Monotone B-splines Research Articles

Forecasting the future burden of primary total shoulder replacement in Australia

To forecast the number of primary total shoulder replacements (TSR) in Australia to the year 2035, and associated costs. De-identified TSR data for 2009-2019 were obtained from the Australian Orthopaedic Association National Joint Replacement Registry. Population data, including population projections to 2035, were obtained from the Australian Bureau of Statistics. Three forecasting scenarios were used: constant TSR rates from 2019 onwards (Scenario 1, conservative); continued growth in TSR rates using negative binomial regression (Scenario 2, exponential); and continued growth using negative binomial regression with monotone B-splines (Scenario 3, moderate). Healthcare costs were estimated using TSR projections and average procedure costs, inflated to 2035 Australian dollars. The use of TSR increased by 242% in Australia from 2009 to 2019 (from 1983 to 6789 procedures for people ≥40 years). Under Scenario 1, the incidence of TSR is conservatively projected to rise to 9676 procedures by 2035 (43% increase from 2019), at a cost of $AUD 312.6 million to the health system. Under Scenario 2, TSR incidence would increase to 45,295 procedures by 2035 (567% increase), costing $AUD 1.46 billion. Under Scenario 3, 28,257 TSR procedures are forecast in 2035 (316% increase) at a cost of $913 million. Recent growth in TSR likely relates to prosthesis improvements, greater surgeon proficiency, and expanded clinical indications. Under moderate and exponential scenarios that consider rising TSR rates and population projections, Australia would face three- to five-fold growth in procedures by 2035. This would have profound implications for the healthcare budget, clinical workforce, and infrastructure.

An efficient penalized estimation approach for semiparametric linear transformation models with interval-censored data.

We consider efficient estimation of flexible transformation models with interval-censored data. To reduce the dimension of semiparametric models, the unknown monotone function is approximated via a monotone B-spline. A penalization technique is used to provide computationally efficient estimation of all parameters. To accomplish model fitting and inference, an easy to implement nested iterative expectation-maximization (EM) algorithm is developed for estimation, and a simple variance-covariance estimation approach is proposed which makes large-sample inference for the regression parameters possible. Theoretically, we show that the estimator of the unknown monotone increasing function achieves the optimal rate of convergence, and the estimators of the regression parameters are asymptotically normal and efficient under the appropriate selection of the order of the smoothing parameter and the knots of the spline space. The proposed penalized procedure is assessed through extensive numerical experiments and implemented in R package PenIC. The proposed methodology is further illustrated via a signal tandmobiel study.

Projections of Primary and Revision Shoulder Arthroplasty until 2040: Facing a Massive Rise in Fracture-Related Procedures

Although the demand for shoulder arthroplasties has reached its highest number worldwide, there remains a lack of epidemiologic data regarding recent and future trends. In this study, data for all shoulder arthroplasties (hemiarthroplasty, reverse/anatomic shoulder arthroplasty) from the nationwide inpatient statistics of Germany (2010–2019) and population forecasts until 2040 were gathered. A Poisson and a negative binomial approach using monotone B-splines were modeled for all types of prostheses to project the annual number and incidence of primary and revision arthroplasty. Additionally, trends in main indicators were also gathered and expected changes were calculated. Overall, the number of primary shoulder replacements is set to increase significantly by 2040, reaching at least 37,000 (95% CI 32,000–44,000) procedures per year. This trend is mainly attributable to an about 10-fold increased use of fracture-related reverse shoulder arthroplasty in patients over 80 years of age, although the number of procedures in younger patients will also rise substantially. In contrast, hemiarthroplasties will significantly decrease. The number of revision procedures is projected to increase subsequently, although the revision burden is forecast to decline. Using these country-specific projection approaches, a massive increase of primary and revision shoulder arthroplasties is expected by 2040, mainly due to a rising number of fracture-related procedures. These growth rates are substantially higher than those from hip or knee arthroplasty. As these trends are similar in most Western countries, this draws attention to the international issue, of: if healthcare systems will be able to allocate human and financial resources adequately, and if future research and fracture-prevention programs may help to temper this rising burden in the upcoming decades.

Efficient estimation of a linear transformation model for current status data via penalized splines.

