Negative Binomial Regression Analysis And other count models

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1 Negatve Bnomal Regresson Analyss And other count models Asst. Prof. Nkom Thanomseng Department of Bostatstcs & Demography Faculty of Publc Health, Khon Kaen Unversty Emal: Web: Negatve Bnomal Regresson Analyss & other count Outlnes: Negatve Bnomal regresson Problem of Zero Counts Zero nflated Posson (zp Zero nflated negatve Bnomal (znb Comparson of Models Test of Comparatve Ft Other count data models 204 Department of Bostatstcs & Demography, Faculty of Publc Health, Khon Kaen Unversty Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB The earlest defntons of the negatve bnomal are based on the bnomal PDF. NB2 (Cameron and Trved, 986, NB2 s derved from a Posson gamma mxture dstrbuton. NB, The NB model can also be derved as a form of Posson gamma mxture, but wth dfferent propertes resultng n a lnear varance. The negatve bnomal model, as a Posson gamma mxture model, s approprate to use when the overdsperson n an otherwse Posson model s thought to take the form of a gamma shape or dstrbuton. A more general class of negatve bnomal models wth mean μ and varance functon (μ + αμ p. NB2 wth p = 2, NB wth p=. Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2 NB2 (Cameron and Trved, 986, NB2 s derved from a Posson gamma mxture dstrbuton. The NB2 model, wth p = 2, s the standard formulaton of the negatve bnomal model NB2 varance functon μ + αμ 2 It has densty. y ( y f ( y, ( ( y 0, y 0,, 2, Ths reduces to the Posson f α = 0 Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2 The log-lkelhood functon for NB2 ln L(, n y ln( j ( y ln( exp( x y ln y x j0 ln y! NB, The NB model can also be derved as a form of Posson gamma mxture, but wth dfferent propertes resultng n a lnear varance. The negatve bnomal model, as a Posson gamma mxture model, s approprate to use when the overdsperson n an otherwse Posson model s thought to take the form of a gamma shape or dstrbuton. Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2: Example A comparson of fnancal performance, organzatonal characterstcs and management strategy among rural & urban faclltes. (Smth, HL., Pland, NF. & Fsher, N. J. Rural Health, 27-40, 992 Sample: Lcensed Nurse n=52 bed = number of beds n home, tdays = annual total patent days (n hundreds pcrev = annual total patent care revenue(n $ mllons nsal = annual nursng salares(n $ mllons fexp = annual facltes expendtures(n $ mllons rural = ( = rural; 0 = nonrural

2 Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2: nbreg. nbreg bed pcrev nsal fexp rural pn pf nf, exp(tdays d(mean Fttng Posson model: Negatve bnomal regresson Number of obs = 52 LR ch2(7 = 7.60 Dsperson = mean Prob > ch2 = Log lkelhood = Pseudo R2 = bed pcrev nsal fexp rural pn pf nf _cons tdays (exposure /lnalpha alpha Lkelhood-rato test of alpha=0: chbar2(0 = Prob>=chbar2 = Neagatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2: glm. glm bed pcrev nsal fexp rural pn pf nf, exp(tdays f(nb l(log Iteraton 0: log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Generalzed lnear models No. of obs = 52 Optmzaton : ML Resdual df = 44 Scale parameter = Devance = (/df Devance = Pearson = (/df Pearson = Varance functon: V(u = u+( u^2 [Neg. Bnomal] Lnk functon : g(u = ln(u [Log] AIC = Log lkelhood = BIC = OIM bed pcrev nsal fexp rural pn pf nf _cons tdays (exposure Neagatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2: glm (Stata +. glm bed pcrev nsal fexp rural pn pf nf, exp(tdays f(nb ml l(log Iteraton 0: log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Generalzed lnear models No. of obs = 52 Optmzaton : ML Resdual df = 44 Scale parameter = Devance = (/df Devance = Pearson = (/df Pearson = Varance functon: V(u = u+(.03u^2 [Neg. Bnomal] Lnk functon : g(u = ln(u [Log] AIC = Log lkelhood = BIC = OIM bed pcrev nsal fexp rural pn pf nf _cons ln(tdays (exposure Note: Negatve bnomal parameter estmated va ML and treated as fxed once estmated. Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2: Interpretaton usng the rate Methods of nterpretaton based on E(y x E( k e IRR E( The nterpretaton For a change of n x k f, the expected count ncreases by a factor of exp( k x, holdng all other varables constant. -For specfc values of Factor change. For a unt change n x k, the expected count changes by a factor of exp( k, holdng all other varables constant. Standardze factor change. For a standard devaton change to x k, the expected count changes by a factor of exp( k x s k, holdng all other varables constant. Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2: Interpretaton usng percentage Alternatvely, the percentage change th the expected count for a unt change n x k, holdng other varables constant. Methods of nterpretaton based on E(y x E E( ( x 00 [exp E( x ( k k ] x 00 The nterpretaton For a factor x k, the expected count ncreases (decreases by n% [exp(k-]x00, holdng all other varables constant. Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2: Interpretaton usng the rate. nbreg bed pcrev nsal fexp rural pn pf nf, exp(tdays d(mean rr Negatve bnomal regresson Number of obs = 52 LR ch2(7 = 7.60 Dsperson = mean Prob > ch2 = Log lkelhood = Pseudo R2 = bed IRR Std. Err. z P> z [95% Conf. Interval] pcrev nsal fexp rural pn pf nf tdays (exposure /lnalpha alpha Lkelhood-rato test of alpha=0: chbar2(0 = Prob>=chbar2 =

3 Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2: Interpretaton usng the rate. nbreg bed pcrev nsal fexp rural pn pf nf, exp(tdays d(mean rr. lstcoef,help nbreg (N=52: Factor Change n Expected Count Observed SD: bed b z P> z e^b e^bstdx SDofX pcrev nsal fexp rural pn pf nf ln alpha alpha SE(alpha = LR test of alpha=0: Prob>=LRX2 = b = raw coeffcent e^b = exp(b = factor change n expected count for unt ncrease n X e^bstdx = exp(b*sd of X = change n expected count for SD ncrease n X SDofX = standard devaton of X Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2: Interpretaton usng the rate. nbreg bed pcrev nsal fexp rural pn pf nf, exp(tdays d(m rr. lstcoef,help percent nbreg (N=52: Percentage Change n Expected Count Observed SD: bed b z P> z % %StdX SDofX pcrev nsal fexp rural pn pf nf ln alpha alpha SE(alpha = LR test of alpha=0: Prob>=LRX2 = b = raw coeffcent % = percent change n expected count for unt ncrease n X %StdX = percent change n expected count for SD ncrease n X SDofX = standard devaton of X Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2: Interpretaton usng the rate Interpretaton based on Incdnce rate rato Beng a annual total patent care revenue decreases the expected number of beds n home by.6792, holdng all other varables constant. Interpreataton based on percentage Beng a annual total patent care revenue decreases the expected number of beds n home by 32.%, holdng all other varables constant. Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB NB, The NB model can also be derved as a form of Posson gamma mxture, but wth dfferent propertes resultng n a lnear varance. The NB model, whch sets p =, s also of nterest because t has the same varance functon, ( + αμ = μ, as that used n the GLM approach. The NB log-lkelhood functon s ln L(, n y ln( j exp( x j0 ln y! ( y exp( x ln( y ln Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB: nbreg. nbreg bed pcrev nsal fexp rural pn pf nf, exp(tdays d(c Fttng Posson model: Iteraton 0: log lkelhood = Iteraton 4: log