Chapter 20 Duration Analysis
|
|
- Alexandra McDowell
- 6 years ago
- Views:
Transcription
1 Chapter 20 Duraton Analyss Duraton: tme elapsed untl a certan event occurs (weeks unemployed, months spent on welfare). Survval analyss: duraton of nterest s survval tme of a subject, begn n an ntal state and observed whether ext from state or censored (stll n ths state). Example: Unemployment Tme to leave UE Medcne: Tme to death untl specfc treatment Traffc: Tme untl accdent Frm: Tme untl frm closes Model dependence of T on covarates!
2 Approaches: ) We could model duraton as Y, where Y mght be censored and censorng pont mght vary between ndvduals Tobt type approach. Why should we use any other methods than Tobt? 2) Instead of modellng the duraton, one often models the hazard rate Ths permts for tme-varyng covarates 3) Also more helpful to extend to competng rsk models multple ext states there not only duraton s of nterest Other ssues: - left and rght censorng - endogenous varables - multple spells
3 8.2 Hazard Functon: Hazard Functons wthout Covarates Notaton T 0 s tme at whch a person leaves ntal state, whch has some dstrbuton n populaton t denotes a partcular value of T. The cdf of T s F(t) = P(T t), t 0. Survval functon: S(t) = - F(t) = P(T > t) - probablty of survvng past tme t. The pdf of T s f(t) = df (t)/dt. Hazard functon gves probablty of leavng ntal state n the nterval [t, t + h) gven survval up untl tme t: λ () t = ( < + ) P t T t ht t lm h 0 h For small h, t follows: P( t T < t+ ht t) h λ ( t)
4 Examples:. Unemployment Duraton T s length of tme unemployed n weeks then λ(20) s probablty of becomng employed between weeks 20 and 2, condtonal on havng been unemployed up to week Recdvsm Duraton T s number of months before a former prsoner s arrested for a crme then λ(2) s probablty of beng arrested durng the 3th month, condtonal on not havng been arrested durng the frst year. The hazard functon can be expressed n terms of the pdf and cdf of T: P( t T < t+ h) F( t+ h) F( t) P. ( t T < t+ ht t) = = P T t F t 2. λ () t ( + ) ( ) ( ) ( ) () ( ) () () ( )/ () F t h F t f t f t ds t dt dlog S() t = lm = = = = h 0 h F() t F t S t S t dt
5 Usng F(0) = 0, we can ntegrate to get t F t s ds t 0 t f t t s ds 0 () = exp λ (), 0 and () = λ() exp λ( ) All probabltes can be computed usng the hazard functon: ( a T < a2) P( T a ) a2 P P( a T < a2 T a) = = exp λ ( s) ds, t 0 a Shape of the hazard functon: duraton dependence ) If the hazard functon s constant, λ() t = λ process drvng T s wthout memory: the probablty of ext n the next nterval does not depend on how much tme has been spent n the ntal state. A constant hazard mples: F( t) exp[ λt] = s the cdf of the exponental dstrbuton
6 2) Webull dstrbuton: α F() t = exp γt, γ 0, α 0 f t = t γ t α and ( ) γα exp ( t) = f ( t) / S( t) = t α λ γα If α =, the Webull dstrbuton reduces to the exponental If α >, the hazard s monotcally ncreasng postve duraton dependence If α <, the hazard s monotcally decreasng negatve duraton dependence α 3) Log-logstc hazard functon: α γαt λ γ γα γ α + γ t α α α () t =, F() t = ( + t ), and f () t = t ( + t ) 2 Accordng to the sgn of α, the hazard exhbts postve or negatve duraton dependence
7 Illustraton of some survvor and hazard functons hazard Webull hazard functons hazard Log-logstc hazard functons t alpha = 0.5 alpha = alpha = t alpha = 0.5 alpha = alpha =.5 Webull Survvor functons Log-logstc Survvor functons Survvor Survvor t alpha = 0.5 alpha = alpha = t alpha = 0.5 alpha = alpha =.5
8 8.2.2 Hazard Functons Condtonal on Tme-Invarant Covarates Condtonal hazard s: P ( t T < t+ ht t, x) λ ( t; x ) = lm h 0 h where x s a vector of explanatory varables ( ) ( x ) f t x λ ( t; x) = F t Important class wth tme-nvarant regressors: proportonal hazard models λ ( t; ) = ( ) λ ( t) x κ x 0 wth κ(.) > 0 of x () and λ 0 t > 0 s the baselne hazard (captures the duraton dependence). Often κ(.) s parameterzed as κ ( x) = exp( x β ) then, log λ( t; x) = x β + log λ0 ( t) wth β j s the elastcty of the hazard w.r.t. z j such that xj = log( zj )
