Inverse probability of treatment weighting with generalized linear outcome models for doubly robust estimation.

There are now many options for doubly robust estimation; however, there is a concerning trend in the applied literature to believe that the combination of a propensity score and an adjusted outcome model automatically results in a doubly robust estimator and/or to misuse more complex established doubly robust estimators. A simple alternative, canonical link generalized linear models (GLM) fit via inverse probability of treatment (propensity score) weighted maximum likelihood estimation followed by standardization (the g $$ g $$ -formula) for the average causal effect, is a doubly robust estimation method. Our aim is for the reader not just to be able to use this method, which we refer to as IPTW GLM, for doubly robust estimation, but to fully understand why it has the doubly robust property. For this reason, we define clearly, and in multiple ways, all concepts needed to understand the method and why it is doubly robust. In addition, we want to make very clear that the mere combination of propensity score weighting and an adjusted outcome model does not generally result in a doubly robust estimator. Finally, we hope to dispel the misconception that one can adjust for residual confounding remaining after propensity score weighting by adjusting in the outcome model for what remains 'unbalanced' even when using doubly robust estimators. We provide R code for our simulations and real open-source data examples that can be followed step-by-step to use and hopefully understand the IPTW GLM method. We also compare to a much better-known but still simple doubly robust estimator.

Estimation of separable direct and indirect effects in a continuous-time illness-death model.

In this article we study the effect of a baseline exposure on a terminal time-to-event outcome either directly or mediated by the illness state of a continuous-time illness-death process with baseline covariates. We propose a definition of the corresponding direct and indirect effects using the concept of separable (interventionist) effects (Robins and Richardson in Causality and psychopathology: finding the determinants of disorders and their cures, Oxford University Press, 2011; Robins et al. in arXiv:2008.06019 , 2021; Stensrud et al. in J Am Stat Assoc 117:175-183, 2022). Our proposal generalizes Martinussen and Stensrud (Biometrics 79:127-139, 2023) who consider similar causal estimands for disentangling the causal treatment effects on the event of interest and competing events in the standard continuous-time competing risk model. Unlike natural direct and indirect effects (Robins and Greenland in Epidemiology 3:143-155, 1992; Pearl in Proceedings of the seventeenth conference on uncertainty in artificial intelligence, Morgan Kaufmann, 2001) which are usually defined through manipulations of the mediator independently of the exposure (so-called cross-world interventions), separable direct and indirect effects are defined through interventions on different components of the exposure that exert their effects through distinct causal mechanisms. This approach allows us to define meaningful mediation targets even though the mediating event is truncated by the terminal event. We present the conditions for identifiability, which include some arguably restrictive structural assumptions on the treatment mechanism, and discuss when such assumptions are valid. The identifying functionals are used to construct plug-in estimators for the separable direct and indirect effects. We also present multiply robust and asymptotically efficient estimators based on the efficient influence functions. We verify the theoretical properties of the estimators in a simulation study, and we demonstrate the use of the estimators using data from a Danish registry study.

Estimation of separable direct and indirect effects in continuous time.

Many research questions involve time-to-event outcomes that can be prevented from occurring due to competing events. In these settings, we must be careful about the causal interpretation of classical statistical estimands. In particular, estimands on the hazard scale, such as ratios of cause-specific or subdistribution hazards, are fundamentally hard to interpret causally. Estimands on the risk scale, such as contrasts of cumulative incidence functions, do have a clear causal interpretation, but they only capture the total effect of the treatment on the event of interest; that is, effects both through and outside of the competing event. To disentangle causal treatment effects on the event of interest and competing events, the separable direct and indirect effects were recently introduced. Here we provide new results on the estimation of direct and indirect separable effects in continuous time. In particular, we derive the nonparametric influence function in continuous time and use it to construct an estimator that has certain robustness properties. We also propose a simple estimator based on semiparametric models for the two cause-specific hazard functions. We describe the asymptotic properties of these estimators and present results from simulation studies, suggesting that the estimators behave satisfactorily in finite samples. Finally, we reanalyze the prostate cancer trial from Stensrud et al. (2020).

Discussion of "A formal causal interpretation of the case-crossover design".