lkelhood = Negatve bnomal regresson Number of obs = 52 LR ch2(7 = 4.50 Dsperson = constant Prob > ch2 = Log lkelhood = Pseudo R2 = bed pcrev nsal fexp rural pn pf nf _cons tdays (exposure /lndelta delta Lkelhood-rato test of delta=0: chbar2(0 = 8.44 Prob>=chbar2 = Negatve Bnomal Regresson Analyss Negatve Bnomal Regresson (NB2: glm,(nb. glm bed pcrev nsal fexp rural pn pf nf, exp(tdays f(nb l(log Iteraton 0: log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Generalzed lnear models No. of obs = 52 Optmzaton : ML Resdual df = 44 Scale parameter = Devance = (/df Devance = Pearson = (/df Pearson = Varance functon: V(u = u+(u^2 [Neg. Bnomal] Lnk functon : g(u = ln(u [Log] AIC =.2560 Log lkelhood = BIC = OIM bed pcrev nsal fexp rural pn pf nf _cons tdays (exposure

4 Problem of Zero n Counts Model Problem of Zero counts Count response models havng for more zeros than expected by dstrbutonal assumptons of Posson and Negatve bnomal models result ncorrect & based. Incorrect parameter estmates Based standard Error. Cause of Overdsperson Zero Inflated Posson Regresson Model Zero Inflated Posson (ZIP Zero-nflated count models were frst ntroduced by Lambert (992 to provde another method of accountng for excessve zero counts. ZIP are two-part models, consstng of both bnary and count model sectons. (provde for the modelng of zero counts usng both bnary and count processes. Let the response Y denote a non-negatve nteger count for the th observaton, =,,N. Zero Inflated Posson Model Probablty of Zero Inflated Posson The probablty of an excess zero s denoted by π, 0, the random varable Y follows a ZIP dstrbuton f E Y ( e, y 0 Pr( Y y y e (, y,2,,! 0 Var Y 2 ( ( ; ( Zero Inflated Negatve Bnomal Model Zero Inflated Negatve Bnomal (ZINB Let the response Y denote a non-negatve nteger count for the th observaton, =,,N. then ZINB dstrbuton k ( (, y 0 k Pr( Y y y k ( k y k (, ( (! k y k k 0 E(Y = ( λ and Var(Y = ( λ (+(κ+ λ, where κ s an overdsperson parameter y 0 Zero Inflated Negatve Bnomal Model Zero Inflated Negatve Bnomal (ZINB Example: Synthetc NB2 data :STATA (Hlbe,20. tab y -> tabulaton of y y Freq. Percent Cum. 0 20, , , , , , Total 50, ZIP & ZINB Model ZIP & ZINB: example Example: Synthetc NB2 data :STATA (Hlbe,20. tab y -> tabulaton of y y Freq. Percent Cum. 0 20, , , , , , Total 50,

5 Zero Inflated Posson Model Zero Inflated Posson Example: zp. zp y x x2, nflate(x x2 Fttng constant-only model: Iteraton 0: log lkelhood = Iteraton 4: log lkelhood = Fttng full model: Iteraton 0: log lkelhood = Iteraton 4: log lkelhood = Zero-nflated Posson regresson Number of obs = Inflaton model = logt LR ch2(2 = Log lkelhood = Prob > ch2 = y x x _cons nflate x x _cons Zero Inflated Negatve Bnomal Model Zero Inflated Negatve Bnomal Example: znb. znb y x x2, nflate(x x2 Zero-nflated negatve bnomal regresson Number of obs = Inflaton model = logt LR ch2(2 = Log lkelhood = Prob > ch2 = y x x _cons nflate x x _cons /lnalpha alpha Zero Inflated Posson Regresson Model Zero nflated Posson Model (ZIP: Interpretaton Interpretaton based on Posson Model Posson Model, contans coeffcents for the factor change n expected count for those n the Not Always Zero group. constant. The coeffcents can be nterpreted n the same way as coeffcent from the Posson Regresson Model. Interpretaton based on Bnary Logt Model Bnary Logt Model, contans coeffcents for the factor change n the odds of beng n the Always Zero group compared wth the Not Always Zero group. The coeffcents nterpreted n the same way as coeffcents for a bnary logt model Zero Inflated Negatve Bnomal Model Zero nflated Negatve Bnomal Model (ZINB: Interpretaton Interpretaton based on Negatve Bnomal Model NB Model, contans coeffcents for the