9 8.2.3 Hazard Functons Condtonal on Tme-Varyng Covarates Let x(t) the vector of regressors at tme t; for t 0, X() t denotes the covarate path up through tme t: { :0 } ( ) ( ) X t x s s t ( ) The condtonal hazard functon at tme t by λ t X () t Strct exogenety of the covarates: ( X( tt+ h) T t+ hx( t) ) = X( tt+ h) X ( t) P,, P, ( ) ; = lm ( < + X( + )) P t T t ht t, t h h 0 h Proportonal Hazard wth tme-varyng covarates: λ ( t; x ( t )) = κ ( x( t) ) λ 0 ( t), wth κ ( x( t) ) = exp x( t) β
10 8.3 Analyss of Sngle-Spell Data wth Tme-Invarant Covarates Populaton of nterest are ndvduals enterng the ntal state durng a gven nterval of tme [0,b], where b > 0 a known constant. We use at most one completed spell per ndvdual sngle-spell data 8.3. Flow Samplng Indvduals enterng the state at some pont durng the nterval [0,b]. Length of tme each ndvdual s n the ntal state s recorded. Data on covarates known at the tme the ndvdual entered the state are collected. Rght censorng: spells are not completed, because stop trackng ndvduals at a fxed tme ML under Flow Samplng and Rght Censorng For a random draw from the populaton, let a [0,b] denote the tme at whch ndvdual enters the ntal state, let t * denote the length of tme n the ntal state (duraton), and let x the vector of observed covarates. ( θ ) f t x ;, t 0 condtonal densty of ( ) * t * Rght censorng: t = mn t, c, where c s the censorng tme for ndvdual and t s the observed duraton
11 8.3 Analyss of Sngle-Spell Data wth Tme-Invarant Covarates ML under Flow Samplng and Rght Censorng (contnued) Condtonal on covarates, true duraton s ndependent of the startng pont * * and the censorng tme c : D ( t x, a, c) = D( t x ) a Under ths assumpton, the dstrbuton of t ( x, a, c ) does not depend on ( a, c ) * gven f duraton not censored, the contrbuton to the lkelhood s the densty: f ( t x ; θ ) f duraton s censored, contrbuton to the lkelhood s the survvor: F( c x ; θ ) Let d be a censorng ndcator ( = f uncensored and = 0 f censored). d Condtonal lkelhood for observaton s: f ( t ; ) ( ; ) x θ F t x θ Then MLE of θ s obtaned by maxmsng: dlog f ( t x; θ ) + ( d) log F( t x ; θ ) MLE s N -consstent and asymptotcally normal. N = d
12 Parameters of nterest are effects of covarates on expected duraton rather than the hazard We can apply a censored Tobt analyss to the log of the duraton. Suppose logt xδ φ σ * 2 λ(, t x) = log( t ) x ~ N( xδ, σ ) Hazard functon s σt logt x δ Φ σ the Hazard s not monotonc and does not have the PH form The estmates of the δ are easy to nterpret, because the model s equvalent to * 2 ( t ) = xδ + e e x N( σ ) log, where ~ 0, These are sem-elastctes (or elastctes f regressors n log form) on the expected duraton Webull model can also be represented n regresson form wth δ j = β j / α (Webull densty s (, ) exp ( ) α exp exp θ = β α ( β) x x x α ) f t t t Resdual n regresson equaton s extreme-value-i dstrbuted. Log-logstc model can also be represented n regresson form wth e has a 0 mean logstc dstrbuton and s ndependent of x δ = α β. (log-logstc hazard s ( ) exp ( ) α λ = β α / + exp( β) t x x α t x t )
13 8.3 Analyss of Sngle-Spell Data wth Tme-Invarant Covarates Stock Samplng Indvduals that are n ntal state are sampled at a gven pont n tme. Now, rght and left censorng are possble Wthout correcton: stock samplng bas Left censorng occurs when some startng tmes a are not observed. The sample selecton problem caused by stock samplng s called length-based samplng. Assumptons: a) startng tmes a for all ndvduals sampled at tme b are observed b) the sampled ndvduals can be observed for a certan length of tme Let ( a, c, x, t) a random draw from the populaton of all spells startng n [0,b]. Ths vector s observed f the person s stll n ntal state at tme b.
