Phd Program in Transportation. Transport Demand Modeling. Session 7
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1 Phd Program n Transportaton Transport Demand Modelng Lus Martínez (based on the Lessons of Anabela Rbero TDM2010) Sesson 7 Generalzed Lnear Models Phd n Transportaton / Transport Demand Modellng 1/48
2 Outlne Generalzed regresson models Generalzed regresson models appled to the count of events Posson regresson model Negatve bnomal regresson model Zero-nflated Posson regresson model Phd n Transportaton / Transport Demand Modellng 2/48
3 Generalzed Lnear Models Dstngushng between GLM (General Lnear Models) and GZLM (Generalzed Lnear Models) GLM (General Lnear Models) varance and regresson models whch analyze normally dstrbuted dependent varables usng an dentty lnk functon (predcton s drectly of the values of the dependent) GZLM (Generalzed Lnear Models) Allows for dependent varables wth non normal dstrbutons and for many other lnk functon other than dentty (flexble generalzaton of GLM) Phd n Transportaton / Transport Demand Modellng 3/48
4 Generalzed Lnear Models Why usng GZLM nstead of GLM? The dependent varable may not be contnuous The effect of ndependent varables may not be lnear Untl half of the XX century normal lnear After: non lnear adjustments: log-log; probt, logt, log-lnear GZLM unfy all non lnear models, used to explan the stuaton were the lnear normal regresson was not able to explan the relaton under analyss Phd n Transportaton / Transport Demand Modellng 4/48
5 Generalzed Lnear Models GZLM have three man components A random component Usually a probablty dstrbuton from the exponental famly A systematc component Usually a lnear predctor such as A lnk functon Usually a lnk functon g X wth a mean as E( Y ) g And where the varance s a functon of the mean such as Var(Y ) V V g 1 1 Phd n Transportaton / Transport Demand Modellng 5/48
6 Probablty dstrbuton Generalzed Lnear Models Random Component N ndependent varables Y (=1,2,...N), wth an average of μ and a probablty functon belongng to the exponental famly b y f y, exp c, a y, y Y dstrbuton ts parameterzed n terms of the mean μ and a scale parameter ϕ (not the canonc θ) where EY b' VarY a b' ' Here and are scalar parameters and a(.), b(.) and c(.) area real functons known and specfc from each dstrbuton Phd n Transportaton / Transport Demand Modellng 6/48
7 Generalzed Lnear Models Random Component Probablty dstrbuton On a GZLM the random dstrbuton of all random varables s the same. The probablty of all random varables Y presents the same shape, not varyng wth, and wth a constant scale parameter Normal, Posson, Gama or other Example: Posson Y follows a Posson dstrbuton f y y exp ln( ) ln( ) ln(!) e y, y 0 Phd n Transportaton / Transport Demand Modellng 7/48
8 Normal Generalzed Lnear Models Random Component The dstrbuton of the dependent takes the form of the famous bellshaped symmetrcal curve centered on the mean. Ths mples the dependent s contnuous numerc Normal dstrbuton (normalzed) Phd n Transportaton / Transport Demand Modellng 8/48
9 Inverse Gaussan (Wald Dstrbuton) Generalzed Lnear Models Random Component Also called the nverse normal or Wald dstrbuton, ths s used for dependents whch are postvely skewed and have values always greater than 0. Values must be greater than 0 or are dropped. It has been used to model dffuson processes, nsurance clams If λ tends to nfnty, the dstrbuton becames smlar to a Normal Phd n Transportaton / Transport Demand Modellng 9/48
