Lecture notes on epidemiology
|
|
- Beverly Pope
- 6 years ago
- Views:
Transcription
1 Bostatstcs 2002, ÅS Lecture otes o epdemology The ams of epdemology: Epdemology s the study of the dstrbuto ad determats of health related states or evets specfed populatos ad the applcato of ths study to cotrol health problems (Mould) Typcally oe tres to fd assocatos betwee rsk factors (exposures) ad dseases by studyg the occurrece of dseases populatos The fal am s to coclude casual relatos Examples of epdemologcal fdgs (from Rothma): Smokg ad lug cacer Smokg ad other health related evets Passve smokg Low-level osg radato ad leukaema Sacchar ad bladder cacer Swe flu ad Gulla-Barré sydrome The effect of dethylstlbestrol o offsprg Tampos ad toxc-shock sydrome Coffee drkg ad pacreatc cacer Epdemologcal studes are ot geerally ameable to beg vestgated by radomsed trals ad observatoal studes are therefore the more practcal to study factors ad exposure whch ca ot be cotrolled by the vestgators (Mould) 1
2 Bostatstcs 2002, ÅS Hll s crtera for dstgushg betwee causal ad o-causal assocatos: 1 Stregth (the assocato should be suffcetly strog) 2 Cosstecy (t should be see dfferet populatos uder dfferet crcumstaces) 3 Specfcty (a cause should be specfc to ts effect, eg smokg leads to a specfc form of lug cacers) 4 Temporalty (cause precedes effect) 5 Bologcal gradet (greater dose should gve larger effect) 6 Plausblty (see amal models) 7 Coherece (ot coflct to kow bology) 8 Expermetal evdece (f possble a radomsed expermet should be carred through) 9 Aalogy (the cause make sese) Type of studes: Studes may be classfed as prospectve or retrospectve I a prospectve study measures of exposures ad covarates are made before lless occur I a retrospectve study these measuremets are made after the case have already occurred Cross-sectoal study A study that cludes all persos, the populato, at the tme of ascertamet, or a represetatve sample of such persos Dsease ad exposure are observed smultaeously 2
3 Bostatstcs 2002, ÅS Cohort study Two groups (or more) groups of people that are free from the dsease ad that dffer accordg to the extet of ther exposure to a potetal cause of the dseases are studed durg a tme terval Case-Cotrol study A case cotrol study cludes people wth a dsease ad a sutable cotrol group of people uaffected by the dsease The occurrece of the possble cause (exposure) s compared betwee cases ad cotrols A relevat stuato s to thk of a case-cotrol study as a cohort study where exposure s vestgated for all cases ad for a sample of the o-cases Measures of dsease frequecy: Prevalece s the umber of cases of a dsease a defed populato The prevalece ca be measured by a prevalece rate, e, the proporto of the populato that has the dsease at a specfc tme Ths ca be measured by a crosssectoal study Icdece s the umber of ew cases of dsease a populato The cdece ca oly be measured by cosderg a populato durg a tme terval It ca be measured by the cdece rate The cdece rate s defed as (The umber of ew cases the populato durg the tme cosdered)/(the total umber of years that people are uder rsk do develop the dsease) Cumulatve cdece s the umber of people a cohort that gets the dsease durg the tme of study The relato betwee prevalece ad cdece s somewhat complcated It depeds o the durato of the dsease state A approxmate relato s P = I D Where D s the mea durato whch dseased people carry the dsease 3
4 Bostatstcs 2002, ÅS If D s creased (eg the mortalty s lowered by ew medces or f the dsease s dagosed at a earler state) the prevalece rate creases eve f the cdece rate s the same Epdemologcal are most ofte cocered wth the study of cdeces The ma am s to compare cdeces for exposed ad o-exposed people Measures of effect ad assocato: Icdece rates exposed ad o-exposed people ca be compared several way Obvous alteratves are dffereces ad ratos Let I 1 be the cdece rate for exposed people ad I 0 be the cdece rate of o-exposed people The the two alteratves are I 1 - I 0 ad I 1 /I 0 If p 1 s the probablty that a exposed perso gets ll durg a tme terval ad p 0 s the correspodg probablty that a uexposed perso gets ll the we ca measure the dfferece ether as a rsk dfferece p 1 - p 0, Or a rsk rato p 1 /p 0 Stll aother alteratve s the odds rato {p 1 /(1-p 1 )} /{p 0 /(1-p 0 )} These measures gves dfferet results all cases other the p 1 = p 0 I that case the dfferece s 0 ad the rsk rato ad the odds rato are both equal to 1 It should be observed that these comparsos say lttle of the absolute values of the rsk Other measures (whch relates to a certa populato) s attrbutable rsk or etologcal fracto 4
