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Head Circumference Growth. Observed head circumference and age for 7040 boys with estimated quantile curves for τ = 0.04, 0.02, 0.1, 0.25, 0.5, 0.75, 0.9, 0.98, 0.996.
Source publication
We propose and study properties of maximum likelihood estimators in the class
of conditional transformation models. Based on a suitable explicit
parameterisation of the unconditional or conditional transformation function,
we establish a cascade of increasingly complex transformation models that can
be estimated, compared and analysed in the maximu...
Context in source publication
Context 1
... fitted the same growth curves by the conditional transformation model (Φ, (a Bs,3 (HC) ⊗ b Bs,3 (age 1/3 ) ) , ϑ) by maximisation of the approx- imate log-likelihood under 3 × 4 linear constraints. Figure 3 shows the data overlaid with quantile curves obtained via inversion of the estimated conditional distributions. The figure Figure 1: Old Faithful Geyser. ...
Citations
... additive models for location, scale, and shape (GAMLSS, Rigby & Stasinopoulos, 2005) and conditional transformation models (CTMs, Hothorn et al., 2014Hothorn et al., , 2018. In GAMLSS, we assume a parametric distribution for the response's conditional distribution given the covariates, and relate one or more of this distribution's parameters to covariates. ...
... The latter also directly solves the problem of overdispersion. Count data PTMs relate to conditional transformation models (CTMs) for count data (Carlan & Kneib, 2022;Siegfried & Hothorn, 2020) in a similar way as continuous PTMs relate to continuous CTMs and transformation additive models (Hothorn et al., 2018;Siegfried et al., 2023): the location and scale parameters of count PTMs act directly on the probability mass function at the response level. This means, changes to these parameters can be thought of as horizontally moving the probability mass to the left or right, and lowering or heightening its variance, respectively. ...
Penalized transformation models (PTMs) are a novel form of location-scale regression. In PTMs, the shape of the response's conditional distribution is estimated directly from the data, and structured additive predictors are placed on its location and scale. The core of the model is a monotonically increasing transformation function that relates the response distribution to a reference distribution. The transformation function is equipped with a smoothness prior that regularizes how much the estimated distribution diverges from the reference distribution. These models can be seen as a bridge between conditional transformation models and generalized additive models for location, scale and shape. Markov chain Monte Carlo inference for PTMs can be conducted with the No-U-Turn sampler and offers straightforward uncertainty quantification for the conditional distribution as well as for the covariate effects. A simulation study demonstrates the effectiveness of the approach. We apply the model to data from the Fourth Dutch Growth Study and the Framingham Heart Study. A full-featured implementation is available as a Python library.
... The MS-TDS has been created and internally validated by integration of retrospective routine clinical, imaging and laboratory data from 65 predictors (Additional file 1: Table S1) collected for deeply characterized 475 MS patients at the "Klinikum rechts der Isar, Technical University of Munich" [4]. To create the MS-TDS, a predictive random forest (RFs) model was implemented through transformation forests based on fully parameterized Cox proportional hazards models to deal with the interval-censored outcome [11,15]. A benchmark study was performed for hyperparameter tuning and to choose the best performing model. ...
Introduction
In Multiple Sclerosis (MS), patients´ characteristics and (bio)markers that reliably predict the individual disease prognosis at disease onset are lacking. Cohort studies allow a close follow-up of MS histories and a thorough phenotyping of patients. Therefore, a multicenter cohort study was initiated to implement a wide spectrum of data and (bio)markers in newly diagnosed patients.
