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1. More efficient approximation of smoothing splines via space-filling basis selection

   https://doi.org/10.1093/biomet/asaa019  

   Meng, Cheng; Zhang, Xinlian; Zhang, Jingyi; Zhong, Wenxuan; Ma, Ping (May 2020, Biometrika)

   null (Ed.)

   Summary We consider the problem of approximating smoothing spline estimators in a nonparametric regression model. When applied to a sample of size $$n$$, the smoothing spline estimator can be expressed as a linear combination of $$n$$ basis functions, requiring $O(n^3)$ computational time when the number $$d$$ of predictors is two or more. Such a sizeable computational cost hinders the broad applicability of smoothing splines. In practice, the full-sample smoothing spline estimator can be approximated by an estimator based on $$q$$ randomly selected basis functions, resulting in a computational cost of $O(nq^2)$. It is known that these two estimators converge at the same rate when $$q$$ is of order $$O\{n^{2/(pr+1)}\}$$, where $$p\in [1,2]$$ depends on the true function and $r > 1$ depends on the type of spline. Such a $$q$$ is called the essential number of basis functions. In this article, we develop a more efficient basis selection method. By selecting basis functions corresponding to approximately equally spaced observations, the proposed method chooses a set of basis functions with great diversity. The asymptotic analysis shows that the proposed smoothing spline estimator can decrease $$q$$ to around $$O\{n^{1/(pr+1)}\}$$ when $$d\leq pr+1$$. Applications to synthetic and real-world datasets show that the proposed method leads to a smaller prediction error than other basis selection methods. 

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2. Statistical inference for time‐to‐event data in non‐randomized cohorts with selective attrition

   https://doi.org/10.1002/sim.9952  

   Wang, Tuo; Mao, Lu; Cocco, Aldo; Kim, KyungMann (January 2024, Statistics in Medicine)

   In multi‐season clinical trials with a randomize‐once strategy, patients enrolled from previous seasons who stay alive and remain in the study will be treated according to the initial randomization in subsequent seasons. To address the potentially selective attrition from earlier seasons for the non‐randomized cohorts, we develop an inverse probability of treatment weighting method using season‐specific propensity scores to produce unbiased estimates of survival functions or hazard ratios. Bootstrap variance estimators are used to account for the randomness in the estimated weights and the potential correlations in repeated events within each patient from season to season. Simulation studies show that the weighting procedure and bootstrap variance estimator provide unbiased estimates and valid inferences in Kaplan‐Meier estimates and Cox proportional hazard models. Finally, data from the INVESTED trial are analyzed to illustrate the proposed method. 

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3. Debiased inference for a covariate-adjusted regression function

   https://doi.org/10.1093/jrsssb/qkae041  

   Takatsu, Kenta; Westling, Ted (May 2024, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   In this article, we study nonparametric inference for a covariate-adjusted regression function. This parameter captures the average association between a continuous exposure and an outcome after adjusting for other covariates. Under certain causal conditions, it also corresponds to the average outcome had all units been assigned to a specific exposure level, known as the causal dose–response curve. We propose a debiased local linear estimator of the covariate-adjusted regression function and demonstrate that our estimator converges pointwise to a mean-zero normal limit distribution. We use this result to construct asymptotically valid confidence intervals for function values and differences thereof. In addition, we use approximation results for the distribution of the supremum of an empirical process to construct asymptotically valid uniform confidence bands. Our methods do not require undersmoothing, permit the use of data-adaptive estimators of nuisance functions, and our estimator attains the optimal rate of convergence for a twice differentiable regression function. We illustrate the practical performance of our estimator using numerical studies and an analysis of the effect of air pollution exposure on cardiovascular mortality. 

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4. High‐dimensional robust inference for Cox regression models using desparsified Lasso

   https://doi.org/10.1111/sjos.12543  

   Kong, Shengchun; Yu, Zhuqing; Zhang, Xianyang; Cheng, Guang (July 2021, Scandinavian Journal of Statistics)

   Abstract We consider high‐dimensional inference for potentially misspecified Cox proportional hazard models based on low‐dimensional results by Lin and Wei (1989). A desparsified Lasso estimator is proposed based on the log partial likelihood function and shown to converge to a pseudo‐true parameter vector. Interestingly, the sparsity of the true parameter can be inferred from that of the above limiting parameter. Moreover, each component of the above (nonsparse) estimator is shown to be asymptotically normal with a variance that can be consistently estimated even under model misspecifications. In some cases, this asymptotic distribution leads to valid statistical inference procedures, whose empirical performances are illustrated through numerical examples. 

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5. Ensemble estimation and variable selection with semiparametric regression models

   https://doi.org/10.1093/biomet/asaa012  

   Shin, Sunyoung; Liu, Yufeng; Cole, Stephen R; Fine, Jason P (April 2020, Biometrika)

   Summary We consider scenarios in which the likelihood function for a semiparametric regression model factors into separate components, with an efficient estimator of the regression parameter available for each component. An optimal weighted combination of the component estimators, named an ensemble estimator, may be employed as an overall estimate of the regression parameter, and may be fully efficient under uncorrelatedness conditions. This approach is useful when the full likelihood function may be difficult to maximize, but the components are easy to maximize. It covers settings where the nuisance parameter may be estimated at different rates in the component likelihoods. As a motivating example we consider proportional hazards regression with prospective doubly censored data, in which the likelihood factors into a current status data likelihood and a left-truncated right-censored data likelihood. Variable selection is important in such regression modelling, but the applicability of existing techniques is unclear in the ensemble approach. We propose ensemble variable selection using the least squares approximation technique on the unpenalized ensemble estimator, followed by ensemble re-estimation under the selected model. The resulting estimator has the oracle property such that the set of nonzero parameters is successfully recovered and the semiparametric efficiency bound is achieved for this parameter set. Simulations show that the proposed method performs well relative to alternative approaches. Analysis of an AIDS cohort study illustrates the practical utility of the method. 

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