5 Bostatstcs 2002, ÅS Statstcal aalyss: Oe strata: I the smplest stuato the outcome of a cohort study ca be summarzed a 2x2- table: exposed ll Not ll A terestg hypothess s that the rsks of gettg ll are the same for exposed ad o-exposed people Ths correspods to a χ 2 -test of homogeety (or Fsher s exact test) If ths hypothess s rejected t s of terest to estmate the effect of the exposure Oe effect measure s the odds rato The odds-rato s estmated by the cross-product rato: Most commo s however to estmate the logarthm of the odds rato The MLestmate s l(11 ) - l(12) - l( 21) + l( 22) If the observatos the cells are large t s possble to prove that ths estmate s asymptotcally ormal dstrbuted wth the true value of the logarthm of the odds rato as mea ad a asymptotc varace that ca be estmated by 1/ /12 1/ 21 1/ 22 Ths ca be used to calculate a approxmate cofdece terval for the logarthm of the odds rato by the followg formula l( + 11) - l(12) - l( 21) + l( 22) z a 1/11 + 1/12 + 1/ 21 1/ 22 From ths a approxmate cofdece rego for the log odds rato ca be derved as 5
6 Bostatstcs 2002, ÅS ( z 1/ + 1/ + 1/ + ) exp 1/ a The reaso for calculate estmates ad cofdece tervals for log odds stead of the odds drectly s that the ormal approxmato ofte works better for the log of the cross-product rato tha for the cross-product rato tself I a case-cotrol study the observatos ca also be summarzed a 2x2 table, but wth a dfferet terpretato: exposed cases cotrols The same aalyss ca be appled Observe that a a case-cotrol study you ca estmate the odds-rato but ot the rsk rato However f the rsk s small these measure are almost detcal Several strata: Sometmes the aalyss has to be stratfed to dfferet groups of people I that case we have to summarze the observatos to several 2x2-tables (oe for each strata) The reaso for ths s that there may be some other factor that relates to the exposure ad to the dsease that we wat to cotrol for Makg a separate aalyss for strata whch such factors are costat does ths Typcal stratfcato varables are sex, age, smokg, drkg ad other lfe style factors Such factor may ether fluece the effect as beg ether effect modfers or cofouders Cofoudg s a systematc bas the estmate of the effect that follows from that the cases ad cotrols dffers wth regard to a rsk factor ot cosdered the study The aalyss have to cosder ths possblty as s see by the followg example Stratum 1 Exposed Cases Cotrols
7 Bostatstcs 2002, ÅS Ths gves the estmated odds rato 40 Stratum 2 Exposed Cases Cotrols Ths gves the estmated odds rato 43 Summg these two tables gves Strata Exposed Cases Cotrols Whch gves the estmated odds rato 33 That t ot possble to do a far result by just summarsg (collapsg) over strata s referred to as o-collapsblty It s easy to costruct other examples that gve eve more devatg estmates If the effect s dfferet dfferet strata ths effect modfcato has to be descrbed ad aalysed Formulatg a model for how the effect s modfed by varous covarates may do ths If the effect s the same all strata there s eed for a statstcal method to do testg ad estmatg smultaeously for several 2x2-tables: The ull hypothess of o effect ay of the strata ca be tested by a so-called Matel-Haeszel test 7
8 Bostatstcs 2002, ÅS Stratum 1: exposed cases a 1 b 1 cotrols c 1 d 1 Stratum exposed cases a b cotrols c d Stratum exposed cases a b cotrols c d The observed umber of exposed cases s O = a The expected umber of exposed cases s (accordg to hypergeometrc dstrbuto each strata) E = ( a + b )( a + c ) a + b + c + d The varace of O s (accordg to the hypergeometrc dstrbuto) V = ( a + b )( c + d )( a + c )( b + d ) 2 t ( t - 1) 8
9 Bostatstcs 2002, ÅS Where t = a + b + c + d The ull hypothess of o effect all strata s rejected f ( O - E) 2 V s large compared to a χ 2 (1)-percetle Matched case-cotrol studes: I order to mmse the possblty of troducg cofouders epdemologcal studes are sometmes matched Ths meas that cotrols are selected specfc to each case (or a group of cases) a way that makes them as close to the cases as possble as regards covarates that are ot uder study A typcal examples of ths s gve Rce secto 135 Ths descrbes a 1-1 matchg It s also possble to have more cotrols, so called 1-m matchg Data from matched studes should be aalysed by methods for smultaeous aalyss of several 2x2-tables Two remarks: I matched studes the precso gaed by creasg the umber of cotrols to each cases decreases rather fast There may be a dsadvatage of matched studes sce t prevets ay study of the effect of the varables that has bee used to defe the matchg Aalyss of matched case-cotrol studes The methods for estmatg matched case-cotrol studes are the same as for aalysg several 2x2-tables where same effect s assumed all strata I a matched study each case defe a stratum A estmate of the commo effect s obtaed through the Matel-Haeszel estmator: a d b c / t / t 9
