Misspecified proportional hazard models

Struthers, C. A.; Kalbfleisch, John D.

doi:10.1093/biomet/73.2.363

Cited by 339 publications

(247 citation statements)

References 5 publications

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“…Then by Theorem 2.1 of Struthers and Kalbfleisch [17] and the mean value theorem, we know that

∣ u_{j} (0) ∣ = ∣ u_{j} (β_{0 j}) - u_{j} (0) ∣ = ∣ u_{j}^{'} (β^{*}) ∣ ∣ β_{0 j} ∣

for some β * between β 0 j and 0, where

u_{j}^{'} (β) = {d u}_{j} (β) / d β

. We will first bound

u_{j}^{'} (β)

and then use Assumption 7 to conclude.…”

Section: Assumptionmentioning

confidence: 99%

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Principled sure independence screening for Cox models with ultra-high-dimensional covariates

Zhao

2012

Journal of Multivariate Analysis

196

164

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It is rather challenging for current variable selectors to handle situations where the number of covariates under consideration is ultra-high. Consider a motivating clinical trial of the drug bortezomib for the treatment of multiple myeloma, where overall survival and expression levels of 44760 probesets were measured for each of 80 patients with the goal of identifying genes that predict survival after treatment. This dataset defies analysis even with regularized regression. Some remedies have been proposed for the linear model and for generalized linear models, but there are few solutions in the survival setting and, to our knowledge, no theoretical support. Furthermore, existing strategies often involve tuning parameters that are difficult to interpret. In this paper we propose and theoretically justify a principled method for reducing dimensionality in the analysis of censored data by selecting only the important covariates. Our procedure involves a tuning parameter that has a simple interpretation as the desired false positive rate of this selection. We present simulation results and apply the proposed procedure to analyze the aforementioned myeloma study.

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“…Then by Theorem 2.1 of Struthers and Kalbfleisch [17] and the mean value theorem, we know that

∣ u_{j} (0) ∣ = ∣ u_{j} (β_{0 j}) - u_{j} (0) ∣ = ∣ u_{j}^{'} (β^{*}) ∣ ∣ β_{0 j} ∣

for some β * between β 0 j and 0, where

u_{j}^{'} (β) = {d u}_{j} (β) / d β

. We will first bound

u_{j}^{'} (β)

and then use Assumption 7 to conclude.…”

Section: Assumptionmentioning

confidence: 99%

“…Following Struthers and Kalbfleisch [17] and under Assumptions 1 and 2 in the Appendix, we know that the β̂ j are consistent for β 0 j . It is therefore natural to ask how accurate these estimates are.…”

Section: Principled Cox Sure Independence Screeningmentioning

confidence: 99%

Principled sure independence screening for Cox models with ultra-high-dimensional covariates

Zhao

2012

Journal of Multivariate Analysis

196

164

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“…Inserting this

{\overset{Λ}{true}}_{N} (β)

in (11), the resulting

{\overset{β}{true}}_{N}

solves

{double-struckP}_{N}^{\overset{π}{true}} Δ true(Z - \frac{{double-struckP}_{N}^{\overset{π}{true}} {italicZe}^{Z^{normalT} β} Y}{{double-struckP}_{N}^{\overset{π}{true}} e^{Z^{normalT} β} Y} (T) true) = 0,

an IPW version of the Cox partial likelihood equations. Similarly, the “true values” of the parameters ( β 0 , ʌ 0 ) = ( β 0 ( P ), ʌ 0 ( P )) are now defined as the solutions to the population version of the estimating equations Ψ β , ʌ = 0, which we write as

\begin{matrix} right Ψ_{1; β, Λ} = & left P \int {ZdM}_{β, Λ} = P Δ Z - Λ (italicPZ e^{Z^{T} β} Y) = 0 \\ right Ψ_{2; β, Λ} h = & left P \int {hdM}_{β, Λ} = P Δ h (T) - Λ (italichP e^{Z^{T} β} Y) = 0, h \in H . \end{matrix}

Taking

h = PZ e^{Z^{normalT} β} Y ∕ {italicPe}^{Z^{normalT} β} Y

, and subtracting, β 0 solves

P Δ true[Z - \frac{PZ e^{Z^{normalT} β} Y}{{italicPe}^{Z^{normalT} β} Y} (T) true] = 0 .

The arguments of Struthers and Kalbfleisch (1986) may be adapted to demonstrate that, under standard regularity conditions,

…”

Section: Applications To Survival Modelsmentioning

confidence: 99%

Z-estimation and stratified samples: application to survival models

Breslow

Wellner

2015

Lifetime Data Anal

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The infinite dimensional Z-estimation theorem offers a systematic approach to joint estimation of both Euclidean and non-Euclidean parameters in probabiity models for data. It is easily adapted for stratified sampling designs. This is important in applications to censored survival data because the inverse probability weights that modify the standard estimating equations often depend on the entire follow-up history. Since the weights are not predictable, they complicate the usual theory based on martingales. This paper considers joint estimation of regression coefficients and baseline hazard functions in the Cox proportional and Lin-Ying additive hazards models. Weighted likelihood equations are used for the former and weighted estimating equations for the latter. Regression coefficients and baseline hazards may be combined to estimate individual survival probabilities. Efficiency is improved by calibrating or estimating the weights using information available for all subjects. Although inefficient in comparison with likelihood inference for incomplete data, which is often difficult to implement, the approach provides consistent estimates of desired population parameters even under model misspecification.

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“…The model (1) may fail in three ways: (i) the time invariance of the hazard ratio does not hold; (ii) the exponential form of the link function for the hazard ratio is inappropriate; (iii) the functional forms of individual covariates in the exponent of the model are misspecified. The model misspecification can have detrimental effects on the validity and efficiency of the partial likelihood inference for the proportional hazards model (Lagakos and Schoenfeld, 1984; Struthers and Kalbfleisch, 1986; Lagakos, 1988(b); Lin and Wei, 1989). …”

Section: Introductionmentioning

confidence: 99%

Goodness-of-fit test of the stratified mark-specific proportional hazards model with continuous mark

Sun

Gilbert

2016

Computational Statistics & Data Analysis

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Motivated by the need to assess HIV vaccine efficacy, previous studies proposed an extension of the discrete competing risks proportional hazards model, in which the cause of failure is replaced by a continuous mark only observed at the failure time. However the model assumptions may fail in several ways, and no diagnostic testing procedure for this situation has been proposed. A goodness-of-fit test procedure for the stratified mark-specific proportional hazards model in which the regression parameters depend nonparametrically on the mark and the baseline hazards depends nonparametrically on both time and the mark is proposed. The test statistics are constructed based on the weighted cumulative mark-specific martingale residuals. The critical values of the proposed test statistics are approximated using the Gaussian multiplier method. The performance of the proposed tests are examined extensively in simulations for a variety of the models under the null hypothesis and under different types of alternative models. An analysis of the ‘Step’ HIV vaccine efficacy trial using the proposed method is presented. The analysis suggests that the HIV vaccine candidate may increase susceptibility to HIV acquisition.

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Misspecified proportional hazard models

Cited by 339 publications

References 5 publications

Principled sure independence screening for Cox models with ultra-high-dimensional covariates

Principled sure independence screening for Cox models with ultra-high-dimensional covariates

Z-estimation and stratified samples: application to survival models

Goodness-of-fit test of the stratified mark-specific proportional hazards model with continuous mark

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