Semiparametric Estimation of Treatment Effect in a Pretest-Posttest Study

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1 Semparametrc Estmaton of Treatment Effect n a Pretest-Posttest Study Mare Davdan Department of Statstcs North Carolna State Unversty Based on: Avalable on my web page Leon, S., Tsats, A.A., and Davdan, M. (2003) Semparametrc estmaton of treatment effect n a pretest-posttest study. Bometrcs 59, Davdan, M., Tsats, A.A., and Leon, S. (2005) Semparametrc estmaton of treatment effect n a pretest-posttest study wth mssng data (wth Dscusson). Statstcal Scence 20, davdan (Jont work wth A.A. Tsats and S. Leon) Greenberg Lecture II: Pretest-Posttest Study 1 Greenberg Lecture II: Pretest-Posttest Study 2 Outlne of popular methods 2. Influence functons 3. Robns, Rotntzky, and Zhao (1994) 4. Estmaton when full data are avalable 5. Estmaton when posttest response s mssng at random (MAR) 6. Full data nfluence functons, revsted 7. Smulaton evdence 8. Applcaton ACTG Dscusson The pretest-posttest study: Ubqutous n research n medcne, publc health, socal scence, etc... Subjects are randomzed to two treatments ( treatment and control ) Response s measured at baselne ( pretest ) and at a pre-specfed follow-up tme ( posttest ) Focus of nference: Dfference n change of mean response from baselne to follow-up between treatment and control Greenberg Lecture II: Pretest-Posttest Study 3 Greenberg Lecture II: Pretest-Posttest Study 4 For example: AIDS Clncal Trals Group patents randomzed to ZDV, ZDV+ddI, ZDV+zalctabne, ddi wth equal probablty (1/4) Prmary analyss (tme-to-event endpont): ZDV nferor to other three (no dfferences) Two groups: ZDV alone ( control ) and other three ( treatment ) Secondary analyses : Compare change n CD4 count (mmunologc status) from () baselne to 20±5 weeks and () baselne to 96±5 weeks between control and treatment Formally: Defne Y 1 Y 2 Z baselne (pretest) response (e.g., count) follow-up (posttest) response (e.g., 20±5 week CD4 count) = 0 f control, = 1 f treatment, P (Z = 1) = By randomzaton, reasonable to assume E(Y 1 Z = 0) = E(Y 1 Z = 1) = E(Y 1) = µ 1 Effect of nterest: β, where {E(Y 2 Z = 1) E(Y 1 Z = 1)} {E(Y 2 Z = 0) E(Y 1 Z = 0)} = {E(Y 2 Z = 1) µ 1} {E(Y 2 Z = 0) µ 1)} = E(Y 2 Z = 1) E(Y 2 Z = 0) = µ (1) 2 µ (0) 2 = β Greenberg Lecture II: Pretest-Posttest Study 5 Greenberg Lecture II: Pretest-Posttest Study 6

2 Basc data: (Y 1, Y 2, Z ), = 1,..., n, d n 1 = Z = I(Z = 1), n 0 = (1 Z ) = I(Z = 0) Popular estmators for β: Two-sample t-test estmator β 2samp = n 1 1 Z Y 2 n 1 0 Pared t-test estmator ( change scores ) β par = D 1 D 0, D c = n 1 c (1 Z )Y 2 I(Z = c)(y 2 Y 1), c = 0, 1 Popular estmators for β: ANCOVA Adjust drectly by fttng the model E(Y 2 Y 1, Z) = α 0 + α 1Y 1 + βz ANCOVA II Include nteracton and estmate β as coeffcent of Z Z n regresson of Y 2 Y 2 on Y 1 Y 1, Z Z, and (Y 1 Y 1)(Z Z) GEE (Y 1, Y 2) T s multvarate response wth mean (µ 1, µ (0) 2 + βz) T and (2 2) unstructured covarance matrx Assume lnear relatonshp between Y 2 and Y 1 Greenberg