Showing papers on "Latent variable model published in 2021"

Action and Perception as Divergence Minimization

Abstract: We introduce a unified objective for action and perception of intelligent agents. Extending representation learning and control, we minimize the joint divergence between the combined system of agent and environment and a target distribution. Intuitively, such agents use perception to align their beliefs with the world, and use actions to align the world with their beliefs. Minimizing the joint divergence to an expressive target maximizes the mutual information between the agent's representations and inputs, thus inferring representations that are informative of past inputs and exploring future inputs that are informative of the representations. This lets us explain intrinsic objectives, such as representation learning, information gain, empowerment, and skill discovery from minimal assumptions. Moreover, interpreting the target distribution as a latent variable model suggests powerful world models as a path toward highly adaptive agents that seek large niches in their environments, rendering task rewards optional. The framework provides a common language for comparing a wide range of objectives, advances the understanding of latent variables for decision making, and offers a recipe for designing novel objectives. We recommend deriving future agent objectives the joint divergence to facilitate comparison, to point out the agent's target distribution, and to identify the intrinsic objective terms needed to reach that distribution.

Affective Processes: stochastic modelling of temporal context for emotion and facial expression recognition

Abstract: Temporal context is key to the recognition of expressions of emotion. Existing methods, that rely on recurrent or self-attention models to enforce temporal consistency, work on the feature level, ignoring the task-specific temporal dependencies, and fail to model context uncertainty. To alleviate these issues, we build upon the framework of Neural Processes to propose a method for apparent emotion recognition with three key novel components: (a) probabilistic contextual representation with a global latent variable model; (b) temporal context modelling using task-specific predictions in addition to features; and (c) smart temporal context selection. We validate our approach on four databases, two for Valence and Arousal estimation (SEWA and AffWild2), and two for Action Unit intensity estimation (DISFA and BP4D). Results show a consistent improvement over a series of strong baselines as well as over state-of-the-art methods.

Evaluating the Viability of Neurocognition as a Transdiagnostic Construct Using Both Latent Variable Models and Network Analysis

Abstract: The relational structure of psychological symptoms and disorders is of crucial importance to mechanistic and causal research Methodologically, factor analytic approaches (latent variable modeling) and network analyses are two dominant approaches Amidst some debate about their relative merits, use of both methods simultaneously in the same data set has rarely been reported in child or adolescent psychopathology A second issue is that the nosological structure can be enriched by inclusion of transdiagnostic constructs, such as neurocognition (eg, executive functions and other processes) These cut across traditional diagnostic boundaries and are rarely included even though they can help map the mechanistic architecture of psychopathology Using a sample enriched for ADHD (n = 498 youth ages 6 to 17 years; M = 108 years, SD = 23 years, 55% male), both approaches were used in two ways: (a) to model symptom structure and (b) to model seven neurocognitive domains hypothesized as important transdiagnostic features in ADHD and associated disorders The structure of psychopathology domains was similar across statistical approaches with internalizing, externalizing, and neurocognitive performance clusters Neurocognition remained a distinct domain according to both methods, showing small to moderate associations with internalizing and externalizing domains in latent variable models and high connectivity in network analyses Overall, the latent variable and network approaches yielded more convergent than discriminant findings, suggesting that both may be complementary tools for evaluating the utility of transdiagnostic constructs for psychopathology research

Variational Prototype Inference for Few-Shot Semantic Segmentation

Abstract: In this paper, we propose variational prototype inference to address few-shot semantic segmentation in a probabilistic framework. A probabilistic latent variable model infers the distribution of the prototype that is treated as the latent variable. We formulate the optimization as a variational inference problem, which is established with an amortized inference network based on an auto-encoder architecture. The probabilistic modeling of the prototype enhances its generalization ability to handle the inherent uncertainty caused by limited data and the huge intra-class variations of objects. Moreover, it offers a principled way to incorporate the prototype extracted from support images into the prediction of the segmentation maps for query images. We conduct extensive experimental evaluations on three benchmark datasets. Ablation studies show the effectiveness of variational prototype inference for few-shot semantic segmentation by probabilistic modeling. On all three benchmarks, our proposal achieves high segmentation accuracy and surpasses previous methods by considerable margins.

Toward a better monitoring statistic for profile monitoring via variational autoencoders

Abstract: Variational autoencoders have been recently proposed for the problem of process monitoring. While these works show impressive results over classical methods, the proposed monitoring statistics ofte...

