Incorporating Survey Weights into Binary and Multinomial Logistic Regression Models

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1 Science Jornal of Applied ahemaics and Saisics 205; 3(6): Pblished online November 20, 205 (hp:// doi: 0.648/j.sjams ISSN: (Prin); ISSN: (Online) Incorporaing Srvey Weighs ino Binary and linomial Logisic Regression odels Kennedy Sakaya Barasa *, Chris chwanj eparmen of Saisics and Acarial Sciences, Jomo Kenyaa Universiy of Science and Technology, Nairobi, Kenya address: (K. S. Barasa), (C. chwanj) To cie his aricle: Kennedy Sakaya Barasa, Chris chwanj. Incorporaing Srvey Weighs ino Binary and linomial Logisic Regression odels. Science Jornal of Applied ahemaics and Saisics. Vol. 3, No. 6, 205, pp doi: 0.648/j.sjams Absrac: Since sampling weighs are no simply eqal o he reciprocal of selecion probabiliies is always challenging o incorporae srvey weighs ino likelihood-based analysis. These weighs are always adjsed for varios characerisics. In cases where logisic regression model is sed o predic caegorical ocomes wih srvey daa, he sampling weighs shold be considered if he sampling design does no give each individal an eqal chance of being seleced in he sample. The weighs are rescaled o sm o an eqivalen sample size since original weighs have small variances. The new weighs are called he adjsed weighs. Qasi-likelihood maximizaion is he mehod ha is sed o make esimaion wih he adjsed weighs b he oher new mehod ha can be creaed is correc likelihood for logisic regression which inclded he adjsed weighs. Adjsed weighs are frher sed o adjs for boh covariaes and inerceps when he correc likelihood mehod was sed. We also looked a he differences and similariies beween he wo mehods. Analysis: Boh binary logisic regression model and mlinomial logisic regression model were sed in parameer esimaion and we applied he mehods o body mass index daa from Nairobi Hospial, which is in Nairobi Cony where a sample of 265 was sed. R-sofware Version was sed in he analysis. Conclsion: The resls from he sdy showed ha here were some similariies and differences beween he qasi-likelihood and correc likelihood mehods in parameer esimaes, sandard errors and saisical p-vales. Keywords: Binary Logisic Regression, linomial Logisic Regression, Adjsed Weighs, Correc Likelihood, Qasi-Likelihood, Nairobi. Inrodcion.. Inrodcion Logisic regression provides a mehod for modeling a binary response variable, which akes vales and 0. For example, we may wish o invesigae how deah () or srvival (0) of paiens can be prediced by he level of one or more meabolic markers. When he response variable is binary (e.g. deah or srvival), hen he probabiliy disribion of he nmber of deahs in a sample of a pariclar size, for given vales of he explanaory variables, is sally assmed o be binomial. (Corvoisier e al, 20) Regression models have become an inegral componen of any daa analysis concerned wih describing he relaionship beween he response variable and one or more explanaory variables. I is ofen he case ha he ocome variable is discree, aking on wo or more possible vales. (Hosmer and Lemeshow, 2000). For example, in a sdy of obesiy for adls, seleced individals can have a high body mass index (BI) or do no have a high BI. In sch a case BI will be he independen variable while he independen variables gender, age and race. The dependen variable has wo possible ocomes: individals having a high BI, no having a high BI. Sbseqenly, we can code hem as and 0, respecively. Binary logisic regression model can be exended o mlinomial logisic regression model, in which he response variable has more han wo levels. The simlaneos increases in obesiy in almos all conries seem o be driven mainly by changes in he global food sysem, which is prodcing more processed, affordable, and effecively markeed food han ever before. This passive overconsmpion of energy leading o obesiy is a predicable ocome of marke economies predicaed on consmpion-based growh (Swinbrn e al, 20) Using an example, in he sdy of obesiy for adls he BI vale can be divided ino for differen levels (obese, overweigh, normal, and nderweigh), hen we bild he

