Pretreatment Covariates Research Articles

Treatment response amplitude and timing in chronic inflammatory demyelinating polyneuropathy with routine care: Study of a UK cohort.

The amplitude, timing, and determinants of improvement with available treatments are uncertain in chronic inflammatory demyelinating polyneuropathy (CIDP). Our primary objective was to quantify categorized outcomes with routine care. We retrospectively studied treatment response within 36 months from initiation in 112 consecutive subjects with CIDP. Response was classified into a proposed new "CIDP treatment-response category" (CT-RC), based on achieved endpoints. Determinants of the CT-RC, of timing of maximum improvement, and of treatment discontinuation were ascertained. The CT-RC demonstrated high concurrent validity with current outcome measures. Thirty-six subjects (32.1%) achieved a "complete response," 37 (33%) a "good partial response," 10 (8.9%) a "moderate partial response," and 15 (13.4%) a "poor partial response." Fourteen subjects (12.5%) were "nonresponsive." The CT-RC was independently predicted only by age. Mean time to maximum improvement was 12.1 months (range = 1-36) and was not associated with any pretreatment covariate. Treatment discontinuation occurred in 24 of 62 (38.2%) partial responders and was only associated with shorter pretreatment disease duration. Nonresponders were older and received a similar number of treatments compared to responders. CT-RC classification indicates persistent disability in >60% of treatment responders in CIDP. Timing of maximum improvement is variable, frequently delayed, and unpredictable. Treatment withdrawal without deterioration is achievable in approximately 40% of subjects and may be more likely with prompt treatment. Treatment withdrawal in partial responders and limited escalation in nonresponders suggest implication of physician- and patient-related factors in suboptimal response. More effective treatments/treatment methods and better understanding of other factors influencing response are needed in CIDP.

The benefit of adjuvant chemotherapy following pancreaticoduodenectomy for pancreatic adenocarcinoma depends on response to neoadjuvant therapy.

The benefit of adjuvant therapy (AT) remains unclear in pancreatic ductal adenocarcinoma (PDAC) after neoadjuvant therapy (NAT) and surgical resection. The 2019 National Cancer Database was queried for patients with non-metastatic PDAC who received NAT followed by pancreaticoduodenectomy. Only patients with data regarding receipt of AT were included. Patients were classified if they had nodal down-staging specifically, or any downstaging (Tumor, Nodal, or overall). Propensity score matching (PSM) adjusted for pretreatment covariate imbalance between groups. The weighted Kaplan-Meier method and log-rank test were used to estimate the cumulative survival. After exclusion criteria and PSM, a total of 2784 patients remained; 1689 (60.7%) received AT and 1095 (39.3%) did not receive AT. Among all, those with additional AT had a significantly improved overall survival (OS) (p < 0.001). Upon evaluation of patients without downstaging after NAT, those who received AT had improved OS (no nodal downstaging or any downstaging; p = 0.002; p = 0.001). When evaluating patients with downstaging after NAT, those receiving AT did not have improved OS (nodal downstaging or any downstaging: p = 0.352; p = 0.99). Response to NAT appears to correlate with the benefit of AT following pancreaticoduodenectomy; patients who have a favorable response to NAT may not benefit from AT.

An Analysis of the Effect of Streaming on Civic Participation Through a Causal Hidden Markov Model

We examine the effect of streaming based on ability levels on individuals’ civic participation throughout their adult life. The hypothesis we test is that ability grouping influences individuals’ general self-concept and, consequently, their civic participation choices across the life course. We employ data from the British National Child Development Study, which follows all UK citizens born during a certain week in 1958. Six binary variables observed at 33, 42, and 51 years of age are considered to measure civic participation. Our approach defines causal estimands with multiple treatments referring to the evolution of civic engagement over time in terms of potential versions of a sequence of latent variables assumed to follow a Markov chain with initial and transition probabilities depending on posttreatment time-varying covariates. The model also addresses partially or entirely missing data on one or more indicators at a given time occasion and missing posttreatment covariate values using dummy indicators. The model is estimated by maximizing a weighted log-likelihood function with weights corresponding to the inverse probability of the received treatment obtained from a multinomial logit model based on pretreatment covariates. Our results show that ability grouping affects the civic participation of high-ability individuals when they are 33 years old with respect to participation in general elections.