10 Bostatstcs 2002, ÅS Logstc regresso I may stuato covarates ca be attach to each dvdual uder study Ths meas that we do ot oly observe the Beroull varable Y whch ca take the values 0 ad 1 accordg to f the dvdual s dseases or ot but we also have other measuremets (x 1,,x k ) whch may fluece the rsk of gettg ll I such cases there s eed for a model that descrbes the fluece of the covarates (depedet varables) o the dstrbuto of Y Such a model whch s smlar to ormal lear regresso assumes that exp Pr(Y = 1) = 1 + exp ( b0 + b1x1 + + bkxk ) ( b + b x + + b x ) k k Ths model s called a logstc regresso model Observe that the log odds s a lear combato of the covarates, e, Pr(Y = 1) l = b0 + b1x1 + K + b Ł Pr(Y = 0) ł k x k Oe of the covarates may be exposure, ether defed as a 0/1-varable (exposed or ot) or as a dose (that measure the amout of exposures I such case the correspodg regresso coeffcet measures the effect of the exposure I fact the estmate of the regresso coeffcet s the logarthm of the odds rato effect of exposure that dffers wth oe ut of the scale whch the exposure s measured Codtoal logstc regresso I matched case cotrol studes we have a case wth correspodg covarates ad oe or more cotrols wth possble dfferet covarates We ca the use a logstc regresso model to calculate a codtoal probablty that says how propable t s that amog a umber of people (the case ad the cotrols) wth a ceta set of covarate values t s that just the case have the dsease uder study Let cosder a 1-1 matchg stuato ad assume that the case has covarates x ad the cotrol covarate z (For smplcty we assume a oe-dmesoal covarate) The we ca calculate the codtoal probablty that the perso wth covarates x have the dsease gve that exactly oe of the two persos have the dsease Elemetary (but ot smple) calculatos gve the result 10
11 Bostatstcs 2002, ÅS exp exp( b1x) ( b x) + exp( b z) 1 1 The statstcal aalyss of the matched data ca be aalysed usg these probabltes to bult up a lkelhood ML estmates ca be foud ad lkelhood rato test derved Possble sources of Bas epdemologcal studes: Cofoudg Selecto bas (eg Healthy worker effect) Recall Bas (cases may report exposures more accurately tha cotrols) Publcato Bas (sgfcat results teds to be publshed easer tha studes wth sgfcace s ot foud) 11
Summary of the lecture in Biostatistics
Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the
More informationLecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model
Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The
More informationLecture Notes Types of economic variables
Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:
More informationChapter 14 Logistic Regression Models
Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as
More informationChapter 8. Inferences about More Than Two Population Central Values
Chapter 8. Ifereces about More Tha Two Populato Cetral Values Case tudy: Effect of Tmg of the Treatmet of Port-We tas wth Lasers ) To vestgate whether treatmet at a youg age would yeld better results tha
More informationSimple Linear Regression
Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato
More informationC. Statistics. X = n geometric the n th root of the product of numerical data ln X GM = or ln GM = X 2. X n X 1
C. Statstcs a. Descrbe the stages the desg of a clcal tral, takg to accout the: research questos ad hypothess, lterature revew, statstcal advce, choce of study protocol, ethcal ssues, data collecto ad
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More informationSTATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1
STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ
More informationOrdinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
More informationENGI 3423 Simple Linear Regression Page 12-01
ENGI 343 mple Lear Regresso Page - mple Lear Regresso ometmes a expermet s set up where the expermeter has cotrol over the values of oe or more varables X ad measures the resultg values of aother varable
