TOBIT REGRESSION AND CENSORED CYTOKINE DATA. Terrence L. O Day. BS, St Vincent College, MBA, University of Phoenix, 2003

Transcription

1 TOBIT REGRESSION AND CENSORED CYTOKINE DATA by Terrence L O Day BS, St Vncent College, 1981 MBA, Unversty of Phoenx, 2003 Submtted to the Graduate Faculty of the Graduate School of Publc Health n partal fulfllment of the requrements for the degree of Master of Scence Unversty of Pttsburgh 2005

2 UNIVERSITY OF PITTSBURGH THE GRADUATE SCHOOL OF PUBLIC HEALTH Ths thess was presented by Terrence L O Day It was defended on Aprl 6, 2005 and approved by Thess Advsor: Lsa Wessfeld, PhD Professor and Assocate Char Department of Bostatstcs Graduate School of Publc Health Unversty of Pttsburgh Commttee Member: Lan Kong, PhD Assstant Professor Department of Bostatstcs Graduate School of Publc Health Unversty of Pttsburgh Commttee Member: John Kellum, MD Assocate Professor Department of Crtcal Care Medcne School of Medcne Unversty of Pttsburgh Commttee Member: Derek Angus, MD Assstant Professor Department of Health Polcy & Management Graduate School of Publc Health Unversty of Pttsburgh

3 Lsa Wessfeld, PhD TOBIT REGRESSION AND CENSORED CYTOKINE DATA Terrence L O Day, MS Unversty of Pttsburgh, 2005 Well desgned clncal studes theoretcally produce accurate data from whch a reasonable concluson(s) may be drawn Data accuracy may be hndered by the measurement tool or devce Addtonally, the data may be n such a form that t s problematc from an analytc and nterpretve pont of vew An example of such a problematc form may be seen n censored, sample-selected, or truncated data Clncal data may be partcularly prone to censorng or truncaton snce varous assays used to measure patent parameters have lmted senstvty Lower and upper lmts of assay senstvty may have a drect mpact on the clncal dagnoss and prognoss of the patent, especally f the patent s a hgh rsk crtcal care patent The am of ths report s to estmate mean cytokne levels usng varous approaches, ncludng the arthmetc and geometrc mean, and mean estmaton from a tobt model The data set s from the Department of Crtcal Care Medcne and contans values for several cytoknes from 1753 patents (dscharge status) or 1610 patents (follow-up status), ncludng Interleukn 6 (IL- 6), Interleukn 10 (IL-10), and Tumor Necross Factor (TNF) A bref overvew of the mmune system and ts relatonshp to cytokne producton wll be presented pror to an explanaton of the estmaton procedures Fnally, recommendatons for estmatng a mean from the censored data set wll be presented

4 Although not specfc to Crtcal Care Medcne, the problem of censored data s evdent n many areas of study, specfcally Publc Health Gudelnes for dealng wth censored data would be a sgnfcant and valuable tool for Publc Health professonals v

5 TABLE OF CONTENTS 1 INTRODUCTION 1 2 BACKGROUND AND REVIEW 3 21 Immunology Overvew of Immune System Overvew of Cytoknes TNF IL IL The Data Censored, Sample-Selected, Truncated data The Tobt Model Maxmum Lkelhood Tobt Model and Maxmum Lkelhood The Delta Method for Standard Error Determnaton Addtonal Analyss Arthmetc Mean Geometrc Mean 21 3 RESULT 22 4 CONCLUSION / RECOMMENDATION 24 5 APPENDIX A: STATA Output 26 6 BIBLIOGRAPHY 46 v

6 LIST OF TABLES Table 1 Defntons of Censored, Sample-Selected, and Truncated Varables 14 Table 2 Frequency of censored cytokne values and dead/alve status at dscharge and follow-up 22 Table 3 Comparson of mean estmates for TNF, IL6, and IL10 for death at dscharge based on the tobt model, the arthmetc and the geometrc mean Note that the populaton s the 1753 ndvduals wth both complete lab and follow up data wth 78 reported deaths at dscharge and 1675 subjects alve at hosptal dscharge Standard errors are gven n parentheses below the mean 23 Table 4 Comparson of mean estmates for TNF, IL6, and IL10 for death at follow-up based on the tobt model, the arthmetc and the geometrc mean Note that the populaton s the 1753 ndvduals wth both complete lab and follow-up data wth 231 reported deaths at followup and 1379 subjects alve at follow up and 143 wth mssng values Standard errors are gven n parentheses below the mean 23 v

7 1 INTRODUCTION Well desgned clncal studes theoretcally produce accurate data from whch a reasonable concluson(s) may be drawn However well ntentoned nvestgators may be, the data may fal to accurately characterze the populaton from whch the parameters were measured Addtonally, the data may be n such a form that t s problematc from an analytc and nterpretve pont of vew An example of such a problematc form may be seen n censored, sample-selected, or truncated data The problem of censored, sample-selected, or truncated data s evdent n busness/economc, socal scence, and medcal lteratures 1 However, t was n the economcs lterature where the semnal paper by Tobn 2 n 1958 lad the foundaton for the contemporary analyss and nterpretaton of censored, sample-selected, or truncated data But no matter what problematc form the data may take, the ultmate goal of the analyss s to provde the best estmate of the populaton parameters Clncal data may be partcularly prone to censorng or truncaton snce varous assays used to measure patent parameters have lmted senstvty Lower and upper lmts of assay senstvty may have a drect mpact on the clncal dagnoss and prognoss of the patent, especally f the patent s a hgh rsk crtcal care patent In ths stuaton t s essental that the clncan have a tool that best estmates the parameter n queston to provde the patent wth a relable dagnoss and effectve treatment plan that wll, n turn, offer the most accurate prognoss One major factor n the management of the Crtcal Care Patent s the assessment of laboratory values One measurable parameter that may be pertnent n Crtcal Care patent management s serum Cytokne level Cytokne levels may be pertnent snce cytokne 1