Total Knee Arthroplasty Versus Education and Exercise for Knee Osteoarthritis: A Propensity-Matched Analysis.

We estimate the treatment effect of total knee arthroplasty (TKA) versus an education and exercise (Edu+Ex) program on pain, function, and quality of life outcomes 3 and 12 months after treatment initiation for knee osteoarthritis (OA). Patients with knee OA who had undergone TKA were matched on a 1:1 ratio with participants in an Edu+Ex program based on a propensity score fitted to a range of pretreatment covariates. After matching, between-group differences in improvement (the treatment effect) in Knee Injury and Osteoarthritis Outcome Score 12-item version (0, worst to 100, best) pain, function, and quality of life from baseline to 3 and 12 months were estimated using linear mixed models, adjusting for unbalanced covariates, if any, after matching. The matched sample consisted of 522 patients (Edu+Ex, n=261; TKA, n=261) who were balanced on all pretreatment characteristics. At 12-month follow-up, TKA resulted in significantly greater improvements in pain (mean difference [MD] 22.8; 95% confidence interval [95% CI] 19.7-25.8), function (MD 21.2; 95% CI 17.7-24.4), and quality of life (MD 18.3; 15.0-21.6). Even so, at least one-third of patients receiving Edu+Ex had a clinically meaningful improvement in outcomes at 12 months compared with 75% of patients with TKA. TKA is associated with greater improvements in pain, function, and quality of life, but these findings also suggest that Edu+Ex may be a viable alternative to TKA in a meaningful proportion of patients, which may reduce overall TKA need. Confirmatory trials are needed.

CTNI-17. APPLICATION OF THE GBM-X DATA PLATFORM TO ESTIMATE TREATMENT EFFECTS OF THE EXPERIMENTAL THERAPIES OF THE RANDOMIZED PHASE 2 INDIVIDUALIZED SCREENING TRIAL OF INNOVATIVE GLIOBLASTOMA THERAPIES

Abstract BACKGROUND Leveraging external data has the potential to increase efficiency and precision of drug development in glioblastoma by complementing or replacing data from in-trial controls. We used an external control arm (ECA) to re-analyze data of the abemaciclib, neratinib and CC-115 arms of the Individualized Screening Trial of Innovative Glioblastoma Therapy (INSIGhT) trial. METHODS INSIGhT (NCT02977780) is an ongoing phase 2 adaptive platform trial in newly diagnosed MGMT unmethylated glioblastoma with a shared control arm of patients receiving radiation therapy with concurrent and maintenance temozolomide. The initial experimental arms (abemaciclib, neratinib, CC-115) have not shown a survival benefit relative to the shared control arm. We re-analyzed the experimental arm data and replaced the shared control arm with an ECA. The ECA was developed from a collection of trial datasets and real-world data available through the Glioblastoma External (GBM-X) Data Platform (). Propensity score matching was used to adjust for possible differences between pretreatment covariates (age, sex, resection, KPS) in the ECA and shared control arm. Cox proportional hazards models were used to estimate treatment effects. RESULTS On INSIGhT, patients were randomized to abemaciclib (n = 74), neratinib (n = 80), and CC-115 (n = 12). There were 687 unmethylated MGMT patients in the GBM-X ECA. After propensity score matching, there was no significant treatment effect associated with abemaciclib (HR 1.091, 95%CI:0.834-1.433, p = 0.536), neratinib (HR = 1.043, 95%CI:0.787-1.383, p = 0.770), or CC-115 (HR 1.088, 95%CI:0.834-1.433, p = 0.794) compared to the ECA. CONCLUSION In comparison to an ECA, there was no significant survival benefit with abemaciclib, neratinib and CC-115. Treatment effects estimates were similar to the primary analyses based on the original shared control arm of INSIGhT. The use of external control arms in early phase testing of new therapies is supported by our findings and could significantly reduce costs and time to conduct trials in newly diagnosed glioblastoma.