Methods
ProVal-MS (Prospective study to validate a multidimensional decision score that predicts treatment outcome at 24 months in untreated patients with clinically isolated syndrome or early Relapsing–Remitting-MS) is a prospective cohort study in patients with clinically isolated syndrome (CIS) or Relapsing–Remitting (RR)-MS (McDonald 2017 criteria), diagnosed within the last two years, conducted at five academic centers in Southern Germany. The collection of clinical, laboratory, imaging, and paraclinical data as well as biosamples is harmonized across centers. The primary goal is to validate (discrimination and calibration) the previously published DIFUTURE MS-Treatment Decision score (MS-TDS). The score supports clinical decision-making regarding the options of early (within 6 months after study baseline) platform medication (Interferon beta, glatiramer acetate, dimethyl/diroximel fumarate, teriflunomide), or no immediate treatment (> 6 months after baseline) of patients with early RR-MS and CIS by predicting the probability of new or enlarging lesions in cerebral magnetic resonance images (MRIs) between 6 and 24 months. Further objectives are refining the MS-TDS score and providing data to identify new markers reflecting disease course and severity. The project also provides a technical evaluation of the ProVal-MS cohort within the IT-infrastructure of the DIFUTURE consortium (Data Integration for Future Medicine) and assesses the efficacy of the data sharing techniques developed.
Perspective
Clinical cohorts provide the infrastructure to discover and to validate relevant disease-specific findings. A successful validation of the MS-TDS will add a new clinical decision tool to the armamentarium of practicing MS neurologists from which newly diagnosed MS patients may take advantage.
Trial registration ProVal-MS has been registered in the German Clinical Trials Register, `Deutsches Register Klinischer Studien` (DRKS)—ID: DRKS00014034, date of registration: 21 December 2018; https://drks.de/search/en/trial/DRKS00014034
Supplementary Information
The online version contains supplementary material available at 10.1186/s42466-024-00310-x.
... The MS-TDS has been created and internally validated by integration of retrospective routine clinical, imaging and laboratory data from 65 predictors (Additional file 1: Table S1) collected for deeply characterized 475 MS patients at the "Klinikum rechts der Isar, Technical University of Munich" [4]. To create the MS-TDS, a predictive random forest (RFs) model was implemented through transformation forests based on fully parameterized Cox proportional hazards models to deal with the interval-censored outcome [11,15]. A benchmark study was performed for hyperparameter tuning and to choose the best performing model. ...
Introduction: In Multiple Sclerosis (MS), patients´ characteristics and (bio)markers that reliably predict the individual disease prognosis at disease onset are lacking. Cohort studies allow a close follow-up of MS histories and a thorough phenotyping of patients. Therefore, a multicenter cohort study was initiated to implement a wide spectrum of data and (bio)markers in newly diagnosed patients.
Methods: ProVal-MS (Prospective study to validate a multidimensional decision score that predicts treatment outcome at 24 months in untreated patients with clinically isolated syndrome or early Relapsing–Remitting-MS) is a prospective cohort study in patients with clinically isolated syndrome (CIS) or Relapsing–Remitting (RR)-MS (McDonald 2017 criteria), diagnosed within the last two years, conducted at five academic centers in Southern Germany. The collection of clinical, laboratory, imaging, and paraclinical data as well as biosamples is harmonized across centers. The primary goal is to validate (discrimination and calibration) the previously published DIFUTURE MS-Treatment Decision score (MS-TDS). The score supports clinical decision-making regarding the options of early (within 6 months after study baseline) platform medication (Interferon beta, glatiramer acetate, dimethyl/diroximel fumarate, teriflunomide), or no immediate treatment (> 6 months after baseline) of patients with early RR-MS and CIS by predicting the probability of new or enlarging lesions in cerebral magnetic resonance images (MRIs) between 6 and 24 months. Further objectives are refining the MS-TDS score and providing data to identify new markers reflecting disease course and severity. The project also provides a technical evaluation of the ProVal-MS cohort within the IT-infrastructure of the DIFUTURE consortium (Data Integration for Future Medicine) and assesses the efficacy of the data sharing techniques developed.
Perspective: Clinical cohorts provide the infrastructure to discover and to validate relevant disease-specific findings. A successful validation of the MS-TDS will add a new clinical decision tool to the armamentarium of practicing MS neurologists from which newly diagnosed MS patients may take advantage.
