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Penalised spline estimation for generalised partially linear single-index models

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robert john tibshirani - Stanford University

George Casella and Roger L. Starting from the basics of probability, the authors develop the theory of statistical inference using techniques, definitions, and concepts that are statistical and.

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The estimator, x , reduces to the Cox partial likelihood estimator if the covariate is discrete. Moreover, Breslow-type estimation of the cumulative baseline hazard function, using the proposed estimator x , is proved to be efficient.

The asymptotic bias and variance are derived under regularity conditions. Computation of the estimator involves an iterative but simple algorithm. Extensive simulation studies provide evidence supporting the theory. The method is illustrated with the Stanford heart transplant data set. The proposed global approach is also extended to a partially linear proportional hazards model and found to provide efficient estimation of the slope parameter. This article has the supplementary materials online.

The Cox proportional hazards model Cox is widely used in the analysis of time-to-failure data in biomedical, economic, and social studies. The covariate effect in the Cox model is usually assumed to be log-linear; that is, the logarithm of the hazard function is a linear function of the covariate.

The regression parameter retains interpretability and can be easily estimated through the partial likelihood method. The assumption of log-linearity may not hold in practice, however. A nonparametric proportional hazards model, in which the form of the covariate effect is unspecified, provides a useful variant. Specifically, let T be the survival time and let X be a one-dimensional covariate. Several statistical methods involving smoothing techniques, including nearest-neighbor, spline, and local polynomial smoothing methods, have been developed for this model see, e.

In particular, Tibshirani and Hastie and Fan, Gijbels, and King applied nearest-neighbor and local polynomial smoothing methods, respectively, and developed a local partial likelihood approach. The main idea of that approach is quite insightful and nontrivial.

The reason for estimating the derivative first is that the link function can be identified only up to an additive constant, so only the derivative can be identified. In essence, the local partial likelihood approach corresponds to a particular smoothing method, but it resembles and inherits the major advantages of the partial likelihood approach.

Fan, Gijbels, and King developed applications of local polynomial smoothing, along with asymptotic theory. This approach enjoys the advantages of the local polynomial smoothing method Fan and Gijbels , such as numerical simplicity and design adaptivity; however, it is not efficient, as we demonstrate in this article. Aiming at the difference has the advantage that the target is identifiable, and this approach may gain efficiency over the local partial likelihood method.

Nevertheless, this approach is still local and similar overall to earlier local partial likelihood methods. Motivated by the deficiency of all local partial likelihood methods, we propose a global version of partial likelihood. In addition, we prove that the Breslow-type estimator of the cumulative baseline hazard function, using the proposed estimator , is semiparametric efficient. The efficiency gain initially seems unusual, because as local smoothing yields optimal procedures for nonparametric regression functions.

But the situation is quite different for a hazard-based model, such as the proportional hazards model. The partial likelihood function for the regression parameter contributed by each observation involves all subjects who are at risk. One advantage of the local polynomial approach is its design-adaptive feature.

For example, when the covariates are concentrated near a few points, our approach, being design-adaptive, reduces naturally to Cox partial likelihood. In contrast, the spline method has difficulty doing this properly, because the observations are too sparse between the few points. The article is organized as follows. In Section 2 we introduce local and global partial likelihood and present an iterative algorithm for computing the proposed estimates. The main idea is to derive estimating functions directly from the partial likelihood, rather than from the local partial likelihood.

In contrast to the local partial likelihood method, this approach reduces to the partial likelihood approach when the covariate has a discrete distribution. Asymptotic properties and semiparametric efficiency of the estimates are presented in Section 3.

Section 4 presents simulation results, evaluating the finite-sample properties of the new method and comparing the global procedure with two local procedures by Fan, Gijbels, and King and Chen and Zhou An analysis of the Stanford heart transplant data set with the global procedure is reported. In Section 5, the methodology developed in Section 2 for a single covariate is extended to a partially linear proportional hazards model.