Convolutional Neural Networks for Valid and Efficient Causal Inference

Convolutional neural networks (CNN) have been successful in machine learning applications. Their success relies on their ability to consider space invariant local features. We consider the use of CNN to fit nuisance models in semiparametric estimation of the average causal effect of a treatment. In this setting, nuisance models are functions of pretreatment covariates that need to be controlled for. In an application where we want to estimate the effect of early retirement on a health outcome, we propose to use CNN to control for time-structured covariates. Thus, CNN is used when fitting nuisance models explaining the treatment and the outcome. These fits are then combined into an augmented inverse probability weighting estimator yielding efficient and uniformly valid inference. Theoretically, we contribute by providing rates of convergence for CNN equipped with the rectified linear unit activation function and compare it to an existing result for feedforward neural networks. We also show when those rates guarantee uniformly valid inference. A Monte Carlo study is provided where the performance of the proposed estimator is evaluated and compared with other strategies. Finally, we give results on a study of the effect of early retirement on hospitalization using data covering the whole Swedish population. Supplementary materials are available online at https://github.com/stat4reg/Causal/_CNN.

Treatment Effect Risk: Bounds and Inference

Because the average treatment effect (ATE) measures the change in social welfare, even if positive, there is a risk of negative effect on, say, some 10% of the population. Assessing such risk is difficult, however, because any one individual treatment effect (ITE) is never observed, so the 10% worst-affected cannot be identified, whereas distributional treatment effects only compare the first deciles within each treatment group, which does not correspond to any 10% subpopulation. In this paper, we consider how to nonetheless assess this important risk measure, formalized as the conditional value at risk (CVaR) of the ITE distribution. We leverage the availability of pretreatment covariates and characterize the tightest possible upper and lower bounds on ITE-CVaR given by the covariate-conditional average treatment effect (CATE) function. We then proceed to study how to estimate these bounds efficiently from data and construct confidence intervals. This is challenging even in randomized experiments as it requires understanding the distribution of the unknown CATE function, which can be very complex if we use rich covariates to best control for heterogeneity. We develop a debiasing method that overcomes this and prove it enjoys favorable statistical properties even when CATE and other nuisances are estimated by black box machine learning or even inconsistently. Studying a hypothetical change to French job search counseling services, our bounds and inference demonstrate a small social benefit entails a negative impact on a substantial subpopulation. This paper was accepted by J. George Shanthikumar, data science. Funding: This work was supported by the Division of Information and Intelligent Systems [Grant 1939704]. Supplemental Material: The data files and online appendices are available at https://doi.org/10.1287/mnsc.2023.4819 .

One-step targeted maximum likelihood estimation for targeting cause-specific absolute risks and survival curves

Summary This paper considers the one-step targeted maximum likelihood estimation methodology for multi-dimensional causal parameters in general survival and competing risk settings where event times take place on the positive real line and are subject to right censoring. We focus on effects of baseline treatment decisions possibly confounded by pretreatment covariates, but remark that our work generalizes to settings with time-varying treatment regimes and time-dependent confounding. We point out two overall contributions of our work. First, our methods can be used to obtain simultaneous inference for treatment effects on multiple absolute risks in competing risk settings. Second, our methods can be used to achieve inference for the full survival curve, or a full absolute risk curve, across time. The one-step targeted maximum likelihood procedure is based on a one-dimensional universal least favourable submodel for each cause-specific hazard that we implement in recursive steps along a corresponding nonuniversal multivariate least favourable submodel. Our empirical study demonstrates the practical use of the methods.

PD44-08 BENEFIT OF WHOLE-PELVIS RADIATION FOR PATIENTS WITH MUSCLE-INVASIVE BLADDER CANCER: AN INVERSE PROBABILITY TREATMENT WEIGHTED ANALYSIS

PD44-08 BENEFIT OF WHOLE-PELVIS RADIATION FOR PATIENTS WITH MUSCLE-INVASIVE BLADDER CANCER: AN INVERSE PROBABILITY TREATMENT WEIGHTED ANALYSIS

Towards Optimal Variance Reduction in Online Controlled Experiments

We study optimal variance reduction solutions for count and ratio metrics in online controlled experiments. Our methods apply flexible machine learning tools to incorporate covariates that are independent from the treatment but have predictive power for the outcomes, and employ the cross-fitting technique to remove the bias in complex machine learning models. We establish CLT-type asymptotic inference based on our estimators under mild convergence conditions. Our procedures are optimal (efficient) for the corresponding targets as long as the machine learning estimators are consistent, without any requirement for their convergence rates. In complement to the general optimal procedure, we also derive a linear adjustment method for ratio metrics as a special case that is computationally efficient and can flexibly incorporate any pre-treatment covariates. We evaluate the proposed variance reduction procedures with comprehensive simulation studies and provide practical suggestions regarding commonly adopted assumptions in computing ratio metrics. When tested on real online experiment data from LinkedIn, the proposed optimal procedure for ratio metrics can reduce up to 80% of variance compared to the standard difference-in-mean estimator and also further reduce up to 30% of variance compared to the CUPED approach by going beyond linearity and incorporating a large number of extra covariates.