14 Under the condtonal ndependence assumpton: D * * ( t,, ) D x a c = ( t x ) * ( t b a x a c) = F( b a x a) P,,, the log-lkelhood functon wth truncated densty and probablty can be wrtten as N = ( θ ) ( ) ( θ ) F( b a x θ ) dlog f t ; + d log F t ; log ; x x where t = c when d = 0 If all unts are rght censored at ntervew date, the prevous log lkelhood does not dentfy θ. Even when all observed duratons are censored at the ntervew date, θ can stll be estmated gven a model for the condtonal dstrbuton of the startng tmes D( a x ) s specfed. D( a x ) s assumed to be contnuous on [0,b) wth densty k (., η ) x. Let s a sample selecton ndcator equal to f a random draw s observed.e. t b a *
15 Estmaton of θ and η can proceed by applyng CMLE to the densty of a condtonal on and s =. x Ths condtonal densty s ( x, = ) = ( x ; η ) ( x ; θ) /P( = x ; θ, η) p a s k a F b a s b where 0 < a < b and P( = x ; θη, ) = ( ; θ) x ( x ; η) s F b u k u du 0
16 8.3 Analyss of Sngle-Spell Data wth Tme-Invarant Covarates Unobserved Heterogenety The key assumptons used n most models that ncorporate unobserved heterogenety are () heterogenety s ndependent of the observed covarates, as well as startng tmes and censorng tmes; (2) heterogenety has a dstrbuton known up to a fnte number of parameters; (3) heterogenety enters the hazard functon multplcatvely. Example: Webull hazard functon condtonal on x and v α ( x ) ( x ) λ t, v = v exp β αt, where x = and v > 0 Identfcaton of α and β requres a normalsaton E( v ) = Integrate out the unobserved effect: ( x, θ, ρ) ( x,, θ) ( ; ρ) G t = F t v h v dv 0 The densty can also be obtaned. So the same methods of secton and can be F t, θ g t, θ, ρ used by replacng Gt ( x, θ, ρ ) by ( x ) and ( x ) by f ( t x, θ )
17 If a gamma-dstrbuted heterogenety, v ~ Gamma ( δ, δ ). The cdf of t * ( x, v ) ( x, ) exp t ( ; ) x exp ( ; ) 0 ξ x F t v = v k s ds = v t where λ ( t x, v) = vk ( t; x ) ( ; x ) t ( ; x ) ξ t = k s ds 0 δ δ ( ) = δ exp ( δ )/ Γ ( δ) hv v v Then, G( t x ) = + ξ( t; x ) / δ δ and g ( t x ) ( ) ( ) k t x t x ( δ ) = ; + ξ ; / δ wth the Webull hazard, the resultng duraton dstrbuton Burr dstrbuton. Importance of unobserved heterogenety because of the duraton dependence: condtonal on x only there can be some duraton dependence whle condtonal on x and v there s no duraton dependence. Example: Tme constant hazards wth dscrete heterogenety!
18 8.4 Analyss of Grouped Data Grouped data arse when each duraton s only known to fall nto a certan tme nterval. Panel data allow to treat grouped duratons. Tmelne s dvded nto M + ntervals, [0, a),[ a, a2),...,[ am, ), where a s are known constants Let c m be a bnary censorng ndcator equal to f the duraton s censored n nterval m. Smlarly, y m s a bnary ndcator equal to f the duraton ends n the mth nterval. {,,...,,, } For each person, we observe ( ) ( M M ) A parametrc hazard functon s specfed as λ ( t, θ ) y c y c x whch s a balanced panel. x. Let T denote the tme untl ext from the ntal state.
19 T s not fully observed, we know whch nterval t falls nto and whether t was censored nto a partcular nterval. We can thus obtan ( ym = ym = x cm = ) ( m m x m ) P 0 0,, 0 P y = y = 0,, c = 0, m=,..., M Under the assumpton that T s ndependent of c,..., c M gven x (random censorng), we have am ( m = m = x m = ) = ( m m m x) = λ( x θ) P y y 0,, c 0 P a T a T a, exp s;, ds a m α m ( x, θ) Therefore, P( ym = 0 ym = 0, x, cm = 0 ) = αm ( x, θ ) We can use these probabltes to construct the lkelhood functon for observaton. m log αh x, θ + log αm x, θ h= ( ) d ( ) ( )
20 To mplement the CMLE, a hazard functon must be specfed. A popular hazard functon s a pecewse-constant PH ( t x, ) = ( x, ) m, am- t < a m where ( x, ) > 0 ( ( x, ) = exp ( x, )) λ θ κ β λ κ β κ β β Wth ths functon, we have α ( x, θ m ) exp exp( x β ) λ m( a a m m ) and β and λ can be estmated. Wthout covarates, MLE of λ m leads to a well-known estmator of the survvor functon: Kaplan-Meer estmator. The survvor functon at tme a m s ( m) = P( > m) = P( > r > r ) S a T a T a T a m r= N r denotes the number of persons n the rsk set for nterval r (who have nether left the state nor been censored at tme r a whch s the begnnng of nterval r) and E r the number of persons observed to leave the state n the rth nterval Therefore, a consstent estmator of the survvor functon at tme a m s m Sˆ ( am) = ( Nr Er) / N r, m=,2,..., M r=
21 8.4 Analyss of Grouped Data Tme-Varyng Covarates Dervng the log-lkelhood functon n ths case s more complcated, especally when the strct exogenety s not assumed. Nevertheless, f the regressors are constant whthn each tme nterval [ am, am), the form of log-lkelhood s same as n secton 8.4. wth replacng x by xm n nterval m. Under the condtonal ndependence assumpton on the censorng ndctor that D TT a, x, c = D TT a, x, m=,..., M ( m m m) ( m m) Under ths assumpton, the probablty of ext (wthout censorng) s am ( m = m = xm m = ) = ( m m m xm) = λ( xm θ) a P y y 0,, c 0 P a T a T a, exp s;, ds m Therefore, the partal log-lkelhood s gven by equaton () wth αh( x, θ ) replaced by αh( x, h θ ) and αm (, θ ) αm x, m, θ. x by ( ) α m ( x, θ) m