Lecture II: Pretest-Posttest Study 7 Greenberg Lecture II: Pretest-Posttest Study 8 ACTG 175: Y 2 = CD4 at 20±5 weeks vs. Y 1 = (control and treatment groups) follow-up CD4 at 20+/-5 weeks follow-up CD4 at 20+/-5 weeks Addtonal data: Baselne and ntermedate covarates X 1 X 2 Baselne (pre-treatment) characterstcs Characterstcs observed after pretest but before posttest, ncludng ntermedate responses In ACTG 175: X 1 ncludes weght, age, gender, Karnofsky score, pror ARV therapy, CD8 count, sexual preference,... X 2 ncludes off treatment ndcator, ntermedate CD4, CD8 Greenberg Lecture II: Pretest-Posttest Study 9 Greenberg Lecture II: Pretest-Posttest Study 10 Addtonal estmators: Incorporate nonlnearty, covarates Fancer regresson models, e.g. E(Y 2 Y 1, Z) = α 0 + α 1Y 1 + α 2Y βz Adjustment for baselne covarates, e.g., Both E(Y 2 X 1, Y 1, Z) = α 0 + α 1Y 1 + α 2X 1 + βz Intutvely, adjustment for ntermedate covarates buys nothng wthout some assumptons (formally exhbted shortly... ) and could be dangerous Whch estmator? Stll confuson among practtoners Is normalty requred? What f the relatonshp sn t lnear? What f a model for E(Y 2 X 1, Y 1, Z) s wrong? Further complcaton: Mssng posttest response Y 2 In ACTG 175, no mssng CD4 for any subject at 20±5 weeks but 797 (37%) of subjects were mssng CD4 at 96±5 weeks (almost entrely due to dropout from study) Common n practce complete case analyss, whch yelds possbly based nference on β unless Y 2 s mssng completely at random Also common Last Observaton Carred Forward, also based Greenberg Lecture II: Pretest-Posttest Study 11 Greenberg Lecture II: Pretest-Posttest Study 12

3 Introducton and revew Introducton and revew Mssng at random (MAR) assumpton: Posttest mssngness assocated wth (X 1, Y 1, X 2, Z) but not Y 2 Often reasonable, but s an assumpton If R = 1 f Y 2 s observed and R = 0 f Y 2 s mssng, the assumpton s P (R = 1 X 1, Y 1, X 2, Y 2, Z) = P (R = 1 X 1, Y 1, X 2, Z) Mssngness s assocated only wth nformaton that s always observed Full data: If no mssngness, observe (X 1, Y 1, X 2, Y 2, Z) Ordnarly: Models for (X 1, Y 1, X 2, Y 2, Z) may nvolve assumptons If Y 2 not mssng, wdespread belef that normalty of (Y 1, Y 2) s requred for valdty of popular estmators When Y 2 s MAR, maxmum lkelhood, mputaton approaches requre assumptons on aspects of the jont dstrbuton of (X 1, Y 1, X 2, Y 2, Z) Consequences? Greenberg Lecture II: Pretest-Posttest Study 13 Greenberg Lecture II: Pretest-Posttest Study 14 Introducton and revew Introducton and revew Semparametrc models: May contan parametrc and nonparametrc components Nonparametrc components unable or unwllng to make specfc assumptons on aspects of (X 1, Y 1, X 2, Y 2, Z) Here: Consder a semparametrc model for (X 1, Y 1, X 2, Y 2, Z) No assumptons on jont dstrbuton of (X 1, Y 1, X 2, Y 2, Z) beyond ndependence of (X 1, Y 1) and Z nduced by randomzaton (nonparametrc) Interested n the functonal of ths dstrbuton β = µ (1) 2 µ (0) 2 = E(Y 2 Z = 1) E(Y 2 Z = 0) Where we are gong: Under ths semparametrc model Fnd a class of