We propose a flexible and computationally efficient penalized estimation method for a semi-parametric linear transformation model with current status data. To facilitate model fitting, the unknown monotone function is approximated by monotone B-splines, and a computationally efficient hybrid algorithm involving the Fisher scoring algorithm and the isotonic regression is developed. A goodness-of-fit test and model diagnostics are also considered. The asymptotic properties of the penalized estimators are established, including the optimal rate of convergence for the function estimator and the semi-parametric efficiency for the regression parameter estimators. An extensive numerical experiment is conducted to evaluate the finite-sample properties of the penalized estimators, and the methodology is further illustrated with two real studies.

Analisis Regresi Kuantil B-Splines Monoton Naik pada Hubungan Rata-Rata Lama Sekolah dan Pengeluaran Rumah Tangga Per Kapita di Provinsi Yogyakarta

The multidimensional problem in various countries that is always become the government's attention is the problem of poverty, Indonesia is no exception. Poverty is often associated with the education sector due to the function of education as a driving force of the transformation of society to break the chain of poverty. The pattern of relations between poverty and the education sector can be seen from the relationship between the level of education (mean years of schooling) and poverty level (per capita household expenditure). DI Yogyakarta is still the province with the largest percentage of poverty on the Java island despite showing a downward trend since 2007. This study aims to look at the relationship between the level of education (mean years of schooling) and poverty level (per capita household expenditure) in DI Yogyakarta Province 2016. The model that is suitable for determining household characteristics is quantile regression with the Increased monotone B-Splines method that links the mean years of schooling and per capita household expenditure. Estimation results based on the quantile regression model with Increased monotone B-Splines method found that households with the lowest education level are said to be very poor households if monthly per capita expenditure is less than 322,205 rupiah and is said to be a poor household if monthly per capita expenditure is between 322,205 rupiah to 426,666 rupiah. Meanwhile, households with the highest level of education are said to be very poor households if monthly per capita expenditure is less than 3,410,965 rupiahs and is said to be a poor household if monthly per capita expenditure is between 3,410,965 rupiahs up to 4,676,718 rupiahs

A two stage $k$-monotone B-spline regression estimator: Uniform Lipschitz property and optimal convergence rate

This paper considers $k$-monotone estimation and the related asymptotic performance analysis over a suitable Hölder class for general $k$. A novel two stage $k$-monotone B-spline estimator is proposed: in the first stage, an unconstrained estimator with optimal asymptotic performance is considered; in the second stage, a $k$-monotone B-spline estimator is constructed (roughly) by projecting the unconstrained estimator onto a cone of $k$-monotone splines. To study the asymptotic performance of the second stage estimator under the sup-norm and other risks, a critical uniform Lipschitz property for the $k$-monotone B-spline estimator is established under the $\ell_{\infty }$-norm. This property uniformly bounds the Lipschitz constants associated with the mapping from a (weighted) first stage input vector to the B-spline coefficients of the second stage $k$-monotone estimator, independent of the sample size and the number of knots. This result is then exploited to analyze the second stage estimator performance and develop convergence rates under the sup-norm, pointwise, and $L_{p}$-norm (with $p\in [1,\infty )$) risks. By employing recent results in $k$-monotone estimation minimax lower bound theory, we show that these convergence rates are optimal.

Penalized estimation for proportional hazards models with current status data.

We provide a simple and practical, yet flexible, penalized estimation method for a Cox proportional hazards model with current status data. We approximate the baseline cumulative hazard function by monotone B-splines and use a hybrid approach based on the Fisher-scoring algorithm and the isotonic regression to compute the penalized estimates. We show that the penalized estimator of the nonparametric component achieves the optimal rate of convergence under some smooth conditions and that the estimators of the regression parameters are asymptotically normal and efficient. Moreover, a simple variance estimation method is considered for inference on the regression parameters. We perform 2 extensive Monte Carlo studies to evaluate the finite-sample performance of the penalized approach and compare it with the 3 competing R packages: C1.coxph, intcox, and ICsurv. A goodness-of-fit test and model diagnostics are also discussed. The methodology is illustrated with 2 real applications.