22 If the covarates are strctly exogenous and f the censorng s strctly exogenous ( m x ) ( m x m) D TT a,, c = D TT a,, m=,..., M Wth tme-varyng covarates, the hazard specfcaton s λ t x, θ = κ x, β λ, a t < a ( ) ( ) m m m m- m Unobserved Heterogenety Wth tme-varyng covarates and unobserved heterogenety, t s dffcult to relax the strct exogenety assumpton. It s assumed that regressors are strctly exogenous condtonal on unobserved heterogenety and that the unobserved heterogenety s ndependent of the regressors. In the leadng case of the pecewse-constant baselne hazard, the hazard becomes λ tv, x, θ = v κ x, β λ, a t< a ( ) ( ) m m m m- m The densty of ( y,..., y M ) gven ( v, x, c) s m d { α, x, θ } α, x, θ 2 ( v ) ( v ) ( ) h h m, m h=
23 8.4 Analyss of Grouped Data Unobserved Heterogenety (contnued) because equaton (2) depends on the unobserved heterogenety, we cannot use t drectly to consstently estmate θ. We can ntegrate out the unobserved effect n equaton (2) to obtan the densty of y s gven the regressors and censorng ndcators. Based on ths densty, the CMLE can be used.
Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Maxmum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models
More informationMaximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Mamum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models for
More informationSee Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition)
Count Data Models See Book Chapter 11 2 nd Edton (Chapter 10 1 st Edton) Count data consst of non-negatve nteger values Examples: number of drver route changes per week, the number of trp departure changes
More informationThe Occurrence and Timing of Events: The Application of Event History Models in Accounting and Finance Research
The Occurrence and Tmng of Events: The Applcaton of Event Hstory Models n Accountng and Fnance Research Marc J. LeClere Assstant Professor Department of Accountng School of Busness Admnstraton Loyola Unversty
More informationAn Introduction to Censoring, Truncation and Sample Selection Problems
An Introducton to Censorng, Truncaton and Sample Selecton Problems Thomas Crossley SPIDA June 2003 1 A. Introducton A.1 Basc Ideas Most of the statstcal technques we study are for estmatng (populaton)
More informationMarginal Effects in Probit Models: Interpretation and Testing. 1. Interpreting Probit Coefficients
ECON 5 -- NOE 15 Margnal Effects n Probt Models: Interpretaton and estng hs note ntroduces you to the two types of margnal effects n probt models: margnal ndex effects, and margnal probablty effects. It
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationLimited Dependent Variables
Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages
More information1 Binary Response Models
Bnary and Ordered Multnomal Response Models Dscrete qualtatve response models deal wth dscrete dependent varables. bnary: yes/no, partcpaton/non-partcpaton lnear probablty model LPM, probt or logt models
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationStat 543 Exam 2 Spring 2016
Stat 543 Exam 2 Sprng 206 I have nether gven nor receved unauthorzed assstance on ths exam. Name Sgned Date Name Prnted Ths Exam conssts of questons. Do at least 0 of the parts of the man exam. I wll score
More informationParametric fractional imputation for missing data analysis. Jae Kwang Kim Survey Working Group Seminar March 29, 2010
Parametrc fractonal mputaton for mssng data analyss Jae Kwang Km Survey Workng Group Semnar March 29, 2010 1 Outlne Introducton Proposed method Fractonal mputaton Approxmaton Varance estmaton Multple mputaton
More informationBinomial Distribution: Tossing a coin m times. p = probability of having head from a trial. y = # of having heads from n trials (y = 0, 1,..., m).
[7] Count Data Models () Some Dscrete Probablty Densty Functons Bnomal Dstrbuton: ossng a con m tmes p probablty of havng head from a tral y # of havng heads from n trals (y 0,,, m) m m! fb( y n) p ( p)
More informationSTAT 405 BIOSTATISTICS (Fall 2016) Handout 15 Introduction to Logistic Regression
STAT 45 BIOSTATISTICS (Fall 26) Handout 5 Introducton to Logstc Regresson Ths handout covers materal found n Secton 3.7 of your text. You may also want to revew regresson technques n Chapter. In ths handout,
More informationNon-Mixture Cure Model for Interval Censored Data: Simulation Study ABSTRACT
Malaysan Journal of Mathematcal Scences 8(S): 37-44 (2014) Specal Issue: Internatonal Conference on Mathematcal Scences and Statstcs 2013 (ICMSS2013) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal
More informationPredictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore
Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.