consstent and asymptotcally normal (CAN ) estmators for β when full data are avalable and dentfy the best (effcent ) estmator n the class As a by-product, show that popular estmators are potentally neffcent members of ths class can do better! When Y 2 s MAR, fnd a class of CAN estmators for β and dentfy the best In both cases, translate the theory nto practcal technques What we wll explot: Theory n a landmark paper by Robns, Rotntzky, and Zhao (1994) Greenberg Lecture II: Pretest-Posttest Study 15 Greenberg Lecture II: Pretest-Posttest Study Influence functons 2. Influence functons Defnton: For functonal β (scalar) n a parametrc or semparametrc model, an estmator β based on d random vectors W, = 1,..., n, s asymptotcally lnear f n 1/2 ( β β 0) = n 1/2 ϕ(w ) + o p(1) for some ϕ(w ) β 0 = true value of β, E{ϕ(W )} = 0, E{ϕ 2 (W )} < ϕ(w ) s called the nfluence functon of β If β s also regular (not pathologcal ), β s CAN wth asymptotc varance E{ϕ 2 (W )} Effcent nfluence functon ϕ eff (W ) has smallest varance and corresponds to the effcent, regular asymptotcally lnear (RAL ) estmator Result: By the Central Lmt Theorem, f β s RAL wth nfluence functon ϕ(w ), then approxmately ( ) β N β 0, n 1 E{ϕ 2 (W )} As we wll see, ϕ(w ) may be vewed as dependng on β and other parameters θ, say So can estmate E{ϕ 2 (W )} by n 1 ϕ 2 (W, β, θ), say Often called a sandwch estmator for (approxmate) varance Greenberg Lecture II: Pretest-Posttest Study 17 Greenberg Lecture II: Pretest-Posttest Study 18

4 2. Influence functons Example: It may be shown drectly by manpulatng the expresson for n 1/2 ( β 2samp β) and usng n 0/n 1, n 1/n as n that β 2samp has nfluence functon of the form Z(Y 2 µ (1) 2 ) (1 Z)(Y2 µ(0) 2 ) 1 Z(Y2 µ(0) = [depends on W = (X 1, Y 1, X 2, Y 2, Z), β, µ (0) 2 ] 2 β) (1 Z)(Y2 µ(0) 2 ) 1 Why s ths useful? There s a correspondence between CAN, RAL estmators and nfluence functons By dentfyng nfluence functons, one can deduce estmators 2. Influence functons General prncple: From the defnton, β β 0 n 1 ϕ(w) = To deduce estmators, solve ϕ(w) = 0 for β For example: Influence functon for β 2samp { } Z (Y 2 µ (0) 2 β) (1 Z)(Y2 µ(0) 2 0 = ) 1 Substtutng µ (0) 2 = n 1 0 (1 Z)Y2 and solvng for β yelds β = n 1 1 Z Y 2 n 1 0 (1 Z )Y 2 In general, closed form may not be possble (ϕ(w ) depends on β 0) Greenberg Lecture II: Pretest-Posttest Study 19 Greenberg Lecture II: Pretest-Posttest Study Robns, Rotntzky, and Zhao (1994) 3. Robns, Rotntzky, and Zhao (1994) What dd RRZ do? Derved asymptotc theory based on nfluence functons for nference on functonals n general semparametrc models where some components of the full data are possbly MAR Observed data: Data observed when some components of the full data are potentally mssng What dd RRZ do, more specfcally? For the functonal of nterest, dstngushed between Full-data nfluence functons correspond to RAL estmators calculable f full data were avalable; functons of the full data Observed-data nfluence functons correspond to RAL estmators calculable from the observed data under