Web-based Supplementary Materials for A Modified Partial Likelihood Score Method for Cox Regression with Covariate Error Under the Internal

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1 Web-based Supplemetary Materials for A Modified Partial Likelihood Score Method for Cox Regressio with Covariate Error Uder the Iteral Validatio Desig by David M. Zucker, Xi Zhou, Xiaomei Liao, Yi Li, ad Doa Spiegelma

2 Itroductio, Notatio, ad Prelimiaries I this documet we preset the details of the asymptotic theory for the modified score estimator. At the ed of the documet we preset tables with the simulatio results for the commo disease case that we referred to i the mai paper. Deote the true value of θ by θ. We assume that θ lies i a suitably large compact set Q whose iterior cotais θ. Two additioal assumptios will be itroduced later, with clear sigpostig. Let τ deote the maximum follow-up time. All assertios below of uiform covergece of fuctios of θ ad/or t refer to θ Q ad t [, τ]. We use the otatio used i the mai paper. From ow o, i the various S s defied i the mai paper we will write θ istead of β, α. We ow defie some additioal otatio. S e t, θ = S 2a t, θ = S 2b t, θ = S 2c t, θ = S 2d t, θ = ω j Y j t X j α expβ T X j t ω j Y j tx j X T j expβ T X j t ω j Y j t Xα Xα T expβ T Xαt ω j Y j t Xα Xα T expβ T Xαt ω j Y j t XαX T expβ T Xt E W t, θ = E[Y t expβ T Xα] E W W t, θ = E[Y t Xα expβ T Xα] XX t, θ = E[Y txx T expβ T X] E W W W t, θ = E[Y t Xα Xα T expβ T Xα] E W XX t, θ = E[Y t XαX T expβ T X] I additio, we defie Nt = i= N it ad N t = E[Nt]. We ow itroduce the followig assumptio: Assumptio : t, θ is bouded below over t ad θ.

3 This assumptio implies that there is a positive probability of reachig the maximum follow-up time τ with either a evet or a cesorig i the ope iterval, τ. I view of the fuctioal strog law of large umbers Aderse ad Gill, 982, Appedix III, the followig covergece relatios hold uiformly i t ad θ almost surely: S a t, θ π t, θ, S b t, θ πe W t, θ, S c t, θ πe W t, θ S d t, θ t, θ, S a t, θ πx t, θ, S b t, θ πe W W t, θ 2 S c t, θ πe W W t, θ, S d t, θ πe W X t, θ, S e t, θ πe W X t, θ 3 S 2a t, θ πxx t, θ, S 2b t, θ πe W W W t, θ 4 S 2c t, θ πe W W W t, θ, S 2d t, θ πe W XX t, θ 5 It follows that the followig covergece relatios hold uiformly i t ad θ almost surely ote that φπ = π: S t, β t, θ, S t, θ s t, θ 6 Also, it is clear that α α almost surely. Recall the coutig process represetatio of the score fuctio U MS θ: U MS θ = i= [ω i X i + ω i X i α]dn i t S t, θ dnt 7 S t, θ We ca write U MS θ = i= [ω i X i + ω i X i α]dn i t s t, θ t, θ dnt + R θ with R θ = S t, θ S t, θ s t, θ dnt t, θ Give the uiform covergece of S t, θ ad S t, θ, the assumptio that t, θ is bouded below, ad the fact that Nτ, we obtai the result that R θ uiformly i θ. Recall ow the defiitio dm i t = dn i t Y i t expβ T X i tλ tdt. Accordig to coutig 2

4 process theory Gill 984, M i t is a mea-zero martigale process w.r.t. the history defied by F t = σx i ; Y i s, N i s, s [, t], i =,...,. Give the oiformative measuremet error assumptio ad the fact that ω i is idepedet of all the other basic radom variables associated with idividual i, M i t is also a martigale w.r.t. the history defied by G t = σx i, W i, ω i, Y i s, N i s, s [, t], i =,...,. Cosistecy We ca write i= ω i X i dn i t = ω i X i dm i t + i= S a β, αλ t dt ad i= ω i X i α dn i t = ω i X i α dm i t + i= S e β, αλ t dt Similarly, s t, θ t, θ dnt = i= s t, θ τ t, θ dm it + s t, θ t, θ S dt, θ λ tdt Hece, U MS θ = S a β, αλ t dt + +R θ + Mθ S e β, αλ t dt s t, θ t, θ S dt, θ λ tdt where Mθ = i= ω i X i + ω i W i s t, θ dm i t t, θ Now, Mθ is the sample mea of i.i.d. mea-zero r.v. s, ad therefore coverges a.s. to, ad sice Q is compact ad t, θ ad s t, θ are cotiuous i θ uiformly over t the covergece is uiform i θ. Usig this result, alog with the covergece results -5 ad 3