More informationSpecial Instructions / Useful Data
JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationChapter 3 Sampling For Proportions and Percentages
Chapter 3 Samplg For Proportos ad Percetages I may stuatos, the characterstc uder study o whch the observatos are collected are qualtatve ature For example, the resposes of customers may marketg surveys
More informationSTK3100 and STK4100 Autumn 2017
SK3 ad SK4 Autum 7 Geeralzed lear models Part III Covers the followg materal from chaters 4 ad 5: Sectos 4..5, 4.3.5, 4.3.6, 4.4., 4.4., ad 4.4.3 Sectos 5.., 5.., ad 5.5. Ørulf Borga Deartmet of Mathematcs
More informationMultiple Linear Regression Analysis
LINEA EGESSION ANALYSIS MODULE III Lecture - 4 Multple Lear egresso Aalyss Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Cofdece terval estmato The cofdece tervals multple
More informationChapter 11 The Analysis of Variance
Chapter The Aalyss of Varace. Oe Factor Aalyss of Varace. Radomzed Bloc Desgs (ot for ths course) NIPRL . Oe Factor Aalyss of Varace.. Oe Factor Layouts (/4) Suppose that a expermeter s terested populatos
More informationCHAPTER VI Statistical Analysis of Experimental Data
Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca
More informationLecture 8: Linear Regression
Lecture 8: Lear egresso May 4, GENOME 56, Sprg Goals Develop basc cocepts of lear regresso from a probablstc framework Estmatg parameters ad hypothess testg wth lear models Lear regresso Su I Lee, CSE
More informationThe TDT. (Transmission Disequilibrium Test) (Qualitative and quantitative traits) D M D 1 M 1 D 2 M 2 M 2D1 M 1
The TDT (Trasmsso Dsequlbrum Test) (Qualtatve ad quattatve trats) Our am s to test for lkage (ad maybe ad/or assocato) betwee a dsease locus D ad a marker locus M. We kow where (.e. o what chromosome,
More informationbest estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best
Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg
More informationModule 7: Probability and Statistics
Lecture 4: Goodess of ft tests. Itroducto Module 7: Probablty ad Statstcs I the prevous two lectures, the cocepts, steps ad applcatos of Hypotheses testg were dscussed. Hypotheses testg may be used to
More informationImproving coverage probabilities of confidence intervals in random effects meta-analysis with publication bias
Improvg coverage probabltes of cofdece tervals radom effects meta-aalyss th publcato bas Masayuk Hem The Isttute of Statstcal Mathematcs, Japa Joh B. Copas Uversty of Warck, UK Itroducto Meta-aalyss: statstcal
More informationTESTS BASED ON MAXIMUM LIKELIHOOD
ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal
More informationLecture 3. Sampling, sampling distributions, and parameter estimation
Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Aalyss of Varace ad Desg of Exermets-I MODULE II LECTURE - GENERAL LINEAR HYPOTHESIS AND ANALYSIS OF VARIANCE Dr Shalabh Deartmet of Mathematcs ad Statstcs Ida Isttute of Techology Kaur Tukey s rocedure
More informationParameter, Statistic and Random Samples
Parameter, Statstc ad Radom Samples A parameter s a umber that descrbes the populato. It s a fxed umber, but practce we do ot kow ts value. A statstc s a fucto of the sample data,.e., t s a quatty whose
More informationChapter 13 Student Lecture Notes 13-1
Chapter 3 Studet Lecture Notes 3- Basc Busess Statstcs (9 th Edto) Chapter 3 Smple Lear Regresso 4 Pretce-Hall, Ic. Chap 3- Chapter Topcs Types of Regresso Models Determg the Smple Lear Regresso Equato
More informationSimulation Output Analysis
Smulato Output Aalyss Summary Examples Parameter Estmato Sample Mea ad Varace Pot ad Iterval Estmato ermatg ad o-ermatg Smulato Mea Square Errors Example: Sgle Server Queueg System x(t) S 4 S 4 S 3 S 5
More informationThe number of observed cases The number of parameters. ith case of the dichotomous dependent variable. the ith case of the jth parameter
LOGISTIC REGRESSION Notato Model Logstc regresso regresses a dchotomous depedet varable o a set of depedet varables. Several methods are mplemeted for selectg the depedet varables. The followg otato s
More informationA New Family of Transformations for Lifetime Data
Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K. A New Famly of Trasformatos for Lfetme Data Lakhaa Watthaacheewakul Abstract A famly of trasformatos s the oe of several
More informationMean is only appropriate for interval or ratio scales, not ordinal or nominal.