MAR; functons of the observed data RRZ characterzed the class of all observed-data nfluence functons for a general semparametrc model, ncludng the effcent one, and showed that observed-data nfluence functons may be expressed n terms of full-data nfluence functons Greenberg Lecture II: Pretest-Posttest Study 21 Greenberg Lecture II: Pretest-Posttest Study Robns, Rotntzky, and Zhao (1994) 3. Robns, Rotntzky, and Zhao (1994) The man result: Generc full data D = (O, M), semparametrc model for D, functonal β O Part of D that s always observed (never mssng ) M R Part of D that may be mssng = 1 f M s observed, = 0 f M s mssng Observed data are (O, R, RM) Let ϕ F (D) be a full data nfluence functon Let π(o) = P (R = 1 D) = P (R = 1 O) > ɛ (MAR assumpton ) All observed-data nfluence functons have form Rϕ F (D) π(o) R π(o) g(o), π(o) g(o) square-ntegrable Result: Strategy for dervng estmators for a semparametrc model 1. Characterze the class of full-data nfluence functons (whch correspond to full-data estmators) 2. Characterze the observed data under the partcular MAR mechansm and the class of observed-data nfluence functons 3. Identfy observed-data estmators wth nfluence functons n ths class Our approach: Follow these steps for the semparametrc pretestposttest model Jont dstrbuton of (X 1, Y 1, X 2, Y 2, Z) unspecfed except (X 1, Y 1) ndependent of Z Greenberg Lecture II: Pretest-Posttest Study 23 Greenberg Lecture II: Pretest-Posttest Study 24

5 4. Estmaton wth full data 4. Estmaton wth full data Full-data nfluence functons: Can show (later) under the semparametrc pretest-posttest model that all full-data nfluence functons are of the form ( Z(Y2 µ (1) 2 ) (Z ) h (1) (X1, Y1) ) ( ) (1 Z)(Y2 µ (0) 2 ) 1 (Z ) + 1 h(0) (X1, Y1), for arbtrary h (c) (X 1, Y 1), c = 0, 1 wth var{h (c) (X 1, Y 1)} < Dfference of nfluence functons for estmators for µ (1) 2 and µ (0) 2 Full-data estmators may depend on X 1 (but not X 2) Effcent full-data nfluence functon: Correspondng to effcent full-data estmator; takes h (c) (X 1, Y 1) = E(Y 2 X 1, Y 1, Z = c) µ (c) 2, c = 0, 1 Popular estmators: Influence functons of β 2samp, β par, ANCOVA, ANCOVA II, and GEE have h (c) (X 1, Y 1) = η c(y 1 µ 1), c = 0, 1, for constants η c E.g., η c = 0, c = 0, 1 for β 2samp; η c = {(1 )} 1 for β par Popular estmators are n the class = are CAN even f (Y 1, Y 2) are not normal or the relatonshp between Y 1 and Y 2 s not lnear ANCOVA estmators ncorporatng baselne covarates are also n the class, e.g., E(Y 1 X 1, Y 1, Z) = α 0 + α 1Y 1 + α 2X 1 + βz Popular estmators are potentally neffcent among class of RAL estmators for semparametrc model How to use all ths? Effcent estmator s best! Greenberg Lecture II: Pretest-Posttest Study 25 Greenberg Lecture II: Pretest-Posttest Study Estmaton wth full data Effcent estmator: Settng sum over of effcent nfluence functon = 0 and replacng by = n 1/n yelds ( X n nx ) β = n 1 1 ZY2 (Z )E(Y2 X1, b Y1, Z = 1) ( X n nx ) n 1 0 (1 Z)Y2 + (Z )E(Y2 X1, b Y1, Z = 0) Practcal use replace E(Y 2 X 1, Y 1, Z = c) by predcted