8 producton s assocated wth mmune response, nflammaton, and sepss, whch are sgnfcant events for hgh rsk patents However, cytokne assay senstvty may hnder the clncan s ablty to effectvely and effcently treat ther patents when the lmted senstvty results n a cytokne level cutoff value The am of ths report s to estmate mean cytokne levels usng varous approaches, ncludng the arthmetc and geometrc mean, and mean estmaton from a tobt model The data set s from the Department of Crtcal Care Medcne and contans values for several cytoknes from 1788 patents, ncludng Interleukn 6 (IL-6), Interleukn 10 (IL-10), and Tumor Necross Factor (TNF) A bref overvew of the mmune system and ts relatonshp to cytokne producton wll be presented pror to an explanaton of the estmaton procedures Fnally, recommendatons for estmatng a mean from the censored data set wll be presented 2

9 2 BACKGROUND AND REVIEW 21 Immunology The human mmune system acts as a protectve mechansm aganst pathogens whch cause dsease, and tssue damage At the core of the mmune system s the body s ablty to recognze non-self molecules Non-self molecules (antgens) are foregn molecules composed of proten(s) and/or carbohydrate(s) that stmulate an mmune response The mmune system s regulated through the nnate and adaptve mmune responses 211 Overvew of Immune System Innate Immunty The nnate mmune response functons through physcal barrers to entry, mechancal actons, bochemcal defenses, and the actons of specalzed cells The epderms and mucous membranes act as physcal barrers aganst pathogen entry nto the host s body, whle the mechancal actons of sneezng, coughng and vomtng physcally expel the pathogens Addtonally, the cla that lne the respratory tract also brush pathogens along wth a sweepng moton to elmnate them from the body The acdc envronment of the stomach and the dgestve enzymes found n mucus and tears provde bochemcal barrers Addtonal bochemcal defense s possble through complement (ant-bacteral) and nterferon (ant-vral) The specalzed cells that afford nnate defense nclude phagocytes, Natural Kller (NK), M, and Langerhan s cells found n the blood and 3

10 varous organ systems Fnally, the nflammatory response attracts leukocytes to the ste of an nfecton The nflammatory response s a seres of events that results n redness, swellng, and occasonally, pan at the nfecton ste When common bacteral antgens are present, macrophages are stmulated to produce sgnalng protens called cytoknes Examples of cytokne effects nclude ncreased producton of leukocytes n the bone marrow, ncreased attracton of leukocytes to the nfecton ste, ncreased expresson of adheson molecules found n the membranes of epthelal cells that lne the blood vessels, and an ncrease n the amount of flud that enters the tssue from the crculaton Leukocytes possess receptors to adheson molecules that place them n a favorable poston to enter the surroundng tssue Excess flud n the tssue results n ncreased levels of antbacteral molecules that are naturally occurrng n lymph flud Adaptve Immunty The adaptve mmune response functons essentally through antbody producton and varous actons of T cells Adaptve mmunty, although not permanent, can be very endurng A characterstc that dfferentates adaptve mmune response from nnate response s that adaptve mmunty s antgen-specfc and provdes an effcent and effectve defense aganst repeat nfecton(s) Exposure and heredty play crucal roles n each ndvdual s adaptve mmune response Exposure may be classfed as ether actve or passve Examples of actve mmunty are the natural nfecton wth the nfluenza vrus, or vaccnaton wth an attenuated nfluenza vrus Actve mmunty requres a couple of weeks after nfecton to become establshed, but once 4

11 establshed may last for years Passve mmunty occurs wth the drect transfer of antbodes or T cells (cellular mmunty) Examples of passve mmunty nclude antbody transfer from mother to chld through breast mlk, mother to fetus through the placenta, the nfuson of anmal antbodes nto humans as a treatment for posonous venom, and bone marrow transplantaton Immunty at the cell and organ level Immune cells are commonly called whte blood cells or leukocytes Leukocytes play a key role n both the nnate and adaptve mmune system Leukocytes of the nnate system are phagocytes and are not antbody specfc Phagocytes are ether macrophages or polymorphonuclear leukocytes Macrophages and polymorphonuclear leukocytes bnd to proten surface molecules that are common on many pathogens, and engulf and kll them Large granular lymphocytes that offer protecton from vrus nfected cells and cancer cells are the Natural Kller (NK) cells of the nnate system NK cells lyse the nfected cell or cancer cell, whereas macrophages engulf them The NK cells are part of the nnate system snce they lack antgen specfcty Macrophages are also responsble for cytokne producton (see below) Leukocytes that are antgen-specfc are part of the adaptve mmune system Leukocytes begn development n the bone marrow, whereas T cells complete ther development n the Thymus Durng development each cell acqures specfc ndvdual antgen receptors and coreceptors, receptors to cytoknes, and receptors to adheson molecules (found on epthelal cell membranes) Antgen-specfc lymphocytes multply when ther receptors encounter the antgen Ther progeny, or clones, contan the same antgen specfcty as the parent cell that was actvated 5

12 The organs of the mmune system are classfed as ether prmary or secondary The prmary organs nclude the bone marrow and the Thymus The secondary organs are meetng places for antgen and leukocyte The secondary organs nclude the spleen and the lymph nodes Specal clusters of epthela cells (M cells) and lymphocytes also occur where exposure to antgen s most lkely Exposure s most lkely at the membranes lnng respratory, dgestve, and urogental systems Antbodes Antbodes, or mmunnoglobulns, are Y-shaped protens that are composed of two dentcal bndng regons at the upper part of the Y, but vary accordng to antgen specfcty, and an nvarable regon on the lower porton of the Y The nvarable lower porton accounts for the fve dfferent sotypes of antbody ncludng, IgA, IgD, IgE, IgG, and IgM Antbodes are ether membrane bound as receptor stes on B cells or unbound molecules that are secreted by B cells after stmulaton from cytoknes from T cells Antgen elmnaton s the prmary functon of the antbodes The bndng of antbodes to antgen results n neutralzaton and opsonzaton Neutralzaton occurs when the antbodes bnd to bacteral toxns or extracellular protens and nactvates them Opsonzaton occurs when antbodes (or complement) form a coat surroundng the pathogen whch then enables bndng to phagocytes and elmnaton (or lysng) Isotypes The IgM and IgD antbodes (mmunoglobulns) are membrane bound and functon as receptors on B cells IgM, the frst unbound antbody secreted by an actvated B cell, s a very effcent 6

