Pointwise Confidence Intervals Research Articles

Adult genitourinary sarcomas: A single-centre experience over 25 years.

e23532 Background: Genitourinary sarcomas are a rare entity. This study is a retrospective review of the patient characteristics, types of adult GU sarcomas, treatments and outcomes in a single institution over the past 25 years. Methods: Survival probabilities were estimated and plotted using the Kaplan-Meier method. Estimates along with 95% pointwise confidence intervals were reported. Recurrence-free survival (RFS) is defined as time from when the patient was disease-free to recurrence. Otherwise, patients were censored at last disease evaluation. OS was defined as time from when the patient was disease-free to death due to any cause. Patients still alive were censored at last known alive. Results: 29 patients presented with various types of GU sarcomas, with the majority being paratesticular tumors (n = 22, 75.9%). Others included bladder leiomyosarcomas (n = 5), prostatic leiomyosarcoma (n = 1), renal leiomyosarcoma (n = 1). Median age for diagnosis was 51 years (range: 21-79) for leiomyosarcoma vs 61 (range: 36-89) for liposarcoma. Among the paratesticular tumors majority were (n = 16) were liposarcoma, remaining 5 being leiomyosarcoma and 1 being rhabdomyosarcoma. Majority tumors were stage III and above (n = 16, 55.17%), high and intermediate grade (65.51%). All tumors were resected ± adjuvant therapy (n = 4). Leiomyosarcomas had a higher stage (66%) and grade (83%) at diagnosis, recurrence rate was also higher at 37.5% (vs 6.6% for liposarcoma) along with a higher mortality (41.6%). Commonest site of recurrence for leiomyosarcoma was the lungs (n = 3, 100%). Treatment for recurrence commonly involved radiotherapy, pembrolizumab etc. Paratesticular liposarcomas were found to be associated with a lipomatosis syndrome with n = 5 (31.25%) having lipomas elsewhere in the body and n = 2 (12.5%) patients having had previously resected spermatic cord lipoma. Paratesticular sarcomas often required (n = 15, 68%) revision surgery due to being initially diagnosed as a hernia, cord lipoma or hydrocele and requiring high ligation of the cord with hemiscrotectomy and radical orchiectomy ± LND on revision surgery, which often demonstrated a focus of residual tumor in liposarcomas (n = 5, 31%). N = 4 patients (25%) had positive margins in spite of revision surgery. Majority of these were MDM2+ positive on FISH (n = 9, 56%). Overall, GU sarcomas had an RFS of 63% (53-94%) and OS of 82% (53-94%) at 36 months since NED was achieved. The median length of follow-up was 19.9 months for leiomyosarcoma, 33.8 months for liposarcoma. Conclusions: Leiomyosarcomas in adults have worse outcomes compared to other GU sarcomas especially if it originates in the bladder. [Table: see text]

Resampling-based confidence intervals and bands for the average treatment effect in observational studies with competing risks

The g-formula can be used to estimate the treatment effect while accounting for confounding bias in observational studies. With regard to time-to-event endpoints, possibly subject to competing risks, the construction of valid pointwise confidence intervals and time-simultaneous confidence bands for the causal risk difference is complicated, however. A convenient solution is to approximate the asymptotic distribution of the corresponding stochastic process by means of resampling approaches. In this paper, we consider three different resampling methods, namely the classical nonparametric bootstrap, the influence function equipped with a resampling approach as well as a martingale-based bootstrap version, the so-called wild bootstrap. For the latter, three sub-versions based on differing distributions of the underlying random multipliers are examined. We set up a simulation study to compare the accuracy of the different techniques, which reveals that the wild bootstrap should in general be preferred if the sample size is moderate and sufficient data on the event of interest have been accrued. For illustration, the resampling methods are further applied to data on the long-term survival in patients with early-stage Hodgkin’s disease.

Evaluating a novel 24-hour rest/activity rhythm marker of preclinical β-amyloid deposition.