factor change n expected count for those n the Not Always Zero group. The coeffcents can be nterpreted n the same way as coeffcent from the Negatve Bnomal Model. Interpretaton based on Bnary Logt Model Bnary Logt Model, contans coeffcents for the factor change n the odds of beng n the Always Zero group compared wth the Not Always Zero group. The coeffcents nterpreted n the same way as coeffcents for a bnary logt model Zero Inflated Posson Regresson Model Zero nflated Posson Model (ZIP: Example Interpretaton. zp y x x2, nflate(x x2. lstcoef, help zp (N=50000: Factor Change n Expected Count Observed SD: Count Equaton: Factor Change n Expected Count for Those Not Always 0 y b z P> z e^b e^bstdx SDofX x x b = raw coeffcent e^b = exp(b = factor change n expected count for unt ncrease n X e^bstdx = exp(b*sd of X = change n expected count for SD ncrease n X SDofX = standard devaton of X Bnary Equaton: Factor Change n Odds of Always 0 Always0 b z P> z e^b e^bstdx SDofX x x b = raw coeffcent e^b = exp(b = factor change n odds for unt ncrease n X e^bstdx = exp(b*sd of X = change n odds for SD ncrease n X SDofX = standard devaton of X Zero Inflated Posson Regresson Model Zero nflated Posson Model (ZIP: Example Interpretaton. lstcoef, help percent zp (N=50000: Percentage Change n Expected Count Observed SD: Count Equaton: Percentage Change n Expected Count for Those Not Always 0 y b z P> z % %StdX SDofX x x b = raw coeffcent % = percent change n expected count for unt ncrease n X %StdX = percent change n expected count for SD ncrease n X SDofX = standard devaton of X Bnary Equaton: Factor Change n Odds of Always 0 Always0 b z P> z % %StdX SDofX x x b = raw coeffcent % = percent change n odds for unt ncrease n X %StdX = percent change n odds for SD ncrease n X SDofX = standard devaton of X 5

6 Zero Inflated Negatve Bnomal Model Zero Inflated Negatve Bnomal Model (ZINB: Example Interpretaton. znb y x x2, nflate(x x2. lstcoef, help znb (N=50000: Factor Change n Expected Count Observed SD: Count Equaton: Factor Change n Expected Count for Those Not Always 0 y b z P> z e^b e^bstdx SDofX x x ln alpha alpha SE(alpha = b = raw coeffcent e^b = exp(b = factor change n expected count for unt ncrease n X e^bstdx = exp(b*sd of X = change n expected count for SD ncrease n X SDofX = standard devaton of X Bnary Equaton: Factor Change n Odds of Always 0 Always0 b z P> z e^b e^bstdx SDofX x x b = raw coeffcent e^b = exp(b = factor change n odds for unt ncrease n X e^bstdx = exp(b*sd of X = change n odds for SD ncrease n X SDofX = standard devaton of X Zero Inflated Negatve Bnomal Model Zero Inflated Negatve Bnomal Model (ZINB: Example Interpretaton. znb y x x2, nflate(x x2. lstcoef, help percent znb (N=50000: Percentage Change n Expected Count Observed SD: Count Equaton: Percentage Change n Expected Count for Those Not Always 0 y b z P> z % %StdX SDofX x x ln alpha alpha SE(alpha = b = raw coeffcent % = percent change n expected count for unt ncrease n X %StdX = percent change n expected count for SD ncrease n X SDofX = standard devaton of X Bnary Equaton: Factor Change n Odds of Always 0 Always0 b z P> z % %StdX SDofX x x b = raw coeffcent % = percent change n odds for unt ncrease n X %StdX = percent change n odds for SD ncrease n X SDofX = standard devaton of X Test of Comparatve Ft Test comparatve: Vuong test The standard ft test for ZINB s the Vuong test (Vuong, Comparatve of Standard Posson & ZIP - Comparatve of ZINB & ZIP nu P ; ln V u SD( u P u the mean & SD( u standard devaton ZIP ZINP ( y x ( y x Test of Comparatve ft Comparatve test: Zero Inflated Posson VS ZIP. zp y x x2, nflate(x x2 vuong Fttng constant-only model: Zero-nflated Posson regresson Number of obs = Inflaton model = logt LR ch2(2 = Log lkelhood = Prob > ch2 = y x x _cons nflate x x _cons Vuong test of zp vs. standard Posson: z = 39.0 Pr>z = Test of Comparatve ft Comparatve test: Zero Inflated Negatve Bnomal VS NB. znb y x x2, nflate(x x2 vuong zp Zero-nflated negatve bnomal regresson Number of obs = Inflaton model = logt LR ch2(2 = Log lkelhood = Prob > ch2 = y x x _cons nflate x x _cons /lnalpha alpha Lkelhood-rato test of alpha=0: chbar2(0 = Pr>=chbar2 = Vuong test of znb vs. standard negatve bnomal: z = 0.86 Pr>z = Comparson of Models Comparson model: Graph & statstcs across models Summary statstcs across models: BIC, AIC, lkelhood Rato Test, Voung test Graph Dfference between the observed and predcted probablty for the PRM, NB2, ZIP & ZINB models (Long & Freese,

7 Comparson of Models Comparson model: countft (Graph & statstcs across models Summary statstcs across models: BIC, AIC, lkelhood Rato Test, Voung test Graph Dfference between the observed and predcted probablty for the PRM, NB2, ZIP & ZINB models. countft y x x2, gen(base_ nflate(x x2 maxcount(0 /// prm nbreg zp znb nodash Comparson of Mean Observed and Predcted Count Maxmum At Mean Model Dfference Value Dff Base_PRM Base_NBRM Base_ZIP Base_ZINB Tests and Ft Statstcs Comparson of Models Comparson model: countft (Graph & statstcs across models Tests and Ft Statstcs Base_PRM BIC= AIC= Prefer Over Evdence --- vs Base_NBRM BIC= df= NBRM PRM Very strong AIC= df= 0.37 NBRM PRM LRX2= prob= NBRM PRM p= vs Base_ZIP BIC= df= ZIP PRM Very strong AIC= df= 0.76 ZIP PRM Vuong= prob= ZIP PRM p= vs Base_ZINB BIC= df= ZINB PRM Very strong AIC= df= ZINB PRM --- Base_NBRM BIC= AIC= Prefer Over Evdence --- vs Base_ZIP BIC= df= NBRM ZIP Very strong AIC= df= -0.4 NBRM ZIP --- vs Base_ZINB BIC= df= NBRM ZINB Very strong AIC= df= NBRM ZINB Vuong= prob= ZINB NBRM p= Base_ZIP BIC= AIC= Prefer Over Evdence --- vs Base_ZINB BIC= df= ZINB ZIP Very strong AIC= df= 0.32 ZINB ZIP LRX2= prob= ZINB ZIP p= Comparson of Models Comparson model: countft (Graph & statstcs across models Comparson of Models Comparson model: znb (Voung test.znb y x x2, nflate(x x2 vuong zp Fttng zp model: Zero-nflated negatve bnomal regresson Number of obs = 500 Nonzero obs = 304 Zero obs = 96 Inflaton model = logt LR ch2(2 = 4.68 Log lkelhood = Prob > ch2 = y x x _cons nflate x x _cons /lnalpha alpha Lkelhood-rato test of alpha=0: chbar2(0 = 68.5 Pr>=chbar2 = Vuong test of znb vs. standard negatve bnomal: z = 0.52 Pr>z = Other Count Data Models Zero& others Count data Model Zero truncated Posson & Zero truncated negatve bnomal Truncated Posson & truncated negatve bnomal Hurdle model (Mullahy, 986 or zero-altered model (zap & zanb Censored Posson & censored negatve bnomal Generalzed Posson Regresson Generalzed Negatve Bnomal etc Reference Reference: Negatve Bnomal & other Count Models Agrest, A. (2002. Categorcal Data Analyss. John Wley & Sons. New York. Cameron A.C. and Trved P.K. (990. Regresson Analyss of Count Data. Cambrdge Unversty Press. New York. Cameron, A.C. and Trved, P.K. (990. Regresson-based tests for overdsperson n the Posson model. J.Econometrcs, 46, Dean, C. B. (992. Testng for overdsperson n Posson and bnomal regresson models. J. Am. Statst. Assoc.,87, Dean, C. and Lawless, J. F. (989. Tests for detectng overdsperson n Posson Regresson models. J. Am. Statst. Assoc., 84, Fless, J.L., Levn, B., & Pak, M.C. (2003. Statstcal methods for rates and proportons. 3 rd edton. John Wley & Sons. New York. Greene, W.H. (2003. Econometrc Analyss 5th. Prentce & Hall. New Jersey. Hlbe, J.M. (2007. Negatve Bnomal Regresson. Cambrdge Unversty Press. New York. Thanomseng, N. (2007. overtest.ado STATA ado fle: Overdsperson test. Avalable at 7