,prob=pr)\} 5. Applicatios to simulated ad real data Four applicatios to simulated ad real data are studied. The first applicatio deals with data simulated from a stadard biomial. The secod applicatio presets data comig from a distributio which has variace less tha the stadard biomial. The third ad fourth applicatios deal with real data with excess-zeros, which usually have more variace tha the stadard biomial. Moreover the useful ad commo-used beta-biomial distributio that is defied as ( ) B(α + x,β + x) f (x;α,β)=, x = 0,1,...,, x B(α,β) is compared with LMBD i both simulated ad real data aalysis. The package VGAM i R- software is used to estimate the parameters α ad β. Published by Atlatis Press 100

10 E.A.H. Elamir Table. Simulated data from biomial distributio with = 4adp = y Total Obs E. LMBD { } X = E. biomial { } X = 0.03 E. Beta-biomial { } X = Obs.: observed ad E.: Expected value. Fig. 5. Profile likelihood ad 95% CI: (ω L,ω U ) for the data i Applicatio Applicatio 1: Simulated data from biomial The data are simulated from a stadard biomial distributio with = 4ad p = The expected values ad Pearso s X statistic are give i Table. From Habib (010) the maximum likelihood estimates for LMBD parameters are ψ = ad ω = The values of X statistic i Table show a suitable fit for the data by all the distributios. Figure 5 shows the profile log-likelihood fuctio for ω with its maximal value of 9.373, ad the 95% cofidece iterval (0.857, 1.116). Clearly, this C.I. suggests that a stadard biomial distributio would be a appropriate model for the data, sice ω = 1 is icluded i the iterval. This ca also be see by the likelihood ratio test, Λ = (0.087)= idicatig the appropriateess of the stadard biomial model. 5.. Applicatio : Simulated data from LMBD The data are simulated from a LMBD with = 6, ψ = 0.5, ω = 1.5, ad variace The expected values ad X statistics are give i Table 3. From Habib (010) the maximum likelihood estimates for LMBD parameters are ψ = ad ω = 1.. The estimate for p is The values of X statistics i Table 3 show a suitable fit for the data by the LMBD but ot by the Published by Atlatis Press 101

11 Multiplicative-Biomial Distributio Table 3. Simulated data from LMBD with = 6, ψ = 0.5, ad ω = 1.5. y Total Obs E. LMBD { } { } X = 0.44 E. Biomial X = 38.8 E. Beta-biomial Not defied Fig. 6. Profile likelihood ad 99% CI: (ω L,ω U ) for the data i Applicatio. biomial model ad the parameters of beta-biomial distributio caot be estimated usig package VGAM. Figure 6 shows the profile log-likelihood fuctio for ω with its maximal value of 14.57, ad the 99% cofidece iterval (1.1, 1.33). Clearly, this C.I. suggests that a stadard biomial distributio would ot be a appropriate model for the data, sice ω = 1 is ot icluded i the iterval. This ca also be see by the likelihood ratio test Λ = (.44) = Applicatio 3: Traffic accidet research Kua et al. (1991) discuss data comig from the Califoria Departmet of Motor Vehicles master driver licese file. The variable of iterest is the umber of accidets per driver. A possible motivatio ca be see i the possibility of fidig risk factors ivolved i the accidets. The data, the expected values ad X are give i Table 4. These data have mea 0.03 ad variace 0.35 ad there is a clear spike of extra zeros (about 83%) represetig the accidet-free drivers. From Habib (010) the maximum likelihood estimates for LMBD parameters are ψ = ad ω = The estimate for p is The values of X statistics i Table 4 show a suitable fit for the data by the LMB ad beta-biomial distributios but ot by the biomial distributio. Figure 7 shows the profile log-likelihood fuctio for ω with its maximal value of 10.10, ad the 95% cofidece iterval (0.539, 0.6). Clearly, this C.I. suggests that a stadard biomial Published by Atlatis Press 10

12 E.A.H. Elamir Table 4. Distributio of the umber of accidets per deriver. umber of accidets Total Number of drivers Relative frequecy E. LMB X = 0.86 E. biomial { } X = E. Beta-biomial X = 1.36 Fig. 7. Profile likelihood fuctio ad 95% CI: (ω L,ω U ) for the traffic accidet research data. distributio would ot be a appropriate model for these data. This ca also be see by the likelihood ratio test Λ = (80.16)= Applicatio 4: Crime Sociology I a study o deviatig behavior Bohig et al. (1997) provide a aalysis of a data set o 4301 persos with crimial behavior. The variable of iterest is the umber of crimial acts per perso. The motivatio for such a study might be to fid factors leadig to deviatig behavior. The data, the expected values ad X statistic are give i Table 5. These data have mea ad variace There is a clear spike of extra zeros (about 94%) represetig acts-free crimials. From Habib (010) the maximum likelihood estimates for LMBD parameters are ψ = 0.03 ad ω = The estimate for p is The values of X statistics i Table 5 show a suitable fit for the data by the LMBD but ot by the biomial model for all level of sigificace ad the beta-biomial distributio at 10% ad 5% level of sigificace. Figure 8 shows the profile log-likelihood fuctio for ω with its maximal value of 13.3, ad the 95% cofidece iterval (0.435, 0.485). Agai, this C.I. suggests that a stadard biomial Published by Atlatis Press 103