... Hothorn et al. (2014) proposed the class of conditional transformation models, which is a general approach to model the distribution function F(t|X i ) = 1 − S(t|X i ) conditional on a set of covariates (including direct survival models as special cases). Of note, Hothorn et al. (2018) developed a likelihood-based approach for the modeling of F(t|X i ) that does not require prespecification of a grid of time points (see also Hothorn 2019 and the references therein). Similar to Garcia et al. (2019), Hothorn et al. (2018) proposed to model the baseline risk and the covariate effects using basis functions. ...
... Of note, Hothorn et al. (2018) developed a likelihood-based approach for the modeling of F(t|X i ) that does not require prespecification of a grid of time points (see also Hothorn 2019 and the references therein). Similar to Garcia et al. (2019), Hothorn et al. (2018) proposed to model the baseline risk and the covariate effects using basis functions. Despite the flexibility of the aforementioned approaches, we emphasize that the building of survival models (in particular, the specification of the model structure) remains a challenging task. ...
This paper presents a semi-parametric modeling technique for estimating the survival function from a set of right-censored time-to-event data. Our method, named pseudo-value regression trees (PRT), is based on the pseudo-value regression framework, modeling individual-specific survival probabilities by computing pseudo-values and relating them to a set of covariates. The standard approach to pseudo-value regression is to fit a main-effects model using generalized estimating equations (GEE). PRT extend this approach by building a multivariate regression tree with pseudo-value outcome and by successively fitting a set of regularized additive models to the data in the nodes of the tree. Due to the combination of tree learning and additive modeling, PRT are able to perform variable selection and to identify relevant interactions between the covariates, thereby addressing several limitations of the standard GEE approach. In addition, PRT include time-dependent effects in the node-wise models. Interpretability of the PRT fits is ensured by controlling the tree depth. Based on the results of two simulation studies, we investigate the properties of the PRT method and compare it to several alternative modeling techniques. Furthermore, we illustrate PRT by analyzing survival in 3,652 patients enrolled for a randomized study on primary invasive breast cancer.
... NPMLEs are desirable because they are fully efficient. For continuous outcomes, more recent developments have used B-splines and Bernstein polynomials to flexibly model the transformation [9,10], but these estimators of the transformation are not fully nonparametric. ...
Regression models for continuous outcomes frequently require a transformation of the outcome, which is often specified a priori or estimated from a parametric family. Cumulative probability models (CPMs) nonparametrically estimate the transformation by treating the continuous outcome as if it is ordered categorically. They thus represent a flexible analysis approach for continuous outcomes. However, it is difficult to establish asymptotic properties for CPMs due to the potentially unbounded range of the transformation. Here we show asymptotic properties for CPMs when applied to slightly modified data where bounds, one lower and one upper, are chosen and the outcomes outside the bounds are set as two ordinal categories. We prove the uniform consistency of the estimated regression coefficients and of the estimated transformation function between the bounds. We also describe their joint asymptotic distribution, and show that the estimated regression coefficients attain the semiparametric efficiency bound. We show with simulations that results from this approach and those from using the CPM on the original data are very similar when a small fraction of the data are modified. We reanalyze a dataset of HIV-positive patients with CPMs to illustrate and compare the approaches.
... transformation (Hothorn et al., 2018): This estimation was embedded in a maximum likelihood framework and it was facilitated by parametrization of the function h. Specifically, the parametrization was conducted by Bernstein polynomials. ...
Introduction
In metabolomics, the investigation of associations between the metabolome and one trait of interest is a key research question. However, statistical analyses of such associations are often challenging. Statistical tools enabling resilient verification and clear presentation are therefore highly desired.
Objectives
Our aim is to provide a contribution for statistical analysis of metabolomics data, offering a widely applicable open-source statistical workflow, which considers the intrinsic complexity of metabolomics data.