More informationTHE ROYAL STATISTICAL SOCIETY 2006 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE
THE ROYAL STATISTICAL SOCIETY 6 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutons to assst canddates preparng for the eamnatons n future years and for
More informationSTK4080/9080 Survival and event history analysis
SK48/98 Survval and event hstory analyss Lecture 7: Regresson modellng Relatve rsk regresson Regresson models Assume that we have a sample of n ndvduals, and let N (t) count the observed occurrences of
More informationMATH 829: Introduction to Data Mining and Analysis The EM algorithm (part 2)
1/16 MATH 829: Introducton to Data Mnng and Analyss The EM algorthm (part 2) Domnque Gullot Departments of Mathematcal Scences Unversty of Delaware Aprl 20, 2016 Recall 2/16 We are gven ndependent observatons
More informationStat 543 Exam 2 Spring 2016
Stat 543 Exam 2 Sprng 2016 I have nether gven nor receved unauthorzed assstance on ths exam. Name Sgned Date Name Prnted Ths Exam conssts of 11 questons. Do at least 10 of the 11 parts of the man exam.
More informationWeb-based Supplementary Materials for Inference for the Effect of Treatment. on Survival Probability in Randomized Trials with Noncompliance and
Bometrcs 000, 000 000 DOI: 000 000 0000 Web-based Supplementary Materals for Inference for the Effect of Treatment on Survval Probablty n Randomzed Trals wth Noncomplance and Admnstratve Censorng by Ne,
More informationComputation of Higher Order Moments from Two Multinomial Overdispersion Likelihood Models
Computaton of Hgher Order Moments from Two Multnomal Overdsperson Lkelhood Models BY J. T. NEWCOMER, N. K. NEERCHAL Department of Mathematcs and Statstcs, Unversty of Maryland, Baltmore County, Baltmore,
More informationA joint frailty-copula model between disease progression and death for meta-analysis
CSA-KSS-JSS Specal Invted Sessons 4 / / 6 A jont fralty-copula model between dsease progresson and death for meta-analyss 3/5/7 Takesh Emura Graduate Insttute of Statstcs Natonal Central Unversty TAIWAN
More informationChapter 13: Multiple Regression
Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to
More informationLimited Dependent Variables and Panel Data. Tibor Hanappi
Lmted Dependent Varables and Panel Data Tbor Hanapp 30.06.2010 Lmted Dependent Varables Dscrete: Varables that can take onl a countable number of values Censored/Truncated: Data ponts n some specfc range
More information6. Stochastic processes (2)
Contents Markov processes Brth-death processes Lect6.ppt S-38.45 - Introducton to Teletraffc Theory Sprng 5 Markov process Consder a contnuous-tme and dscrete-state stochastc process X(t) wth state space
More information6. Stochastic processes (2)
6. Stochastc processes () Lect6.ppt S-38.45 - Introducton to Teletraffc Theory Sprng 5 6. Stochastc processes () Contents Markov processes Brth-death processes 6. Stochastc processes () Markov process
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationCredit Card Pricing and Impact of Adverse Selection
Credt Card Prcng and Impact of Adverse Selecton Bo Huang and Lyn C. Thomas Unversty of Southampton Contents Background Aucton model of credt card solctaton - Errors n probablty of beng Good - Errors n
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationTime to dementia onset: competing risk analysis with Laplace regression
Tme to dementa onset: competng rsk analyss wth Laplace regresson Gola Santon, Debora Rzzuto, Laura Fratglon 4 th Nordc and Baltc STATA Users group meetng, Stockholm, November 20 Agng Research Center (ARC),
More informationLOGIT ANALYSIS. A.K. VASISHT Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi
LOGIT ANALYSIS A.K. VASISHT Indan Agrcultural Statstcs Research Insttute, Lbrary Avenue, New Delh-0 02 amtvassht@asr.res.n. Introducton In dummy regresson varable models, t s assumed mplctly that the dependent
More informationAnalysis of Discrete Time Queues (Section 4.6)
Analyss of Dscrete Tme Queues (Secton 4.6) Copyrght 2002, Sanjay K. Bose Tme axs dvded nto slots slot slot boundares Arrvals can only occur at slot boundares Servce to a job can only start at a slot boundary
More informationBasic R Programming: Exercises
Basc R Programmng: Exercses RProgrammng John Fox ICPSR, Summer 2009 1. Logstc Regresson: Iterated weghted least squares (IWLS) s a standard method of fttng generalzed lnear models to data. As descrbed
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationj) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1
Random varables Measure of central tendences and varablty (means and varances) Jont densty functons and ndependence Measures of assocaton (covarance and correlaton) Interestng result Condtonal dstrbutons
More informationPrimer on High-Order Moment Estimators
Prmer on Hgh-Order Moment Estmators Ton M. Whted July 2007 The Errors-n-Varables Model We wll start wth the classcal EIV for one msmeasured regressor. The general case s n Erckson and Whted Econometrc
More informationDiagnostics in Poisson Regression. Models - Residual Analysis
Dagnostcs n Posson Regresson Models - Resdual Analyss 1 Outlne Dagnostcs n Posson Regresson Models - Resdual Analyss Example 3: Recall of Stressful Events contnued 2 Resdual Analyss Resduals represent
More informationRELIABILITY ASSESSMENT
CHAPTER Rsk Analyss n Engneerng and Economcs RELIABILITY ASSESSMENT A. J. Clark School of Engneerng Department of Cvl and Envronmental Engneerng 4a CHAPMAN HALL/CRC Rsk Analyss for Engneerng Department
More information1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 -Davd Klenfeld - Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys
More informationSample Size Calculation Based on the Semiparametric Analysis of Short-term and Long-term Hazard Ratios. Yi Wang
Sample Sze Calculaton Based on the Semparametrc Analyss of Short-term and Long-term Hazard Ratos Y Wang Submtted n partal fulfllment of the requrements for the degree of Doctor of Phlosophy under the Executve
More information1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands
Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of
More informationLecture 3: Probability Distributions
Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More informationMaximum Likelihood Estimation
Maxmum Lkelhood Estmaton INFO-2301: Quanttatve Reasonng 2 Mchael Paul and Jordan Boyd-Graber MARCH 7, 2017 INFO-2301: Quanttatve Reasonng 2 Paul and Boyd-Graber Maxmum Lkelhood Estmaton 1 of 9 Why MLE?