Discussion on "A formal causal interpretation of the case-crossover design" by Zach Shahn, Miguel A. Hernan, and James M. Robins.

Instrumental variable estimation of the causal hazard ratio.

Cox's proportional hazards model is one of the most popular statistical models to evaluate associations of exposure with a censored failure time outcome. When confounding factors are not fully observed, the exposure hazard ratio estimated using a Cox model is subject to unmeasured confounding bias. To address this, we propose a novel approach for the identification and estimation of the causal hazard ratio in the presence of unmeasured confounding factors. Our approach is based on a binary instrumental variable, and an additional no-interaction assumption in a first-stage regression of the treatment on the IV and unmeasured confounders. We propose, to the best of our knowledge, the first consistent estimator of the (population) causal hazard ratio within an instrumental variable framework. A version of our estimator admits a closed-form representation. We derive the asymptotic distribution of our estimator and provide a consistent estimator for its asymptotic variance. Our approach is illustrated via simulation studies and a data application.

Rejoinder to discussions on "Instrumental variable estimation of the causal hazard ratio".

In this paper, we respond to comments on our paper, "Instrumental variable estimation of the causal hazard ratio."

Instrumental variables estimation with competing risk data.

Time-to-event analyses are often plagued by both-possibly unmeasured-confounding and competing risks. To deal with the former, the use of instrumental variables (IVs) for effect estimation is rapidly gaining ground. We show how to make use of such variables in competing risk analyses. In particular, we show how to infer the effect of an arbitrary exposure on cause-specific hazard functions under a semi-parametric model that imposes relatively weak restrictions on the observed data distribution. The proposed approach is flexible accommodating exposures and IVs of arbitrary type, and enabling covariate adjustment. It makes use of closed-form estimators that can be recursively calculated, and is shown to perform well in simulation studies. We also demonstrate its use in an application on the effect of mammography screening on the risk of dying from breast cancer.

Analysis of time-to-event for observational studies: Guidance to the use of intensity models.

This paper provides guidance for researchers with some mathematical background on the conduct of time-to-event analysis in observational studies based on intensity (hazard) models. Discussions of basic concepts like time axis, event definition and censoring are given. Hazard models are introduced, with special emphasis on the Cox proportional hazards regression model. We provide check lists that may be useful both when fitting the model and assessing its goodness of fit and when interpreting the results. Special attention is paid to how to avoid problems with immortal time bias by introducing time-dependent covariates. We discuss prediction based on hazard models and difficulties when attempting to draw proper causal conclusions from such models. Finally, we present a series of examples where the methods and check lists are exemplified. Computational details and implementation using the freely available R software are documented in Supplementary Material. The paper was prepared as part of the STRATOS initiative.

Instrumental variables estimation under a structural Cox model.

Instrumental variable (IV) analysis is an increasingly popular tool for inferring the effect of an exposure on an outcome, as witnessed by the growing number of IV applications in epidemiology, for instance. The majority of IV analyses of time-to-event endpoints are, however, dominated by heuristic approaches. More rigorous proposals have either sidestepped the Cox model, or considered it within a restrictive context with dichotomous exposure and instrument, amongst other limitations. The aim of this article is to reconsider IV estimation under a structural Cox model, allowing for arbitrary exposure and instruments. We propose a novel class of estimators and derive their asymptotic properties. The methodology is illustrated using two real data applications, and using simulated data.

Ovarian reserve markers after discontinuing long-term use of combined oral contraceptives.