To compare sleep and 24-hour rest/activity rhythms (RARs) between cognitively normal older adults who are β-amyloid-positive (Aβ+) or Aβ- and replicate a novel time-of-day-specific difference between these groups identified in a previous exploratory study. We studied 82 cognitively normal participants from the Baltimore Longitudinal Study of Aging (aged 75.7 ± 8.5 years, 55% female, 76% white) with wrist actigraphy data and Aβ+ versus Aβ- status measured by [11C] Pittsburgh compound B positron emission tomography. RARs were calculated using epoch-level activity count data from actigraphy. We used novel, data-driven function-on-scalar regression analyses and standard RAR metrics to cross-sectionally compare RARs between 25 Aβ+ and 57 Aβ- participants. Compared to Aβ- participants, Aβ+ participants had higher mean activity from 1:00 p.m. to 3:30 p.m. when using less conservative pointwise confidence intervals (CIs) and from 1:30 p.m. to 2:30 p.m. using more conservative, simultaneous CIs. Furthermore, Aβ+ participants had higher day-to-day variability in activity from 9:00 a.m. to 11:30 a.m. and lower variability from 1:30 p.m. to 4:00 p.m. and 7:30 p.m. to 10:30 p.m. according to pointwise CIs, and lower variability from 8:30 p.m. to 10:00 p.m. using simultaneous CIs. There were no Aβ-related differences in standard sleep or RAR metrics. Findings suggest Aβ+ older adults have higher, more stable day-to-day afternoon/evening activity than Aβ- older adults, potentially reflecting circadian dysfunction. Studies are needed to replicate our findings and determine whether these or other time-of-day-specific RAR features have utility as markers of preclinical Aβ deposition and if they predict clinical dementia and agitation in the afternoon/evening (i.e. "sundowning").

Longitudinal varying coefficient single-index model with censored covariates.

It is of interest to health policy research to estimate the population-averaged longitudinal medical cost trajectory from initial cancer diagnosis to death, and understand how the trajectory curve is affected by patient characteristics. This research question leads to a number of statistical challenges because the longitudinal cost data are often non-normally distributed with skewness, zero-inflation, and heteroscedasticity. The trajectory is nonlinear, and its length and shape depend on survival, which are subject to censoring. Modeling the association between multiple patient characteristics and nonlinear cost trajectory curves of varying lengths should take into consideration parsimony, flexibility, and interpretation. We propose a novel longitudinal varying coefficient single-index model. Multiple patient characteristics are summarized in a single-index, representing a patient's overall propensity for healthcare use. The effects of this index on various segments of the cost trajectory depend on both time and survival, which is flexibly modeled by a bivariate varying coefficient function. The model is estimated by generalized estimating equationswith an extended marginal mean structure to accommodate censored survival time as a covariate. We established the pointwise confidence interval of the varying coefficient and a test for the covariate effect. The numerical performance was extensively studied in simulations. We applied the proposed methodology to medical cost data of prostate cancer patients from the Surveillance, Epidemiology, and End Results-Medicare-Linked Database.

Empirical likelihood and estimation in a partially linear varying coefficient model with right censored data

In this paper, we study the empirical likelihood and estimation of parameters of interest in a partially linear varying coefficient model with right censored data. Two cases are considered: censoring is independent of the covariates, and censoring depends on the covariates. The bias-corrected empirical log-likelihood ratios for the regression parameters are presented, Wilks' theorem is proved, and thus the confidence regions of the regression parameters are constructed. The estimators of parametric and nonparametric components are constructed, their asymptotic distributions are obtained, and the consistent estimators of the asymptotic variances are also given. Furthermore, the partial profile empirical log-likelihood ratios for each component of the regression parameters and the coefficient functions are constructed, and show they are asymptotically chi-squared. The obtained results can be directly used to construct the confidence regions/intervals for the regression parameters and the pointwise confidence intervals for the coefficient functions. Our approach is to directly calibrate the empirical log-likelihood ratio so that the resulting ratio is asymptotically chi-squared, undersmoothing of the coefficient functions is avoided, and existing data-driven methods can be effectively used to select the optimal bandwidth. Simulation studies compare the empirical likelihood method with one based on normal approximation and perform real data analysis.