13 antgen-bnder and a complement actvator, but s also a very large molecule Its large sze lmts ts effcency of nfltraton nto varous tssues IgG and IgA are secreted by the B cells after they are actvated by cytoknes from the T cells IgG and IgA neutralzes toxns and vruses by blockng the host cells bndng capabltes IgG s predomnantly n the serum, thus neutralzaton and opsonzaton by IgG prevents the toxns and vruses from enterng the host s cells IgA s present n mucus secretons, ncludng breast mlk and those found n the lnng of the dgestve, respratory, and urogental systems IgE s the mmunoglobuln responsble for hstamne release that occurs wth allergc reactons by bndng to mast cells IgE also functons as protecton agan antgens assocated wth parastes 212 Overvew of Cytoknes A key component n the mmune response s cell-to-cell sgnalng Cell-to-cell sgnalng s acheved through cytokne secreton Cytokne s a generc term for a famly of small protens wth short half lves that regulate and modulate the mmune system They are predomnately produced by helper T cells and macrophages and result n actvaton, nhbton, nflammaton, dfferentaton, prolferaton, and cell death Specfc cytoknes nclude lymphoknes, monoknes, chemoknes, and nterleukn Lymphoknes and monoknes are produced by lymphocytes and monocytes, respectvely Chemoknes produce chemotactce actvty, e they attract leukocytes to the ste of an nfecton Interleukns are cytoknes produced by one type of leukocyte that causes a response n a dfferent type of leukocyte Cytoknes are capable of producng a varety of actons, ncludng autocrne, paracrne and endocrne actons Autocrne, paracrne and endocrne refer to actons upon, the same cells that 7

14 secrete the cytokne, cells n close proxmty, and dstant cells, respectvely Addtonally, cytoknes exhbt redundancy and pleotropy Redundancy refers to dfferent cytoknes producng smlar functons or effects Pleotropy refers to one cytokne actvatng many dfferent types of target cells, or for many dfferent cytokne-producng cells to produce the same cytokne Other characterstcs of cytoknes are ther ablty to behave n a synergstc or antagonstc fashon, and for the producton of one cytokne to ntate producton of another cytokne n a dfferent cell Cytoknes may also be categorzed by functon Immune cell prolferaton and dfferentaton characterzes the largest group of cytoknes Ths group ncludes TNF, IL-1, IL-2, IL-3, IL-4, IL-5, IL-6, IL-7, nterferon gamma (IFNγ), and Granulocyte Monocyte Colony-Stmulatng Factor (GM-CSF) Another group s characterzed by ts ablty to nhbt vral replcaton n nfected cells and smultaneously stmulatng antgen-presentng MHC expresson Ths group ncludes Interferon alpha (IFNα) and Interferon beta (IFNβ) Interleukn-8, MCP-1, MIP-1α, MIP-1β, Lemphotactn, and Fractalkne represent the chemokne group Fnally, there s also a group that s composed of cytoknes that nhbt nflammatory cytokne producton by macrophages and ncludes IL-10 and IL-13 Addtonal detals concernng IL-6, IL-10, and TNF are provded snce they are the focus of ths analyss and report 2121 TNF TNF s a superfamly of molecules frst recognzed when certan cancer patents exhbted regresson of tumors followng nfecton Tumor Necross Factor alpha (α) and beta (β) are only two types of the many specfc molecules found wthn ths superfamly 8

15 TNFα s a boactve nonglycosylated transmembrane or soluble polypeptde The transmembrane form s 233 amno acds long, has a mass of 26 KDa and conssts of cytoplasmc, transmembrane, and extracellular regons composed of 29, 28 and 176 amno acds respectvely 3 A soluble homotrmerc molecule of 157 amno acds s formed when a proteolytc TNFα convertng enzyme cleaves the membrane bound proten 4 The soluble, crculatng form s reported to be more potent than the membrane bound form and normal levels are pg/ml 5 6 Many dfferent cells express TNFα The varous cell types nclude T helper cells 7 (CD4 and CD8), macrophages, mast cells, 8 osteoblasts, 9 pancreatc cells, 10 fat cells, 11 dendrtc cells, 12 neurons, 13 astrocytes, 14 and monocytes 15 The effect of the TNFα expressed by these cells s cytotoxc to tumor cells and nduces other cell types to secrete other cytoknes nvolved wth the nflammatory response TNFβ s a crculatng 25 kda glycosylated polypeptde; one composed of 171 amno acds, the other of 194 amno acds 16 TNFβ bnds to the same receptors as TNFα and crculatng levels are reported to be ~150pg/mL 17 Unlke TNFα, TNFβ s not a transmembrane proten and only has a crculatng or membrane-assocated form NK cell, and T and B cells express TNFβ 18 Whle the effects of TNFβ are smlar to those of TNFα, TNFβ also ncreases the phagocytc actvty of macrophages and neutrophls 2122 IL-6 IL-6 s a relatvely recent name gven to a cytokne that was assocated wth many dfferent functons and havng equally many dfferent names It was only after a common gene was dentfed that these dfferently named molecules based on dfferent functons became known collectvely as IL-6 19 Ths phenomenon s the bass of the pleotrophc characterstc of ths 9