values ê h(c), say, c = 0, 1, from parametrc or nonparametrc regresson modelng Can lead to substantal ncrease n precson over popular estmators Even f E(Y 2 X 1, Y 1, Z = c) are modeled ncorrectly, β s stll consstent Observed data: (X 1, Y 1, X 2, Z) are never mssng, Y 2 may be mssng for some subjects R = 1 f Y 2 observed, R = 0 f Y 2 mssng Observed data are (X 1, Y 1, X 2, Z, R, RY 2) MAR assumpton P (R = 1 X 1, Y 1, X 2, Y 2, Z) = P (R = 1 X 1, Y 1, X 2, Z) = π(x 1, Y 1, X 2, Z) ɛ > 0 π(x 1, Y 1, X 2, Z) = Zπ (1) (X 1, Y 1, X 2) + (1 Z)π (0) (X 1, Y 1, X 2), π (c) (X 1, Y 1, X 2) = π(x 1, Y 1, X 2, c), c = 0, 1 Greenberg Lecture II: Pretest-Posttest Study 27 Greenberg Lecture II: Pretest-Posttest Study 28 Recall: Generc form of observed-data nfluence functons Rϕ F (D) π(o) R π(o) g(o) π(o) For smplcty: Focus on nfluence functons for estmators for µ (1) 2 Those for estmators for µ (0) 2 smlar Influence functons for estmators for β: take the dfference Full-data nfluence functons for estmators for µ (1) 2 : Recall Z(Y 2 µ (1) 2 ) (Z ) h (1) (X 1, Y 1), var{h (1) (X 1, Y 1)} < Thus: Observed-data nfluence functons for estmators for µ (1) 2 have form R{Z(Y 2 µ (1) 2 ) (Z )h(1) (X 1, Y 1)} R π(x1, Y1, X2, Z) g (1) (X 1, Y 1, X 2, Z) π(x 1, Y 1, X 2, Z) π(x 1, Y 1, X 2, Z) var{h (1) (X 1, Y 1)} <, var{g (1) (X 1, Y 1, X 2, Z)} < Choce of h (1) leadng to the effcent observed-data nfluence functon need not be the same as that leadng to the effcent full-data nfluence functon n general Turns out that the optmal h (1) s the same n the specal case of the pretest-posttest problem... Greenberg Lecture II: Pretest-Posttest Study 29 Greenberg Lecture II: Pretest-Posttest Study 30

6 Re-wrtng: Equvalently, observed-data nfluence functons are RZ(Y 2 µ (1) 2 ) π(x 1, Y 1, X 2, Z) ) (Z h (1) (X 1, Y 1) R π(x1, Y1, X2, Z) g (1) (X 1, Y 1, X 2, Z) π(x 1, Y 1, X 2, Z) Optmal choces leadng to the effcent observed-data nfluence functon are h eff(1) (X 1, Y 1) = E(Y 2 X 1, Y 1, Z = 1) µ (1) 2 g eff(1) (X 1, Y 1, X 2, Z) = Z{E(Y 2 X 1, Y 1, X 2, Z) µ (1) 2 } = Z{E(Y 2 X 1, Y 1, X 2, Z = 1) µ (1) 2 } Result: Wth these optmal choces, algebra yelds { µ (1) 2 = (n) 1 R Z Y 2 π (1) (X 1, Y 1, X (Z )E(Y 2 X 1, Y 1, Z = 1) 2) } {R π (1) (X 1, Y 1, X 2)}Z E(Y 2 X 1, Y 1, X 2, Z π (1) = 1) (X 1, Y 1, X 2) Smlarly for µ (0) 2 dependng on π (0), E(Y 2 X 1, Y 1, Z = 0), E(Y 2 X 1, Y 1, X 2, Z = 0) Estmator for β take the dfference Practcal use replace these quanttes by predcted values from regresson modelng (comng up) Greenberg Lecture II: Pretest-Posttest Study 31 Greenberg Lecture II: Pretest-Posttest Study 32 Complcaton 1: π (c) (X 1, Y 1, X 2) are not known, c = 0, 1 Common strategy: adopt parametrc models (e.g. logstc regresson) dependng on parameter γ (c) π (c) (X 1, Y 1, X 2; γ (c) ) Imposes an addtonal assumpton on semparametrc model for (X 1, Y 1, X 2, Y 1, Z) Substtute the MLE γ (c) for γ (c), obtan predcted values π (c) As long as ths model s correct, resultng estmators wll be CAN Greenberg Lecture II: Pretest-Posttest Study 33 