Methods
We combined selected R packages tailored for all properties of heterogeneous metabolomics datasets, where metabolite parameters typically (i) are analyzed in different matrices, (ii) are measured on different analytical platforms with different precision, (iii) are analyzed by targeted as well as non-targeted methods, (iv) are scaled variously, (v) reveal heterogeneous variances, (vi) may be correlated, (vii) may have only few values or values below a detection limit, or (viii) may be incomplete.
Results
The code is shared entirely and freely available. The workflow output is a table of metabolites associated with a trait of interest and a compact plot for high-quality results visualization. The workflow output and its utility are presented by applying it to two previously published datasets: one dataset from our own lab and another dataset taken from the repository MetaboLights.
Conclusion
Robustness and benefits of the statistical workflow were clearly demonstrated, and everyone can directly re-use it for analysis of own data.
... We propose an approach based on Bernstein-Polynomial Normalizing Flows (BNFs) for short-term density forecasting of LV loads, which have been used in the statistics community for a while [43] and were brought to deep learning by [39]. Compared to existing methods, BNF allow full density forecasts without strong parametric assumptions on the distribution and flexible modelling of explanatory variables. ...
... Alternatively, there are approaches that use a single flexible transformation, such as sum-ofsquares polynomials [46], or splines [47], [48]. Our approach benefits from Bernstein polynomials introduced recently in the statistics community [43] and combined with NN in [39], [49]. Compared to other methods Bernstein-Polynomial Normalizing Flow (BNF) have several advantages, like 1) robustness against initial and round-off errors, 2) higher interpretability, 3) a theoretical upper bound for the approximation error, 4) ability to increase the flexibility at no cost to the training stability. ...
... Generated by the M + 1 density of the Beta distribution [51] for details of the proof and further properties of the Bernstein polynomials). Higher degree Bernstein polynomials increase the expressiveness with no cost to the training stability [43], [49]. Empirically, M 10 polynomials are often sufficient in typical regression settings [43]. ...
The transition to a fully renewable energy grid requires better forecasting of demand at the low-voltage level to increase efficiency and ensure reliable control. However, high fluctuations and increasing electrification cause huge forecast variability, not reflected in traditional point estimates. Probabilistic load forecasts take uncertainties into account and thus allow more informed decision-making for the planning and operation of low-carbon energy systems. We propose an approach for flexible conditional density forecasting of short-term load based on Bernstein polynomial normalizing flows, where a neural network controls the parameters of the flow. In an empirical study with 3639 smart meter customers, our density predictions for 24h-ahead load forecasting compare favorably against Gaussian and Gaussian mixture densities. Furthermore, they outperform a non-parametric approach based on the pinball loss, especially in low-data scenarios.
... Properties and extensions. Hothorn et al. (2014) prove consistency of boosted conditional transformation models, whereas the monotonicity constraints are practically ignored.Hothorn et al. (2018) prove consistency and asympotic normality of the MLE estimator. ...
Flexible modeling of how an entire distribution changes with covariates is an important yet challenging generalization of mean-based regression that has seen growing interest over the past decades in both the statistics and machine learning literature. This review outlines selected state-of-the-art statistical approaches to distributional regression, complemented with alternatives from machine learning. Topics covered include the similarities and differences between these approaches, extensions, properties and limitations, estimation procedures, and the availability of software. In view of the increasing complexity and availability of large-scale data, this review also discusses the scalability of traditional estimation methods, current trends, and open challenges. Illustrations are provided using data on childhood malnutrition in Nigeria and Australian electricity prices.
... Hothorn et al. (2014) proposed a conditional transformation function for taking into account covariate effects on a single response variable for continuous, binary, or censored data. A maximum likelihood estimator for MLT was also proposed (Hothorn et al., 2018) and extended to MCTM inference, which provides consistent and asymptotically normal estimators. Therefore, for a multivariate variable with a distribution that depends on a covariate vector , MCTM estimation is given by the conditional transformation function ℎ( | ). ...
