Robert Tibshirani

Siamak K. Sorooshyari, Manuel A. Rivas, Robert Tibshirani

Despite being ubiquitous in science, clustering remains a technique whose results are not quantitatively scrutinized via a framework. We present an analysis called evaluating replicability via iterative clustering assignments (ERICA) that is applied to a dataset to determine whether clusters are identified in a replicable manner. The pipeline computes a statistic that describes whether structure is found in a dataset. Quantitative visualization methods are presented to answer important questions such as the similarity between clusters, and the identity of points that may be outliers. When tested on synthetic data, the findings show clusters being discovered in a replicable manner. However, we note a possibility for non-replicable results when the pipeline is applied to three gene expression datasets for breast cancer subtype validation. The study underscores the need for rigorous inspection and offers a practical tool for doing so.

Xuelin Yang, Louis Abraham, Sejin Kim et al.

The Cox proportional hazards model is a canonical method in survival analysis for prediction of the life expectancy of a patient given clinical or genetic covariates -- it is a linear model in its original form. In recent years, several methods have been proposed to generalize the Cox model to neural networks, but none of these are both numerically correct and computationally efficient. We propose FastCPH, a new method that runs in linear time and supports both the standard Breslow and Efron methods for tied events. We also demonstrate the performance of FastCPH combined with LassoNet, a neural network that provides interpretability through feature sparsity, on survival datasets. The final procedure is efficient, selects useful covariates and outperforms existing CoxPH approaches.

Erin Craig, Robert Tibshirani

In-context learning with attention enables large neural networks to make context-specific predictions by selectively focusing on relevant examples. Here, we adapt this idea to supervised learning procedures such as lasso regression and gradient boosting, for tabular data. Our goals are to (1) flexibly fit personalized models for each prediction point and (2) retain model simplicity and interpretability. Our method fits a local model for each test observation by weighting the training data according to attention, a supervised similarity measure that emphasizes features and interactions that are predictive of the outcome. Attention weighting allows the method to adapt to heterogeneous data in a data-driven way, without requiring cluster or similarity pre-specification. Further, our approach is uniquely interpretable: for each test observation, we identify which features are most predictive and which training observations are most relevant. We then show how to use attention weighting for time series and spatial data, and we present a method for adapting pretrained tree-based models to distributional shift using attention-weighted residual corrections. Across real and simulated datasets, attention weighting improves predictive performance while preserving interpretability, and theory shows that attention-weighting linear models attain lower mean squared error than the standard linear model under mixture-of-models data-generating processes with known subgroup structure.

Maximilian Schuessler, Erik Sverdrup, Robert Tibshirani et al.

Quasi-experimental evaluations are central for generating real-world causal evidence and complementing insights from randomized trials. The regression discontinuity design (RDD) is a quasi-experimental design that can be used to estimate the causal effect of treatments that are assigned based on a running variable crossing a threshold. Such threshold-based rules are ubiquitous in healthcare, where predictive and prognostic biomarkers frequently guide treatment decisions. However, standard RD estimators rely on complete outcome data, an assumption often violated in time-to-event analyses where censoring arises from loss to follow-up. To address this issue, we propose a nonparametric approach that leverages doubly robust censoring corrections and can be paired with existing RD estimators. Our approach can handle multiple survival endpoints, long follow-up times, and covariate-dependent variation in survival and censoring. We discuss the relevance of our approach across multiple areas of applications and demonstrate its usefulness through simulations and the prostate component of the Prostate, Lung, Colorectal and Ovarian (PLCO) Cancer Screening Trial where our new approach offers several advantages, including higher efficiency and robustness to misspecification. We have also developed an open-source software package, $\texttt{rdsurvival}$, for the $\texttt{R}$ language.