16 cytokne Membershp n the IL-6 famly s related to ts helcal structure and receptor characterstcs IL-6 s a secreted glycoproten wth a mass of kda IL-6 s orgnally translated nto a molecule of 212 amno acds, endng as a mature molecule of 184 amno acds 22 Normal crculatng levels of IL-6 are reported to be ~1 pg/ml 23 Elevatons of IL-6 are also reported n assocaton wth menstruaton, 24 melanoma, 25 and post-surgery 26 As seen wth TNF, varous cell types express IL-6 Expresson of IL-6 s seen n T cells (CD8), 27 fat cells, 28 fbroblasts, 29 osteoblasts, 30 mast cells, 31 astrocytes, pgment cells n the retna, 34 cerebral cortex and sympathetc neurons, eosnophls, 37 neutrophls, 38 monocytes, 39 epthelal cells of the large ntestne, 40 synovocytes, 41 pancreatc cells, 42 and others IL-6 has been assocated wth varous actvtes IL-6 s descrbed as both a pro- and antnflammatory molecule 43 as well as a co-stmulator (wth IL-1) n T cell actvaton and plasma cell prolferaton 44 IL-6 also ncreases hematopoess 45 and modulates resorpton of bone IL-10 IL-10 s a nonglycosylated molecule that weghs 18 kda and s composed of 178 amno acds wth a mature length of 160 amno acds 47 Normal crculatng levels of IL-10 are reported to be ~05 pg/ml As seen wth TNF and IL-6, varous cell types express IL-10 Expresson of IL-10 s seen n T cells (CD8), 48 macrophages that have been actvated, 49 B cells, 50 melanoma cells, 51 NK cells, 52 eosnophls, 53 dendrtc cells, 54 and others The secreton of IL-10 suppresses cytokne producton by the T H 1 subset of T helper cells IL-10 modulates functon n varous cell types ncludng neutrophls, monocytes, dendrtc cells, and mast cells and NK cells, as well as T and B cells 10

17 IL-10 has an ant-nflammatory effect on neutrophls: Blocks IL-1β and TNFα and nhbts secreton of IL-8, MIP-1α, MIP-1 β 55 Results n less superoxde beng produced, whch hnders ant-body dependent cytotoxcty 56 IL-10 s effect on monocytes: Results n decreased IL-8 producton 57 Results n ncreased hyaluronectn n connectve tssue (whch s assocated wth decreased mgraton of metastatc tumor cells) 58 Reduces MHC-II expresson on the cell surface 59 IL-10 has an mmunosuppressve effect on dendrtc cells IL-10 results n: Increases n macrophages (whle ths may appear benefcal, macrophages are not good antgen presentng cells) 60 Dendrtc cells beng less effcent at stmulatng T cells 61 Immoble dendrtc cells 62 In NK cells, IL-10: Promotes TNFα and GM-CSF producton 63 May ncrease IL-2 nduced NK cell prolferaton 64 Increases NK cells cytotoxcty wth the ad of IL-12 and IL IL-10 has the opposte effect n mast cells compared to NK cells In mast cells, IL-10 results n: Blocked producton of TNFα and GM-CSF 66 Increased release of hstamne IL-10 effects T cell functon by: Suppressng IL

18 Inhbtng T cell apoptoss 68 Inducton of CD8 chemotaxs 69 IL-10 effects B cell functon by: Intatng dfferentaton/growth and promotes plasma cell formaton 70 Inducng the producton of IgA 71 Inductng the producton of IgG1 and IgG3 (n the absence of TGFβ) 72 12

19 22 The Data Censored, Sample-Selected, Truncated data Many nvestgators use ordnary least squares (OLS) regresson n the event of a contnuous outcome varable However, devaton from the assumptons of ordnary least squares wll result n based estmates Unnformed nvestgators may also perform OLS on lmted datasets, or datasets wth mssng values, whle others may sort ther data to elmnate groups wth mssng values and then perform OLS These types of datasets often have a problem commonly referred to as a censorng problem, but may be categorzed as ether censored, sample selected, or truncated To delneate the dfferences between these types of data see Table

20 Table 1 Defntons of Censored, Sample-Selected, and Truncated Varables Data/Sample Dependent Varable Independent Varable(s) Censored Sample selected Truncated y s known exactly only f some crteron defned n terms of the value of y s met, such as y > c (or y < c) y s a truncated random varable y s observed only f some crteron defned n terms of another random varable, z, s met such as f z = 1 y s a truncated random varable y s observed only f some crteron defned n terms of the value of y s met, such as y > c (or y < c) y s a truncated random varable x varable values are observed for all of the sample, regardless of whether y s known exactly x and w are observed for all the sample, regardless of whether y s observed or not Independent varables are observed only f y s observed The dataset n ths report s classfed as censored from below A varable s left censored (censored from below) f for some value y, the exact value of y taken s y > c For other values of y t s only know that y c A varable may be rght censored (censored from above) f for 14

21 some value y, the exact value of y taken s less than some threshold, y < d For other values of y t s only know that y d A fnal example of censored data s where the data s censored from the left and the rght (e from below and above) In ths nstance, c < y < d; where the exact values of y are known between the lower and upper lmts However, f outsde of the specfed range, t s only known that y c and/or y d 221 The Tobt Model The smplest method for analyss for censored data s the Tobt model 74 The Tobt model may be nterpreted n terms of an underlyng latent varable, y*, of whch y s the realzed observaton Another way of sayng ths, s that y* s the true value, and y s the value that s observed (rememberng that the value s lmted or censored) The model may be wrtten n terms of the latent varable y*: y * = T x β + u where the error term u s assumed to be ndependent and normally dstrbuted wth a mean of zero and a constant varance, σ 2 The observed and latent varables are related by the followng relatonshp: y = y * f y * > c y = c f y * c where c s the censorng threshold In ths report, the censorng thresholds for tnf, IL6 and IL10, are 39, 49, and 49, respectvely The model wrtten n terms of the observed varable y, usng tnf for example s: y = x β + u f > 39 T y = 39 otherwse 15