To Adjust or not to Adjust? Estimating the Average Treatment Effect in Randomized Experiments with Missing Covariates

Randomized experiments allow for consistent estimation of the average treatment effect based on the difference in mean outcomes without strong modeling assumptions. Appropriate use of pretreatment covariates can further improve the estimation efficiency. Missingness in covariates is nevertheless common in practice, and raises an important question: should we adjust for covariates subject to missingness, and if so, how? The unadjusted difference in means is always unbiased. The complete-covariate analysis adjusts for all completely observed covariates, and is asymptotically more efficient than the difference in means if at least one completely observed covariate is predictive of the outcome. Then what is the additional gain of adjusting for covariates subject to missingness? To reconcile the conflicting recommendations in the literature, we analyze and compare five strategies for handling missing covariates in randomized experiments under the design-based framework, and recommend the missingness-indicator method, as a known but not so popular strategy in the literature, due to its multiple advantages. First, it removes the dependence of the regression-adjusted estimators on the imputed values for the missing covariates. Second, it does not require modeling the missingness mechanism, and yields consistent estimators even when the missingness mechanism is related to the missing covariates and unobservable potential outcomes. Third, it ensures large-sample efficiency over the complete-covariate analysis and the analysis based on only the imputed covariates. Lastly, it is easy to implement via least squares. We also propose modifications to it based on asymptotic and finite sample considerations. Importantly, our theory views randomization as the basis for inference, and does not impose any modeling assumptions on the data-generating process or missingness mechanism. Supplementary materials for this article are available online.

A flexible approach for causal inference with multiple treatments and clustered survival outcomes.

When drawing causal inferences about the effects of multiple treatments on clustered survival outcomes using observational data, we need to address implications of the multilevel data structure, multiple treatments, censoring, and unmeasured confounding for causal analyses. Few off-the-shelf causal inference tools are available to simultaneously tackle these issues. We develop a flexible random-intercept accelerated failure time model, in which we use Bayesian additive regression trees to capture arbitrarily complex relationships between censored survival times and pre-treatment covariates and use the random intercepts to capture cluster-specific main effects. We develop an efficient Markov chain Monte Carlo algorithm to draw posterior inferences about the population survival effects of multiple treatments and examine the variability in cluster-level effects. We further propose an interpretable sensitivity analysis approach to evaluate the sensitivity of drawn causal inferences about treatment effect to the potential magnitude of departure from the causal assumption of no unmeasured confounding. Expansive simulations empirically validate and demonstrate good practical operating characteristics of our proposed methods. Applying the proposed methods to a dataset on older high-risk localized prostate cancer patients drawn from the National Cancer Database, we evaluate the comparative effects of three treatment approaches on patient survival, and assess the ramifications of potential unmeasured confounding. The methods developed in this work are readily available in the package .

Solutions for surrogacy validation with longitudinal outcomes for a gene therapy.

Valid surrogate endpoints S can be used as a substitute for a true outcome of interest T to measure treatment efficacy in a clinical trial. We propose a causal inference approach to validate a surrogate by incorporating longitudinal measurements of the true outcomes using a mixed modeling approach, and we define models and quantities for validation that may vary across the study period using principal surrogacy criteria. We consider a surrogate-dependent treatment efficacy curve that allows us to validate the surrogate at different time points. We extend these methods to accommodate a delayed-start treatment design where all patients eventually receive the treatment. Not all parameters are identified in the general setting. We apply a Bayesian approach for estimation and inference, utilizing more informative prior distributions for selected parameters. We consider the sensitivity of these prior assumptions as well as assumptions of independence among certain counterfactual quantities conditional on pretreatment covariates to improve identifiability. We examine the frequentist properties (bias of point and variance estimates, credible interval coverage) of a Bayesian imputation method. Our work is motivated by a clinical trial of a gene therapy where the functional outcomes are measured repeatedly throughout thetrial.

Posttreatment confounding in causal mediation studies: A cutting-edge problem and a novel solution via sensitivity analysis.