Alejandro Lozano, Min Woo Sun, James Burgess et al. · stanford

The development of vision-language models (VLMs) is driven by large-scale and diverse multimodal datasets. However, progress toward generalist biomedical VLMs is limited by the lack of annotated, publicly accessible datasets across biology and medicine. Existing efforts are restricted to narrow domains, missing the full diversity of biomedical knowledge encoded in scientific literature. To address this gap, we introduce BIOMEDICA, a scalable, open-source framework to extract, annotate, and serialize the entirety of the PubMed Central Open Access subset into an easy-to-use, publicly accessible dataset. Our framework produces a comprehensive archive with over 24 million unique image-text pairs from over 6 million articles. Metadata and expert-guided annotations are also provided. We demonstrate the utility and accessibility of our resource by releasing BMCA-CLIP, a suite of CLIP-style models continuously pre-trained on the BIOMEDICA dataset via streaming, eliminating the need to download 27 TB of data locally. On average, our models achieve state-of-the-art performance across 40 tasks - spanning pathology, radiology, ophthalmology, dermatology, surgery, molecular biology, parasitology, and cell biology - excelling in zero-shot classification with a 6.56% average improvement (as high as 29.8% and 17.5% in dermatology and ophthalmology, respectively), and stronger image-text retrieval, all while using 10x less compute. To foster reproducibility and collaboration, we release our codebase and dataset for the broader research community.

Alejandro Lozano, Min Woo Sun, James Burgess et al. · stanford

Despite the excitement behind biomedical artificial intelligence (AI), access to high-quality, diverse, and large-scale data - the foundation for modern AI systems - is still a bottleneck to unlocking its full potential. To address this gap, we introduce Biomedica, an open-source dataset derived from the PubMed Central Open Access subset, containing over 6 million scientific articles and 24 million image-text pairs, along with 27 metadata fields (including expert human annotations). To overcome the challenges of accessing our large-scale dataset, we provide scalable streaming and search APIs through a web server, facilitating seamless integration with AI systems. We demonstrate the utility of the Biomedica dataset by building embedding models, chat-style models, and retrieval-augmented chat agents. Notably, all our AI models surpass previous open systems in their respective categories, underscoring the critical role of diverse, high-quality, and large-scale biomedical data.

Min Woo Sun, Alejandro Lozano, Javier Gamazo Tejero et al. · stanford

Embedding vision-language models (VLMs) are typically pretrained with short text windows (<77 tokens), which forces the truncation of long-format captions. Yet, the distribution of biomedical captions from large-scale open source literature reveals that a huge portion of captions far exceed 77 tokens. To this end, we investigate the impact of pretraining on long-format biomedical captions by extending the context length of text encoders in VLMs. We find that longer context (thus, enabling additional supervision provided in long-format captions) correlates with better retrieval and classification performance. Given this finding, we introduce BIOMEDICA-LongCAP, a dataset of 1M image-caption pairs enriched with context-aware descriptions from full-text articles, providing longer and additional textual supervision. Using BIOMEDICA-LongCAP, we train BMC-LongCLIP, a long-context biomedical VLM with a text encoder supporting windows of up to 512 tokens. Our model extends context capacity by 6.6x, reducing token waste from 55% to just 2.2%. On long-caption retrieval benchmarks, BMC-LongCLIP achieves up to +30% absolute gains in Recall@1 and +2% average improvements in classification, while also converging faster than short-context. Our results demonstrate that long-context modeling is a promising direction for advancing biomedical VLMs.

Jing Shang, James Bannon, Benjamin Haibe-Kains et al.

Random forests are a statistical learning technique that use bootstrap aggregation to average high-variance and low-bias trees. Improvements to random forests, such as applying Lasso regression to the tree predictions, have been proposed in order to reduce model bias. However, these changes can sometimes degrade performance (e.g., an increase in mean squared error). In this paper, we show in theory that the relative performance of these two methods, standard and Lasso-weighted random forests, depends on the signal-to-noise ratio. We further propose a unified framework to combine random forests and Lasso selection by applying adaptive weighting and show mathematically that it can strictly outperform the other two methods. We compare the three methods through simulation, including bias-variance decomposition, error estimates evaluation, and variable importance analysis. We also show the versatility of our method by applications to a variety of real-world datasets.