Mea Same as ordary average Sum all the data values ad dvde by the sample sze. x = ( x + x +... + x Usg summato otato, we wrte ths as x = x = x = = ) x Mea s oly approprate for terval or rato scales, ot
More informationChapter 5 Properties of a Random Sample
Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample
More informationAnalysis of Variance with Weibull Data
Aalyss of Varace wth Webull Data Lahaa Watthaacheewaul Abstract I statstcal data aalyss by aalyss of varace, the usual basc assumptos are that the model s addtve ad the errors are radomly, depedetly, ad
More informationTHE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE
THE ROYAL STATISTICAL SOCIETY 00 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for the
More informationChapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance
Chapter, Part A Aalyss of Varace ad Epermetal Desg Itroducto to Aalyss of Varace Aalyss of Varace: Testg for the Equalty of Populato Meas Multple Comparso Procedures Itroducto to Aalyss of Varace Aalyss
More informationPoint Estimation: definition of estimators
Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.
More informationBootstrap Method for Testing of Equality of Several Coefficients of Variation
Cloud Publcatos Iteratoal Joural of Advaced Mathematcs ad Statstcs Volume, pp. -6, Artcle ID Sc- Research Artcle Ope Access Bootstrap Method for Testg of Equalty of Several Coeffcets of Varato Dr. Navee
More informationExample: Multiple linear regression. Least squares regression. Repetition: Simple linear regression. Tron Anders Moger
Example: Multple lear regresso 5000,00 4000,00 Tro Aders Moger 0.0.007 brthweght 3000,00 000,00 000,00 0,00 50,00 00,00 50,00 00,00 50,00 weght pouds Repetto: Smple lear regresso We defe a model Y = β0
More informationSimple Linear Regression
Correlato ad Smple Lear Regresso Berl Che Departmet of Computer Scece & Iformato Egeerg Natoal Tawa Normal Uversty Referece:. W. Navd. Statstcs for Egeerg ad Scetsts. Chapter 7 (7.-7.3) & Teachg Materal
More informationTHE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA
THE ROYAL STATISTICAL SOCIETY EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA PAPER II STATISTICAL THEORY & METHODS The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for
More information( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model
Chapter 3 Asmptotc Theor ad Stochastc Regressors The ature of eplaator varable s assumed to be o-stochastc or fed repeated samples a regresso aalss Such a assumpto s approprate for those epermets whch
More informationLINEAR REGRESSION ANALYSIS
LINEAR REGRESSION ANALYSIS MODULE V Lecture - Correctg Model Iadequaces Through Trasformato ad Weghtg Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Aalytcal methods for
More informationSTA 105-M BASIC STATISTICS (This is a multiple choice paper.)