Radiomics Model for Predicting Pulmonary Metastasis in Sarcoma Patients

The lung is the most common site of metastasis in localized extremity soft tissue sarcomas (STS). Indeterminant lung nodules leads to challenges for both patients and providers. As such, accuracy in the diagnosis of pulmonary metastasis (PM) is important for clinical decision making. The purpose of this is study is to develop and evaluate a CT-based radiomic model to predict PM in sarcoma patients with indeterminate pulmonary nodules. Fifty-seven stage I-III sarcoma patients with indeterminate lung nodules on unenhanced CT that were approved for this study by our institution's internal review board (IRB). Twenty-two developed and categorized as PM and 35 without PM. A total number of 322 indeterminate nodules were investigated, where 165 & 157 nodules were from patients with and without PM, respectively. A total of 96 radiomic features (RF) (including first order statistics, 2d shape features, gray level co-occurrence, gray level size zone, gray level run length, neighboring gray tone difference & gray level dependence matrices) were taken from unenhanced CT images of each patient. We included 2D shape features since the size and shape of indeterminate lung nodules may have some correlation with PM. Spearman correlation were used to rule out redundant features (r2>0.9). T-test statistics was used to determine feature with significant differences that can differentiate responders with and without PM. Support Vector Machine (SVM) model was fitted to combinations of groups of 2-3 radiomic features. The AUC of the ROC curve was used to evaluate the model performance. Pointwise confidence intervals (CI) for AUC were calculated using 100 bootstrap replicates. Four features were found significant after Spearman correlation and T-test statistical testing: 1. Maximum (MAX); 2. Minor Axis Length (MAL); 3. Gray Level Non-uniformity (GLNU); and 4. Interquartile Range (IR). The best performing 2-feature radiomic SVM model had an AUC of 0.69 (95% CI 0.61,0.74) for the MAX and GLNU features. The best performing 3-feature radiomic SVM model had an AUC of 0.74 (95% CI 0.70,0.80) for the MAX, MAL and IR features. CT-based radiomics model has the potential to be used to predict PM in sarcoma patients presenting with indeterminate lung nodules. Additional research will focus on investigating derived parameters for existing RF, such as the ratio of the max RF to mean RF, to see if they improve the predictive performance of the model, as well as testing the models on external patient cohort. These radiomic models could be useful tools in guiding PM screening and providing a patient specific monitoring of sarcoma patients.

CANKADO PRO-React eHealth support in patients with HR+ HER2- metastatic breast cancer receiving palbociclib and endocrine therapy and the affect on time to deterioration of quality of life: Primary outcome analysis of the multicenter randomized PreCycle trial.

1008 Background: The multicenter, randomized phase IV Intergroup PreCycle trial (NCT03220178) evaluated the impact of CANKADO-based ePRO assessment on quality of life (QoL) in HR+/HER2- locally advanced or metastatic breast cancer (MBC) patients (pts) treated with palbociclib (P) and an aromatase inhibitor or P+fulvestrant. CANKADO PRO-React, an EU- registered medical device, is an autonomous application reacting to changes in pt self- reported QoL. Methods: Between 2017 and 2021, 499 pts (median age 59y) from 71 centers were randomized (2:1, stratified by therapy line) between an active (A; CANKADO PRO-React) or an inactive arm (B; inform; limited CANKADO functionality). 412 pts (271 A; 141 B) were available for analysis of the primary endpoint, i.e. time to deterioration (TTD) of QoL (10-point drop on FACT-G), using an Aalen-Johansen estimator for cumulative incidence function of TTD DQoL with 95% pointwise confidence intervals (CI). Secondary endpoints included PFS, OS, and DQoL (QoL deterioration). Results: In all pts (ITT-ePRO), a significantly decreased risk (HR 0.6982) for the CANKADO active arm A with 95% CI [0.5059, 0.9635] regarding DQoL was observed (p=0.03). In 1stL pts (n=295), the decreased risk for arm A was 0.7162 (0.4839, 1.06; p=0.09), in 2ndL pts (n=117) 0.6614 (0.3744, 1.168; p=0.2). Absolute pt numbers declined in later visits; until about visit 30, FACT- G completion rates were 80% and higher; mean total scores were similar between arms; the change from baseline showed a linear decline and a nearly constant offset in favor of arm A, particularly in 1st L. No significant differences in clinical outcome were observed between arms: Median PFS (ITT population) was 21.4 (95% CI 19.4-23.7) (A) and 18.7 (15.1.-23.5) months (B); median OS was not reached (A) and 42.6 months (B). Conclusions: PreCycle is the first multicenter randomized eHealth trial demonstrating a significant benefit of an interactive autonomous patient empowerment application on QoL in MBC pts receiving oral tumor therapy. Clinical trial information: NCT03220178 .