Complcaton 2: Modelng E(Y 2 X 1, Y 1, Z = c), E(Y 2 X 1, Y 1, X 2, Z = c), c = 0, 1 MAR = E(Y 2 X 1, Y 1, X 2, Z) = E(Y 2 X 1, Y 1, X 2, Z, R = 1) (can base modelng/fttng on complete cases only) Obtan predcted values ê q(c), c = 0, 1 However, deally requre compatblty,.e. E(Y 2 X 1, Y 1, Z) = E{E(Y 2 X 1, Y 1, X 2, Z) X 1, Y 1, Z} and no longer vald to ft usng only complete cases Practcally go ahead and model drectly and ft usng complete cases, obtan predcted values ê h(c) Estmaton of parameters n these models does not affect (asymptotc) varance of β as long as π (c) models are correct Greenberg Lecture II: Pretest-Posttest Study 34 Estmator: Wth = n 1/n { R Z Y 2 (Z π (1) )ê h(1) β = n 1 1 n 1 0 { R (1 Z )Y 2 π (0) + (Z )ê h(0) (R π (1) π (1) (R π (0) } )Z ê q(1) Effcent f modelng done correctly; otherwse, close to optmal performance Takng ê h(c) = ê q(c) = 0 yelds the smple nverse-weghted complete case estmator (neffcent ) )(1 Z )ê q(0) π (0) Modelng E(Y 2 X 1, Y 1, Z = c), E(Y 2 X 1, Y 1, X 2, Z = c) augments ths, takng advantage of relatonshps among varables to mprove precson } Double Robustness: Stll consstent f π (c) are correctly modeled but E(Y 2 X 1, Y 1, Z = c) and E(Y 2 X 1, Y 1, X 2, Z = c) aren t E(Y 2 X 1, Y 1, Z = c) and E(Y 2 X 1, Y 1, X 2, Z = c) are correctly modeled but π (c) aren t No longer effcent If both sets of models ncorrect, nconsstent n general Standard errors: Use the sandwch formula (follows from nfluence functon) Greenberg Lecture II: Pretest-Posttest Study 35 Greenberg Lecture II: Pretest-Posttest Study 36

7 6. Full data, revsted Recap: Ths approach requres one to make an assumpton about π (c) (X 1, Y 1, X 2), c = 0, 1 No assumpton s made about E(Y 2 X 1, Y 1, X 2, Z = c), E(Y 2 X 1, Y 1, Z = c) Model s stll semparametrc... and double robustness holds Alternatve approach: Make an assumpton nstead about the E(Y 2 X 1, Y 1, X 2, Z = c), E(Y 2 X 1, Y 1, Z = c) Effcent estmator s maxmum lkelhood Don t need to even worry about π (c) (X 1, Y 1, X 2) But no double robustness property! For dscusson see: Bang and Robns (2005, Bometrcs) Greenberg Lecture II: Pretest-Posttest Study 37 How dd we get the full-data nfluence functons? One way use classcal semparametrc theory Another way Vew as a fake mssng data problem by castng the full-data problem n terms of counterfactuals Counterfactual representaton: Y (0) 2, Y (1) 2 are potental posttest responses f a subject were assgned to control or treatment We observe Y 2 = ZY (1) 2 + (1 Z)Y (0) 2 Fake full data (X 1, Y 1, X 2, Y (0) 2, Y (1) 2, Z) Fake observed data (X 1, Y 1, X 2, Z, ZY (1) 2, (1 Z)Y (0) 2 ) Apply the RRZ theory Greenberg Lecture II: Pretest-Posttest Study Smulaton evdence 7. Smulaton evdence Full-data problem: Substantal gans n effcency over popular methods, especally when there are nonlnear relatonshps among varables Parametrc and nonparametrc regresson modelng work well Vald standard errors, confdence ntervals Observed-data problem: Popular methods wth complete cases or LOCF can exhbt substantal bases Inverse-weghted