Ji Won Park, Robert Tibshirani, Kyunghyun Cho

Many risk-sensitive applications require well-calibrated prediction sets over multiple, potentially correlated target variables, for which the prediction algorithm may report correlated errors. In this work, we aim to construct the conformal prediction set accounting for the joint correlation structure of the vector-valued non-conformity scores. Drawing from the rich literature on multivariate quantiles and semiparametric statistics, we propose an algorithm to estimate the $1-α$ quantile of the scores, where $α$ is the user-specified miscoverage rate. In particular, we flexibly estimate the joint cumulative distribution function (CDF) of the scores using nonparametric vine copulas and improve the asymptotic efficiency of the quantile estimate using its influence function. The vine decomposition allows our method to scale well to a large number of targets. As well as guaranteeing asymptotically exact coverage, our method yields desired coverage and competitive efficiency on a range of real-world regression problems, including those with missing-at-random labels in the calibration set.

Erica Zhang, Ryunosuke Goto, Naomi Sagan et al.

We introduce LLM-Lasso, a novel framework that leverages large language models (LLMs) to guide feature selection in Lasso $\ell_1$ regression. Unlike traditional methods that rely solely on numerical data, LLM-Lasso incorporates domain-specific knowledge extracted from natural language, enhanced through a retrieval-augmented generation (RAG) pipeline, to seamlessly integrate data-driven modeling with contextual insights. Specifically, the LLM generates penalty factors for each feature, which are converted into weights for the Lasso penalty using a simple, tunable model. Features identified as more relevant by the LLM receive lower penalties, increasing their likelihood of being retained in the final model, while less relevant features are assigned higher penalties, reducing their influence. Importantly, LLM-Lasso has an internal validation step that determines how much to trust the contextual knowledge in our prediction pipeline. Hence it addresses key challenges in robustness, making it suitable for mitigating potential inaccuracies or hallucinations from the LLM. In various biomedical case studies, LLM-Lasso outperforms standard Lasso and existing feature selection baselines, all while ensuring the LLM operates without prior access to the datasets. To our knowledge, this is the first approach to effectively integrate conventional feature selection techniques directly with LLM-based domain-specific reasoning.

Thomas Le Menestrel, Erin Craig, Robert Tibshirani et al.

Recent genome-wide association studies (GWAS) have uncovered the genetic basis of complex traits, but show an under-representation of non-European descent individuals, underscoring a critical gap in genetic research. Here, we assess whether we can improve disease prediction across diverse ancestries using multiomic data. We evaluate the performance of Group-LASSO INTERaction-NET (glinternet) and pretrained lasso in disease prediction focusing on diverse ancestries in the UK Biobank. Models were trained on data from White British and other ancestries and validated across a cohort of over 96,000 individuals for 8 diseases. Out of 96 models trained, we report 16 with statistically significant incremental predictive performance in terms of ROC-AUC scores (p-value < 0.05), found for diabetes, arthritis, gall stones, cystitis, asthma and osteoarthritis. For the interaction and pretrained models that outperformed the baseline, the PRS score was the primary driver behind prediction. Our findings indicate that both interaction terms and pre-training can enhance prediction accuracy but for a limited set of diseases and moderate improvements in accuracy

Maximilian Schuessler, Erik Sverdrup, Robert Tibshirani

Robust estimation of heterogeneous treatment effects is a fundamental challenge for optimal decision-making in domains ranging from personalized medicine to educational policy. In recent years, predictive machine learning has emerged as a valuable toolbox for causal estimation, enabling more flexible effect estimation. However, accurately estimating conditional average treatment effects (CATE) remains a major challenge, particularly in the presence of many covariates. In this article, we propose pretraining strategies that leverage a phenomenon in real-world applications: factors that are prognostic of the outcome are frequently also predictive of treatment effect heterogeneity. In medicine, for example, components of the same biological signaling pathways frequently influence both baseline risk and treatment response. Specifically, we demonstrate our approach within the R-learner framework, which estimates the CATE by solving individual prediction problems based on a residualized loss. We use this structure to incorporate side information and develop models that can exploit synergies between risk prediction and causal effect estimation. In settings where these synergies are present, this cross-task learning enables more accurate signal detection, yields lower estimation error, reduced false discovery rates, and higher power for detecting heterogeneity.

Erin Craig, Timothy Keyes, Jolanda Sarno et al.