2 Science Jornal of Applied ahemaics and Saisics 205; 3(6): mlinomial logisic regression model wih gender, age and race as covariaes. We labeled he levels as, 2, 3 and 4, respecively Weighs always make sre he sample is represenaive of he poplaion of ineres and ha oher objecives are me and are pariclarly imporan when over-sampling occrs. The sampling weighs shold be considered if he sampling design does no give each individal an eqal chance of being seleced. Sampling weighs can be hogh as he nmber of observaions represened by a ni in he poplaion if hey are scaled o sm o he poplaion size. According o Gelman (2007) sampling weigh is a mess. I is no easy o esimae anyhing more complicaed sing weighs han a simple mean or raio, and sandard errors are ricky even wih simple weighed means. Conrary o wha is assmed by many researchers, srvey weighs are no in general eqal o he inverse of probabiliies selecion, b raher are consrced based on a combinaion of probabiliy calclaions and nonresponsive adjsmens. In his research since he variance was oo small, we rescaled he sampling weighs o sm o an eqivalen sample size. These new weighs are called he adjsed weighs and are he ones ha were incorporaed ino he logisic regression model o esimae he parameers..2. Qasi-Likelihood and Adjsed Weighs Qasi-Likelihood ehod can be sed o esimae parameers in he logisic regression model wih adjsed weighs since he variance fncion and mean fncion varies independenly. This model esimaes he variance fncion from he daa direcly wiho normal disribional assmpion..2.. Probabiliy Weighs Weighs may vary for several reasons. The esimaor of oal will be eqal o, where is he overall probabiliy ha he h elemen is seleced. We can define he sampling weigh for he h elemen as. Since Smaller selecion probabiliies may be assigned o he elemens wih high daa collecion coss and a high selecion probabiliies may be assigned o he elemens wih larger variances. (Kner and Nachsheim, 2004).2.2. Adjsed Weighs Here we don rely on condiioning on model elemens sch as covariaes o adjs for design effecs. B we rescale sample weighs o sm o he eqivalen sample size so as o obain esimaors The eqivalen sample size is smaller han he sample size b in oher cases he eqivalen sample size cold be larger, b we resric aenion o simple random sampling In he sper poplaion model, le denoe he response variable for he h ni in he sample. Here, are assmed o be independen random variables. Le s define he mean for h ni ( ) and variance ( ),,, where is he sample size. The mean and variance of he sper poplaion model are ( The esimae of and variance of mean are ( ) ) () ( ). Le s consider anoher se of weighs defined by ( # $ #%& # $ ) ' $ ' #%& # ), where is ( #%& #. We call as he adjsed weighs and as an eqivalen sample size (Pohoff, Woodbry and anon 992). The eqivalen sample size is smaller han he poplaion size. We rescale he sampling weighs o sm o an eqivalen sample size becase he original variance is oo small o inclde enogh informaion. These new weighs are called he adjsed weighs. We can rewrie he esimaors sing as ( ) ( ) ( ) () ( ).3. Binary Logisic Regression odel When have a binary op variable Y, and we wan o model he condiional probabiliy Pr(Y X x) as a fncion of x; any nknown parameers in he fncion are o be esimaed by maximm likelihood.. Firs le p( x) be a linear fncion of x. Every incremen of a componen of x wold add or sbrac so mch o he probabiliy, p ms be beween 0 and, and linear fncions are nbonded. 2. Le log p(x) be a linear fncion of x, so ha changing an inp variable mliplies he probabiliy by a fixed amon. Remember ha logarihms are nbonded in only one direcion, whereas linear fncions are no. 3. Finally, we make he logisic (or logi) ransformaion, p log (Which has an nbonded range) a linear p fncion ofx. Formally, he model logisic regression model is ha Solving for phis gives p( x) log β0 + x. β p( x) β0 + βx + β2x β pxp e p(y X x) + e + e β0 + βx + β2x βpxp β0 + βx + β2x βpxp ( ) Where p he probabiliy of individal i who has a high BI wih x se of predicors, e he base of naral logarihms