13 Multiplicative-Biomial Distributio Table 5. Distributio of the umber of crimial acts. #ofcrimial acts Total # of persos Relative frequecy E. LMBD { } X =.55 E. Biomial { } X = E. Beta-biomial { } X = 8.1 Fig. 8. Profile likelihood fuctio ad 95% CI: (ω L,ω U ) for the crime sociology data. distributio would ot be a appropriate model for these data. This ca also be see by the likelihood ratio test Λ = (114.95) = Coclusio It has bee show that the LMB distributio ca be very useful i a variety of applicatios sice it allows for over-dispersio ad uder-dispersio relative to the stadard biomial distributio. The first four o-cetral ad cetral momets are obtaied. Moreover, the LMBD has bee used for testig the appropriateess of the stadard biomial distributio to the data usig the likelihood ratio test ad the cofidece iterval has bee obtaied for its parameters usig the profile likelihood. The simulatio procedures for geeratig data from multiplicative-biomial distributio usig R-software are illustrated. It is show that the LMB distributio could outperform the stadard biomial ad beta-biomial distributios usig four applicatios to simulated ad real data. Ackowledgmet The author is greatly grateful to two reviewers for careful readig, valuable commets ad highly costructive suggestios that have led to clarity, better presetatio ad improvemets i the mauscript. Published by Atlatis Press 104

14 E.A.H. Elamir Refereces [1] Altham, P. (1978). Two geeralizatios of the biomial distributio. Applied Statistics, 7, [] Bohig, D., Dietz, E. ad Schlattma (1997). Zero-iflated cout models ad their applicatios i public health ad social sciece. I: Applicatios of latet trait ad latet class models i the social scieces, ed. by Jurge R. ad Lageheie, WAXMANN Verlag. [3] Baik, S. ad Kibria, B.M.G. (009). O some discrete distributios ad their applicatios with real life data. Joural of Moder Applied Statistical Methods, 8, [4] Bowma, D. ad George, E.O. (1995). A saturated model for aalyzig exchageable biary data: applicatio to cliical ad developmet toxicity studies. Joural of America Statistical Associatio, 90, [5] Cox, D.R. (197). The aalysis of multivariate biary data. Applied Statistics, 1, [6] Habib E.A. (010). Estimatio of log-liear biomial distributio with applicatios. Joural of Probability ad Statistics, 10, [7] Kolev, N. ad Paiva, D. (008). Radom sums of exchageable variables ad actuarial applicatios. Isurace: Mathematics ad Ecoomics, 14, [8] Kibria, B.M.G. (006). Applicatios of some discrete regressio models for cout data. Pakista Joural of Statistics ad Operatio Research,, [9] Kua, J., Peck, R.C. ad Jake, M.K. (1991). Statistical methods for traffic accidet research, i: Proceedigs of the 1990 Taipei Symposium i Statistics, Jue 8-30, 1990, ed. by Mi-Te Chao ad Philip E. Cheg. Taipei, Istitute of Statistical Sciece. [10] Lambert, D. (199). Zero-iflated Poisso regressio, with applicatio to defects i maufacturig. Techometrics, 34, [11] Loviso, G. (1998). A alterative represetatio of Altham s multiplicative-biomial distributio. Statistics & Probability Letters, 36, [1] Pawita, Y. (001). I all likelihood: Statistical modelig ad iferece usig likelihood. (1 st Ed.), Claredo Press, Oxford. [13] Severii, T.A. (000). Likelihood methods i statistics. (1 st Ed.), Oxford Uiversity Press. [14] Siegel, S., ad Castella, N.J. (988). Noparametric statistics for the behavioral scieces( d Ed.), New York, NY: McGraw-Hill. [15] Vezo, D.J. ad Moolgavkar, H. (1988). A Method for computig profile-likelihood-based cofidece itervals. Applied Statistics, 37, [16] Yu, C. ad Zelterma, D. (00). Sums of depedet Beroulli radom variables ad disease clusterig. Statistics & Probability Letters, 57, [17] Yu, C. ad Zelterma, D. (008). Sums of exchageable Beroulli radom variables for family ad litter frequecy data. Computatioal Statistics & Data Aalysis, 5, Published by Atlatis Press 105