Single-cell datasets often lack individual cell labels, making it challenging to identify cells associated with disease. To address this, we introduce Mixture Modeling for Multiple Instance Learning (MMIL), an expectation maximization method that enables the training and calibration of cell-level classifiers using patient-level labels. Our approach can be used to train e.g. lasso logistic regression models, gradient boosted trees, and neural networks. When applied to clinically-annotated, primary patient samples in Acute Myeloid Leukemia (AML) and Acute Lymphoblastic Leukemia (ALL), our method accurately identifies cancer cells, generalizes across tissues and treatment timepoints, and selects biologically relevant features. In addition, MMIL is capable of incorporating cell labels into model training when they are known, providing a powerful framework for leveraging both labeled and unlabeled data simultaneously. Mixture Modeling for MIL offers a novel approach for cell classification, with significant potential to advance disease understanding and management, especially in scenarios with unknown gold-standard labels and high dimensionality.

Min Woo Sun, Robert Tibshirani

Cross-validation (CV) is one of the most widely used techniques in statistical learning for estimating the test error of a model, but its behavior is not yet fully understood. It has been shown that standard confidence intervals for test error using estimates from CV may have coverage below nominal levels. This phenomenon occurs because each sample is used in both the training and testing procedures during CV and as a result, the CV estimates of the errors become correlated. Without accounting for this correlation, the estimate of the variance is smaller than it should be. One way to mitigate this issue is by estimating the mean squared error of the prediction error instead using nested CV. This approach has been shown to achieve superior coverage compared to intervals derived from standard CV. In this work, we generalize the nested CV idea to the Cox proportional hazards model and explore various choices of test error for this setting.

Daisy Yi Ding, Shuangning Li, Balasubramanian Narasimhan et al.

We propose a new method for supervised learning with multiple sets of features ("views"). The multiview problem is especially important in biology and medicine, where "-omics" data such as genomics, proteomics and radiomics are measured on a common set of samples. Cooperative learning combines the usual squared error loss of predictions with an "agreement" penalty to encourage the predictions from different data views to agree. By varying the weight of the agreement penalty, we get a continuum of solutions that include the well-known early and late fusion approaches. Cooperative learning chooses the degree of agreement (or fusion) in an adaptive manner, using a validation set or cross-validation to estimate test set prediction error. One version of our fitting procedure is modular, where one can choose different fitting mechanisms (e.g. lasso, random forests, boosting, neural networks) appropriate for different data views. In the setting of cooperative regularized linear regression, the method combines the lasso penalty with the agreement penalty, yielding feature sparsity. The method can be especially powerful when the different data views share some underlying relationship in their signals that can be exploited to boost the signals. We show that cooperative learning achieves higher predictive accuracy on simulated data and a real multiomics example of labor onset prediction. Leveraging aligned signals and allowing flexible fitting mechanisms for different modalities, cooperative learning offers a powerful approach to multiomics data fusion.

Stephen Bates, Trevor Hastie, Robert Tibshirani

Cross-validation is a widely-used technique to estimate prediction error, but its behavior is complex and not fully understood. Ideally, one would like to think that cross-validation estimates the prediction error for the model at hand, fit to the training data. We prove that this is not the case for the linear model fit by ordinary least squares; rather it estimates the average prediction error of models fit on other unseen training sets drawn from the same population. We further show that this phenomenon occurs for most popular estimates of prediction error, including data splitting, bootstrapping, and Mallow's Cp. Next, the standard confidence intervals for prediction error derived from cross-validation may have coverage far below the desired level. Because each data point is used for both training and testing, there are correlations among the measured accuracies for each fold, and so the usual estimate of variance is too small. We introduce a nested cross-validation scheme to estimate this variance more accurately, and we show empirically that this modification leads to intervals with approximately correct coverage in many examples where traditional cross-validation intervals fail.

J. Kenneth Tay, Nima Aghaeepour, Trevor Hastie et al.

In some supervised learning settings, the practitioner might have additional information on the features used for prediction. We propose a new method which leverages this additional information for better prediction. The method, which we call the feature-weighted elastic net ("fwelnet"), uses these "features of features" to adapt the relative penalties on the feature coefficients in the elastic net penalty. In our simulations, fwelnet outperforms the lasso in terms of test mean squared error and usually gives an improvement in true positive rate or false positive rate for feature selection. We also apply this method to early prediction of preeclampsia, where fwelnet outperforms the lasso in terms of 10-fold cross-validated area under the curve (0.86 vs. 0.80). We also provide a connection between fwelnet and the group lasso and suggest how fwelnet might be used for multi-task learning.