Shared Linear Encoder-Based Multikernel Gaussian Process Latent Variable Model for Visual Classification

Abstract: Multiview learning has been widely studied in various fields and achieved outstanding performances in comparison to many single-view-based approaches. In this paper, a novel multiview learning method based on the Gaussian process latent variable model (GPLVM) is proposed. In contrast to existing GPLVM methods which only assume that there are transformations from the latent variable to the multiple observed inputs, our proposed method simultaneously takes a back constraint into account, encoding multiple observations to the latent variable by enjoying the Gaussian process (GP) prior. Particularly, to overcome the difficulty of the covariance matrix calculation in the encoder, a linear projection is designed to map different observations to a consistent subspace first. The obtained variable in this subspace is then projected to the latent variable in the manifold space with the GP prior. Furthermore, different from most GPLVM methods which strongly assume that the covariance matrices follow a certain kernel function, for example, radial basis function (RBF), we introduce a multikernel strategy to design the covariance matrix, being more reasonable and adaptive for the data representation. In order to apply the presented approach to the classification, a discriminative prior is also embedded to the learned latent variables to encourage samples belonging to the same category to be close and those belonging to different categories to be far. Experimental results on three real-world databases substantiate the effectiveness and superiority of the proposed method compared with state-of-the-art approaches.

Harmonized Multimodal Learning with Gaussian Process Latent Variable Models

Abstract: Multimodal learning aims to discover the relationship between multiple modalities. It has become an important research topic due to extensive multimodal applications such as cross-modal retrieval. This paper attempts to address the modality heterogeneity problem based on Gaussian process latent variable models (GPLVMs) to represent multimodal data in a common space. Previous multimodal GPLVM extensions generally adopt individual learning schemes on latent representations and kernel hyperparameters, which ignore their intrinsic relationship. To exploit strong complementarity among different modalities and GPLVM components, we develop a novel learning scheme called Harmonization , where latent representations and kernel hyperparameters are jointly learned from each other. Beyond the correlation fitting or intra-modal structure preservation paradigms widely used in existing studies, the harmonization is derived in a model-driven manner to encourage the agreement between modality-specific GP kernels and the similarity of latent representations. We present a range of multimodal learning models by incorporating the harmonization mechanism into several representative GPLVM-based approaches. Experimental results on four benchmark datasets show that the proposed models outperform the strong baselines for cross-modal retrieval tasks, and that the harmonized multimodal learning method is superior in discovering semantically consistent latent representation.

Latent representation learning in biology and translational medicine.

Abstract: Current data generation capabilities in the life sciences render scientists in an apparently contradicting situation. While it is possible to simultaneously measure an ever-increasing number of systems parameters, the resulting data are becoming increasingly difficult to interpret. Latent variable modeling allows for such interpretation by learning non-measurable hidden variables from observations. This review gives an overview over the different formal approaches to latent variable modeling, as well as applications at different scales of biological systems, such as molecular structures, intra- and intercellular regulatory up to physiological networks. The focus is on demonstrating how these approaches have enabled interpretable representations and ultimately insights in each of these domains. We anticipate that a wider dissemination of latent variable modeling in the life sciences will enable a more effective and productive interpretation of studies based on heterogeneous and high-dimensional data modalities.

Patchwise Generative ConvNet: Training Energy-Based Models from a Single Natural Image for Internal Learning

Abstract: Exploiting internal statistics of a single natural image has long been recognized as a significant research paradigm where the goal is to learn the internal distribution of patches within the image without relying on external training data. Different from prior works that model such a distribution implicitly with a top-down latent variable model (e.g., generator), this paper proposes to explicitly represent the statistical distribution within a single natural image by using an energy-based generative framework, where a pyramid of energy functions, each parameterized by a bottom-up deep neural network, are used to capture the distributions of patches at different resolutions. Meanwhile, a coarse-to-fine sequential training and sampling strategy is presented to train the model efficiently. Besides learning to generate random samples from white noise, the model can learn in parallel with a self-supervised task (e.g., recover the input image from its corrupted version), which can further improve the descriptive power of the learned model. The proposed model is simple and natural in that it does not require an auxiliary model (e.g., discriminator) to assist the training. Besides, it also unifies internal statistics learning and image generation in a single framework. Experimental results presented on various image generation and manipulation tasks, including super-resolution, image editing, harmonization, style transfer, etc, have demonstrated the effectiveness of our model for internal learning.