3 245 Kennedy Sakaya Barasa and Chris chwanj: Incorporaing Srvey Weighs ino Binary and linomial Logisic Regression odels β 0 he consan of he eqaion and, x he coefficien of he predicor variables (Hosmer and Lemeshow, 2000).4. liple Logisic Regression odel The simple logisic regression model can be easily exended, for i o have o more han one predicor variable. efiniion, *, * * + / * -. We shall have / 2. 3 /2,-. 4 * *, + * + + * * *, + * * -. 3,-. Therefore 89 : ; <-(> # 4?).5. linomial Logisic Regression odel - When he response variables have more han wo levels, we sill se logisic regression model. We divide he response ino A response caegories, he variables will be9,..., 9 B. Then, le A be he baseline, he logi for he C h comparison is: ; B EFG < ; ; B * B C, 2,, A ; J3K 4 * L B. + J3( 4 * ) (Andersson e al. 200).6. aximm Likelihood C, 2,, A aximm likelihood, also called he maximm likelihood mehod, is he procedre of finding he vale of one or more parameers for a given saisic which makes he known likelihood disribion a maximm. (Einicke e al, 202) aximm likelihood mehods are sed o esimae and make inference abo he parameers and hese mehods are efficien and aracive when he model follows he normal disribion assmpion. Since no all he disribions are normal, sch as a Poisson disribion, in which he variance is same as he mean he mehod will no always apply. The mean and variance parameers do no vary independenly. (Balgobin and Choi, 200) We can recall ha, he join probabiliy fncion for binary logisic regression is: P G(9,, 9 ) N O (9 ) N ; # ( ; ).P # EFG < G(9,, 9 ) EFG < N O (9 ) EFG < N ; P # ( ; ).P # Q9 EFG < ; + ( 9 )EFG < ( ; )R S9 EFG < ( ; )T + EFG ; < ( ; ) Since we know ha UV? & # ) And EFG < W X # Y *.X, + * 3 # Therefore, EFG < Z(*,, * ) 9 (*, + * 3 ) EFG < Q + J3(*, + * 3 )Rhere we are rying o find *, and * o maximize he log-likelihood fncion: E EFG < Z(*,, * ) efine: 9 (*, + * ) EFG < Q + exp(*, + * )R e f i f ~` ^ + / b d h d h a c f a g The model is 9 f *. The esimaor of j is *k ( f b. b b ). f b. b b, where m l a diagonal marix wih h diagonal elemen n. (Nandram and Choi, 2002).7. Qasi-Likelihood We analyze he binary logisic regression wih sampling weighs. The qasi likelihood is O ( ) # ; # `# ( ; ) (.`#) # E EFG < G(9,, 9 ) EFG < N O ( ) # EFG < N ; `# # ( ; ) # (.`# ) Q EFG < ; + ( )EFG < ( ; )R W 3 Y * EFG < K + J #? L Le p' qrsq p '? Le pqrsq p? ( 3 ) # # 0 $ #w #?W@< # Y. #?< # #%& x W@< # Y ' We find esimaors o maximize he qasi log-likelihood

4 Science Jornal of Applied ahemaics and Saisics 205; 3(6): fncion: Z() Z( ). The esimaor of *is*k ( f b y b.8. Correc Likelihood. b. b. b ). b f y b b When we incorporae he weighs ino he probabiliy disribion fncion (pdf), in order o keep he new fncion sill o be a pdf, we need o normalize i. For he discree disribion, he new fncion becomesh(3) {() {(}), and for he coninos disribion, he new probabiliy disribion fncion becomesh(3) {(). We inrodce he sampling ~ {(}) } weighs in he probabiliy disribion fncion, h(3) O(3). Here are some of he disribions and heir normalizaion wih sampling weighs. We compared heir similariies and differences sing he mean and variance. Example Le 3~ (, n )wih sampling weighs (), The densiy fncion of normal disribion is: O(3, n ) ( ) ' ƒ X J. ' ', < 3 < Inrodcing he sampling weighs we have: O(3,, n ) J.( ) ' ' ~ J. ( )' ' ' Š3. ( ) ' ƒ X J. ' ' ( ƒ X J. ' '. ~ S ( ) ' J. ' ' ƒ 2; T Š3 2; n J. ( ) ' ' ' Here, 3 ~ (, ƒ' ) We see ha he mean of normal disribion is(3), and he mean of normalized disribion wih sampling weighs is (3 ). There, he mean of normal disribion does no change afer normalizaion. Similarly, he variance of normal disribion is (3) n,and he variance of normalized disribion wih sampling weighs is(3 ) ƒ '. The variance of he normal disribion changes afer normalizaion. Example 2 Le 3~jJF EE ()wih sampling weighs (), The densiy fncion of Bernolli disribion is: Ž( 3 ) ( ). 