J. Kenneth Tay, Robert Tibshirani

Sparse generalized additive models (GAMs) are an extension of sparse generalized linear models which allow a model's prediction to vary non-linearly with an input variable. This enables the data analyst build more accurate models, especially when the linearity assumption is known to be a poor approximation of reality. Motivated by reluctant interaction modeling (Yu et al. 2019), we propose a multi-stage algorithm, called $\textit{reluctant generalized additive modeling (RGAM)}$, that can fit sparse generalized additive models at scale. It is guided by the principle that, if all else is equal, one should prefer a linear feature over a non-linear feature. Unlike existing methods for sparse GAMs, RGAM can be extended easily to binary, count and survival data. We demonstrate the method's effectiveness on real and simulated examples.

Ismael Lemhadri, Feng Ruan, Louis Abraham et al.

Much work has been done recently to make neural networks more interpretable, and one obvious approach is to arrange for the network to use only a subset of the available features. In linear models, Lasso (or $\ell_1$-regularized) regression assigns zero weights to the most irrelevant or redundant features, and is widely used in data science. However the Lasso only applies to linear models. Here we introduce LassoNet, a neural network framework with global feature selection. Our approach enforces a hierarchy: specifically a feature can participate in a hidden unit only if its linear representative is active. Unlike other approaches to feature selection for neural nets, our method uses a modified objective function with constraints, and so integrates feature selection with the parameter learning directly. As a result, it delivers an entire regularization path of solutions with a range of feature sparsity. On systematic experiments, LassoNet significantly outperforms state-of-the-art methods for feature selection and regression. The LassoNet method uses projected proximal gradient descent, and generalizes directly to deep networks. It can be implemented by adding just a few lines of code to a standard neural network.

Jonathan Johannemann, Robert Tibshirani

Nonlinear dimensionality reduction methods are a popular tool for data scientists and researchers to visualize complex, high dimensional data. However, while these methods continue to improve and grow in number, it is often difficult to evaluate the quality of a visualization due to a variety of factors such as lack of information about the intrinsic dimension of the data and additional tuning required for many evaluation metrics. In this paper, we seek to provide a systematic comparison of dimensionality reduction quality metrics using datasets where we know the ground truth manifold. We utilize each metric for hyperparameter optimization in popular dimensionality reduction methods used for visualization and provide quantitative metrics to objectively compare visualizations to their original manifold. In our results, we find a few methods that appear to consistently do well and propose the best performer as a benchmark for evaluating dimensionality reduction based visualizations.

J. Kenneth Tay, Jerome Friedman, Robert Tibshirani

We propose a new method for supervised learning, especially suited to wide data where the number of features is much greater than the number of observations. The method combines the lasso ($\ell_1$) sparsity penalty with a quadratic penalty that shrinks the coefficient vector toward the leading principal components of the feature matrix. We call the proposed method the "principal components lasso" ("pcLasso"). The method can be especially powerful if the features are pre-assigned to groups (such as cell-pathways, assays or protein interaction networks). In that case, pcLasso shrinks each group-wise component of the solution toward the leading principal components of that group. In the process, it also carries out selection of the feature groups. We provide some theory for this method and illustrate it on a number of simulated and real data examples.

Alejandro Schuler, Michael Baiocchi, Robert Tibshirani et al.

Practitioners in medicine, business, political science, and other fields are increasingly aware that decisions should be personalized to each patient, customer, or voter. A given treatment (e.g. a drug or advertisement) should be administered only to those who will respond most positively, and certainly not to those who will be harmed by it. Individual-level treatment effects can be estimated with tools adapted from machine learning, but different models can yield contradictory estimates. Unlike risk prediction models, however, treatment effect models cannot be easily evaluated against each other using a held-out test set because the true treatment effect itself is never directly observed. Besides outcome prediction accuracy, several metrics that can leverage held-out data to evaluate treatment effects models have been proposed, but they are not widely used. We provide a didactic framework that elucidates the relationships between the different approaches and compare them all using a variety of simulations of both randomized and observational data. Our results show that researchers estimating heterogenous treatment effects need not limit themselves to a single model-fitting algorithm. Instead of relying on a single method, multiple models fit by a diverse set of algorithms should be evaluated against each other using an objective function learned from the validation set. The model minimizing that objective should be used for estimating the individual treatment effect for future individuals.