10 Gamma Generalzed Lnear Models Random Component Ths s an alternatve for postvely skewed dependent varables. It s hghly senstve to the shape parameter. When the shape parameter s greater than 1, the gamma dstrbuton s mounded but postvely skewed as shown n the fgure below. When the shape parameter s 1, the gamma dstrbuton s exponentally declnng. When the shape parameter s less than 1, the gamma dstrbuton s also exponentally declnng and asymptotc to the axes The gamma dstrbuton has been used n survval analyss and modelng duraton-to-event data Phd n Transportaton / Transport Demand Modellng 10/48
11 Generalzed Lnear Models Random Component Probablty dstrbuton Multnomal Ths dstrbuton s used when the dependent has a fnte number of categores, has text strng values, or s ordnal. The dstrbuton among categores, not shown, s arbtrary Phd n Transportaton / Transport Demand Modellng 11/48
12 Bnomal Generalzed Lnear Models Random Component Used when the dependent s bnary. The count of events n a fxed number of trals also has a bnary dstrbuton. Examples of bnomal data are attrbute present/not present, nnovaton adopted/not adopted, or success/falure data. It s assumed that the two values have a fxed rather than changng probablty of occurrence (as n conflppng), even f that probablty s not known Phd n Transportaton / Transport Demand Modellng 12/48
13 Posson Generalzed Lnear Models Random Component The Posson dstrbuton s also used for count data and s preferred when events are rare, as n modelng accdents, wars, or epdemcs. A rule of thumb s to use a Posson rather than bnomal dstrbuton when n s 100 or more and the probablty s.05 or less. Where the bnomal dstrbuton s used where the varable of nterest s count of successes per gven number of trals, the Posson dstrbuton s used for count of successes per gven number of tme unts Phd n Transportaton / Transport Demand Modellng 13/48
14 Generalzed Lnear Models Random Component Posson The Posson dstrbuton s also used when "event occurs" can be counted but non-occurrence cannot be counted. In Posson dstrbutons, the mean equals the varance. All values are nonnegatve ntegers (thus count data, whch cannot be negatve, are better represented by Posson than normal dstrbutons) Phd n Transportaton / Transport Demand Modellng 14/48
15 Generalzed Lnear Models Random Component Negatve bnomal Ths s smlar to the Posson dstrbuton, also used for count data, but s used when the varance s larger than the mean. Typcally ths s characterzed by "there beng too many 0's." It s not assumed all cases have an equal probablty of experencng the rare event, but rather that events may cluster. The negatve bnomal model s therefore sometmes called the "over dspersed Posson model". Values must stll be non-negatve ntegers The negatve bnomal s specfed by an ancllary /dsperson parameter, k. When k=0, the negatve bnomal s dentcal to the Posson dstrbuton Phd n Transportaton / Transport Demand Modellng 15/48
16 Generalzed Lnear Models Systematc component The lnear predctor Quantty wtch ncorporates the nformaton about the depended varables nto the model A matrx of N observatons and of p varables A lnear predctor p x j 1 j j Where each the j x j s the value of the j varable for the observaton s a vector of unknown parameters to be estmated Phd n Transportaton / Transport Demand Modellng 16/48
17 Generalzed Lnear Models Lnk functon The lnk functon s the transformaton of the dependent varable whch s modeled Predctors have a non lnear effect on the dependent OLS and GLM n general use an dentty functon, modelng the actual value of the dependent varable n a lnear fashon Phd n Transportaton / Transport Demand Modellng 17/48
18 Generalzed Lnear Models Lnk functon Provdes the relatonshp between the lnear predctor and the mean of the dstrbuton functon The way the two prevous components relate to each other It s a monotonous and dfferentable functon that transforms g n (one functon for each observaton ) The same functon for all observatons E 1 Y g 0 1X 1... p X p e Phd n Transportaton / Transport Demand Modellng 18/48