RESEARCH QUESTION: How early do the ovarian reserve markers anti-Müllerian hormone (AMH) and antral follicle count (AFC) normalize after discontinuation of the long-term use of combined oral contraceptives (COC). DESIGN: This was a prospective cohort study of 68 women with a history of long-term COC use. Serum AMH concentrations, ovarian volume and AFC were measured during COC use and serially in a 4-month period after discontinuing COC: 1 and 2 weeks after discontinuation, and on cycle day 2-5 during three consecutive menstrual cycles. Changes in AMH and AFC were investigated using linear mixed models of repeated measurements adjusted for relevant covariates. RESULTS: Mean age was 29.4 years and mean duration of COC use 8.0 years. Baseline median AMH concentrations during COC use of 13 pmol/l (interquartile range [IQR] 8.4-22 pmol/l) increased to a median of 22.5 pmol/l (IQR 11-37 mol/l) 3 months after discontinuation. The estimated average increase was 53% (95% confidence interval [CI] 1.40-1.68, P < 0.001). AFC increased from a median value of 17 (IQR 11-25) to 24 (IQR 17-34). The estimated average increase was 41% (95% CI 1.30-1.52, P < 0.001). Ovarian volume increased from 2.4 to 5.8 ml (P < 0.001). The ovarian reserve markers increased continuously from baseline measurements until 2 months after discontinuation. Thereafter a plateau was reached. CONCLUSION: After discontinuation of COC, AMH increased by 53% and AFC by 41%, with values returning to normal within 2 months. This study provides clinicians with the highly relevant knowledge that AMH and AFC can be measured 2 months after discontinuation of COC without having to account for their influence.

S100B and NSE in Cluster Headache - Evidence for Glial Cell Activation?

OBJECTIVE: Neuronal-specific enolase (NSE) and protein S100B have gained considerable interest as the markers of CNS injury, glial cell activation, and/or blood-brain barrier (BBB) disruption. No studies have investigated NSE and S100B in cluster headache (CH), but these biomarkers could contribute to the understanding of CH. METHODS: Patients with episodic CH in bout (eCHa), in remission (eCHr), and chronic CH (cCH) were included in this randomized, double-blind, placebo-controlled, 2-way cross-over provocation study carried out at the Danish Headache Center. The primary endpoints included (1) differences of NSE and S100B in between groups (eCHa, eCHr, and cCH) at baseline; (2) differences over time in plasma concentrations of NSE and S100B between patient developing an attack and those who did not; (3) differences in plasma concentrations over time of NSE and S100B between active day and placebo day. Baseline findings were compared to the historical data on migraine patients and healthy controls and presented with means ± SD. RESULTS: Nine eCHa, 9 eCHr, and 13 cCH patients completed the study and blood samples from 11 CGRP-induced CH attacks were obtained. There were no differences in NSE levels between CH groups at baseline, but CH patients in active disease phase had higher levels compared with 32 migraine patients (9.1 ± 2.2 µg/L vs 6.0 ± 2.2 µg/L, P < .0001) and 6 healthy controls (9.1 ± 2.2 µg/L vs 7.3 ± 2.0 µg/L, P = .007). CGRP-infusion caused no NSE changes and, but a slight, non-significant, increase in NSE was seen in patients who reported a CGRP-induced CH attack (2.39 µg/L, 95% Cl [-0.26, 3.85], P = .061). At baseline S100B levels in eCHa patients were higher compared to cCH patients (0.06 ± 0.02 µg/L vs 0.04 ± 0.02 µg/L, P = .018). Infusion of CGRP and CGRP-induced attacks did not change S100B levels. Apart from induced CH-attacks no other adverse events were noted. CONCLUSIONS: At baseline eCHa patients had higher S100B plasma levels than cCH patients and there was a slight, however not significant, NSE increase in response to CGRP-induced CH attack. Our findings suggest a possible role of an ictal activation of glial cells in CH pathophysiology, but further studies are warranted.

Subtleties in the interpretation of hazard contrasts.

The hazard ratio is one of the most commonly reported measures of treatment effect in randomised trials, yet the source of much misinterpretation. This point was made clear by Hernán (Epidemiology (Cambridge, Mass) 21(1):13-15, 2010) in a commentary, which emphasised that the hazard ratio contrasts populations of treated and untreated individuals who survived a given period of time, populations that will typically fail to be comparable-even in a randomised trial-as a result of different pressures or intensities acting on different populations. The commentary has been very influential, but also a source of surprise and confusion. In this note, we aim to provide more insight into the subtle interpretation of hazard ratios and differences, by investigating in particular what can be learned about a treatment effect from the hazard ratio becoming 1 (or the hazard difference 0) after a certain period of time. We further define a hazard ratio that has a causal interpretation and study its relationship to the Cox hazard ratio, and we also define a causal hazard difference. These quantities are of theoretical interest only, however, since they rely on assumptions that cannot be empirically evaluated. Throughout, we will focus on the analysis of randomised experiments.