More informationBIO Lab 2: TWO-LEVEL NORMAL MODELS with school children popularity data
Lab : TWO-LEVEL NORMAL MODELS wth school chldren popularty data Purpose: Introduce basc two-level models for normally dstrbuted responses usng STATA. In partcular, we dscuss Random ntercept models wthout
More informationChapter 9: Statistical Inference and the Relationship between Two Variables
Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,
More informationInterval Regression with Sample Selection
Interval Regresson wth Sample Selecton Géraldne Hennngsen, Arne Hennngsen, Sebastan Petersen May 3, 07 Ths vgnette s largely based on Petersen et al. 07. Model Specfcaton The general specfcaton of an nterval
More informationJanuary Examinations 2015
24/5 Canddates Only January Examnatons 25 DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR STUDENT CANDIDATE NO.. Department Module Code Module Ttle Exam Duraton (n words)
More informationsince [1-( 0+ 1x1i+ 2x2 i)] [ 0+ 1x1i+ assumed to be a reasonable approximation
Econ 388 R. Butler 204 revsons Lecture 4 Dummy Dependent Varables I. Lnear Probablty Model: the Regresson model wth a dummy varables as the dependent varable assumpton, mplcaton regular multple regresson
More informationAndreas C. Drichoutis Agriculural University of Athens. Abstract
Heteroskedastcty, the sngle crossng property and ordered response models Andreas C. Drchouts Agrculural Unversty of Athens Panagots Lazards Agrculural Unversty of Athens Rodolfo M. Nayga, Jr. Texas AMUnversty
More information9 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 - Chapter 9R -Davd Klenfeld - Fall 2005 9 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys a set
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationComputing MLE Bias Empirically
Computng MLE Bas Emprcally Kar Wa Lm Australan atonal Unversty January 3, 27 Abstract Ths note studes the bas arses from the MLE estmate of the rate parameter and the mean parameter of an exponental dstrbuton.
More informationRockefeller College University at Albany
Rockefeller College Unverst at Alban PAD 705 Handout: Maxmum Lkelhood Estmaton Orgnal b Davd A. Wse John F. Kenned School of Government, Harvard Unverst Modfcatons b R. Karl Rethemeer Up to ths pont n
More informationStat 642, Lecture notes for 01/27/ d i = 1 t. n i t nj. n j
Stat 642, Lecture notes for 01/27/05 18 Rate Standardzaton Contnued: Note that f T n t where T s the cumulatve follow-up tme and n s the number of subjects at rsk at the mdpont or nterval, and d s the
More information8 Derivation of Network Rate Equations from Single- Cell Conductance Equations
Physcs 178/278 - Davd Klenfeld - Wnter 2015 8 Dervaton of Network Rate Equatons from Sngle- Cell Conductance Equatons We consder a network of many neurons, each of whch obeys a set of conductancebased,
More informationHydrological statistics. Hydrological statistics and extremes
5--0 Stochastc Hydrology Hydrologcal statstcs and extremes Marc F.P. Berkens Professor of Hydrology Faculty of Geoscences Hydrologcal statstcs Mostly concernes wth the statstcal analyss of hydrologcal
More informationEconomics 130. Lecture 4 Simple Linear Regression Continued
Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do
More informationThe young are not forever young:
The young are not forever young: the assmlaton of young persons n the labour market n France Stephen Bazen and Khald Maman Wazr Paper presented at AMSE-Banque de France Labour Economcs Conference December
More informationFirst Year Examination Department of Statistics, University of Florida
Frst Year Examnaton Department of Statstcs, Unversty of Florda May 7, 010, 8:00 am - 1:00 noon Instructons: 1. You have four hours to answer questons n ths examnaton.. You must show your work to receve
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More informationSampling Theory MODULE VII LECTURE - 23 VARYING PROBABILITY SAMPLING
Samplng heory MODULE VII LECURE - 3 VARYIG PROBABILIY SAMPLIG DR. SHALABH DEPARME OF MAHEMAICS AD SAISICS IDIA ISIUE OF ECHOLOGY KAPUR he smple random samplng scheme provdes a random sample where every