19 Generalzed Lnear Models Lnk functon Same functon for all observatons E 1 Y g 0 1X 1... p X p e Where s the expected value of Y Where X j are the predctors or explcatve varables =1,, N j = 1,,p Phd n Transportaton / Transport Demand Modellng 19/48
20 Generalzed Lnear Models Lnk functon It s used to mantan a lnear relatonshp between the coeffcents and predctors on the rght hand sde of the model equaton and the dependent transformed by the lnk functon on the left hand sde of the equaton The choces on the lnk functon depend on the dstrbuton selected Phd n Transportaton / Transport Demand Modellng 20/48
21 Generalzed Lnear Models Lnk functon Normal, Gamma, Inverse Gaussan, Posson Identty, log, power (exponental) Phd n Transportaton / Transport Demand Modellng 21/48
22 Generalzed Lnear Models Lnk functon Normal - Identty Phd n Transportaton / Transport Demand Modellng 22/48
23 Generalzed Lnear Models Lnk functon Posson - Log Loglnear models: assume a Posson dstrbuton and use a log lnk functon Phd n Transportaton / Transport Demand Modellng 23/48
24 Generalzed Lnear Models Lnk functon All - Power Phd n Transportaton / Transport Demand Modellng 24/48
25 Negatve Bnomal Identty, log, power (exponental) Or...negatve bnomal lnk functons (only for bnomal negatve) Log(x/(x+k-1)) Generalzed Lnear Models Lnk functon By default, some programs keep log lnk for negatve bnomal n order to make t comparable wth Posson Models Bnomal Identty, log, power (exponental) Or several other lnk functons Phd n Transportaton / Transport Demand Modellng 25/48
26 Generalzed Lnear Models Lnk functon Logt Probablty of sucess F(x)=log(x/(1-x)) Probt Rate data F(x)=F-1(x) Complementary log-log Survval data Log-log. F(x)=log(-log(1-x)) Negatve log log Assymetrc probablty F(x)=-log(-log(x)) Log complement Health rsk F(x)=log(1-x) Odds power F(x)=[(x/(1-x))a-1]/a, f a 0 e f(x)=log(x), f a=0 Phd n Transportaton / Transport Demand Modellng 26/48
27 Multnomal Identty, log, power (exponental) Or several other lnk functons Cumulatve logt Ordnal logstcal regresson F(x)=ln(x/(1-x)) Cumulatve probt Ordnal probt models F(x)=F-1(x) Cumulatve Caucht Many extreme values F(x)=tan(p(x-0,5)) Cumulatve complementary log-log Hgher values n the dependent more probable; F(x)=ln(-ln(1-x)) Cumulatve negatve log-log - Lower values n the dependent more probable F(x)=-ln(-ln(x)) Generalzed Lnear Models Lnk functon Phd n Transportaton / Transport Demand Modellng 27/48
28 Generalzed Lnear Models Lnk functon Common relatons between dstrbutons and lnk functons Canoncal Lnk Functons Dstrbuton Name Lnk Functon Mean Functon Normal Exponental Gamma Inverse Gaussan Posson Bnomal Multnomal Identty Inverse Inverse squared Log Logt Phd n Transportaton / Transport Demand Modellng 28/48
29 Generalzed Lnear Models Summary Random Component Lnk Functon Systematc Component Common model types Phd n Transportaton / Transport Demand Modellng 29/48
30 Generalzed Lnear Models Common model types Scale/nterval Lnear Regresson, Anova, GLM Gamma wth log lnk Alternatve to lnear when normalty assumpton s volated Gamma famly varatons (wth other lnks) To dentfy the lowest devance (AIC or BIC) Identty-gamma regresson Duraton data Gamma wth nverse log lnk Phd n Transportaton / Transport Demand Modellng 30/48
31 Generalzed Lnear Models Common model types Ordnal dependents Ordnal Logstc Ordnal Probt Loglnear models Chekng for overdsperson negatve bnomal Posson regresson for count data Negatve bnomal wth log lnk When overdsperson s present n count data More stable than Posson for small data sets Phd n Transportaton / Transport Demand Modellng 31/48
32 Generalzed Lnear Models Common model types Bnary or events/trals dependents Bnary logstc Logstc regresson Bnary probt Probt Regresson Interval censored survval Phd n Transportaton / Transport Demand Modellng 32/48