5 the fact that α α a.s., we fid that U MS β, α coverges uiformly almost surely to uβ = s t, θ s t, β, α t, β, α t, θ λ t dt. We have that times the matrix of derivatives of U MS θ with respect to β is give by D ββ θ = Ṡt, θ S t, θ S t, θ S t, θ T Ṡ t, θ dnt 8 S t, θ where Sa t, θ Ṡ t, θ = S a t, θ + S b t, θ S c t, θ Sb t, θ Ṡ t, θ = S 2a t, θ + S 2b t, θ + S c t, θ + S d t, θ S c t, θ [ Sa t, θ + S b t, θ S c t, θ S 2d t, θ S 2c t, θ [ Sb t, θ S c t, θ Sb t, θ S c t, θ Sa t, θ S c t, θ ] T Sc t, θ S c t, θ ] Sc t, θ S c t, θ The limitig value of D ββ θ is give by [ τ XX t, θ d ββ θ = π t, θ + π EXX t, θ t, θ [ E W XX t, θ t, θ ] T EXX t, θ t, θ λ tdt t, θ EW X t, θ t, θ ] T EXX t, θ t, θ λ tdt t, θ which is times the matrix of derivatives of uθ with respect to β. The first term i the above expressio is a positive defiite matrix for all θ. At this poit, we itroduce a additioal assumptio: Assumptio 2: d ββ θ is osigular. By ispectio of uθ, we see that uβ, α =. Give this fact, the assumed osigularity of d ββ θ, the covergece of α to α, ad the uiform covergece of U MS θ to uθ ad D ββ θ to d ββ θ, it follows from the argumets of Foutz 977 that there exists a uique root of the score equatio U MS β, α = that coverges to β. 4

6 Asymptotic Distributio of βms The backgroud for the derivatio of the asymptotic distributio of β MS is preseted i Sectio 2.2 of the mai paper. We start here with the derivatio of U θ. As i Zucker ad Spiegelma 28, we use the argumet of Li ad Wei 989. I what follows, the symbol. = will deote equality up to egligible terms. Let us recall the coutig process represetatio of U θ: U θ = i= ω i X i + ω i X i αdn i t S t, θ S t, θ dnt We ow proceed to aalyze this quatity. For brevity, we will omit the argumets t ad θ. We ca write U θ = U a θ U θ with U a θ = U b θ = b i= S S ŝ ω i X i + ω i X i α ŝ dn =. S ŝ S dn i dn where ŝ t, β, α = πx t, β, α + πe W X t, β, α. Regardig U a θ, after some maipulatio we obtai U a θ = ω i i= + X i X ω i i= dn i π X i E W X dn i + π E W X X dn i dn E W X X dn i dn We ow tur to U b θ. We eed to work o the term S t, θ/s t, θ. We ca write S ŝ = S S ŝ S S E X. = E X = E X S ŝ S E X S ŝ ŝ S 5

7 Now, Sa S = S a + S b S c. = S a π + S b πe W + E φs a π φ S c πe W W E W = π ω j Y j e βt X j + ω j Y j e βt Xj E W E W φ ω j Y j e βt Xj E W By a similar argumet, S ŝ. = E W ω j Y j X j e βt X j X + + φ + E W E W X E W W ω j Y j Xj e βt Xj E W W ω j Y j Xj e βt X j E W X φ ω j Y j Xj e βt Xj E W W ω j Y j e βt Xj E W φ ω j Y j e βt Xj E W As a result, we ca write U θ. = U θ with where Z i = U θ = π m X i X φ φ E X τ i= ω i Z i dn i π Y i Xi e βt X i E W X E X dn + φ + π m i= E W X X dn i dn ω i Z 2 i Y i Xi e βt Xi E W W E X dn E W E W X E W W Y i e βt Xi E W E X dn + φŝ E W Y i e βt Xi E W E X dn ŝ π Y i X i e βt X i X E X dn Y i e βt X i E X dn 6