3 0, 0 Inrodcing he sampling weighs we have: (3, ) - (.-) & 3 0, - Here, 3 ~jjf EE W Y We see ha he mean of Bernolli disribion is(3). The mean of normalized Bernolli disribion wih sampling - weighs is (3 ) &. So ha (3 ) < (3) or (3 ) > (3) or (3 ) (3)when. Example 3 Le 3~ E FE () wih sampling weighs (), The densiy fncion of linomial disribion is: ( ~ ) >,, where he ni is C, oherwise 0. Inrodcing he sampling weighs we have: KL z > > B > Q R > N Here, > ~ ~ E (, ), where - - %& C, 2, š > The mean of independen Bernolli disribions is eqal o wiho any sampling weighs; i has changed o - C,2,, š in he presence of he sampling - %& weighs. Example 4 < Le ~jj Y wih sampling weighs (), The densiy fncion of binary logisic regression is: (9 *) ` ( ).` J? + J? `.` S + J?T Inrodcing he sampling weighs we have: (9 *) w x ` w x ` J?` Q + J? R` J? + J? w w x + w K@< L ` + J? ` Q + J? R J? + x J?` + J?.` S + J?T, 0, `.` S + J?T The mean of binary logisic regression The mean of normalized binary logisic regression wih sampling < weighs Sampling weighs will be sed o adjs he covariaes and inerceps in he normalized binary logisic regression. When we esimae binary logisic regression coefficiens, we mliply sampling weighs wih <

5 247 Kennedy Sakaya Barasa and Chris chwanj: Incorporaing Srvey Weighs ino Binary and linomial Logisic Regression odels boh covariaes and inerceps o creae new covariaes and new inerceps, and se correc likelihood esimaion mehods. In he new mehod, we normalize he binary logisic regression wih adjsed weighs and se he correc likelihood o make esimaion and inferences. Clearly, here are some differences beween correc likelihood of normalized binary logisic regression wih adjsed weighs and qasi-likelihood of binary logisic regression wih adjsed weighs. Example 5 Le ~ E œ &,, œ œ & wih sampling weighs(), The densiy fncion of mlinomial logisic regressions is: W ~ E, œ &,, œ œ & Inrodcing he sampling weighs we have: (9 *) W œ & `œ! œ & `œ!.!.! Y œ & S `. œ & `. T œ & œ & S T + œ & œ &. Y J ~?` J ~? + N W + J ~? `œ ~ Y. J ~?`œ Y W + J? `œ + J ~?.? & J ~ + J ~?4. œ `œ +. J ~?4 œ. `œ For mlinomial logisic regression, we ake one caegory as he reference caegory, hen compare ohers wih i. We se sampling weighs o adjs he covariaes and inerceps in he normalized mlinomial logisic regressions. When we esimae mlinomial logisic regressions coefficiens, we mliply boh covariaes and inerceps wih sampling weighs o creae new covariaes and new inerceps, and se correc likelihood esimaion mehods. In he new mehod, we normalize mlinomial logisic regression wih adjsed weighs and se he correc likelihood o make esimaion and inferences. There are some differences beween correc likelihood of normalized mlinomial logisic regressions wih adjsed weighs and qasi-likelihood of mlinomial logisic regressions wih adjsed weighs. (Grilli and Praesi, 2004).9. Smmary The sampling weighs of he correc likelihood mehod are he same as hose in he qasi-likelihood mehod b boh of hem are adjsed weighs. Qasi likelihood mehod has he inerceps as, and covariaes are he reglar covariaes. However, in he correc likelihood mehod, he sampling weighs are frher sed o adjs for boh covariaes and inerceps. In correc likelihood mehod, we mliply adjsed weighs wih boh inerceps and covariaes; he inerceps are eqal o adjsed and also he covariaes are eqal o reglar covariaes. Table. ifferences beween qasi-likelihood and correc likelihood. Qasi-likelihood Covariaes Reglar Covariaes Inerceps Correc Likelihood Adjsed Covariaes Adjsed Weighs Likelihood h(3) O(3) No Normalized h(3) {() {(}) Normalized 2. Analysis and Inerpreaion 2.. Analysis Using Binary Logisic Regression A binary logisic regression model was sed o analyze he daa. We had for levels of BI,, 2, 3 and 4 in which we compared nderweigh wih no nderweigh or normal weigh wih no normal weigh and so on. In he daase of Nairobi Cony we ook each variable as, and call he ohers 0, o perform he binary regression. We call BI eqals o as, and call BI eqals o 2 o 4 as 0 in Nairobi Cony, did he firs