Alejandro Schuler, Ken Jung, Robert Tibshirani et al.

Many decisions in healthcare, business, and other policy domains are made without the support of rigorous evidence due to the cost and complexity of performing randomized experiments. Using observational data to answer causal questions is risky: subjects who receive different treatments also differ in other ways that affect outcomes. Many causal inference methods have been developed to mitigate these biases. However, there is no way to know which method might produce the best estimate of a treatment effect in a given study. In analogy to cross-validation, which estimates the prediction error of predictive models applied to a given dataset, we propose synth-validation, a procedure that estimates the estimation error of causal inference methods applied to a given dataset. In synth-validation, we use the observed data to estimate generative distributions with known treatment effects. We apply each causal inference method to datasets sampled from these distributions and compare the effect estimates with the known effects to estimate error. Using simulations, we show that using synth-validation to select a causal inference method for each study lowers the expected estimation error relative to consistently using any single method.

Scott Powers, Junyang Qian, Kenneth Jung et al.

When devising a course of treatment for a patient, doctors often have little quantitative evidence on which to base their decisions, beyond their medical education and published clinical trials. Stanford Health Care alone has millions of electronic medical records (EMRs) that are only just recently being leveraged to inform better treatment recommendations. These data present a unique challenge because they are high-dimensional and observational. Our goal is to make personalized treatment recommendations based on the outcomes for past patients similar to a new patient. We propose and analyze three methods for estimating heterogeneous treatment effects using observational data. Our methods perform well in simulations using a wide variety of treatment effect functions, and we present results of applying the two most promising methods to data from The SPRINT Data Analysis Challenge, from a large randomized trial of a treatment for high blood pressure.

Scott Powers, Trevor Hastie, Robert Tibshirani

We propose the nuclear norm penalty as an alternative to the ridge penalty for regularized multinomial regression. This convex relaxation of reduced-rank multinomial regression has the advantage of leveraging underlying structure among the response categories to make better predictions. We apply our method, nuclear penalized multinomial regression (NPMR), to Major League Baseball play-by-play data to predict outcome probabilities based on batter-pitcher matchups. The interpretation of the results meshes well with subject-area expertise and also suggests a novel understanding of what differentiates players.

Xiaotong Suo, Victor Minden, Bradley Nelson et al.

Canonical correlation analysis was proposed by Hotelling [6] and it measures linear relationship between two multidimensional variables. In high dimensional setting, the classical canonical correlation analysis breaks down. We propose a sparse canonical correlation analysis by adding l1 constraints on the canonical vectors and show how to solve it efficiently using linearized alternating direction method of multipliers (ADMM) and using TFOCS as a black box. We illustrate this idea on simulated data.

Stefan Wager, Wenfei Du, Jonathan Taylor et al.

We study the problem of treatment effect estimation in randomized experiments with high-dimensional covariate information, and show that essentially any risk-consistent regression adjustment can be used to obtain efficient estimates of the average treatment effect. Our results considerably extend the range of settings where high-dimensional regression adjustments are guaranteed to provide valid inference about the population average treatment effect. We then propose cross-estimation, a simple method for obtaining finite-sample-unbiased treatment effect estimates that leverages high-dimensional regression adjustments. Our method can be used when the regression model is estimated using the lasso, the elastic net, subset selection, etc. Finally, we extend our analysis to allow for adaptive specification search via cross-validation, and flexible non-parametric regression adjustments with machine learning methods such as random forests or neural networks.

William Fithian, Jonathan Taylor, Robert Tibshirani et al.