33 Generalzed Lnear Models Common model types for count data (the case of accdents) The Posson Regresson Model The Truncated Posson Regresson The Zero Inflated Posson The Negatve Bnomal Model Phd n Transportaton / Transport Demand Modellng 33/48
34 GZLM for count of events Posson Regresson Model The Posson Dstrbuton s commonly used to descrbe the count of events occurrng at random n tme or space Examples: The number of cars passng through an ntersecton durng a certan hour The number of calls for emergency ambulance servce durng a tour of duty The number of fres arsng n a neghborhood Number of vehcles watng n a queue Auto breakdowns n an express way n rush hour Number of heart attack deaths per week n a county Number of homes destroyed by a fre durng the summer Number of accdents n a road secton or ntersecton Phd n Transportaton / Transport Demand Modellng 34/48
35 GZLM for count of events Posson Regresson Model Varables wth dscrete values between zero and nfnte Lmted case of bnomal when the total number of Bernoull trals (an experment whose outcome s random and can be ether of two possble outcomes, "success" and "falure ) approaches nfnty and the chance of success n any ndvdual tral approaches zero Applcaton to plannng stuatons (1) Large number of ndvdual cases (2) Roughly the same low probablty of success n any ndvdual tral (3) Lttle or no dependence among trals Phd n Transportaton / Transport Demand Modellng 35/48
36 GZLM for count of events Posson Regresson Model Probablty of ntersecton havng y accdents per year (y s a non nteger) P( y ) y! Where P(y ) s the probablty of the ntersecton havng y accdents (per year) and λ s the Posson parameter for the ntersecton, whch s equal to the expected number of accdents per year at ntesecton, E(y ) Parameter s estmated as a functon of explanatory varables e (geometrc condtons, sgnalzaton, pavement types, vsblty, and so on) y Phd n Transportaton / Transport Demand Modellng 36/48
37 GZLM for count of events Posson Regresson Model The most common relatonshp between the explanatory and the Posson parameter s the log-lnear model (because the logarthm of ths functon produces the lnear combnaton of explanatory varables) or e Ln( ) ( X) X p E[ Y] exp 0 x j j 1 The expected number of accdents per perod s gven by E[ y ] e ( X) Phd n Transportaton / Transport Demand Modellng 37/48
38 GZLM for count of events Posson Regresson Model Posson model Example of a smple model wth only one explanatory varable E[ Y Xj ] E[ Y Xj] (exp( j)) One ncrement of δ on the explanaory varable (X + δ) has an multplcatve mpact on the expected value of X j For the case of accdents (Q s unt of exposure vehcles per year) p E[ Y] Q exp 0 x j j j 1 Phd n Transportaton / Transport Demand Modellng 38/48
39 GZLM for count of events Posson Regresson Model Posson model Estmaton by standard maxmum lkelhood methods, wth the lkelhood functon gven as EXP[ EXP( X )][ EXP( X L( ) y! or LL( ) n 1 [ EXP( X ) y X Maxmum lkelhood estmates produce Posson parameters wtch are consstent, asymptotcally normal, and asymptotcally effcent )] y LN( y!)] Phd n Transportaton / Transport Demand Modellng 39/48
40 GZLM for count of events Posson Regresson Model Posson model Elastcty s computed on the parameter estmaton to evaluate the margnal effects of the ndependent varables Effect of a 1% change n the varable on the expected frequency λ Computed for each observaton and then a sngle average s reported Contnuous varables xk Ex k k x x Count data E xk k EXP( k ) 1 EXP( ) k k Phd n Transportaton / Transport Demand Modellng 40/48
41 GZLM for count of events Truncated Posson Regresson Truncated Posson Regresson Helpful f the data must be truncated Example: commutng to work lmted to fve days n a week: truncated to fve.... Phd n Transportaton / Transport Demand Modellng 41/48