binary regression, hen we repeaed he process, call BI eqals o 2 as, and call BI eqals o, 3 and 4 as 0 in Nairobi Cony did he second binary regression. We kep repeaing nil he forh binary regression analyses.the following able shows he resls of he binary regression. I shows he differences and similariies beween he QL and CL. In his sdy we analyzed he binary and he mlinomial logisic regression one by one, basing on he wo mehods o compare heir differences and similariies. We inclded p-vales (Pr), esimaes, Wald Chi-Sqare (WS) saisics and sandard errors (SE). The variables age, race and gender are he independen variables for regression of BI. From able 2 above we were comparing binary logisic regressions in Nairobi Cony; we can see differences and similariies beween qasi-likelihood and correc likelihood mehods. Specific abo differences, when BI eqals o hree compared o he ohers, he p-vale of race is differen; he qasi-likelihood mehods vale was (>0.05), b when sing he correc likelihood mehod we obained a vale of 0.08 (<0.05). Also in his analysis, we fond o ha, he p-vale of gender was also differen; he qasi-likelihood mehods vale was (<0.05), b new mehods (correc likelihood) vale was (>0.05). When BI eqals o for compared o he ohers, he p-vale of age is differen; he qasi-likelihood mehods vale was (<0.05), b he new was (>0.05). The p-vale of gender is also differen; he qasi-likelihood mehods vale was (>0.05), while he correc likelihoods mehod vale was (<0.05). Also from able 2, we can see ha here were some similariies, for example, when he BI was eqal o wo

6 Science Jornal of Applied ahemaics and Saisics 205; 3(6): compared o he ohers, he p-vale of age was he same; he qasi-likelihood mehods vale was and sing he correc likelihood mehod, we obained 0.567, a clear indicaion ha boh vales obained were greaer ha he Qasi-Likelihood Table 2. Coefficiens of binary logisic regression model (n265). p-vale (0.05) When BI eqals o hree compared o he ohers, he p-vale of age for he qasi-likelihood mehod and correc likelihood mehods were (>0.05), and (>0.05) respecively. Correc-Likelihood Esimae SE WS Pr Esimae SE WS Pr vs 234 Inercep Age Race Gender vs 34 Inercep Age Race Gender vs 24 Inercep Age Race * Gender * 4 vs 23 Inercep Age * Race Gender * *Vales wih aserisk indicae differences, vales wiho aserisk indicae similariies 2.2. Analysis Using linomial Logisic Regression We consrced a mlinomial logisic regression model o analyze he for levels of BI ogeher. The for levels were labeled as, 2, 3 and 4 where we compared nderweigh, normal weigh, overweigh and obese a he same ime. We sed BI as he reference caegory and compared i wih he oher hree levels of BI all ogeher. I is he same o se BI 2 as he reference caegory. In he sdy, normal weighs, overweigh and obese were compared basing on he nderweigh. Table 3. Coefficiens of mlinomial logisic regression model (n265). Qasi-Likelihood Correc-Likelihood Esimae SE WS Pr Esimae SE WS Pr Inercep * Inercep Inercep * Age Age Age Race Race * Race Gender Gender * Gender Vales wih aserisk indicae differences, vales wiho aserisk indicae similariies From able 3 above we can see coefficiens of mlinomial logisic regression, here were differences and similariies beween qasi-likelihood and correc likelihood mehods. Specific abo differences, when BI eqals o hree, he p-vale of gender is differen; he qasi-likelihood mehods vale was (<0.05), whereas ha of correc likelihood mehod was (>0.05). The p-vale of race is differen; he qasi-likelihood mehods vale was (>0.05), b ha of correc likelihood mehod is 0.03 (<0.05). The similariies we see from able 3 above shows ha when BI was eqals o wo, he p-vale of age is he same beween hese wo mehods, in he sense ha hey were boh greaer