Survivor bias in Mendelian randomization analysis.

Mendelian randomization studies employ genotypes as experimental handles to infer the effect of genetically modified exposures (e.g. vitamin D exposure) on disease outcomes (e.g. mortality). The statistical analysis of these studies makes use of the standard instrumental variables framework. Many of these studies focus on elderly populations, thereby ignoring the problem of left truncation, which arises due to the selection of study participants being conditional upon surviving up to the time of study onset. Such selection, in general, invalidates the assumptions on which the instrumental variables analysis rests. We show that Mendelian randomization studies of adult or elderly populations will therefore, in general, return biased estimates of the exposure effect when the considered genotype affects mortality; in contrast, standard tests of the causal null hypothesis that the exposure does not affect the mortality rate remain unbiased, even when they ignore this problem of left truncation. To eliminate "survivor bias" or "truncation bias" from the effect of exposure on mortality, we next propose various simple strategies under a semi-parametric additive hazard model. We examine the performance of the proposed methods in simulation studies and use them to infer the effect of vitamin D on all-cause mortality based on the Monica10 study with the genetic variant filaggrin as instrumental variable.

Calcitonin-gene related peptide and disease activity in cluster headache.

OBJECTIVE: To investigate the role of calcitonin gene-related peptide, pituitary adenylate cyclase-activating polypeptide-38 (PACAP38) and vasoactive intestinal polypeptide in cluster headache, we measured these vasoactive peptides interictally and during experimentally induced cluster headache attacks. METHODS: We included patients with episodic cluster headache in an active phase (n = 9), episodic cluster headache patients in remission (n = 9) and patients with chronic cluster headache (n = 13). Cluster headache attacks were induced by infusion of calcitonin gene-related peptide (1.5 µg/min) in a randomized, double-blind, placebo controlled, two-way cross-over study. At baseline, we collected interictal blood samples from all patients and during 11 calcitonin gene-related peptide-induced cluster headache attacks. RESULTS: At baseline, episodic cluster headache patients in remission had higher plasma levels of calcitonin gene-related peptide, 100.6 ± 36.3 pmol/l, compared to chronic cluster headache patients, 65.9 ± 30.5 pmol/l, ( p = 0.011). Episodic cluster headache patients in active phase had higher PACAP38 levels, 4.0 ± 0.8 pmol/l, compared to chronic cluster headache patients, 3.3 ± 0.7 pmol/l, ( p = 0.033). Baseline levels of vasoactive intestinal polypeptide did not differ between cluster headache groups. We found no attack-related increase in calcitonin gene-related peptide, PACAP38 or vasoactive intestinal polypeptide levels during calcitonin gene-related peptide-induced cluster headache attacks. CONCLUSIONS: This study suggests that cluster headache disease activity is associated with alterations of calcitonin gene-related peptide expression. Future studies should investigate the potential of using calcitonin gene-related peptide measurements in monitoring of disease state and predicting response to preventive treatments, including response to anti-calcitonin gene-related peptide monoclonal antibodies.

On doubly robust estimation of the hazard difference.

The estimation of conditional treatment effects in an observational study with a survival outcome typically involves fitting a hazards regression model adjusted for a high-dimensional covariate. Standard estimation of the treatment effect is then not entirely satisfactory, as the misspecification of the effect of this covariate may induce a large bias. Such misspecification is a particular concern when inferring the hazard difference, because it is difficult to postulate additive hazards models that guarantee non-negative hazards over the entire observed covariate range. We therefore consider a novel class of semiparametric additive hazards models which leave the effects of covariates unspecified. The efficient score under this model is derived. We then propose two different estimation approaches for the hazard difference (and hence also the relative chance of survival), both of which yield estimators that are doubly robust. The approaches are illustrated using simulation studies and data on right heart catheterization and mortality from the SUPPORT study.

Efficacy of laser treatment for onychomycotic nails: a systematic review and meta-analysis of prospective clinical trials.