22 The purpose of regresson s to estmate an ntercept (α), a regresson coeffcent (β), and the standard error of the ndependent error term, σ (assumed normally dstrbuted) In some crcumstances (where certan assumptons met- no correlaton between u and x, ndependence of u s, zero expectaton of u, and homoscedastcty) ordnary least squares (OLS) provde estmates that are the best lnear unbased estmators (BLUE), e the estmates have the smallest samplng varance (makng them the most effcent) of all the lnear unbased estmators However, n the case of censored data, maxmum lkelhood estmaton s used to estmate α, β, and σ 222 Maxmum Lkelhood The goal of maxmum lkelhood s to fnd the set of parameters that would have generated the observed sample most often, f the parameters are true of the populaton Maxmum lkelhood s applcable n both the dscrete and contnuous case Regardless of type of varable, the frst step s to formulate the lkelhood functon Formulatng a lkelhood functon frst starts wth the jont probablty dstrbuton f(y 1,y 2, y N ) And snce the sample observatons are assumed to be ndependent, the jont probablty s equal to the product of the margnal probabltes f(y 1,f(y 2 ) f(y N ) N ( ) = f y 1 Although ths equaton s dentcal to the jont probablty dstrbuton of the sample, t s somethng completely dfferent n terms of maxmum lkelhood estmaton In the case of a jont probablty dstrbuton, the parameters of the dstrbuton are fxed, and the y values are varable In the case of lkelhood, the y values are fxed and the parameters are allowed to vary 16

23 For smplfyng calculatons, the natural logarthm of the lkelhood functon s then taken, and maxmzed wth respect to each parameter The dependent varables n ths report (tnf, IL6, and IL10) are contnuous, and therefore the probablty densty functon s used to formulate the lkelhood functon (as opposed to probablty mass functon) Assumng the populaton y s are normally dstrbuted the densty functon s [( y µ )/ ] 2 1 σ f ( y ) = exp 2 2πσ 2 Takng the product of the denstes of the y s and then takng the natural logarthm of ths functon results n the log-lkelhood functon N log ( µ ) 2 2 y 2πσ 2σ Let µ = α + βx, and substtute nto the equaton above Ths substtuton s made snce µ vares over the sample resultng n 1 1 T ( y ( α + x β ) 2 N log σ 2πσ Parameter estmates of α, β, and σ are derved by maxmzng the above equaton 75 Ths s the log-lkelhood for the normal error regresson model However, n the tobt model, censored and uncensored observatons make separate contrbutons to the log-lkelhood functon 223 Tobt Model and Maxmum Lkelhood Let y be the serum cytokne level (tnf, IL6, IL10) of the th patent n the populaton of study patents, and let x be the value of dead or alve status at dscharge or follow-up The goal s to estmate the vector β, whch s the set of populaton regresson parameters relatng x to the level 17

24 of crculatng cytokne The sample s composed of N patents, of whch N 0 have truncated cytokne (censored) values, and N 1 (=N-N 0 ) wth observed (uncensored) values To formulate the lkelhood functon for the tobt model t s assumed that u has a normal dstrbuton, the error terms of each observaton are ndependent of each other, the error term s ndependent of the ndependent varable(s) n the model 76 We also have cytokne values for all patents (day 1), and that for uncensored (N 1 ) observatons, the exact value s known Contrbutons to the lkelhood come from censored and uncensored observatons The lkelhood contans the product of N 0 observatons that are censored and N 1 observatons that are uncensored The product of the N 0 observatons s n o = 1 y + (1 Φ σ T ( α x β ) where Ф (unless stated otherwse) denotes the standard normal dstrbuton functon (mean = 0, varance = 1) The product of the N 1 observatons s T ( α x β ) N Φ = no + 1 σ y + However, for the N 1 observatons, the exact cytokne values are known, therefore the followng term becomes part of the lkelhood, 1 ) T [( y ( α + x β ))/ σ ] T y ( α + x β ) 1 Φ σ Φ σ 18

25 When the three product terms are multpled, the Ф ( ) term (n the second and thrd productterms) cancels Φ σ The result s the lkelhood functon: T T T ' y ( α β ) x y ( α + β ) x 1 Φ[ ( y ) ] Φ x β / σ 0 y + 1 Φ σ 1 σ T ( α x β ) The natural logarthm of the lkelhood functon s: y + T ( α x β ) 1 σ [ ] T Φ ( y ( α + x β ))/ σ 1 Φ T ( y x β ) 2 log 1 + log Φ σ 1 1 2σ 1 2πσ 1 Notce that the crcled porton (uncensored contrbuton) of the tobt log-lkelhood functon s same as the log-lkelhood for the normal error regresson model n the prevous secton (122) 224 The Delta Method for Standard Error Determnaton The predctnl command s mplemented as an ado-fle followng an estmaton command (eg tobt) n STATA The quanttes generated by predctnl are not scalars, but functons of the data, and are therefore vectors over the observatons wthn the data For general predcton, g(θ,x ) for = 1,,n where θ are model parameters and x are data for the th observaton (and are assumed to be fxed) In STATA, g(θ,x ) s estmated by g ˆ, θ x ) ( 19