Self-Supervised Learning with Data Augmentations Provably Isolates Content from Style

Abstract: Self-supervised representation learning has shown remarkable success in a number of domains. A common practice is to perform data augmentation via hand-crafted transformations intended to leave the semantics of the data invariant. We seek to understand the empirical success of this approach from a theoretical perspective. We formulate the augmentation process as a latent variable model by postulating a partition of the latent representation into a content component, which is assumed invariant to augmentation, and a style component, which is allowed to change. Unlike prior work on disentanglement and independent component analysis, we allow for both nontrivial statistical and causal dependencies in the latent space. We study the identifiability of the latent representation based on pairs of views of the observations and prove sufficient conditions that allow us to identify the invariant content partition up to an invertible mapping in both generative and discriminative settings. We find numerical simulations with dependent latent variables are consistent with our theory. Lastly, we introduce Causal3DIdent, a dataset of high-dimensional, visually complex images with rich causal dependencies, which we use to study the effect of data augmentations performed in practice.

Learning a Generative Motion Model From Image Sequences Based on a Latent Motion Matrix

Abstract: We propose to learn a probabilistic motion model from a sequence of images for spatio-temporal registration. Our model encodes motion in a low-dimensional probabilistic space – the motion matrix – which enables various motion analysis tasks such as simulation and interpolation of realistic motion patterns allowing for faster data acquisition and data augmentation. More precisely, the motion matrix allows to transport the recovered motion from one subject to another simulating for example a pathological motion in a healthy subject without the need for inter-subject registration. The method is based on a conditional latent variable model that is trained using amortized variational inference. This unsupervised generative model follows a novel multivariate Gaussian process prior and is applied within a temporal convolutional network which leads to a diffeomorphic motion model. Temporal consistency and generalizability is further improved by applying a temporal dropout training scheme. Applied to cardiac cine-MRI sequences, we show improved registration accuracy and spatio-temporally smoother deformations compared to three state-of-the-art registration algorithms. Besides, we demonstrate the model’s applicability for motion analysis, simulation and super-resolution by an improved motion reconstruction from sequences with missing frames compared to linear and cubic interpolation.

The Poisson-Lognormal Model as a Versatile Framework for the Joint Analysis of Species Abundances

Abstract: Joint Species Distribution Models (JSDM) provide a general multivariate framework to study the joint abundances of all species from a community. JSDM account for both structuring factors (environmental characteristics or gradients, such as habitat type or nutrient availability) and potential interactions between the species (competition, mutualism, parasitism, etc.), which is instrumental in disentangling meaningful ecological interactions from mere statistical associations. Modeling the dependency between the species is challenging because of the count-valued nature of abundance data and most JSDM rely on Gaussian latent layer to encode the dependencies between species in a covariance matrix. The multivariate Poisson-lognormal (PLN) model is one such model, which can be viewed as a multivariate mixed Poisson regression model. Inferring such models raises both statistical and computational issues, many of which were solved in recent contributions using variational techniques and convex optimization tools. The PLN model turns out to be a versatile framework, within which a variety of analyses can be performed, including multivariate sample comparison, clustering of sites or samples, dimension reduction (ordination) for visualization purposes, or inferring interaction networks. This paper presents the general PLN framework and illustrates its use on a series a typical experimental datasets. All the models and methods are implemented in the R package PLNmodels, available from cran.r-project.org.

Using multivariate generalized linear latent variable models to measure the difference in event count for stranded marine animals

Abstract: BACKGROUND AND OBJECTIVES: The classification of marine animals as protected species makes data and information on them to be very important. Therefore, this led to the need to retrieve and understand the data on the event counts for stranded marine animals based on location emergence, number of individuals, behavior, and threats to their presence. Whales are generally often stranded in very shallow areas with sloping sea floors and sand. Data were collected in this study on the incidence of stranded marine animals in 20 provinces of Indonesia from 2015 to 2019 with the focus on animals such as Balaenopteridae, Delphinidae, Lamnidae, Physeteridae and Rhincodontidae.METHODS:Multivariate latent generalized linear model was used to compare several distributions to analyze the diversity of event counts. Two optimization models including Laplace and Variational approximations were also applied.RESULTS: The best theta parameter in the latent multivariate latent generalized linear latent variable model was found in the Akaike Information Criterion, Akaike Information Criterion Corrected and Bayesian Information Criterion values, andthe information obtained was used to create a spatial cluster. Moreover, there was a comprehensive discussion on ocean-atmosphere interaction and the reasons the animals were stranded.CONCLUSION: The changes in marine ecosystems due to climate change, pollution, overexploitation, changes in sea use, and the existence of invasive alien species deserve serious attention.