In causal mediation studies that decompose an average treatment effect into indirect and direct effects, examples of posttreatment confounding are abundant. In the presence of treatment-by-mediator interactions, past research has generally considered it infeasible to adjust for a posttreatment confounder of the mediator-outcome relationship due to incomplete information: for any given individual, a posttreatment confounder is observed under the actual treatment condition while missing under the counterfactual treatment condition. This paper proposes a new sensitivity analysis strategy for handling posttreatment confounding and incorporates it into weighting-based causal mediation analysis. The key is to obtain the conditional distribution of the posttreatment confounder under the counterfactual treatment as a function of not only pretreatment covariates but also its counterpart under the actual treatment. The sensitivity analysis then generates a bound for the natural indirect effect and that for the natural direct effect over a plausible range of the conditional correlation between the posttreatment confounder under the actual and that under the counterfactual conditions. Implemented through either imputation or integration, the strategy is suitable for binary as well as continuous measures of posttreatment confounders. Simulation results demonstrate major strengths and potential limitations of this new solution. A reanalysis of the National Evaluation of Welfare-to-Work Strategies (NEWWS) Riverside data reveals that the initial analytic results are sensitive to omitted posttreatment confounding.

Observational Study of the Effect of the Juvenile Stay-At-Home Order on SARS-CoV-2 Infection Spread in Saline County, Arkansas

We assess the treatment effect of juvenile stay-at-home orders (JSAHO) on reducing the rate of SARS-CoV-2 infection spread in Saline County (”Saline”), Arkansas, by examining the difference between Saline’s and control Arkansas counties’ changes in daily and mean log infection rates of pretreatment (March 28–April 5, 2020) and treatment periods (April 6–May 6, 2020). Synthetic control county is constructed based on the parallel-trends assumption, least-squares fitting on pretreatment and socio-demographic covariates, and elastic-net-based methods, from which the counterfactual outcome is predicted and the treatment effect is estimated using the difference-in-differences, the synthetic control, and the changes-in-changes methodologies. Both the daily and average treatment effects of JSAHO are shown to be significant. Despite its narrow scope and lack of enforcement for compliance, JSAHO reduced the rate of the infection spread in Saline.

Comparative effectiveness of common treatments for new-onset atrial fibrillation within the ICU: Accounting for physiological status.

BackgroundNew-onset atrial fibrillation (NOAF) is common in patients on an intensive care unit (ICU). Evidence guiding treatments is limited, though recent reports suggest beta blocker (BB) therapy is associated with reduced mortality.MethodsWe conducted a multicentre cohort study of adult patients admitted to 3 ICUs in the UK and 5 ICUs in the USA. We analysed the haemodynamic changes associated with NOAF. We analysed rate control, rhythm control, and hospital mortality associated with common NOAF treatments. We balanced admission and post-NOAF, pre-treatment covariates across treatment groups.ResultsNOAF was followed by a systolic blood pressure reduction of 5 mmHg (p < 0.001). After adjustment, digoxin therapy was associated with inferior rate control versus amiodarone (adjusted hazard ratio (aHR) 0.56, [95% CI 0.34–0.92]). Calcium channel blocker (CCB) therapy was associated with inferior rhythm control versus amiodarone (aHR 0.59 (0.37–0.92). No difference was detected between BBs and amiodarone in rate control (aHR 1.15 [0.91–1.46]), rhythm control (aHR 0.85, [0.69–1.05]), or hospital mortality (aHR 1.03 [0.53–2.03]).ConclusionsNOAF in ICU patients is followed by decreases in blood pressure. BBs and amiodarone are associated with similar cardiovascular control and appear superior to digoxin and CCBs. Accounting for key confounders removes previously reported mortality benefits associated with BB treatment.

Movement-based patient-therapist attunement in psychological therapy and its association with early change.