DCDM BUSINESS SCHOOL September Mock Eamatos STA 0-M BASIC STATISTICS (Ths s a multple choce paper.) Tme: hours 0 mutes INSTRUCTIONS TO CANDIDATES Do ot ope ths questo paper utl you have bee told to do
More information: At least two means differ SST
Formula Card for Eam 3 STA33 ANOVA F-Test: Completely Radomzed Desg ( total umber of observatos, k = Number of treatmets,& T = total for treatmet ) Step : Epress the Clam Step : The ypotheses: :... 0 A
More informationWu-Hausman Test: But if X and ε are independent, βˆ. ECON 324 Page 1
Wu-Hausma Test: Detectg Falure of E( ε X ) Caot drectly test ths assumpto because lack ubased estmator of ε ad the OLS resduals wll be orthogoal to X, by costructo as ca be see from the momet codto X'
More informationSTK3100 and STK4100 Autumn 2018
SK3 ad SK4 Autum 8 Geeralzed lear models Part III Covers the followg materal from chaters 4 ad 5: Cofdece tervals by vertg tests Cosder a model wth a sgle arameter β We may obta a ( α% cofdece terval for
More informationMultiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades
STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos
More informationX ε ) = 0, or equivalently, lim
Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece
More informationENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections
ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty
More informationContinuous Distributions
7//3 Cotuous Dstrbutos Radom Varables of the Cotuous Type Desty Curve Percet Desty fucto, f (x) A smooth curve that ft the dstrbuto 3 4 5 6 7 8 9 Test scores Desty Curve Percet Probablty Desty Fucto, f
More informationTHE ROYAL STATISTICAL SOCIETY 2009 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULAR FORMAT MODULE 2 STATISTICAL INFERENCE
THE ROYAL STATISTICAL SOCIETY 009 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULAR FORMAT MODULE STATISTICAL INFERENCE The Socety provdes these solutos to assst caddates preparg for the examatos future
More informationObjectives of Multiple Regression
Obectves of Multple Regresso Establsh the lear equato that best predcts values of a depedet varable Y usg more tha oe eplaator varable from a large set of potetal predctors {,,... k }. Fd that subset of
More informationESS Line Fitting
ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here
More information22 Nonparametric Methods.
22 oparametrc Methods. I parametrc models oe assumes apror that the dstrbutos have a specfc form wth oe or more ukow parameters ad oe tres to fd the best or atleast reasoably effcet procedures that aswer
More informationf f... f 1 n n (ii) Median : It is the value of the middle-most observation(s).
CHAPTER STATISTICS Pots to Remember :. Facts or fgures, collected wth a defte pupose, are called Data.. Statstcs s the area of study dealg wth the collecto, presetato, aalyss ad terpretato of data.. The
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationPermutation Tests for More Than Two Samples
Permutato Tests for ore Tha Two Samples Ferry Butar Butar, Ph.D. Abstract A F statstc s a classcal test for the aalyss of varace where the uderlyg dstrbuto s a ormal. For uspecfed dstrbutos, the permutato
More informationGenerative classification models
CS 75 Mache Learg Lecture Geeratve classfcato models Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square Data: D { d, d,.., d} d, Classfcato represets a dscrete class value Goal: lear f : X Y Bar classfcato
More informationA Study of the Reproducibility of Measurements with HUR Leg Extension/Curl Research Line
HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) A Study of the Reproducblty of Measuremets wth HUR Leg Eteso/Curl Research Le A mportat property of measuremets s that the results should
More informationEstimation of Stress- Strength Reliability model using finite mixture of exponential distributions
Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur
More informationStatistics MINITAB - Lab 5
Statstcs 10010 MINITAB - Lab 5 PART I: The Correlato Coeffcet Qute ofte statstcs we are preseted wth data that suggests that a lear relatoshp exsts betwee two varables. For example the plot below s of
More informationChapter 8: Statistical Analysis of Simulated Data
Marquette Uversty MSCS600 Chapter 8: Statstcal Aalyss of Smulated Data Dael B. Rowe, Ph.D. Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 08 by Marquette Uversty MSCS600 Ageda 8. The Sample
More informationECONOMETRIC THEORY. MODULE VIII Lecture - 26 Heteroskedasticity
ECONOMETRIC THEORY MODULE VIII Lecture - 6 Heteroskedastcty Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur . Breusch Paga test Ths test ca be appled whe the replcated data
More informationMedian as a Weighted Arithmetic Mean of All Sample Observations
Meda as a Weghted Arthmetc Mea of All Sample Observatos SK Mshra Dept. of Ecoomcs NEHU, Shllog (Ida). Itroducto: Iumerably may textbooks Statstcs explctly meto that oe of the weakesses (or propertes) of
More informationLikelihood Ratio, Wald, and Lagrange Multiplier (Score) Tests. Soccer Goals in European Premier Leagues
Lkelhood Rato, Wald, ad Lagrage Multpler (Score) Tests Soccer Goals Europea Premer Leagues - 4 Statstcal Testg Prcples Goal: Test a Hpothess cocerg parameter value(s) a larger populato (or ature), based
More informationBayes (Naïve or not) Classifiers: Generative Approach
Logstc regresso Bayes (Naïve or ot) Classfers: Geeratve Approach What do we mea by Geeratve approach: Lear p(y), p(x y) ad the apply bayes rule to compute p(y x) for makg predctos Ths s essetally makg
More informationQualifying Exam Statistical Theory Problem Solutions August 2005
Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),
More informationSPECIAL CONSIDERATIONS FOR VOLUMETRIC Z-TEST FOR PROPORTIONS
SPECIAL CONSIDERAIONS FOR VOLUMERIC Z-ES FOR PROPORIONS Oe s stctve reacto to the questo of whether two percetages are sgfcatly dfferet from each other s to treat them as f they were proportos whch the
More informationTo use adaptive cluster sampling we must first make some definitions of the sampling universe:
8.3 ADAPTIVE SAMPLING Most of the methods dscussed samplg theory are lmted to samplg desgs hch the selecto of the samples ca be doe before the survey, so that oe of the decsos about samplg deped ay ay
More information2.28 The Wall Street Journal is probably referring to the average number of cubes used per glass measured for some population that they have chosen.