Context-Aware Human Trajectories Prediction via Latent Variational Model

Abstract: Understanding human-contextual interaction to predict human trajectories is a challenging problem. Most of previous trajectory prediction approaches focused on modeling the human-human interaction located in a near neighborhood and neglected the influence of individuals which are farther in the scene as well as the scene layout. To alleviate these limitations, in this article we propose a model to address pedestrian trajectory prediction using a latent variable model aware of the human-contextual interaction. Our proposal relies on contextual information that influences the trajectory of pedestrians to encode human-contextual interaction. We model the uncertainty about future trajectories via latent variational model and captures relative interpersonal influences among all the subjects within the scene and their interaction with the scene layout to decode their trajectories. In extensive experiments, on publicly available datasets, it is shown that using contextual information and latent variational model, our trajectory prediction model achieves competitive results compared to state-of-the-art models.

ALMOND: Adaptive Latent Modeling and Optimization via Neural Networks and Langevin Diffusion

Abstract: Latent variable models cover a broad range of statistical and machine learning models, such as Bayesian models, linear mixed models, and Gaussian mixture models. Existing methods often suffer from ...

Contrastive latent variable modeling with application to case-control sequencing experiments

Abstract: High-throughput RNA-sequencing (RNA-seq) technologies are powerful tools for understanding cellular state. Often it is of interest to quantify and summarize changes in cell state that occur between experimental or biological conditions. Differential expression is typically assessed using univariate tests to measure gene-wise shifts in expression. However, these methods largely ignore changes in transcriptional correlation. Furthermore, there is a need to identify the low-dimensional structure of the gene expression shift to identify collections of genes that change between conditions. Here, we propose contrastive latent variable models designed for count data to create a richer portrait of differential expression in sequencing data. These models disentangle the sources of transcriptional variation in different conditions, in the context of an explicit model of variation at baseline. Moreover, we develop a model-based hypothesis testing framework that can test for global and gene subset-specific changes in expression. We test our model through extensive simulations and analyses with count-based gene expression data from perturbation and observational sequencing experiments. We find that our methods can effectively summarize and quantify complex transcriptional changes in case-control experimental sequencing data.

Different Roles of Prior Distributions in the Single Mediator Model with Latent Variables

Abstract: In manifest variable models, Bayesian methods for mediation analysis can have better statistical properties than commonly used frequentist methods. However, with latent variables, Bayesian mediatio...

A Deep Learning Algorithm for High-Dimensional Exploratory Item Factor Analysis

Abstract: Marginal maximum likelihood (MML) estimation is the preferred approach to fitting item response theory models in psychometrics due to the MML estimator's consistency, normality, and efficiency as the sample size tends to infinity. However, state-of-the-art MML estimation procedures such as the Metropolis-Hastings Robbins-Monro (MH-RM) algorithm as well as approximate MML estimation procedures such as variational inference (VI) are computationally time-consuming when the sample size and the number of latent factors are very large. In this work, we investigate a deep learning-based VI algorithm for exploratory item factor analysis (IFA) that is computationally fast even in large data sets with many latent factors. The proposed approach applies a deep artificial neural network model called an importance-weighted autoencoder (IWAE) for exploratory IFA. The IWAE approximates the MML estimator using an importance sampling technique wherein increasing the number of importance-weighted (IW) samples drawn during fitting improves the approximation, typically at the cost of decreased computational efficiency. We provide a real data application that recovers results aligning with psychological theory across random starts. Via simulation studies, we show that the IWAE yields more accurate estimates as either the sample size or the number of IW samples increases (although factor correlation and intercepts estimates exhibit some bias) and obtains similar results to MH-RM in less time. Our simulations also suggest that the proposed approach performs similarly to and is potentially faster than constrained joint maximum likelihood estimation, a fast procedure that is consistent when the sample size and the number of items simultaneously tend to infinity.