26 where θˆ are estmated model parameters stored as e(b) followng the estmaton command In STATA, predctnl generates the estmated predcton, g ˆ, θ x ), but also generates the standard ( error of g ˆ, θ x ), usng the delta method ( 77 The delta method expands a functon of a random varable about ts mean wth a one-step Taylor approxmaton, and then takes the varance 78 When usng predctnl, the transformaton g(θ,x ), s estmated by g ˆ, θ x ), for 1x k parameter vector θ and the data x (whch s assumed ( fxed) The varance of g ˆ, θ x ) ( s estmated by { g( ˆ, x )} = GVG Var ˆ θ where G s the vector of dervatves G ( θ, ) g x = θ =θˆ θ and V s the estmated varance-covarance matrx of θˆ 79 ( 1xk ) For the nstance presented here the mean s estmated by exp(x T βˆ ) Usng the delta method, the estmated varance s gven by: ˆ ˆ ( ) ( ) β ( ) ( ) β β β ˆ β ˆ x x ˆ x ˆ V ( 0) C 0, β1 e e, xe ˆ ˆ β β ˆ 0, β ˆ x C 1 V β1 xe ( ( ) ( ) ) x ˆ β x ˆ β ˆ β ˆ β ˆ β + ˆ β ˆ x β ˆ β ˆ x , β1 + ( ˆ x e C, xe, C e V β1) xe x ˆ β xe = V ( ˆ β ) e = ˆ 2 ˆ 2 ˆ 2 ( ) + ( ) ( ) + ( ) ( ) + ˆ 2 ˆ xβ ˆ ˆ xβ ˆ ˆ xβ ˆ 2 xβ V ( β ) e C β, β x e C β, β x e V ( β x ( e ) ) ˆ 2 2 = ( e ) ( V ( ˆ ) 2C( ˆ, ˆ ) x V ( ˆ ) x ) x β β β β + β

27 23 Addtonal Analyss In addton to tobt estmates of mean cytokne levels n the study populaton, the arthmetc and geometrc mean are also provded 231 Arthmetc Mean The arthmetc mean of a set of numbers s the sum of all the members of the set dvded by the number of tems n the set If the data set s denoted by X = {x 1, x 2,, x n } The arthmetc mean s calculated as: or alternatvely ( x + x + x ) n x = n / The arthmetc mean s greatly nfluenced by outlers Geometrc Mean The geometrc mean s to multplcaton as the arthmetc mean s to addton Just as addng n terms all equal to the arthmetc mean yelds the sum x x n, so multplyng n factors all equal to the geometrc mean yelds the product x 1 x n (these n numbers must be non-negatve) The geometrc mean s or 1/ ( x x ) n 1 2 x n n x 1 x2 x n The geometrc mean s less affected by extreme values than the arthmetc mean and s useful for some postvely skewed dstrbutons 81 21

28 3 RESULT The data set conssts of serum cytokne levels measured on day 1 for 1753 crtcal care patents Tumor necross factor, Interleukn-6, and Interleukn-10 are coded as tnf, l6, and l10 respectvely Tobt estmates of mean cytokne levels are for dead or alve status at dscharge and follow up Dead and alve are coded as 0 and 1 respectvely Dscharge and follow up are coded as dc and fu, respectvely Cytokne values were avalable for 1753 patents at dscharge However, 143 patents were lost to follow up Laboratory values for tnf, l6 and l10 are left censored The lower lmt for tnf, l6 and l10 are 4, 4, and 5 respectvely The number of left censored cytokne values for tnf, l6, and l10 at dscharge, were 670 (3822%), 248 (1415%), and 854 (4872%), respectvely The number of left censored cytokne values for tnf, l6, and l10 at follow up, were 610 (3889%), 230 (1429%), and 788 (4894%), respectvely Mcrosoft Excel, Mcrosoft Access, and STATA 80 SE were used for all data analyses A summary table for mean estmates of tnf, IL6 and IL10 for dscharge status and follow-up status are presented below The STATA output may be found n Appendx A Table 2 Frequency of censored cytokne values and dead/alve status at dscharge and follow-up Dscharge Status (N=1753) Followup Status (N=1610) Frequency Percent Frequency Percent tnf IL IL dead alve N=1610 at follow up due to dead/alve status mssng for 143 patents 22

29 Table 3 Comparson of mean estmates for TNF, IL6, and IL10 for death at dscharge based on the tobt model, the arthmetc and the geometrc mean Note that the populaton s the 1753 ndvduals wth both complete lab and follow up data wth 78 reported deaths at dscharge and 1675 subjects alve at hosptal dscharge Standard errors are gven n parentheses below the mean Cytokne Estmated tobt mean for dead at dscharge TNF (10021) IL (383026) IL (22705) Estmated arthmetc mean for dead at dscharge Estmated geometrc mean for dead at dscharge (39608) ( ) (132713) 1564 Estmated tobt mean for alve at dscharge (01472) (18622) (02483) Estmated arthmetc mean for alve at dscharge Estmated geometrc mean for alve at dscharge (07426) (334703) (16798) 961 Table 4 Comparson of mean estmates for TNF, IL6, and IL10 for death at follow-up based on the tobt model, the arthmetc and the geometrc mean Note that the populaton s the 1753 ndvduals wth both complete lab and follow-up data wth 231 reported deaths at follow-up and 1379 subjects alve at follow up and 143 wth mssng values Standard errors are gven n parentheses below the mean Cytokne Estmated tobt mean for dead at follow-up TNF (05058) IL (104984) IL (09265) Estmated arthmetc mean for dead at follow-up Estmated geometrc mean for dead at follow-up (15252) ( ) (51314) 1217 Estmated tobt mean for alve at follow-up (01565) (19688) (2623) Estmated arthmetc mean for alve at follow-up Estmated geometrc mean for alve at follow-up (07933) (330279) (18900)

30 4 CONCLUSION / RECOMMENDATION Three dfferent methods were presented to estmate the mean cytokne levels of 1753 crtcal care patents An accurate estmate of serum levels of TNF, IL6 and IL10 s mportant snce elevated cytokne levels have been assocated wth tssue damage and a heghtened mmune response However, when quantfyng assays have a lmted range of senstvty, a large porton of the serum samples tested may fall below or above the accurate range of the assay The tobt regresson provdes a method for estmaton when dealng wth censored data The arthmetc and geometrc means are presented for comparson I would not recommend usng the arthmetc mean to estmate cytokne levels wth ths dataset snce the data s hghly skewed The arthmetc mean does not take censored laboratory values nto account and has a larger standard error assocated wth ts mean when compared to the tobt estmate Although the geometrc mean s less affected by extreme values than the arthmetc mean and s useful for some postvely skewed dstrbutons, 82 I also would not recommend usng the geometrc mean for estmatng the mean cytokne levels wth ths cytokne dataset The geometrc mean s smlar to the arthmetc mean n that t does not account for the censored values Of the three estmaton methods presented, I would recommend usng the tobt model for estmaton purposes I would also recommend that the data be log transformed before the analyss and that the delta method be used to estmate the standard error Although the tobt model accounts for censored values, t s assumed that the underlyng latent varable of the model: 24