More informationApplied Stochastic Processes
STAT455/855 Fall 23 Appled Stochastc Processes Fnal Exam, Bref Solutons 1. (15 marks) (a) (7 marks) The dstrbuton of Y s gven by ( ) ( ) y 2 1 5 P (Y y) for y 2, 3,... The above follows because each of
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear
More informationFinancing Innovation: Evidence from R&D Grants
Fnancng Innovaton: Evdence from R&D Grants Sabrna T. Howell Onlne Appendx Fgure 1: Number of Applcants Note: Ths fgure shows the number of losng and wnnng Phase 1 grant applcants over tme by offce (Energy
More information9. Binary Dependent Variables
9. Bnar Dependent Varables 9. Homogeneous models Log, prob models Inference Tax preparers 9.2 Random effects models 9.3 Fxed effects models 9.4 Margnal models and GEE Appendx 9A - Lkelhood calculatons
More informationHere is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y)
Secton 1.5 Correlaton In the prevous sectons, we looked at regresson and the value r was a measurement of how much of the varaton n y can be attrbuted to the lnear relatonshp between y and x. In ths secton,
More information8 Derivation of Network Rate Equations from Single- Cell Conductance Equations
Physcs 178/278 - Davd Klenfeld - Wnter 2019 8 Dervaton of Network Rate Equatons from Sngle- Cell Conductance Equatons Our goal to derve the form of the abstract quanttes n rate equatons, such as synaptc
More information,, MRTS is the marginal rate of technical substitution
Mscellaneous Notes on roducton Economcs ompled by eter F Orazem September 9, 00 I Implcatons of conve soquants Two nput case, along an soquant 0 along an soquant Slope of the soquant,, MRTS s the margnal
More information8 : Learning in Fully Observed Markov Networks. 1 Why We Need to Learn Undirected Graphical Models. 2 Structural Learning for Completely Observed MRF
10-708: Probablstc Graphcal Models 10-708, Sprng 2014 8 : Learnng n Fully Observed Markov Networks Lecturer: Erc P. Xng Scrbes: Meng Song, L Zhou 1 Why We Need to Learn Undrected Graphcal Models In the
More informationConvergence of random processes
DS-GA 12 Lecture notes 6 Fall 216 Convergence of random processes 1 Introducton In these notes we study convergence of dscrete random processes. Ths allows to characterze phenomena such as the law of large
More informationExam. Econometrics - Exam 1
Econometrcs - Exam 1 Exam Problem 1: (15 ponts) Suppose that the classcal regresson model apples but that the true value of the constant s zero. In order to answer the followng questons assume just one
More information4DVAR, according to the name, is a four-dimensional variational method.
4D-Varatonal Data Assmlaton (4D-Var) 4DVAR, accordng to the name, s a four-dmensonal varatonal method. 4D-Var s actually a drect generalzaton of 3D-Var to handle observatons that are dstrbuted n tme. The
More informationA Comparative Study for Estimation Parameters in Panel Data Model
A Comparatve Study for Estmaton Parameters n Panel Data Model Ahmed H. Youssef and Mohamed R. Abonazel hs paper examnes the panel data models when the regresson coeffcents are fxed random and mxed and
More informationCIE4801 Transportation and spatial modelling Trip distribution
CIE4801 ransportaton and spatal modellng rp dstrbuton Rob van Nes, ransport & Plannng 17/4/13 Delft Unversty of echnology Challenge the future Content What s t about hree methods Wth specal attenton for
More informatione i is a random error
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown
More information3/3/2014. CDS M Phil Econometrics. Vijayamohanan Pillai N. CDS Mphil Econometrics Vijayamohan. 3-Mar-14. CDS M Phil Econometrics.
Dummy varable Models an Plla N Dummy X-varables Dummy Y-varables Dummy X-varables Dummy X-varables Dummy varable: varable assumng values 0 and to ndcate some attrbutes To classfy data nto mutually exclusve
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationANSWERS CHAPTER 9. TIO 9.2: If the values are the same, the difference is 0, therefore the null hypothesis cannot be rejected.