Comparison of different scoring methods based on latent variable models of the PHQ-9: an individual participant data meta-analysis

Abstract: Background Previous research on the depression scale of the Patient Health Questionnaire (PHQ-9) has found that different latent factor models have maximized empirical measures of goodness-of-fit. The clinical relevance of these differences is unclear. We aimed to investigate whether depression screening accuracy may be improved by employing latent factor model-based scoring rather than sum scores. Methods We used an individual participant data meta-analysis (IPDMA) database compiled to assess the screening accuracy of the PHQ-9. We included studies that used the Structured Clinical Interview for DSM (SCID) as a reference standard and split those into calibration and validation datasets. In the calibration dataset, we estimated unidimensional, two-dimensional (separating cognitive/affective and somatic symptoms of depression), and bi-factor models, and the respective cut-offs to maximize combined sensitivity and specificity. In the validation dataset, we assessed the differences in (combined) sensitivity and specificity between the latent variable approaches and the optimal sum score (⩾10), using bootstrapping to estimate 95% confidence intervals for the differences. Results The calibration dataset included 24 studies (4378 participants, 652 major depression cases); the validation dataset 17 studies (4252 participants, 568 cases). In the validation dataset, optimal cut-offs of the unidimensional, two-dimensional, and bi-factor models had higher sensitivity (by 0.036, 0.050, 0.049 points, respectively) but lower specificity (0.017, 0.026, 0.019, respectively) compared to the sum score cut-off of ⩾10. Conclusions In a comprehensive dataset of diagnostic studies, scoring using complex latent variable models do not improve screening accuracy of the PHQ-9 meaningfully as compared to the simple sum score approach.

Learning Deep Latent Variable Models by Short-Run MCMC Inference with Optimal Transport Correction

Abstract: Learning latent variable models with deep top-down architectures typically requires inferring the latent variables for each training example based on the posterior distribution of these latent variables. The inference step typically relies on either time-consuming long-run Markov chain Monte Carlo (MCMC) sampling or a separate inference model for variational learning. In this paper, we propose to use a shortrun MCMC, such as a short-run Langevin dynamics, as an approximate flow-based inference engine. The bias existing in the output distribution of the non-convergent short-run Langevin dynamics is corrected by the optimal transport (OT), which aims at transforming the biased distribution produced by the finite-step MCMC to the prior distribution with a minimum transport cost. Our experiments not only verify the effectiveness of the OT correction for the short-run MCMC, but also demonstrate that the latent variable model trained by the proposed strategy performs better than the variational auto-encoder (VAE) in terms of image reconstruction/generation and anomaly detection.

Variational Knowledge Distillation for Disease Classification in Chest X-Rays.

Abstract: Disease classification relying solely on imaging data attracts great interest in medical image analysis. Current models could be further improved, however, by also employing Electronic Health Records (EHRs), which contain rich information on patients and findings from clinicians. It is challenging to incorporate this information into disease classification due to the high reliance on clinician input in EHRs, limiting thepossibility for automated diagnosis. In this paper, we propose variational knowledge distillation (VKD), which is a new probabilistic inference framework for disease classification based on X-rays that leverages knowledge from EHRs. Specifically, we introduce a conditional latent variable model, where we infer the latent representation of the X-ray image with the variational posterior conditioning on the associated EHR text. By doing so, the model acquires the ability to extract the visual features relevant to the disease during learning and can therefore perform more accurate classification for unseen patients at inference based solely on their X-ray scans. We demonstrate the effectiveness of our method on three public benchmark datasets with paired X-ray images and EHRs. The results show that the proposed variational knowledge distillation can consistently improve the performance of medical image classification and significantly surpasses current methods.

R-squared change in structural equation models with latent variables and missing data.

Abstract: Researchers frequently wish to make incremental validity claims, suggesting that a construct of interest significantly predicts a given outcome when controlling for other overlapping constructs and potential confounders Once the significance of such an effect has been established, it is good practice to also assess and report its magnitude In OLS regression, this is easily accomplished by calculating the change in R-squared, ΔR2, between one's full model and a reduced model that omits all but the target predictor(s) of interest Because observed variable regression methods ignore measurement error, however, their estimates are prone to bias and inflated type I error rates As a result, researchers are increasingly encouraged to switch from observed variable modeling conducted in the regression framework to latent variable modeling conducted in the structural equation modeling (SEM) framework Standard SEM software packages provide overall R2 measures for each outcome, yet calculation of ΔR2 is not intuitive in models with latent variables Omitting all indicators of a latent factor in a reduced model will alter the overidentifying constraints imposed on the model, affecting parameter estimation and fit Furthermore, omitting variables in a reduced model may affect estimation under missing data, particularly when conditioning on those variables is essential to meeting the MAR assumption In this article, I describe four approaches to calculating ΔR2 in SEMs with latent variables and missing data, compare their performance via simulation, describe a set of extensions to the methods, and provide a set of R functions for calculating ΔR2 in SEM