.5 x 54.5 a. x 7. 786 7 b. The raked observatos are: 7.4, 7.5, 7.7, 7.8, 7.9, 8.0, 8.. Sce the sample sze 7 s odd, the meda s the (+)/ 4 th raked observato, or meda 7.8 c. The cosumer would more lkely
More informationPart I: Background on the Binomial Distribution
Part I: Bacgroud o the Bomal Dstrbuto A radom varable s sad to have a Beroull dstrbuto f t taes o the value wth probablt "p" ad the value wth probablt " - p". The umber of "successes" "" depedet Beroull
More informationENGI 4421 Propagation of Error Page 8-01
ENGI 441 Propagato of Error Page 8-01 Propagato of Error [Navd Chapter 3; ot Devore] Ay realstc measuremet procedure cotas error. Ay calculatos based o that measuremet wll therefore also cota a error.
More informationSTA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1
STA 08 Appled Lear Models: Regresso Aalyss Sprg 0 Soluto for Homework #. Let Y the dollar cost per year, X the umber of vsts per year. The the mathematcal relato betwee X ad Y s: Y 300 + X. Ths s a fuctoal
More information12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model
1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed
More informationChapter 11 Systematic Sampling
Chapter stematc amplg The sstematc samplg techue s operatoall more coveet tha the smple radom samplg. It also esures at the same tme that each ut has eual probablt of cluso the sample. I ths method of
More informationMS exam problems Fall 2012
MS exam problems Fall 01 (From: Rya Mart) 1. (Stat 401) Cosder the followg game wth a box that cotas te balls two red, three blue, ad fve gree. A player selects two balls from the box at radom, wthout
More informationSTATISTICAL INFERENCE
(STATISTICS) STATISTICAL INFERENCE COMPLEMENTARY COURSE B.Sc. MATHEMATICS III SEMESTER ( Admsso) UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITY P.O., MALAPPURAM, KERALA, INDIA -
More informationDiscrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b
CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package
More informationGOALS The Samples Why Sample the Population? What is a Probability Sample? Four Most Commonly Used Probability Sampling Methods
GOLS. Epla why a sample s the oly feasble way to lear about a populato.. Descrbe methods to select a sample. 3. Defe ad costruct a samplg dstrbuto of the sample mea. 4. Epla the cetral lmt theorem. 5.