7 249 Kennedy Sakaya Barasa and Chris chwanj: Incorporaing Srvey Weighs ino Binary and linomial Logisic Regression odels han 0.05 since he qasi-likelihood mehods vale was and correc likelihood mehod vale The p-vale of gender was 0.53 and 0.24 for he qasi-likelihood and correc likelihood respecively. 3. iscssions and Conclsion In his sdy we sed qasi-likelihood mehod as he old mehod for binary logisic regression model and mlinomial logisic regression model. The maximm likelihood mehods o make esimaion and inference are no longer sefl especially when he logisic regression fails o mee normal disribion assmpion. As Pfeffermann e al (998) poined o maximm likelihood esimaion will prodce some bias. The conribion of his research is o se he correc likelihood mehod as he new mehod for binary logisic regressions model and mlinomial logisic regression model. We p weighs in he pdf, and in order o keep he new fncion sill a pdf, we shold divide i by he inegral or sm of disribion wih weighs (i.e., we accommodae he weighs by normalizaion). In he new mehod (correc likelihood), he weighs are frher sed o adjs he covariaes and inerceps. The qasi-likelihood mehod is he n-normalized disribion wih sampling weighs, b he correc likelihood mehod is he normalized disribion wih sampling weighs. By comparing he resls of he wo mehods from he analysis, we conclde ha here are similariies and differences. The pracical examples we sed was o diagnose overweigh and obesiy for adls. The dependen variable is BI wih for levels, nderweigh, normal weigh, overweigh and obese. We bil binary logisic regression models and mlinomial logisic regression models o show he differences and similariies beween n-normalized disribion wih sampling weighs and normalized disribion wih sampling weighs. We believe sing he normalized disribion, he correc likelihood, is he righ hing o do, alhogh he se of srvey weighs is a conroversial area. Gelman (2007). I wold be nice o compare or mehods wih he mehod of pos sraificaion as described by Gelman (2007). One may wan o pos-sraify he srvey weighs o ge approximaely eqal srvey weighs wihin sraa. Nomenclare BI CL Pdf QL SE WS Body ass Index Correc Likelihood ehod Probabiliy densiy fncion Qasi-Likelihood ehod Sandard Error Wald Chi-sqare References [] Andersson J.C, Verkilen, J., and Peyon, B. L. (200).odeling Polyomos Iem Responses sing Simlaneosly Esimaed linomial Logisic Regression odel. Jornal of Edcaional and Behavioral Saisics, 422. [2] Balgobin Nandram and Jai Won Choi. (200). A Bayesian Analysis of Body ass Index aa from Small omains Under Nonignorable Nonresponse and Selecion. Jornal of American Saisical Associaion, 05, [3] Corvoisier,.S., C. Combescre, T. Agorisas, A. Gaye-Ageron and T.V. Perneger (20) Performance of logisic regression modeling: beyond he nmber of evens per variable, he role of daa srcre. Jornal of Clinical Epidemiology 64: [4] Einicke, G.A.; Falco, G.; nn,.t.; Reid,.C. (ay 202). Ieraive Smooher-Based Variance Esimaion. IEEE Signal Processing Leers 9 (5) [5] Gelman, Andrew. (2007) Srggles wih Srvey Weighing and Regression odeling. Saisical Science, 22, [6] Grilli, L., and Praesi,. (2004). Weighed Esimaion in linomial Ordinal and Binary odels in he Presence of Informaive Sampling esigns. Srvey ehodology, 30, [7] Hosmer.W and Lemeshow S, (2000), Applied Logisic Regression 2nd Ed. John Wiley & Sons, Inc. Canada. PP-7 [8] ichael, H., Kner, C J., and Nachsheim, J. N. (2004).Applied Linear Regression odels. cgraw-hill/irwin [9] Nandram, B., and Choi, J. W. (2002), A Hierarchical Bayesian Nonresponse odel for Binary aa Wih Uncerainy Abo Ignorabiliy, Jornal of he American Saisical Associaion, 97, [0] Pfeffermann,., Skinner, C. J., Holmes,. J., Goldsein, H., and Rasbash, J. (998).Weighing for Uneqal Selecion Probabiliies in linomial odels. Jornal of he Royal Saisical Sociey, 60, [] Pohoff R, Woodbry,. A., and anon, K. G. (992). Eqivalen Sample Size and Eqivalen egrees of Freedom Refinemens for Inference sing Srvey Weighs nder Sper poplaion odels. Jornal of he American Saisical Associaion, 87, [2] Swinbrn BA, Sacks G, Hall K, e al. The global obesiy pandemic: shaped by global drivers and local environmens. Lance. 20; 378(9793) [3] Wang YC, cpherson K, arsh T, Gormaker SL, Brown. Healh and economic brden of he projeced obesiy rends in he USA and he UK. Lance. 20; 378(9793).