THE APPLIED OF FUNCTIONAL DATA ANALYSIS TO COMPARISON 100 TIMES SIMULATED MONTHLY RAINFALL USING SOME TWO PARAMETERS DISTRIBUTION

Three probability models of mothly rainfall such as Gamma, Weibull and Log Normal distribution are evaluated in terms of their ability to reproduce the mean statistics derived from 100 times simulation of monthly rainfall in the Pekanbaru City, Indonesia. One of the important studies is to investigate and understand the simulate mean monthly rainfall patterns that occur throughout the year. To identify the pattern, it requires a rainfall curve to represent monthly observation of rainfall received during the year. Functional data analysis (FDA) methods are capable to convert discrete data into a function that can represent the rainfall curve and as a result, try to describe the hidden patterns of the rainfall. This study is focused on investigating 100 curve average monthly rainfall simulatated by three different quantile functions using the FDA. The mean and standard deviation of FDA for average monthly precipitation are obtained. Through these two statistics a the confidence interval curves of the mean curve are presented represent 95% pointwise confidence intervals. In this study, most of the monthly average rainfall from the actual data were around the FDA mean and the monthly average rainfall was within the FDA confidence interval. In this study, 100 times monthly rainfall simulations using the quantile function of the gamma and log normal distributions found that the mean FDA can capture most of the mean monthly rainfall from historical data, and within the FDA interval. The contradictory results shown by the monthly rainfall simulations using the Weibull distribution, most of the monthly average rainfall historical data cannot be captured by the FDA mean and are outside the FDA confidence interval. Based on the Mean Absolute Error (MAE) value of the average monthly rainfall of historical data and the monthly average of the FDA, it can be concluded that the gamma distribution can produce simulated rain better than the log normal distribution.

Significantly longer time to deterioration of quality of life due to CANKADO PRO-React eHealth support in HR+ HER2− metastatic breast cancer patients receiving palbociclib and endocrine therapy: primary outcome analysis of the multicenter randomized AGO-B WSG PreCycle trial

The multicenter, randomized, phase IV, intergroup AGO-B WSG PreCycle trial (NCT03220178) evaluated the impact of CANKADO-based electronic patient-reported outcome (ePRO) assessment on quality of life (QoL) in hormone receptor-positive, human epidermal growth factor receptor 2-negative locally advanced or metastatic breast cancer (MBC) patients receiving palbociclib and an aromatase inhibitor or palbociclib+ fulvestrant. CANKADO PRO-React, a European Union-registered medical device, is an interactive autonomous application reacting to patient self-reported observations. Between 2017 and 2021, 499 patients (median age 59 years) from 71 centers were randomized (2 : 1, stratified by therapy line) between an active version of CANKADO PRO-React (CANKADO-active arm) and a version with limited functionality (CANKADO-inform arm). A total of 412 patients (271 CANKADO-active; 141 CANKADO-inform) were available for analysis of the primary endpoint, time to deterioration (TTD) of QoL [10-point drop on the Functional Assessment of Cancer Therapy-General (FACT-G) score], using an Aalen-Johansen estimator for cumulative incidence function of TTD DQoL (QoL deterioration) with 95% pointwise confidence intervals (CIs). Secondary endpoints included progression-free survival (PFS), overall survival (OS), and DQoL. In all patients [intention-to-treat (ITT)-ePRO], cumulative incidence of DQoL was significantly more favorable (lower) in the CANKADO-active arm (hazard ratio 0.698, 95% CI 0.506-0.963). Among first-line patients (n= 295), the corresponding hazard ratio was 0.716 (0.484-1.060; P= 0.09), and in second-line patients (n= 117) it was 0.661 (0.374-1.168; P= 0.2). Absolute patient numbers declined in later visits; FACT-G completion rates were 80% and higher until about visit 30. Mean FACT-G scores showed a steady decline from baseline and an offset in favor of CANKADO-active. No significant differences in clinical outcome were observed between arms: median PFS (ITT population) was 21.4 (95% CI 19.4-23.7) (CANKADO-active) and 18.7 (15.1-23.5) months (CANKADO-inform); median OS was not reached (CANKADO-active) and 42.6 months (CANKADO-inform). PreCycle is the first multicenter randomized eHealth trial demonstrating a significant benefit for MBC patients receiving oral tumor therapy when using an interactive autonomous patient empowerment application.