ANSWERS CHAPTER 9 THINK IT OVER thnk t over TIO 9.: χ 2 k = ( f e ) = 0 e Breakng the equaton down: the test statstc for the ch-squared dstrbuton s equal to the sum over all categores of the expected frequency
More informationEquilibrium Analysis of the M/G/1 Queue
Eulbrum nalyss of the M/G/ Queue Copyrght, Sanay K. ose. Mean nalyss usng Resdual Lfe rguments Secton 3.. nalyss usng an Imbedded Marov Chan pproach Secton 3. 3. Method of Supplementary Varables done later!
More informationEstimating a Semi-Parametric Duration Model without Specifying Heterogeneity
Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety Jerry A. Hausman and Temen M. Woutersen MIT and Johns Hopkns Unversty Draft, August 2005 Abstract. Ths paper presents a new estmator
More informationEstimation of the Mean of Truncated Exponential Distribution
Journal of Mathematcs and Statstcs 4 (4): 84-88, 008 ISSN 549-644 008 Scence Publcatons Estmaton of the Mean of Truncated Exponental Dstrbuton Fars Muslm Al-Athar Department of Mathematcs, Faculty of Scence,
More informationProperties of Least Squares
Week 3 3.1 Smple Lnear Regresson Model 3. Propertes of Least Squares Estmators Y Y β 1 + β X + u weekly famly expendtures X weekly famly ncome For a gven level of x, the expected level of food expendtures
More informationChapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
More information6 Supplementary Materials
6 Supplementar Materals 61 Proof of Theorem 31 Proof Let m Xt z 1:T : l m Xt X,z 1:t Wethenhave mxt z1:t ˆm HX Xt z 1:T mxt z1:t m HX Xt z 1:T + mxt z 1:T HX We consder each of the two terms n equaton
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More informationMarkov Chain Monte Carlo Lecture 6
where (x 1,..., x N ) X N, N s called the populaton sze, f(x) f (x) for at least one {1, 2,..., N}, and those dfferent from f(x) are called the tral dstrbutons n terms of mportance samplng. Dfferent ways
More informationBiostatistics 360 F&t Tests and Intervals in Regression 1
Bostatstcs 360 F&t Tests and Intervals n Regresson ORIGIN Model: Y = X + Corrected Sums of Squares: X X bar where: s the y ntercept of the regresson lne (translaton) s the slope of the regresson lne (scalng
More informationSemiparametric Methods of Time Scale Selection
1 Semparametrc Methods of Tme Scale Selecton Therry Duchesne Unversty of Toronto, Toronto, Canada Abstract: In several relablty applcatons, there may not be a unque plausble scale n whch to analyze falure
More informationAn Experiment/Some Intuition (Fall 2006): Lecture 18 The EM Algorithm heads coin 1 tails coin 2 Overview Maximum Likelihood Estimation
An Experment/Some Intuton I have three cons n my pocket, 6.864 (Fall 2006): Lecture 18 The EM Algorthm Con 0 has probablty λ of heads; Con 1 has probablty p 1 of heads; Con 2 has probablty p 2 of heads
More informationComparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method
Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method
More informationData Abstraction Form for population PK, PD publications
Data Abstracton Form for populaton PK/PD publcatons Brendel K. 1*, Dartos C. 2*, Comets E. 1, Lemenuel-Dot A. 3, Laffont C.M. 3, Lavelle C. 4, Grard P. 2, Mentré F. 1 1 INSERM U738, Pars, France 2 EA3738,
More informationHidden Markov Models & The Multivariate Gaussian (10/26/04)
CS281A/Stat241A: Statstcal Learnng Theory Hdden Markov Models & The Multvarate Gaussan (10/26/04) Lecturer: Mchael I. Jordan Scrbes: Jonathan W. Hu 1 Hdden Markov Models As a bref revew, hdden Markov models
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationStat260: Bayesian Modeling and Inference Lecture Date: February 22, Reference Priors
Stat60: Bayesan Modelng and Inference Lecture Date: February, 00 Reference Prors Lecturer: Mchael I. Jordan Scrbe: Steven Troxler and Wayne Lee In ths lecture, we assume that θ R; n hgher-dmensons, reference
More informationECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2017 Instructor: Victor Aguirregabiria
ECOOMETRICS II ECO 40S Unversty of Toronto Department of Economcs Wnter 07 Instructor: Vctor Agurregabra SOLUTIO TO FIAL EXAM Tuesday, Aprl 8, 07 From :00pm-5:00pm 3 hours ISTRUCTIOS: - Ths s a closed-book
More informationMarkov chains. Definition of a CTMC: [2, page 381] is a continuous time, discrete value random process such that for an infinitesimal
Markov chans M. Veeraraghavan; March 17, 2004 [Tp: Study the MC, QT, and Lttle s law lectures together: CTMC (MC lecture), M/M/1 queue (QT lecture), Lttle s law lecture (when dervng the mean response tme
More informationProbability and Random Variable Primer
B. Maddah ENMG 622 Smulaton 2/22/ Probablty and Random Varable Prmer Sample space and Events Suppose that an eperment wth an uncertan outcome s performed (e.g., rollng a de). Whle the outcome of the eperment
More information