complete case estmator s unbased but neffcent Substantal gans n effcency possble through modelng Vald standard errors, confdence ntervals Full data scenaro: 5000 Monte Carlo data sets = 0.5, Y 1 N (0, 1), Y 2 = µ (0) 2 +βz+β1y1+β2y 2 1 +ɛ, ɛ N (0, 1) µ (0) 2 = 0.25, (β 1, β 2) = (0.5, 0.4), β = 0.5 Estmator MC Mean MC SD Est SE MSE Rato Coverage n New-LOESS New-Quadratc ANCOVA-Interact ANCOVA Pared samp New-LOESS New-Quadratc ANCOVA-Interact ANCOVA Pared samp Greenberg Lecture II: Pretest-Posttest Study 39 Greenberg Lecture II: Pretest-Posttest Study Smulaton evdence Mssng data scenaro: 1000 Monte Carlo data sets = 0.5, Y 1 N (0, 1), ɛ N (0, 1) P (X 2 = 1 Y 1, Z) = (1 Z)/{1+exp( Y 1)}+Z/[1+exp( Y 1)} P (R = 1 Y 1, X 2, Z) = (1 Z)/{1 + exp( 0.2 2Y 1 0.1X 2 0.1X 2Y 1)} + Z/{1 + exp( 1.0Y X 2 0.1X 2Y 1)} yeldng 20% (Z = 0) and 34% (Z = 1) mssng Y 2 = 0.5Z + [ {X 2 E(X 2)}] exp(y 1) + 2.0{X 2 E(X 2)} + ɛ mples a model for E(Y 2 Y 1, Z) Overall, mples β = Smulaton evdence For n = 250 Estmator MC Mean MC SD Est SE MSE Rato Coverage Benchmark New-LOESS New-Quadratc New-Quadratc-π wrong IWCC GEE ANCOVA-Interact ANCOVA Pared samp For n = 1000 Any approach to modelng regresson (LOESS, Quadratc) acheves almost full effcency relatve to Benchmark (MSE ratos > 0.98) Popular estmators on complete cases stll horrbly based Greenberg Lecture II: Pretest-Posttest Study 41 Greenberg Lecture II: Pretest-Posttest Study 42

8 8. Applcaton ACTG Applcaton ACTG 175 Recall: Y 2 = CD4 at 20±5 weeks vs. Y 1 = (control and treatment groups) Apparent curvature follow-up CD4 at 20+/-5 weeks follow-up CD4 at 20+/-5 weeks Results: Models for E(Y 2 X 1, Y 1, Z = c), c = 0, 1 Estmator β SE Parametrc modelng (quadratc n Y 1) Nonparametrc modelng (GAM) ANCOVA Pared t Two-sample t Greenberg Lecture II: Pretest-Posttest Study 43 Greenberg Lecture II: Pretest-Posttest Study Applcaton ACTG Applcaton ACTG 175 Complete cases: Y 2 = CD4 at 96±5 weeks vs. Y 1 = (control and treatment groups) Results: Logstc regresson for π (c), c = 0, 1; parametrc regresson modelng of E(Y 2 X 1, Y 1, X 2, Z = c), E(Y 2 X 1, Y 1, Z = c) 37% mssng Y 2 follow-up CD4 at 96 ± 5 weeks follow-up CD4 at 96 ± 5 weeks Estmator β SE Parametrc modelng (quadratc n Y 1) Smple nverse-weghtng ANCOVA Pared t Greenberg Lecture II: Pretest-Posttest Study 45 Greenberg Lecture II: Pretest-Posttest Study Dscusson 9. Dscusson RRZ theory appled to a standard problem For more than you ever wanted to know: General framework for pretest-posttest analyss llumnatng how relatonshps among varables may be frutfully exploted Practcal estmators Can be extended to censored covarate nformaton Results are equally applcable to baselne covarate adjustment n comparson of two means (Y 1 s just another baselne covarate) Lots of methods for ths problem (lkelhood, mputaton combnatons thereof,... ); semparametrc theory provdes a framework for understandng commonaltes and dfferences among them Avalable June 2006! Greenberg Lecture II: Pretest-Posttest Study 47 Greenberg Lecture II: Pretest-Posttest Study 48