31 y * = x β + u, T s the correct functonal form for the relatonshp between the latent cytokne level, and dscharge or follow-up status However, other relevant varables may have been omtted from the specfcaton 83 and further research may conclude that a more complex model, than the one presented here, may reveal a more clncally meanngful estmate 25

32 5 APPENDIX A: STATA Output log: C:\Documents and Settngs\RSITLO\Desktop\TOD\Tobt_LWDeltaSElog log type: text opened on: 24 Feb 2005, 14:13:51 do "C:\DOCUME~1\RSITLO\LOCALS~1\Temp\STD030000tmp" * estmatng mean tnf and the std error of the predcted value - from STATA manual for "predctnl" > pg 225) **Make sure to check that path for data fle s correct before runnng on dfferent computers **** FOR TNF **** FOR TNF **** FOR TNF * FOR TNF DISCHARGE STATUS clear use "C:\Documents and Settngs\RSITLO\Desktop\TOD\fullvalueslogtransformeddta" tobt lntnf dc, ll Tobt estmates Number of obs = 1753 LR ch2(1) = 1356 Prob > ch2 = Log lkelhood = Pseudo R2 = lntnf Coef Std Err t P> t [95% Conf Interval] dc _cons _se (Ancllary parameter) Obs summary: 670 left-censored observatons at lntnf<= uncensored observatons predct lntnfdcxb, xb sort dc by dc: summarze lntnfdcxb 26

33 -> dc = 0 lntnfdcxb > dc = 1 lntnfdcxb generate esttnfdcxb=exp(lntnfdcxb) sort dc by dc: summarze esttnfdcxb -> dc = 0 esttnfdcxb > dc = 1 esttnfdcxb **Calculaton of standard error s based on Lsa DELTA Method when usng predctnl predctnl pxb=(xb()), se(pxb_se) sort dc by dc: summarze pxb_se -> dc = 0 pxb_se > dc = 1 pxb_se generate se=pxb_se*(exp(pxb)) sort dc 27

34 by dc: summarze se -> dc = 0 se > dc = 1 se ** FOR TNF FOLLOWUP STATUS clear use "C:\Documents and Settngs\RSITLO\Desktop\TOD\fullvalueslogtransformeddta" tobt lntnf fu, ll Tobt estmates Number of obs = 1610 LR ch2(1) = 2041 Prob > ch2 = Log lkelhood = Pseudo R2 = lntnf Coef Std Err t P> t [95% Conf Interval] fu _cons _se (Ancllary parameter) Obs summary: 610 left-censored observatons at lntnf<= uncensored observatons predct lntnffuxb, xb (143 mssng values generated) sort fu by fu: summarze lntnffuxb -> fu = 0 lntnffuxb > fu = 1 28

35 lntnffuxb > fu = lntnffuxb 0 generate esttnffuxb=exp(lntnffuxb) (143 mssng values generated) sort fu by fu: summarze esttnffuxb -> fu = 0 esttnffuxb > fu = 1 esttnffuxb > fu = esttnffuxb 0 **Calculaton of standard error s based on Lsa DELTA Method when usng predctnl predctnl pxb=(xb()), se(pxb_se) (143 mssng values generated) sort fu by fu: summarze pxb_se -> fu = 0 pxb_se > fu = 1 pxb_se

36 -> fu = pxb_se 0 generate se=pxb_se*(exp(pxb)) (143 mssng values generated) sort fu by fu: summarze se -> fu = 0 se > fu = 1 se > fu = se 0 ***** IL6 ***** IL6 ***** IL6 * FOR IL6 DISCHARGE STATUS clear use "C:\Documents and Settngs\RSITLO\Desktop\TOD\fullvalueslogtransformeddta" tobt lnl6 dc, ll Tobt estmates Number of obs = 1753 LR ch2(1) = 4072 Prob > ch2 = Log lkelhood = Pseudo R2 = lnl6 Coef Std Err t P> t [95% Conf Interval] dc _cons _se (Ancllary parameter)

37 Obs summary: 248 left-censored observatons at lnl6<= uncensored observatons predct lnl6dcxb, xb sort dc by dc: summarze lnl6dcxb -> dc = 0 lnl6dcxb > dc = 1 lnl6dcxb generate estl6dcxb=exp(lnl6dcxb) sort dc by dc: summarze estl6dcxb -> dc = 0 estl6dcxb > dc = 1 estl6dcxb **Calculaton of standard error s based on Lsa DELTA Method when usng predctnl predctnl pxb=(xb()), se(pxb_se) sort dc by dc: summarze pxb_se -> dc = 0 pxb_se

38 -> dc = 1 pxb_se generate se=pxb_se*(exp(pxb)) sort dc by dc: summarze se -> dc = 0 se > dc = 1 se ** FOR IL6 FOLLOWUP STATUS clear use "C:\Documents and Settngs\RSITLO\Desktop\TOD\fullvalueslogtransformeddta" tobt lnl6 fu, ll Tobt estmates Number of obs = 1610 LR ch2(1) = 2935 Prob > ch2 = Log lkelhood = Pseudo R2 = lnl6 Coef Std Err t P> t [95% Conf Interval] fu _cons _se (Ancllary parameter) Obs summary: 230 left-censored observatons at lnl6<= uncensored observatons predct lnl6fuxb, xb (143 mssng values generated) sort fu by fu: summarze lnl6fuxb 32