A brief discussion is given in Section 6. All proofs are relegated to the Appendix. In the presence of censoring, let C be the censoring variable, and assume that T and C are conditionally independent given X , but that the distribution of C may depend on X. The global partial likelihood, due to Cox , is. This is essentially the partial likelihood 2 , restricted to observations for which the covariates are near x. A much more refined version of local partial likelihood estimation through local polynomial smoothing was proposed and studied by Fan, Gijbels, and King With a local polynomial smoother of order p , the logarithm of the local partial likelihood of Fan, Gijbels, and King is given by.

They also provided a comprehensive theoretical justification of this method and closed-form expressions for asymptotic bias and variance of the derivative estimates. Asymptotic properties, including asymptotic bias and variance, were derived. Because this procedure still utilizes a local partial likelihood, improvement is possible by using a global partial likelihood method instead, as we demonstrate herein.

The global approach is motivated as follows. This is a crude nearest-neighbor global partial likelihood, analogous to 3 but it also uses the information available at data outside the neighborhood region B n x. The proposed estimate reduces to the partial likelihood estimate when the covariate assumes only finitely many distinct values, say a 1 , …, a K.

As a result, 7 is always satisfied, and the limit of 6 reduces to. This observation lends support to the optimality of the proposed global method of estimation. We note that the estimates of Tibshirani and Hastie , Fan, Gijbels, and King , and Chen and Zhou do not reduce to the Cox partial likelihood estimates in this case, and neither do any spline methods.

In this section we assume that the random variable X is bounded with compact support. Without loss of generality, let the support be [0, 1]. Additional regularity conditions are stated in the Appendix. We start with the uniform consistency of x. Suppose that the regularity conditions C1 — C7 stated in the Appendix hold. Then 9 can be written as. To evaluate the optimality of the global partial likelihood approach, in the following theorem we provide a justification via semiparametric efficiency.

Then for any function , we have. See Klaassen, Lee, and Ruymgaart for a related type of efficiency based on functionals.

In addition, it is noteworthy that Theorem 3 implies the semiparametric efficiency of the cumulative hazard function as presented in Theorem 4. In this section we report simulation studies regarding the finite-sample performance of the global partial likelihood method designated GPL hereinafter.

We did not include decreasing link functions, because the sign of X can be flipped in Models 1—4 to make the link function decreasing. Whereas it is unlikely that a link function will have both a local maximum and a local minimum, as in the settings of Models 5 and 6, we included these to allow comparison of all three methods under a nonmonotone link function.

The performance of the various estimators x was assessed via the weighted mean integrated squared error WMISE ,. This confirms the benefit of the GPL approach compared with local approaches.

We also see that GPL performs better when the covariate x has a uniform distribution than when it has a normal mixture distribution.

The optimal bandwidth h for CZ depends on the choice of the prespecified bandwidth h 0 , however. This is likely because the local methods need to enlarge the included range of data to compensate for the use of less data information than the global procedure. The performance of CZ depends heavily on the bandwidth choice; the values given in parentheses in Table 3 reflect this variation.

Figure 1 shows the biases of the different estimates based on the optimal bandwidths of Tables 1 — 3. It can be seen that the estimated curves based on all three methods are close to the true curves, reflecting the effectiveness of the global and local methods. To appreciate the sample variability of the estimated nonparametric functions at each point, we also calculated the pointwise standard errors at some grid points based on replications.

To save space, we do not report them here. Biases of the estimators based on replications in Models 1—6. In summary, FGK is usually less efficient than CZ, which requires the choice of prespecified h 0 and x 0 ; different choices may yield very different final estimates. In contrast, GPL does not require such prespecified choices and is generally more stable and efficient than CZ. Therefore, the simulation results clearly demonstrate the advantages of the GPL method and also support the theoretical findings of its efficiency.

In another simulation not reported here, we compared the GPL method with the regression spline smoother of Leblanc and Crowley The two procedures are comparable with GPL slightly better than the regression spline method in some cases. We now illustrate the proposed method with the well-known Stanford Heart Transplant data set. In this data, of the patients received heart transplants from October to February Patients alive beyond February were considered censored.

More details about this data and some related work in the literature can Crowley and Hu ; Miller and Halpern Previous analyses have included quadratic functions of age in years at transplantation. Instead of speculating which order of polynomials or other parametric functions would work for the data, a nonarametric link function on age, such as the one proposed in this article, is a good way to explore the data structure. Fan and Gijbels and Cai also reanalyzed the data using nonparametric regression.