Additive hazards model with time-varying coefficients and imaging predictors.

Conventional hazard regression analyses frequently assume constant regression coefficients and scalar covariates. However, some covariate effects may vary with time. Moreover, medical imaging has become an increasingly important tool in screening, diagnosis, and prognosis of various diseases, given its information visualization and quantitative assessment. This study considers an additive hazards model with time-varying coefficients and imaging predictors to examine the dynamic effects of potential scalar and imaging risk factors for the failure of interest. We develop a two-stage approach that comprises the high-dimensional functional principal component analysis technique in the first stage and the counting process-based estimating equation approach in the second stage. In addition, we construct the pointwise confidence intervals for the proposed estimators and provide a significance test for the effects of scalar and imaging covariates. Simulation studies demonstrate the satisfactory performance of the proposed method. An application to the Alzheimer's disease neuroimaging initiative study further illustrates the utility of the methodology.

A Semiparametric Approach to Model-Based Sensitivity Analysis in Observational Studies.

When drawing causal inference from observational data, there is almost always concern about unmeasured confounding. One way to tackle this is to conduct a sensitivity analysis. One widely-used sensitivity analysis framework hypothesizes the existence of a scalar unmeasured confounder U and asks how the causal conclusion would change were U measured and included in the primary analysis. Work along this line often makes various parametric assumptions on U, for the sake of mathematical and computational convenience. In this article, we further this line of research by developing a valid sensitivity analysis that leaves the distribution of U unrestricted. Compared to many existing methods in the literature, our method allows for a larger and more flexible family of models, mitigates observable implications (Franks et al., 2019), and works seamlessly with any primary analysis that models the outcome regression parametrically. We construct both pointwise confidence intervals and confidence bands that are uniformly valid over a given sensitivity parameter space, thus formally accounting for unknown sensitivity parameters. We apply our proposed method on an influential yet controversial study of the causal relationship between war experiences and political activeness using observational data from Uganda.

Efficient Estimation of the Marginal Mean of Recurrent Events

AbstractRecurrent events are often encountered in clinical and epidemiological studies where a terminal event is also observed. With recurrent events data it is of great interest to estimate the marginal mean of the cumulative number of recurrent events experienced prior to the terminal event. The standard nonparametric estimator was suggested in Cook and Lawless and further developed in Ghosh and Lin. We here investigate the efficiency of this estimator that, surprisingly, has not been studied before. We rewrite the standard estimator as an inverse probability of censoring weighted estimator. From this representation we derive an efficient augmented estimator using efficient estimation theory for right-censored data. We show that the standard estimator is efficient in settings with no heterogeneity. In other settings with different sources of heterogeneity, we show theoretically and by simulations that the efficiency can be greatly improved when an efficient augmented estimator based on dynamic predictions is employed, at no extra cost to robustness. The estimators are applied and compared to study the mean number of catheter-related bloodstream infections in heterogeneous patients with chronic intestinal failure who can possibly die, and the efficiency gain is highlighted in the resulting point-wise confidence intervals.