39 -> fu = 0 lnl6fuxb > fu = 1 lnl6fuxb > fu = lnl6fuxb 0 generate estl6fuxb=exp(lnl6fuxb) (143 mssng values generated) sort fu by fu: summarze estl6fuxb -> fu = 0 estl6fuxb > fu = 1 estl6fuxb > fu = estl6fuxb 0 **Calculaton of standard error s based on Lsa DELTA Method when usng predctnl predctnl pxb=(xb()), se(pxb_se) (143 mssng values generated) sort fu by fu: summarze pxb_se -> fu = 0 33

40 pxb_se > fu = 1 pxb_se > fu = pxb_se 0 generate se=pxb_se*(exp(pxb)) (143 mssng values generated) sort fu by fu: summarze se -> fu = 0 se > fu = 1 se > fu = se 0 ***** IL10 ***** IL10 ***** IL10 * FOR IL10 DISCHARGE STATUS clear use "C:\Documents and Settngs\RSITLO\Desktop\TOD\fullvalueslogtransformeddta" tobt lnl10 dc, ll Tobt estmates Number of obs =

41 LR ch2(1) = 1641 Prob > ch2 = Log lkelhood = Pseudo R2 = lnl10 Coef Std Err t P> t [95% Conf Interval] dc _cons _se (Ancllary parameter) Obs summary: 854 left-censored observatons at lnl10<= uncensored observatons predct lnl10dcxb, xb sort dc by dc: summarze lnl10dcxb -> dc = 0 lnl10dcxb > dc = 1 lnl10dcxb generate estl10dcxb=exp(lnl10dcxb) sort dc by dc: summarze estl10dcxb -> dc = 0 estl10dcxb > dc = 1 estl10dcxb **Calculaton of standard error s based on Lsa DELTA Method when usng predctnl predctnl pxb=(xb()), se(pxb_se) sort dc 35

42 by dc: summarze pxb_se -> dc = 0 pxb_se > dc = 1 pxb_se generate se=pxb_se*(exp(pxb)) sort dc by dc: summarze se -> dc = 0 se > dc = 1 se ** FOR IL10 FOLLOWUP STATUS clear use "C:\Documents and Settngs\RSITLO\Desktop\TOD\fullvalueslogtransformeddta" tobt lnl10 fu, ll Tobt estmates Number of obs = 1610 LR ch2(1) = 1478 Prob > ch2 = Log lkelhood = Pseudo R2 = lnl10 Coef Std Err t P> t [95% Conf Interval] fu _cons _se (Ancllary parameter) Obs summary: 788 left-censored observatons at lnl10<=

43 predct lnl10fuxb, xb (143 mssng values generated) sort fu by fu: summarze lnl10fuxb 822 uncensored observatons -> fu = 0 lnl10fuxb > fu = 1 lnl10fuxb > fu = lnl10fuxb 0 generate estl10fuxb=exp(lnl10fuxb) (143 mssng values generated) sort fu by fu: summarze estl10fuxb -> fu = 0 estl10fuxb > fu = 1 estl10fuxb > fu = estl10fuxb 0 **Calculaton of standard error s based on Lsa DELTA Method when usng predctnl predctnl pxb=(xb()), se(pxb_se) 37

44 (143 mssng values generated) sort fu by fu: summarze pxb_se -> fu = 0 pxb_se > fu = 1 pxb_se > fu = pxb_se 0 generate se=pxb_se*(exp(pxb)) (143 mssng values generated) sort fu by fu: summarze se -> fu = 0 se > fu = 1 se > fu = se 0 end of do-fle log close log: C:\Documents and Settngs\RSITLO\Desktop\TOD\Tobt_LWDeltaSElog 38

45 log type: text closed on: 24 Feb 2005, 14:14:07 STATA output STATA 80 from dataset fulllabvaluesday1_recodecutoffsdta tabulate tnf tnf Freq Percent Cum Total 1, tabulate l6 l6 Freq Percent Cum Total 1, tabulate l10 l10 Freq Percent Cum Total 1, FOR dc and fu, dead = 0, alve=1 tabulate dc dc Freq Percent Cum , Total 1, tabw dc Varable **** dc

46 tabulate fu fu Freq Percent Cum , Total 1, tabw fu Varable **** fu FOR dc and fu, dead = 0, alve=1 bysort dc: means tnf l6 l10 -> dc = 0 Varable Type Obs Mean [95% Conf Interval] -- tnf Arthmetc Geometrc Harmonc l6 Arthmetc Geometrc Harmonc l10 Arthmetc Geometrc Harmonc > dc = 1 Varable Type Obs Mean [95% Conf Interval] -- tnf Arthmetc Geometrc Harmonc l6 Arthmetc Geometrc Harmonc l10 Arthmetc Geometrc Harmonc bysort fu: means tnf l6 l10 -> fu = 0 40

47 Varable Type Obs Mean [95% Conf Interval] -- tnf Arthmetc Geometrc Harmonc l6 Arthmetc Geometrc Harmonc l10 Arthmetc Geometrc Harmonc > fu = 1 Varable Type Obs Mean [95% Conf Interval] -- tnf Arthmetc Geometrc Harmonc l6 Arthmetc Geometrc Harmonc l10 Arthmetc Geometrc Harmonc more-- sort dc by dc: summarze tnf l6 l10 -> dc = 0 tnf l l > dc = 1 tnf l l bysort dc: centle tnf l6 l10, centle( ) -> dc = 0 -- Bnom Interp -- Varable Obs Percentle Centle [95% Conf Interval] tnf l

48 l > dc = 1 -- Bnom Interp -- Varable Obs Percentle Centle [95% Conf Interval] tnf l l bysort fu: centle tnf l6 l10, centle( ) -> fu = 0 -- Bnom Interp -- Varable Obs Percentle Centle [95% Conf Interval] tnf l l > fu = 1 -- Bnom Interp -- Varable Obs Percentle Centle [95% Conf Interval] tnf l l > fu = -- Bnom Interp -- Varable Obs Percentle Centle [95% Conf Interval] tnf l l