Many model selection algorithms produce a path of fits specifying a sequence of increasingly complex models. Given such a sequence and the data used to produce them, we consider the problem of choosing the least complex model that is not falsified by the data. Extending the selected-model tests of Fithian et al. (2014), we construct p-values for each step in the path which account for the adaptive selection of the model path using the data. In the case of linear regression, we propose two specific tests, the max-t test for forward stepwise regression (generalizing a proposal of Buja and Brown (2014)), and the next-entry test for the lasso. These tests improve on the power of the saturated-model test of Tibshirani et al. (2014), sometimes dramatically. In addition, our framework extends beyond linear regression to a much more general class of parametric and nonparametric model selection problems. To select a model, we can feed our single-step p-values as inputs into sequential stopping rules such as those proposed by G'Sell et al. (2013) and Li and Barber (2015), achieving control of the familywise error rate or false discovery rate (FDR) as desired. The FDR-controlling rules require the null p-values to be independent of each other and of the non-null p-values, a condition not satisfied by the saturated-model p-values of Tibshirani et al. (2014). We derive intuitive and general sufficient conditions for independence, and show that our proposed constructions yield independent p-values.

Xiaotong Suo, Robert Tibshirani

We consider regression scenarios where it is natural to impose an order constraint on the coefficients. We propose an order-constrained version of L1-regularized regression for this problem, and show how to solve it efficiently using the well-known Pool Adjacent Violators Algorithm as its proximal operator. The main application of this idea is time-lagged regression, where we predict an outcome at time t from features at the previous K time points. In this setting it is natural to assume that the coefficients decay as we move farther away from t, and hence the order constraint is reasonable. Potential applications include financial time series and prediction of dynamic patient out- comes based on clinical measurements. We illustrate this idea on real and simulated data.

Samuel M. Gross, Robert Tibshirani

We consider the scenario where one observes an outcome variable and sets of features from multiple assays, all measured on the same set of samples. One approach that has been proposed for dealing with this type of data is ``sparse multiple canonical correlation analysis'' (sparse mCCA). All of the current sparse mCCA techniques are biconvex and thus have no guarantees about reaching a global optimum. We propose a method for performing sparse supervised canonical correlation analysis (sparse sCCA), a specific case of sparse mCCA when one of the datasets is a vector. Our proposal for sparse sCCA is convex and thus does not face the same difficulties as the other methods. We derive efficient algorithms for this problem, and illustrate their use on simulated and real data.

Nadine Hussami, Robert Tibshirani

We propose a new sparse regression method called the component lasso, based on a simple idea. The method uses the connected-components structure of the sample covariance matrix to split the problem into smaller ones. It then solves the subproblems separately, obtaining a coefficient vector for each one. Then, it uses non-negative least squares to recombine the different vectors into a single solution. This step is useful in selecting and reweighting components that are correlated with the response. Simulated and real data examples show that the component lasso can outperform standard regression methods such as the lasso and elastic net, achieving a lower mean squared error as well as better support recovery.

Noah Simon, Robert Tibshirani

To date, testing interactions in high dimensions has been a challenging task. Existing methods often have issues with sensitivity to modeling assumptions and heavily asymptotic nominal p-values. To help alleviate these issues, we propose a permutation-based method for testing marginal interactions with a binary response. Our method searches for pairwise correlations which differ between classes. In this manuscript, we compare our method on real and simulated data to the standard approach of running many pairwise logistic models. On simulated data our method finds more significant interactions at a lower false discovery rate (especially in the presence of main effects). On real genomic data, although there is no gold standard, our method finds apparent signal and tells a believable story, while logistic regression does not. We also give asymptotic consistency results under not too restrictive assumptions.

Jacob Bien, Jonathan Taylor, Robert Tibshirani

We add a set of convex constraints to the lasso to produce sparse interaction models that honor the hierarchy restriction that an interaction only be included in a model if one or both variables are marginally important. We give a precise characterization of the effect of this hierarchy constraint, prove that hierarchy holds with probability one and derive an unbiased estimate for the degrees of freedom of our estimator. A bound on this estimate reveals the amount of fitting "saved" by the hierarchy constraint. We distinguish between parameter sparsity - the number of nonzero coefficients - and practical sparsity - the number of raw variables one must measure to make a new prediction. Hierarchy focuses on the latter, which is more closely tied to important data collection concerns such as cost, time and effort. We develop an algorithm, available in the R package hierNet, and perform an empirical study of our method.