Laser therapy for onychomycosis is emerging but its efficacy remains unestablished. To examine current evidence on efficacy of laser treatment of onychomycosis. A systematic review and one-arm meta-analysis, including all prospective clinical trials, identified on PubMed, Cochrane Library, and EMBASE databases. Trials with participants as unit of analysis (UOA), n = 13, were analyzed separately from trials with nails as UOA, n = 7. Summary proportions and 95% confidence intervals (95% CI) were calculated. Outcomes were mycological cure, clinical improvement, or complete cure. Twenty-two prospective trials (four randomized controlled trials and 18 uncontrolled trials) with a total of 755 participants were analyzed. Summary proportions with 95% CI for participants as UOA were mycological cure 70.4%, 95% CI 52.2-83.8%; clinical improvement 67.2%, 95% CI 43.2-84.7%; and complete cure 7.2%, 95% CI 1.9-23.5%. High statistical heterogeneity was detected (mycological cure I2 = 88%, P < 0.01; clinical improvement I2 = 69%, P < 0.01; complete cure I2 = 60%, P = 0.11). The current level of evidence is limited and with high heterogeneity, making it difficult to assess the true efficacy of laser treatment for onychomycosis. Larger randomized controlled trials with well-defined methodology are warranted.

Goodness of fit tests for estimating equations based on pseudo-observations.

We study regression models for mean value parameters in survival analysis based on pseudo-observations. Such parameters include the survival probability and the cumulative incidence in a single point as well as the restricted mean life time and the cause-specific number of years lost. Goodness of fit techniques for such models based on cumulative sums of pseudo-residuals are derived including asymptotic results and Monte Carlo simulations. Practical examples from liver cirrhosis and bone marrow transplantation are also provided.

A causal proportional hazards estimator under homogeneous or heterogeneous selection in an IV setting.

In this paper we present a framework to do estimation in a structural Cox model when there may be unobserved confounding. The model is phrased in terms of a selection bias function and a baseline model that describes how covariates affect the survival time in a scenario without exposure. In this way model congeniality is ensured. The method uses an instrumental variable. Interestingly, the formulated model turns out to have similarities to the so-called Cox-Aalen survival model for the observed data. We exploit this to enhance estimation of the unknown parameters. This also allows us to derive large sample properties of the proposed estimator.

Instrumental variables estimation of exposure effects on a time-to-event endpoint using structural cumulative survival models.

The use of instrumental variables for estimating the effect of an exposure on an outcome is popular in econometrics, and increasingly so in epidemiology. This increasing popularity may be attributed to the natural occurrence of instrumental variables in observational studies that incorporate elements of randomization, either by design or by nature (e.g., random inheritance of genes). Instrumental variables estimation of exposure effects is well established for continuous outcomes and to some extent for binary outcomes. It is, however, largely lacking for time-to-event outcomes because of complications due to censoring and survivorship bias. In this article, we make a novel proposal under a class of structural cumulative survival models which parameterize time-varying effects of a point exposure directly on the scale of the survival function; these models are essentially equivalent with a semi-parametric variant of the instrumental variables additive hazards model. We propose a class of recursive instrumental variable estimators for these exposure effects, and derive their large sample properties along with inferential tools. We examine the performance of the proposed method in simulation studies and illustrate it in a Mendelian randomization study to evaluate the effect of diabetes on mortality using data from the Health and Retirement Study. We further use the proposed method to investigate potential benefit from breast cancer screening on subsequent breast cancer mortality based on the HIP-study.

Estimation of average causal effect using the restricted mean residual lifetime as effect measure.

Although mean residual lifetime is often of interest in biomedical studies, restricted mean residual lifetime must be considered in order to accommodate censoring. Differences in the restricted mean residual lifetime can be used as an appropriate quantity for comparing different treatment groups with respect to their survival times. In observational studies where the factor of interest is not randomized, covariate adjustment is needed to take into account imbalances in confounding factors. In this article, we develop an estimator for the average causal treatment difference using the restricted mean residual lifetime as target parameter. We account for confounding factors using the Aalen additive hazards model. Large sample property of the proposed estimator is established and simulation studies are conducted in order to assess small sample performance of the resulting estimator. The method is also applied to an observational data set of patients after an acute myocardial infarction event.