Following the common literature, we limit our analysis to the patients who had completed tissue typing. Whereas further tests are needed to confirm these effects, a quadratic function of age is not suitable to model the age effects.

Instead, a piecewise linear function with breakpoints at age 20 and 40 the breakpoints could be estimated as well, but we have not done so and zero risks between the breakpoints might be a better alternative parametric model.

In many applications, there will be more than one covariate. The global approach developed in previous sections can be extended to many important cases with high-dimensional covariates, for instance, partially linear proportional hazards models, single index proportional hazards models and partially linear additive models and so on.

In this regard, the global approach can be viewed as a building block for general semiparametric or nonparametric approaches.

Penalised spline estimation for generalised partially linear single-index models

In statistics , a generalized additive model GAM is a generalized linear model in which the linear response variable depends linearly on unknown smooth functions of some predictor variables, and interest focuses on inference about these smooth functions. GAMs were originally developed by Trevor Hastie and Robert Tibshirani [1] to blend properties of generalized linear models with additive models. The model relates a univariate response variable, Y , to some predictor variables, x i. An exponential family distribution is specified for Y for example normal , binomial or Poisson distributions along with a link function g for example the identity or log functions relating the expected value of Y to the predictor variables via a structure such as. The functions f i may be functions with a specified parametric form for example a polynomial, or an un-penalized regression spline of a variable or may be specified non-parametrically, or semi-parametrically, simply as 'smooth functions', to be estimated by non-parametric means. So a typical GAM might use a scatterplot smoothing function, such as a locally weighted mean, for f 1 x 1 , and then use a factor model for f 2 x 2. This flexibility to allow non-parametric fits with relaxed assumptions on the actual relationship between response and predictor, provides the potential for better fits to data than purely parametric models, but arguably with some loss of interpretability.

Hastie • Tibshirani • Friedman. Second Edition Trevor Hastie would like to thank the statis- We have not attempted to write a comprehensive catalog of learning on high variance (Stone and Brooks, ; Frank and Friedman, ​). In.

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Generalised linear models are frequently used in modeling the relationship of the response variable from the general exponential family with a set of predictor variables, where a linear combination of predictors is linked to the mean of the response variable. We propose a penalised spline P-spline estimation for generalised partially linear single-index models, which extend the generalised linear models to include nonlinear effect for some predictors. We investigate the P-spline profile likelihood estimation using the readily available R package mgcv , leading to straightforward computation. Simulation studies are considered under various link functions. In addition, we examine different choices of smoothing parameters.

In C, you create structures and function to process data and handle algorithms you write. It covers the common algorithms, algorithmic paradigms, and data structures used to solve these problems. The course emphasizes the relationship between algorithms and programming, and introduces basic performance measures and analysis techniques for these problems. Offered by University of California San Diego. This specialization is a mix of theory and practice: you will learn algorithmic techniques for solving various computational problems and will implement about algorithmic coding problems in a programming language of your choice.

Hastie, T. Jacob Bien and Robert Tibshirani. Hierarchical Clustering with Prototypes via Minimax Linkage. Tibshirani, R.

Generalized additive model

An Introduction to Statistical Learning pp Cite as.

robert john tibshirani - Stanford University

Prediction and classification are two very active areas in modern data analysis. In this paper, prediction with nonlinear optimal scaling transformations of the variables is reviewed, and extended to the use of multiple additive components, much in the spirit of statistical learning techniques that are currently popular, among other areas, in data mining. This is a preview of subscription content, access via your institution. Rent this article via DeepDyve.

Home About Wiki Tools Contacts. Laboratory for Applied Spatial Analysis, SIUE Project based geoprocessing operations of vector and raster dataset in fields such as archeology, anthropology, geology and agriculture. The Laboratory for Landscape Characterization and Spatial Analysis is a research facility dedicated to the study of biophysical and human landscapes. Series, Springer. Randall Pearson, who is also a faculty member of the Geography Department. Sergii Yakunin.

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