Shape-constrained estimation in functional regression with Bernstein polynomials

Shape restrictions on functional regression coefficients such as non-negativity, monotonicity, convexity or concavity are often available in the form of a prior knowledge or required to maintain a structural consistency in functional regression models. A new estimation method is developed in shape-constrained functional regression models using Bernstein polynomials. Specifically, estimation approaches from nonparametric regression are extended to functional data, properly accounting for shape-constraints in a large class of functional regression models such as scalar-on-function regression (SOFR), function-on-scalar regression (FOSR), and function-on-function regression (FOFR). Theoretical results establish the asymptotic consistency of the constrained estimators under standard regularity conditions. A projection based approach provides point-wise asymptotic confidence intervals for the constrained estimators. A bootstrap test is developed facilitating testing of the shape constraints. Numerical analysis using simulations illustrates improvement in efficiency of the estimators from the use of the proposed method under shape constraints. Two applications include i) modeling a drug effect in a mental health study via shape-restricted FOSR and ii) modeling subject-specific quantile functions of accelerometry-estimated physical activity in the Baltimore Longitudinal Study of Aging (BLSA) as outcomes via shape-restricted quantile-function on scalar regression (QFOSR). R software implementation and illustration of the proposed estimation method and the test is provided.

Confidence bands and hypothesis tests for hit enrichment curves

In virtual screening for drug discovery, hit enrichment curves are widely used to assess the performance of ranking algorithms with regard to their ability to identify early enrichment. Unfortunately, researchers almost never consider the uncertainty associated with estimating such curves before declaring differences between performance of competing algorithms. Uncertainty is often large because the testing fractions of interest to researchers are small. Appropriate inference is complicated by two sources of correlation that are often overlooked: correlation across different testing fractions within a single algorithm, and correlation between competing algorithms. Additionally, researchers are often interested in making comparisons along the entire curve, not only at a few testing fractions. We develop inferential procedures to address both the needs of those interested in a few testing fractions, as well as those interested in the entire curve. For the former, four hypothesis testing and (pointwise) confidence intervals are investigated, and a newly developed EmProc approach is found to be most effective. For inference along entire curves, EmProc-based confidence bands are recommended for simultaneous coverage and minimal width. While we focus on the hit enrichment curve, this work is also appropriate for lift curves that are used throughout the machine learning community. Our inferential procedures trivially extend to enrichment factors, as well.

Modelling the driving speed on expressway ramps based on floating car data

Driving speed is one of the key concepts and risk factors in transportation research. The insights into operational and safety aspects of driving can be provided by floating car data (FCD), collecting information about speed, position and time by vehicles themselves. FCD are often recorded at high frequency, representing a continuous phenomenon. As such, convenient approach can be functional data analysis (FDA). Speed trajectories are investigated, identifying sections with rapid changes of speed and sections where drives in central and auxiliary lanes cannot be distinguished in terms of their speed. The effect of curvature and auxiliary lanes on driving speed was shown by functional regression models. Significant influence of road shape on speed was found for ramps with complex shape. The accuracy of regression models was assessed by RMSE, NRMSE, and precision of estimators by point-wise confidence intervals. The analysis was performed on an expressway interchange in Brno, Czech Republic.

Time-varying coefficient additive hazards model with latent variables.

This study considers a time-varying coefficient additive hazards model with latent variables to examine potential observed and latent risk factors for survival of interest. The model consists of two parts: confirmatory factor analysis to measure each latent factor through multiple observable variables and a varying coefficient additive hazards model to examine the time-varying effects of the observed and latent risk factors on the hazard function. A hybrid estimation procedure that combines the expectation-maximum algorithm and corrected estimating equation method is developed to estimate the unknown parameters and nonparametric cumulative hazard functions. The consistency and asymptotic normality of the proposed estimators are established, and the pointwise confidence intervals and general confidence bands of the nonparametric functions are constructed accordingly. A satisfactory performance of the proposed method is demonstrated through simulation studies. An application to a study of chronic kidney disease for Chinese type 2 diabetes patients is presented.

Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards

It is often important to account for the effects of a competing risk when estimating the risk of a particular event of interest by estimating its absolute risk. Available methodology for interval estimation of the absolute risk using the proportional regression of cause-specific hazards (CSH) has been limited to situations with time-invariant covariates and a single random censoring mechanism, without accommodation of cohort sampling study designs. Here we derive asymptotic pointwise confidence intervals in closed form for the absolute risk of an event at a specified time (the value of the cumulative incidence function) in the presence of competing risks using proportional CSH regression, accommodating external time-dependent covariates, cohort sampling study designs and multiple censoring mechanisms. Different covariates may be used for the event of interest and the various competing risks. Consistent with the definition of absolute risk, the CSH method produces absolute risk estimates that are less than or approximately equal to corresponding “conditional” risk estimates that do not account for competing risks. An example shows that this property is not necessarily shared by methods based on subdistribution hazard regression. Simulation studies indicate that the CSH method confidence intervals computed on the log cumulative hazard or the risk scale have coverage probabilities that approximate the nominal level for small and moderate samples, provided that the number of events per covariate is at least 10 and, when using cohort sampling, the ratio of patients without events to patients with events is at least 2:1.

Estimation and inference for partial linear regression surfaces using monotone warped-plane splines

Methods are proposed for spline estimation of monotone regression surfaces, without additivity assumptions and allowing for linear covariate effects. The surfaces are estimated by a continuous piece-wise warped-plane spline with linear inequality constraints. Surface and covariate effects are estimated simultaneously with cone projection, leading to useful inference methods. For surfaces in two predictors, a cube-root convergence rate is attained, and conditions for square-root convergence of the linear terms are derived. Point-wise confidence intervals for the surfaces incorporate mixtures of covariance matrices, with the mixing parameters estimation by simulations. The methods are implemented in the R package cgam.

TWISTER PLOTS FOR TIME-TO-EVENT STUDIES.

Results of randomized trials and observational studies can be difficult to communicate. Results are often presented as risk or survival functions stratified by the treatment or exposure (1, 2). However, a contrast between the stratified risk functions is often of primary interest. Here we propose a “twister” plot to visualize contrasts in risk over the duration of a study. The twister plot is a −90-degree rotation of a typical contrast measure plot (e.g., Figure 3 in Cole et al. (3)), whereby the contrast measure is instead on the abscissa (x-axis) and time on the ordinate (y-axis). Pointwise confidence intervals are similarly added as a shaded region that typically widens as follow-up duration increases, giving twister plots their characteristic shape that resembles their namesake. To ease application, we provide SAS, R, and Python code on GitHub (4). In Figure 1A...

Confidence intervals for multiple isotonic regression and other monotone models

We consider the problem of constructing pointwise confidence intervals in the multiple isotonic regression model. Recently, Han and Zhang (2020) obtained a pointwise limit distribution theory for the so-called block max–min and min–max estimators (Fokianos, Leucht and Neumann (2020); Deng and Zhang (2020)) in this model, but inference remains a difficult problem due to the nuisance parameter in the limit distribution that involves multiple unknown partial derivatives of the true regression function. In this paper, we show that this difficult nuisance parameter can be effectively eliminated by taking advantage of information beyond point estimates in the block max–min and min–max estimators. Formally, let uˆ(x0) (resp. vˆ(x0)) be the maximizing lower-left (resp. minimizing upper-right) vertex in the block max–min (resp. min–max) estimator, and fˆn be the average of the block max–min and min–max estimators. If all (first-order) partial derivatives of f0 are nonvanishing at x0, then the following pivotal limit distribution theory holds: nuˆ,vˆ(x0)(fˆn(x0)−f0(x0))⇝σ·L1d. Here nuˆ,vˆ(x0) is the number of design points in the block [uˆ(x0),vˆ(x0)], σ is the standard deviation of the errors, and L1d is a universal limit distribution free of nuisance parameters. This immediately yields confidence intervals for f0(x0) with asymptotically exact confidence level and oracle length. Notably, the construction of the confidence intervals, even new in the univariate setting, requires no more efforts than performing an isotonic regression once using the block max–min and min–max estimators, and can be easily adapted to other common monotone models including, for example, (i) monotone density estimation, (ii) interval censoring model with current status data, (iii) counting process model with panel count data, and (iv) generalized linear models. Extensive simulations are carried out to support our theory.