... Thus, the shape of the distribution of the response variables and the effects of the covariates are approximated via a known basis function and unknown parameters that need to be estimated from the data. It is beyond the scope of the present work to give a full overview of transformation models (for further details, see Hothorn et al., 2014Hothorn et al., , 2018Klein et al., 2019). ...
... For a continuous response,h should be smooth in , so any polynomial or spline basis is a suitable choice for . Hothorn et al. (2018) proposed the use of Bernstein polynomials basis of order (with + 1 parameters). Under this representation, ( ) results from evaluating densities of beta distributions (characterized by two parameters; and ), a choice that is computationally convenient because strict monotonicity can be formulated as a set of linear constraints on the parameters < +1 for all = 0, … , (Hothorn et al., 2018;McKay Curtis & Ghosh, 2011). ...
The reference interval is the most widely used medical decision-making, constituting a central tool in determining whether an individual is healthy or not. When the results of several continuous diagnostic tests are available for the same patient, their clinical interpretation is more reliable if a multivariate reference region (MVR) is available rather than multiple univariate reference intervals. MVRs, defined as regions containing 95% of the results of healthy subjects, extend the concept of the reference interval to the multivariate setting. However, they are rarely used in clinical practice owing to difficulties associated with their interpretability and the restrictions inherent to the assumption of a Gaussian distribution. Further statistical research is thus needed to make MVRs more applicable and easier for physicians to interpret. Since the joint distribution of diagnostic test results may well change with patient characteristics independent of disease status, MVRs adjusted for covariates are desirable. The present work introduces a novel formulation for MVRs based on multivariate conditional transformation models (MCTMs). Additionally, we take into account the estimation uncertainty of such MVRs by means of tolerance regions. These conditional MVRs imply no parametric restriction on the response, and potentially nonlinear continuous covariate effects can be estimated. MCTMs allow the estimation of the effects of covariates on the joint distribution of multivariate response variables and on these variables' marginal distributions, via the use of most likely transformation estimation. Our contributions proved reliable when tested with simulated data and for a real data application with two glycemic markers.
... We employ transformation models to jointly estimate the transformation function and regression parameters. 12,13 This approach specifies a parametric model for the ROC curve but remains distribution-free because we do not impose any strong assumptions about the transformation function. Using the estimated parameters, we show how to evaluate covariate effects on the discriminatory performance of diagnostic tests. ...
... Equation (2) represents a general class of models called transformation models. 22,13 The transformation function h uniquely characterizes the distribution of Y , similar to the density or distribution function. Plugging in this conditional CDF of Y into equation (1), h cancels out and the covariate-specific ROC curve is given by ...
... Denote the complete parameter vector as = ( ⊤ , ⊤ ) ⊤ , where = ( , ⊤ , ⊤ ) ⊤ ∈ ℝ 2P+1 are the vector of regression coefficients parameterizing the function d from Section 2.1 and ∈ ℝ M+1 are the basis coefficients. We follow the maximum likelihood approach proposed by Hothorn et al. 13 to jointly estimate and . The advantages of embedding the model in the likelihood framework are as follows. ...
Receiver operating characteristic analysis is one of the most popular approaches for evaluating and comparing the accuracy of medical diagnostic tests. Although various methodologies have been developed for estimating receiver operating characteristic curves and their associated summary indices, there is no consensus on a single framework that can provide consistent statistical inference while handling the complexities associated with medical data. Such complexities might include non-normal data, covariates that influence the diagnostic potential of a test, ordinal biomarkers or censored data due to instrument detection limits. We propose a regression model for the transformed test results which exploits the invariance of receiver operating characteristic curves to monotonic transformations and accommodates these features. Simulation studies show that the estimates based on transformation models are unbiased and yield coverage at nominal levels. The methodology is applied to a cross-sectional study of metabolic syndrome where we investigate the covariate-specific performance of weight-to-height ratio as a non-invasive diagnostic test. Software implementations for all the methods described in the article are provided in the tram add-on package to the R system for statistical computing and graphics.