Efficient estimation of partially linear additive Cox model under monotonicity constraint

We provide a simple and practical, yet flexible spline estimation method for partially linear additive Cox model with monotonicity constraint. We propose to approximate the unknown monotone function by monotone B-splines, and employ a hybrid numerical approach based on the Newton–Raphson algorithm and the isotonic regression to compute the spline estimates. We show that the spline estimators of the nonparametric components achieve the optimal rate of convergence under the smooth condition, and that the estimators of the regression parameters are asymptotically normal and efficient. Moreover, a direct variance estimation method based on the least-squares estimation is proposed. The finite-sample performance of the spline estimates is evaluated by a Monte Carlo study. The methodology is illustrated with a synovial sarcoma dataset.

Modeling Regression Quantile Process Using Monotone B-Splines

ABSTRACTQuantile regression as an alternative to conditional mean regression (i.e., least-square regression) is widely used in many areas. It can be used to study the covariate effects on the entire response distribution by fitting quantile regression models at multiple different quantiles or even fitting the entire regression quantile process. However, estimating the regression quantile process is inherently difficult because the induced conditional quantile function needs to be monotone at all covariate values. In this article, we proposed a regression quantile process estimation method based on monotone B-splines. The proposed method can easily ensure the validity of the regression quantile process and offers a concise framework for variable selection and adaptive complexity control. We thoroughly investigated the properties of the proposed procedure, both theoretically and numerically. We also used a case study on wind power generation to demonstrate its use and effectiveness in real problems. Supplementary materials for this article are available online.

Monotonic Metamodels for Deterministic Computer Experiments

In deterministic computer experiments, it is often known that the output is a monotonic function of some of the inputs. In these cases, a monotonic metamodel will tend to give more accurate and interpretable predictions with less prediction uncertainty than a nonmonotonic metamodel. The widely used Gaussian process (GP) models are not monotonic. A recent article in Biometrika offers a modification that projects GP sample paths onto the cone of monotonic functions. However, their approach does not account for the fact that the GP model is more informative about the true function at locations near design points than at locations far away. Moreover, a grid-based method is used, which is memory intensive and gives predictions only at grid points. This article proposes the weighted projection approach that more effectively uses information in the GP model together with two computational implementations. The first is isotonic regression on a grid while the second is projection onto a cone of monotone splines, which alleviates problems faced by a grid-based approach. Simulations show that the monotone B-spline metamodel gives particularly good results. Supplementary materials for this article are available online.

Semiparametric Single-Index Model for Estimating Optimal Individualized Treatment Strategy.

Different from the standard treatment discovery framework which is used for finding single treatments for a homogenous group of patients, personalized medicine involves finding therapies that are tailored to each individual in a heterogeneous group. In this paper, we propose a new semiparametric additive single-index model for estimating individualized treatment strategy. The model assumes a flexible and nonparametric link function for the interaction between treatment and predictive covariates. We estimate the rule via monotone B-splines and establish the asymptotic properties of the estimators. Both simulations and an real data application demonstrate that the proposed method has a competitive performance.

Signed-rank analysis of a partial linear model with B-splines estimated monotone non parametric function

ABSTRACTFor a partially linear regression model, we propose a signed-rank estimation method. The proposed estimator for the linear unknown parameter vector is shown to have -consistency, while monotone B-splines are used to estimate the unknown monotone non parametric function. Finite-sample simulations are carried out to evaluate the performance of the proposed estimation method, and a practical application on the study of air pollution data is given.

Spline-based sieve estimation in monotone constrained varying-coefficient partially linear EV model

This paper studies a monotone constrained varying-coefficient partially linear model with errors in variables using a monotone B-spline approach based on adjusted least squares. The proposed estimator is consistent, asymptotically normal, and performs well in numerical studies.

Parameter estimation of partial linear model under monotonicity constraints with censored data

We propose a weighted least square method for estimation in the partial linear model with monotonicity constraints and right-censored data. This method uses the Kaplan–Meier weights to account for censoring and monotone B-splines to approximate the unknown monotone function. We show that the proposed estimator of regression coefficients is root-n consistent and asymptotically normal under appropriate assumptions. One advantage is that our method can be easily computed using existing software. A simulation study is conducted to evaluate the finite sample performance of the proposed method.

Monotone B-Spline Smoothing for a Generalized Linear Model Response

Various methods have been proposed for smoothing under the monotonicity constraint. We review the literature and implement an approach of monotone smoothing with B-splines for a generalized linear model response. The approach is expressed as a quadratic programming problem and is easily solved using the statistical software R. In a simulation study, we find that the approach performs better than other approaches with much faster computation time. The approach can also be used for smoothing under other shape constraints or mixed constraints. Supplementary materials of the appendices and R code to implement the developed approach is available online.