Exploring State Transition Uncertainty in Variational Reinforcement Learning

Abstract: Model-free agent in reinforcement learning (RL) generally performs well but inefficient in training process with sparse data. A practical solution is to incorporate a model-based module in model-free agent. State transition can be learned to make desirable prediction of next state based on current state and action at each time step. This paper presents a new learning representation for variational RL by introducing the so-called transition uncertainty critic based on the variational encoder-decoder network where the uncertainty of structured state transition is encoded in a model-based agent. In particular, an action-gating mechanism is carried out to learn and decode the trajectory of actions and state transitions in latent variable space. The transition uncertainty maximizing exploration (TUME) is performed according to the entropy search by using the intrinsic reward based on the uncertainty measure corresponding to different states and actions. A dedicate latent variable model with a penalty using the bias of state-action value is developed. Experiments on Cart Pole and dialogue system show that the proposed TUME considerably performs better than the other exploration methods for reinforcement learning.

Latent Variable Modeling and Adaptive Testing for Experimental Cognitive Psychopathology Research.

Abstract: The adaptation of experimental cognitive tasks into measures that can be used to quantify neurocognitive outcomes in translational studies and clinical trials has become a key component of the stra...

Linear latent variable regression (LVR)-based process monitoring

Abstract: Fast-paced developments in data acquisition, instrumentation technology and the era of the Internet-of-Things have resulted in large amounts of data produced by modern industrial processes. The ability to extract useful information from these multivariate datasets has vital benefits that could be utilized in process monitoring. In the absence of a physics-based process model, data-driven approaches such as latent variable modeling have proved to be practical for process monitoring over the past four decades. The aim of this chapter is to review and show the challenges in multivariate process monitoring based on linear models. Specifically, after presenting the limitations of the full-rank regression model, we provide a brief overview of linear latent variable models such as principal component analysis, principal component regression, and partial least squares regression. To deal with dynamic systems, we present dynamic extensions of these methods that capture both static and dynamic features in multivariate processes. We then provide a brief overview of univariate monitoring schemes, such as exponentially-weighted moving average and cumulative sum and generalized likelihood ratio monitoring schemes and their multivariate counterparts. To apply such tools to multivariate data, we employ appropriate multivariate dimension-reduction techniques according to the features of a process, and we use monitoring schemes to monitor more informative variables in a lower dimension. Next, we aim to identify which process variables contribute to abnormal change; conventional contribution plots and radial visualization tool are briefed. Lastly, the effectiveness of the presented inferential modeling techniques is assessed using simulated data. We also present a study on monitoring influent measurements at a water resource recovery facility. Finally, we discuss limitations of the presented monitoring approaches and give some possible directions to rectify these limitations.

Employing a latent variable framework to improve efficiency in composite endpoint analysis

Abstract: Composite endpoints that combine multiple outcomes on different scales are common in clinical trials, particularly in chronic conditions. In many of these cases, patients will have to cross a predefined responder threshold in each of the outcomes to be classed as a responder overall. One instance of this occurs in systemic lupus erythematosus, where the responder endpoint combines two continuous, one ordinal and one binary measure. The overall binary responder endpoint is typically analysed using logistic regression, resulting in a substantial loss of information. We propose a latent variable model for the systemic lupus erythematosus endpoint, which assumes that the discrete outcomes are manifestations of latent continuous measures and can proceed to jointly model the components of the composite. We perform a simulation study and find that the method offers large efficiency gains over the standard analysis, the magnitude of which is highly dependent on the components driving response. Bias is introduced when joint normality assumptions are not satisfied, which we correct for using a bootstrap procedure. The method is applied to the Phase IIb MUSE trial in patients with moderate to severe systemic lupus erythematosus. We show that it estimates the treatment effect 2.5 times more precisely, offering a 60% reduction in required sample size.