8 ad Z 2 i = X i E W X dn i + π E W X X dn i dn + E W E W X E W W Y i e βt Xi E W E X dn + Y i Xi e βt Xi E W W E X dn ŝ E W Y i e βt Xi E W E X dn I computig Ĉ as described i the mai paper, we substitute θ for θ, dnu for dn u, S for, S for ŝ, S b + S c for E W, π S a for X, π S d for E W X, ad S b + S c for E W W. We ow tur to the derivatio of the relevat compoets of Jacobia matrix Dθ. derivative of U θ with respect to β is give by 8. Let etry equals X ir / α t. The matrix ad whose remaiig rows are filled with s. We the have The X i deote the matrix whose r, t X i is a p p+p matrix whose first p rows are I p w T i U θ D βα = α = i= δ i ω i X i + i= δ i H2 S S H S S where ad H = S α = H 2 = S α = Sa i= S c i= ω i Y i β T X i e βt Xi ω i Y i X i e βt Xi + + S c S d S c + Sb S c Sb S 2 c i= S d S c i= i= Sa S b S 2 c i= ω i Y i Xi β T X i e βt Xi ω i Y i β T X i e βt Xi ω i Y i X i e βt X i i= i= ω i Y i X i e βt Xi ω i Y i β T X i e βt Xi ω i Y i β T X i e βt Xi i= ω i Y i Xi β T X i e βt Xi Fially, U2 θ D αα = α = ω i I p w i w T i i= 7

9 Table S: Simulatio results for the sigle-covariate commo disease case with idepedet measuremet error. β is the true value of β. Bias% is the relative bias, i.e. Bias%= ˆβ β /β. IQR is.74 times the iterquartile rage of the ˆβ values. SE is the mea of the estimated stadard error of ˆβ. SD is the empirical stadard deviatio of the ˆβ values. CR is the empirical coverage rate of the asymptotic 95% cofidece iterval. Methods cosidered: MS = modified score, CH = Che, RC = regressio calibratio, CC = complete case, NA = aive. Mea Media CorrX, W expβ β Method ˆβ Bias% ˆβ Bias% IQR SE SD CR MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA

10 Table S2: Simulatio results for the sigle-covariate commo disease case with depedet measuremet error. β is the true value of β. Bias% is the relative bias, i.e. Bias%= ˆβ β /β. IQR is.74 times the iterquartile rage of the ˆβ values. SE is the mea of the estimated stadard error of ˆβ. SD is the empirical stadard deviatio of the ˆβ values. CR is the empirical coverage rate of the asymptotic 95% cofidece iterval. Methods cosidered: MS = modified score, CH = Che, RC = regressio calibratio, CC = complete case, NA = aive. Mea Media CorrX, W expβ β Method ˆβ Bias% ˆβ Bias% IQR SE SD CR MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA

11 Table S3: Simulatio results for the multiple-covariate commo disease case with idepedet covariates ad idepedet measuremet error. β is the true value of β. Bias% is the relative bias, i.e. Bias%= ˆβ β /β. IQR is.74 times the iterquartile rage of the ˆβ values. SE is the mea of the estimated stadard error of ˆβ. SD is the empirical stadard deviatio of the ˆβ values. CR is the empirical coverage rate of the asymptotic 95% cofidece iterval. Methods cosidered: MS = modified score, CH = Che, RC = regressio calibratio, CC = complete case, NA = aive. Mea Media CorrX, W expβ β Method ˆβ Bias% ˆβ Bias% IQR SE SD CR MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA

12 Table S4: Simulatio results for the multiple-covariate commo disease case with idepedet covariates ad depedet measuremet error. β is the true value of β. Bias% is the relative bias, i.e. Bias%= ˆβ β /β. IQR is.74 times the iterquartile rage of the ˆβ values. SE is the mea of the estimated stadard error of ˆβ. SD is the empirical stadard deviatio of the ˆβ values. CR is the empirical coverage rate of the asymptotic 95% cofidece iterval. Methods cosidered: MS = modified score, CH = Che, RC = regressio calibratio, CC = complete case, NA = aive. Mea Media CorrX, W expβ β Method ˆβ Bias% ˆβ Bias% IQR SE SD CR MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA

13 Table S5: Simulatio results for the multiple-covariate commo disease case with depedet covariates ad idepedet measuremet error. β is the true value of β. Bias% is the relative bias, i.e. Bias%= ˆβ β /β. IQR is.74 times the iterquartile rage of the ˆβ values. SE is the mea of the estimated stadard error of ˆβ. SD is the empirical stadard deviatio of the ˆβ values. CR is the empirical coverage rate of the asymptotic 95% cofidece iterval. Methods cosidered: MS = modified score, CH = Che, RC = regressio calibratio, CC = complete case, NA = aive. Mea Media CorrX, W expβ β Method ˆβ Bias% ˆβ Bias% IQR SE SD CR MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA

14 Table S6: Simulatio results for the multiple-covariate commo disease case with depedet covariates ad depedet measuremet error. β is the true value of β. Bias% is the relative bias, i.e. Bias%= ˆβ β /β. IQR is.74 times the iterquartile rage of the ˆβ values. SE is the mea of the estimated stadard error of ˆβ. SD is the empirical stadard deviatio of the ˆβ values. CR is the empirical coverage rate of the asymptotic 95% cofidece iterval. Methods cosidered: MS = modified score, CH = Che, RC = regressio calibratio, CC = complete case, NA = aive. Mea Media CorrX, W expβ β Method ˆβ Bias% ˆβ Bias% IQR SE SD CR MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA MS CH RC CC NA

15 Table S7: Simulatio results for the multiple-covariate itermediate disease rate case with idepedet covariates ad idepedet measuremet error. β is the true value of β. Bias% is the relative bias, i.e. Bias%= ˆβ β /β. IQR is.74 times the iterquartile rage of the ˆβ values. SE is the mea of the estimated stadard error of ˆβ. SD is the empirical stadard deviatio of the ˆβ values. CR is the empirical coverage rate of the asymptotic 95% cofidece iterval. Mea Media CorrX, W expβ β Method ˆβ Bias% ˆβ Bias% IQR SE SD CR MS CH MS CH MS CH Table S8: Simulatio results for various sample sizes for the multiple-covariate commo disease case with idepedet measuremet error. β is the true value of β. NN is the sample size of the mai study ad NV is the sample size of the iteral validatio sample. Bias% is the relative bias, i.e. Bias%= ˆβ β /β. IQR is.74 times the iterquartile rage of the ˆβ values. SE is the mea of the estimated stadard error of ˆβ. SD is the empirical stadard deviatio of the ˆβ values. CR is the empirical coverage rate of the asymptotic 95% cofidece iterval. Methods cosidered: MS = modified score, CH = Che, RC = regressio calibratio, CC = complete case, NA = aive. Mea Media CorrX, W expβ β Sample Size ˆβ Bias% ˆβ Bias% IQR SE SD CR NN=5, NV= NN=25, NV= NN=5, NV= NN=5, NV= NN=25, NV= NN=5, NV= NN=5, NV= NN=25, NV= NN=5, NV= NN=5, NV= NN=25, NV= NN=5, NV= NN=5, NV= NN=25, NV= NN=5, NV= NN=5, NV= NN=25, NV= NN=5, NV= NN=5, NV= NN=25, NV= NN=5, NV= NN=5, NV= NN=25, NV= NN=5, NV= NN=5, NV= NN=25, NV= NN=5, NV=

16 Table S9: Simulatio results for various sample sizes for the multiple-covariate commo disease case with idepedet measuremet error. β is the true value of β. NN is the sample size of the mai study ad NV is the sample size of the iteral validatio sample. Bias% is the relative bias, i.e. Bias%= ˆβ β /β. IQR is.74 times the iterquartile rage of the ˆβ values. SE is the mea of the estimated stadard error of ˆβ. SD is the empirical stadard deviatio of the ˆβ values. CR is the empirical coverage rate of the asymptotic 95% cofidece iterval. Methods cosidered: MS = modified score, CH = Che, RC = regressio calibratio, CC = complete case, NA = aive. Mea Media CorrX, W expβ β Sample Size ˆβ Bias% ˆβ Bias% IQR SE SD CR NN=, NV= NN=, NV= NN=5, NV= NN=, NV= NN=, NV= NN=, NV= NN=5, NV= NN=, NV= NN=, NV= NN=, NV= NN=5, NV= NN=, NV= NN=, NV= NN=, NV= NN=5, NV= NN=, NV= NN=, NV= NN=, NV= NN=5, NV= NN=, NV= NN=, NV= NN=, NV= NN=5, NV= NN=, NV= NN=, NV= NN=, NV= NN=5, NV= NN=, NV= NN=, NV= NN=, NV= NN=5, NV= NN=, NV= NN=, NV= NN=, NV= NN=5, NV= NN=, NV=

Convergence of random variables. (telegram style notes) P.J.C. Spreij

Convergence of random variables. (telegram style notes) P.J.C. Spreij Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space