Spline estimation of generalised monotonic regression

We develop a simple and practical, yet flexible spline estimation method for semiparametric generalised linear models with monotonicity constraints. We propose to approximate the unknown monotone function by monotone B-splines, and employ generalised Rosen algorithm to compute the estimates. We show that the spline estimate of the nonparametric component achieves the optimal rate of convergence under the smooth condition, and that the estimates of regression parameters are asymptotically normal and efficient. The spline-based semiparametric likelihood ratio test (LRT) is also established. Moreover, a direct variance estimation method based on least-squares estimation is proposed. The finite sample performance of the spline estimates is evaluated by a Monte Carlo study. The methodology is illustrated on an air pollution study.

Formula omitted]-estimation for the partially linear regression model under monotonic constraints

In this paper, we study M-estimation for the partially linear model under monotonic constraints. We use monotone B-splines to approximate the monotone nonparametric function. We show the large sample properties of the resulting estimators. The proposed estimator of parameter part is root-n consistent, and asymptotically normal and the estimator for the nonparametric component achieves the optimal convergence rate. A simulation study is conducted to evaluate the finite sample performance of the method. The proposed procedure is illustrated by an air pollution study.

A Spline-Based Semiparametric Maximum Likelihood Estimation Method for the Cox Model with Interval-Censored Data

Abstract. We propose a spline-based semiparametric maximum likelihood approach to analysing the Cox model with interval-censored data. With this approach, the baseline cumulative hazard function is approximated by a monotone B-spline function. We extend the generalized Rosen algorithm to compute the maximum likelihood estimate. We show that the estimator of the regression parameter is asymptotically normal and semiparametrically efficient, although the estimator of the baseline cumulative hazard function converges at a rate slower than root-n. We also develop an easy-to-implement method for consistently estimating the standard error of the estimated regression parameter, which facilitates the proposed inference procedure for the Cox model with interval-censored data. The proposed method is evaluated by simulation studies regarding its finite sample performance and is illustrated using data from a breast cosmesis study.

Semiparametric Estimation Methods for Panel Count Data Using Monotone B-Splines

We study semiparametric likelihood-based methods for panel count data with proportional mean model E[ℕ(t)|Z]=Λ0(t)exp(β0TZ), where Z is a vector of covariates and Λ0(t) is the baseline mean function. We propose to estimate Λ0(t) and β0 jointly with Λ0(t) approximated by monotone B-splines and to compute the estimators using generalized Rosen algorithm proposed by Jamshidian (2004). We show that the proposed spline-based likelihood estimators of Λ0(t) are consistent with a possibly better than n1/3 convergence rate if Λ0(t) is sufficiently smooth. The normality of the estimators of β0 is also established. Comparisons between the proposed estimators and their alternatives studied in Wellner and Zhang (2007) are made through simulations studies, regarding their finite sample performance and computational complexity. A real example from a bladder tumor clinical trial is used to illustrate the methods.

Monotone fitting for developmental variables

In order to study developmental variables, for example, neuromotor development of children and adolescents, monotone fitting is typically needed. Most methods, to estimate a monotone regression function non-parametrically, however, are not straightforward to implement, a difficult issue being the choice of smoothing parameters. In this paper, a convenient implementation of the monotone B-spline estimates of Ramsay [Monotone regression splines in action (with discussion), Stat. Sci. 3 (1988), pp. 425–461] and Kelly and Rice [Montone smoothing with application to dose-response curves and the assessment of synergism, Biometrics 46 (1990), pp. 1071–1085] is proposed and applied to neuromotor data. Knots are selected adaptively using ideas found in Friedman and Silverman [Flexible parsimonous smoothing and additive modelling (with discussion), Technometrics 31 (1989), pp. 3–39] yielding a flexible algorithm to automatically and accurately estimate a monotone regression function. Using splines also simultaneously allows to include other aspects in the estimation problem, such as modeling a constant difference between two groups or a known jump in the regression function. Finally, an estimate which is not only monotone but also has a ‘levelling-off’ (i.e. becomes constant after some point) is derived. This is useful when the developmental variable is known to attain a maximum/minimum within the interval of observation.