More informationFundamentals of Regression Analysis
Fdametals of Regresso Aalyss Regresso aalyss s cocered wth the stdy of the depedece of oe varable, the depedet varable, o oe or more other varables, the explaatory varables, wth a vew of estmatg ad/or
More informationBias Correction in Estimation of the Population Correlation Coefficient
Kasetsart J. (Nat. Sc.) 47 : 453-459 (3) Bas Correcto Estmato of the opulato Correlato Coeffcet Juthaphor Ssomboothog ABSTRACT A estmator of the populato correlato coeffcet of two varables for a bvarate
More informationChapter 2 Supplemental Text Material
-. Models for the Data ad the t-test Chapter upplemetal Text Materal The model preseted the text, equato (-3) s more properl called a meas model. ce the mea s a locato parameter, ths tpe of model s also
More informationLECTURE - 4 SIMPLE RANDOM SAMPLING DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR
amplg Theory MODULE II LECTURE - 4 IMPLE RADOM AMPLIG DR. HALABH DEPARTMET OF MATHEMATIC AD TATITIC IDIA ITITUTE OF TECHOLOGY KAPUR Estmato of populato mea ad populato varace Oe of the ma objectves after
More informationLecture 1 Review of Fundamental Statistical Concepts
Lecture Revew of Fudametal Statstcal Cocepts Measures of Cetral Tedecy ad Dsperso A word about otato for ths class: Idvduals a populato are desgated, where the dex rages from to N, ad N s the total umber
More informationSTA302/1001-Fall 2008 Midterm Test October 21, 2008
STA3/-Fall 8 Mdterm Test October, 8 Last Name: Frst Name: Studet Number: Erolled (Crcle oe) STA3 STA INSTRUCTIONS Tme allowed: hour 45 mutes Ads allowed: A o-programmable calculator A table of values from
More informationThird handout: On the Gini Index
Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The
More informationLogistic regression (continued)
STAT562 page 138 Logstc regresso (cotued) Suppose we ow cosder more complex models to descrbe the relatoshp betwee a categorcal respose varable (Y) that takes o two (2) possble outcomes ad a set of p explaatory
More informationClass 13,14 June 17, 19, 2015
Class 3,4 Jue 7, 9, 05 Pla for Class3,4:. Samplg dstrbuto of sample mea. The Cetral Lmt Theorem (CLT). Cofdece terval for ukow mea.. Samplg Dstrbuto for Sample mea. Methods used are based o CLT ( Cetral
More informationMEASURES OF DISPERSION
MEASURES OF DISPERSION Measure of Cetral Tedecy: Measures of Cetral Tedecy ad Dsperso ) Mathematcal Average: a) Arthmetc mea (A.M.) b) Geometrc mea (G.M.) c) Harmoc mea (H.M.) ) Averages of Posto: a) Meda
More information9.1 Introduction to the probit and logit models
EC3000 Ecoometrcs Lecture 9 Probt & Logt Aalss 9. Itroducto to the probt ad logt models 9. The logt model 9.3 The probt model Appedx 9. Itroducto to the probt ad logt models These models are used regressos
More informationCorrelation and Simple Linear Regression
Correlato ad Smple Lear Regresso Berl Che Departmet of Computer Scece & Iformato Egeerg Natoal Tawa Normal Uverst Referece:. W. Navd. Statstcs for Egeerg ad Scetsts. Chapter 7 (7.-7.3) & Teachg Materal
More informationLecture Note to Rice Chapter 8
ECON 430 HG revsed Nov 06 Lecture Note to Rce Chapter 8 Radom matrces Let Y, =,,, m, =,,, be radom varables (r.v. s). The matrx Y Y Y Y Y Y Y Y Y Y = m m m s called a radom matrx ( wth a ot m-dmesoal dstrbuto,
More informationIntroduction to local (nonparametric) density estimation. methods
Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest
More informationSTRATIFIED SAMPLING IN AGRICULTURAL SURVEYS
3 STRATIFIED SAMPLIG I AGRICULTURAL SURVEYS austav Adtya Ida Agrcultural Statstcs Research Isttute, ew Delh-00 3. ITRODUCTIO The prme objectve of a sample survey s to obta fereces about the characterstc
More informationChapter -2 Simple Random Sampling
Chapter - Smple Radom Samplg Smple radom samplg (SRS) s a method of selecto of a sample comprsg of umber of samplg uts out of the populato havg umber of samplg uts such that every samplg ut has a equal
More informationModule 7. Lecture 7: Statistical parameter estimation
Lecture 7: Statstcal parameter estmato Parameter Estmato Methods of Parameter Estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method Populato Parameter Sample Parameter Ubased estmato
More informationChapter -2 Simple Random Sampling
Chapter - Smple Radom Samplg Smple radom samplg (SRS) s a method of selecto of a sample comprsg of umber of samplg uts out of the populato havg umber of samplg uts such that every samplg ut has a equal
More information