42 GZLM for count of events Zero Inflated Posson Zero-Inflated Posson (ZIP) For example: related wth the very low probablty of occurrence n certan road segments (more zeros than expected n a zero process) Ths mples a two state count of data (one wth zero probablty and the other wthout) Y=0 p ( 1 p) EXP( ) Y=y (1 p) EXP( ) y! y Not applcable to the case study of accdents snce t s not possble to guarantee a road secton totally free of accdents Phd n Transportaton / Transport Demand Modellng 42/48
43 GZLM for count of events Negatve Bnomal Regresson Negatve Bnomal When the Posson condton (mean equals the varance) s volated E[y]=Var[y] E[y]>Var[y] (dspersed) E[y]<Var[y] (over dspersed) Rewrtng EXP( X) for each observaton EXP( X ) Phd n Transportaton / Transport Demand Modellng 43/48
44 GZLM for count of events Negatve Bnomal Regresson Negatve Bnomal Therefore the varance dffer from the mean through the addton of a quadratc term to the varance that represents over dsperson var( Y ) K( ) 2 The Posson model s regarded as a lmted model of the negatve bnomal as K approaches 0. Ths parameter s called the over dsperson parameter Phd n Transportaton / Transport Demand Modellng 44/48
45 Phd n Transportaton / Transport Demand Modellng 45/48 GZLM for count of events Negatve Bnomal Regresson Negatve Bnomal where Г s a Gamma Functon and K s a estmated parameter representatve of dsperson Correspondent lkelhood functon K y K K K K y K y y P (1 1! 1 ) ( K y K K K K y K y L (1 1! 1 ) (
46 GZLM for count of events Negatve Bnomal Regresson Negatve Bnomal wth log lnk Specfes a negatve bnomal dstrbuton (wth the ancllary K parameter = 1) wth a log lnk functon Is used when modelng count data that volates the Posson assumpton of equalty of mean and varance. Also, negatve bnomal regresson s thought to be more stable than Posson regresson for small datasets. An error term ξ of Gamma dstrbuton and varance K 2 s added to the Posson Regresson log( ) p 0 x j j 1 j Phd n Transportaton / Transport Demand Modellng 46/48
47 GZLM for count of events Negatve Bnomal Regresson Tests on over dsperson Lagrange Multpler test (SPSS) The Lagrange multpler test may be used to test f a negatve bnomal model s sgnfcantly dfferent from a Posson model Snce the negatve bnomal model s the same as the Posson model when the bnomal model's ancllary (dsperson) parameter, k, equals 0, the Lagrange multpler test s a test of the null hypothess that k = 0 A non-sgnfcant Lagrange test coeffcent ndcates that k cannot be assumed to be dfferent from 0, and hence a Posson model would be preferred over a negatve bnomal model (negatve bnomal models have one more parameter, k) Phd n Transportaton / Transport Demand Modellng 47/48
48 References References McCullagh, Peter; Nelder, John (1989). Generalzed Lnear Models, Second Edton. Boca Raton: Chapman and Hall/CRC. ISBN J. B. S. Haldane, "On a Method of Estmatng Frequences", Bometrka, Vol. 33, No. 3 (Nov., 1945), pp JSTOR Hlbe, Joseph M., Negatve Bnomal Regresson, Cambrdge, UK: Cambrdge Unversty Press (2007) Negatve Bnomal Regresson - Cambrdge Unversty Press Washngton, Smon P., Karlafts, Mathew G. e Mannerng (2003) Statstcal and Econometrc Methods for Transportaton Data Analyss, CRC Lord, D., Washngton, S. P., & Ivan, J. N. (2005). Posson, Posson-Gamma and zero-nflated regresson models of motor vehcle crashes: balancng statstcal ft and theory. Accdent Analyss and Preventon, pp Fernandes, A. (2010) Programas de manutenção de característcas da superfíce de pavmentos assocados a crtéros de segurança rodovára. Tese de Doutoramento em Engenhara Cvl. Insttuto Superor Técnco, Unversdade Técnca de Lsboa. Phd n Transportaton / Transport Demand Modellng 48/48
y and the total sum of
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