ObjectiveAttunement is a novel measure of nonverbal synchrony reflecting the duration of the present moment shared by two interaction partners. This study examined its association with early change in outpatient psychotherapy.MethodsAutomated video analysis based on motion energy analysis (MEA) and cross-correlation of the movement time-series of patient and therapist was conducted to calculate movement synchrony for N = 161 outpatients. Movement-based attunement was defined as the range of connected time lags with significant synchrony. Latent change classes in the HSCL-11 were identified with growth mixture modeling (GMM) and predicted by pre-treatment covariates and attunement using multilevel multinomial regression.ResultsGMM identified four latent classes: high impairment, no change (Class 1); high impairment, early response (Class 2); moderate impairment (Class 3); and low impairment (Class 4). Class 2 showed the strongest attunement, the largest early response, and the best outcome. Stronger attunement was associated with a higher likelihood of membership in Class 2 (b = 0.313, p = .007), Class 3 (b = 0.251, p = .033), and Class 4 (b = 0.275, p = .043) compared to Class 1. For highly impaired patients, the probability of no early change (Class 1) decreased and the probability of early response (Class 2) increased as a function of attunement.ConclusionsAmong patients with high impairment, stronger patient-therapist attunement was associated with early response, which predicted a better treatment outcome. Video-based assessment of attunement might provide new information for therapists not available from self-report questionnaires and support therapists in their clinical decision-making.

In Search of Inspiration: External Mobility and the Emergence of Technology Intrapreneurs

Recent scholarship has established several ways in which external hiring—versus filling a role with a comparable internal candidate—is detrimental to firms. Yet, organizational learning theory suggests that external hires benefit firms: by importing knowledge that is unavailable or obscured to insiders and applying it toward experimentation and risky recombination. Accordingly and consistent with studies of learning by hiring and innovation, we predict that external hires are at greater risk of intrapreneurship than internal hires. We test this prediction via a study of product managers in large technology companies. We use machine learning to operationalize intrapreneurship by comparing product manager job descriptions with the founding statements of venture-backed technology entrepreneurs. Our research design employs coarsened exact matching to balance pretreatment covariates between product managers who arrived at their roles internally versus externally. The results of our analysis indicate that externally hired product managers are substantially more intrapreneurial than observably equivalent internal hires. However, we also find that intrapreneurial product managers have a higher turnover rate, an effect that is primarily driven by external hires. This suggests that hiring for intrapreneurship may be a difficult strategy to sustain.Funding: The authors acknowledge financial support from London Business School, the National University of Singapore [Grant WBS R-313-000-128-133], and a dissertation grant from the Ewing Kauffman Foundation, Kansas City, Missouri.Supplemental Material: The online appendix is available at https://doi.org/10.1287/orsc.2021.1530 .

Optimal Covariate Balancing Conditions in Propensity Score Estimation

Inverse probability of treatment weighting (IPTW) is a popular method for estimating the average treatment effect (ATE). However, empirical studies show that the IPTW estimators can be sensitive to the misspecification of the propensity score model. To address this problem, researchers have proposed to estimate propensity score by directly optimizing the balance of pretreatment covariates. While these methods appear to empirically perform well, little is known about how the choice of balancing conditions affects their theoretical properties. To fill this gap, we first characterize the asymptotic bias and efficiency of the IPTW estimator based on the covariate balancing propensity score (CBPS) methodology under local model misspecification. Based on this analysis, we show how to optimally choose the covariate balancing functions and propose an optimal CBPS-based IPTW estimator. This estimator is doubly robust; it is consistent for the ATE if either the propensity score model or the outcome model is correct. In addition, the proposed estimator is locally semiparametric efficient when both models are correctly specified. To further relax the parametric assumptions, we extend our method by using a sieve estimation approach. We show that the resulting estimator is globally efficient under a set of much weaker assumptions and has a smaller asymptotic bias than the existing estimators. Finally, we evaluate the finite sample performance of the proposed estimators via simulation and empirical studies. An open-source software package is available for implementing the proposed methods.

Functional additive models for optimizing individualized treatment rules.

A novel functional additive model is proposed, which is uniquely modified and constrained to model nonlinear interactions between a treatment indicator and a potentially large number of functional and/or scalar pretreatment covariates. The primary motivation for this approach is to optimize individualized treatment rules based on data from a randomized clinical trial. We generalize functional additive regression models by incorporating treatment-specific components into additive effect components. A structural constraint is imposed on the treatment-specific components in order to provide a class of additive models with main effects and interaction effects that are orthogonal to each other. If primary interest is in the interaction between treatment and the covariates, as is generally the case when optimizing individualized treatment rules, we can thereby circumvent the need to estimate the main effects of the covariates, obviating the need to specify their form and thus avoiding the issue of model misspecification. The methods are illustrated with data from a depression clinical trial with electroencephalogram functional data as patients' pretreatmentcovariates.