Disagreement remains on what the target estimand should be for population-adjusted indirect treatment comparisons. This debate is of central importance for policy-makers and applied practitioners in health technology assessment. Misunderstandings are based on properties inherent to estimators, not estimands, and on generalizing conclusions based on linear regression to non-linear models. Estimators of marginal estimands need not be unadjusted and may be covariate-adjusted. The population-level interpretation of conditional estimates follows from collapsibility and does not necessarily hold for the underlying conditional estimands. For non-collapsible effect measures, neither conditional estimates nor estimands have a population-level interpretation. Estimators of marginal effects tend to be more precise and efficient than estimators of conditional effects where the measure of effect is non-collapsible. In any case, such comparisons are inconsequential for estimators targeting distinct estimands. Statistical efficiency should not drive the choice of the estimand. On the other hand, the estimand, selected on the basis of relevance to decision-making, should drive the choice of the most efficient estimator. Health technology assessment agencies make reimbursement decisions at the population level. Therefore, marginal estimands are required. Current pairwise population adjustment methods such as matching-adjusted indirect comparison are restricted to target marginal estimands that are specific to the comparator study sample. These may not be relevant for decision-making. Multilevel network meta-regression (ML-NMR) can potentially target marginal estimands in any population of interest. Such population could be characterized by decision-makers using increasingly available ``real-world'' data sources. Therefore, ML-NMR presents new directions and abundant opportunities for evidence synthesis.
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Let $X$ and $Y$ be two real-valued random variables. Let $(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots$ be independent identically distributed copies of $(X,Y)$. Suppose there are two players A and B. Player A has access to $X_{1},X_{2},\ldots$ and player B has access to $Y_{1},Y_{2},\ldots$. Without communication, what joint probability distributions can players A and B jointly simulate? That is, if $k,m$ are fixed positive integers, what probability distributions on $\{1,\ldots,m\}^{2}$ are equal to the distribution of $(f(X_{1},\ldots,X_{k}),\,g(Y_{1},\ldots,Y_{k}))$ for some $f,g\colon\mathbb{R}^{k}\to\{1,\ldots,m\}$? When $X$ and $Y$ are standard Gaussians with fixed correlation $\rho\in(-1,1)$, we show that the set of probability distributions that can be noninteractively simulated from $k$ Gaussian samples is the same for any $k\geq m^{2}$. Previously, it was not even known if this number of samples $m^{2}$ would be finite or not, except when $m\leq 2$. Consequently, a straightforward brute-force search deciding whether or not a probability distribution on $\{1,\ldots,m\}^{2}$ is within distance $0<\epsilon<|\rho|$ of being noninteractively simulated from $k$ correlated Gaussian samples has run time bounded by $(5/\epsilon)^{m(\log(\epsilon/2) / \log|\rho|)^{m^{2}}}$, improving a bound of Ghazi, Kamath and Raghavendra. A nonlinear central limit theorem (i.e. invariance principle) of Mossel then generalizes this result to decide whether or not a probability distribution on $\{1,\ldots,m\}^{2}$ is within distance $0<\epsilon<|\rho|$ of being noninteractively simulated from $k$ samples of a given finite discrete distribution $(X,Y)$ in run time that does not depend on $k$, with constants that again improve a bound of Ghazi, Kamath and Raghavendra.
Observational studies can play a useful role in assessing the comparative effectiveness of competing treatments. In a clinical trial the randomization of participants to treatment and control groups generally results in well-balanced groups with respect to possible confounders, which makes the analysis straightforward. However, when analysing observational data, the potential for unmeasured confounding makes comparing treatment effects much more challenging. Causal inference methods such as the Instrumental Variable and Prior Even Rate Ratio approaches make it possible to circumvent the need to adjust for confounding factors that have not been measured in the data or measured with error. Direct confounder adjustment via multivariable regression and Propensity score matching also have considerable utility. Each method relies on a different set of assumptions and leverages different data. In this paper, we describe the assumptions of each method and assess the impact of violating these assumptions in a simulation study. We propose the prior outcome augmented Instrumental Variable method that leverages data from before and after treatment initiation, and is robust to the violation of key assumptions. Finally, we propose the use of a heterogeneity statistic to decide two or more estimates are statistically similar, taking into account their correlation. We illustrate our causal framework to assess the risk of genital infection in patients prescribed Sodium-glucose Co-transporter-2 inhibitors versus Dipeptidyl Peptidase-4 inhibitors as second-line treatment for type 2 diabets using observational data from the Clinical Practice Research Datalink.
The effect of combined, generating R- and Q-factors of measured variables on the loadings resulting from R-factor analysis was investigated. It was found algebraically that a model based on the combination of R- and Q-factors results in loading indeterminacy beyond rotational indeterminacy. Although R-factor analysis of data generated by a combination of R- and Q-factors is nevertheless possible, this may lead to model error. Accordingly, even in the population, the resulting R-factor loadings are not necessarily close estimates of the original loadings of the generating R-factors. This effect was also shown in a simulation study at the population level. Moreover, the simulation study based on samples drawn from the populations revealed that the R-factor loadings averaged across samples were larger than the population loadings of the generating R-factors. Overall, the results indicate that -- in data that are generated by a combination of R- and Q-factors -- the Q-factors may lead to substantial loading indeterminacy and loading bias in R-factor analysis.
Ensemble methods based on subsampling, such as random forests, are popular in applications due to their high predictive accuracy. Existing literature views a random forest prediction as an infinite-order incomplete U-statistic to quantify its uncertainty. However, these methods focus on a small subsampling size of each tree, which is theoretically valid but practically limited. This paper develops an unbiased variance estimator based on incomplete U-statistics, which allows the tree size to be comparable with the overall sample size, making statistical inference possible in a broader range of real applications. Simulation results demonstrate that our estimators enjoy lower bias and more accurate confidence interval coverage without additional computational costs. We also propose a local smoothing procedure to reduce the variation of our estimator, which shows improved numerical performance when the number of trees is relatively small. Further, we investigate the ratio consistency of our proposed variance estimator under specific scenarios. In particular, we develop a new "double U-statistic" formulation to analyze the Hoeffding decomposition of the estimator's variance.
Dyadic data is often encountered when quantities of interest are associated with the edges of a network. As such it plays an important role in statistics, econometrics and many other data science disciplines. We consider the problem of uniformly estimating a dyadic Lebesgue density function, focusing on nonparametric kernel-based estimators taking the form of dyadic empirical processes. Our main contributions include the minimax-optimal uniform convergence rate of the dyadic kernel density estimator, along with strong approximation results for the associated standardized and Studentized $t$-processes. A consistent variance estimator enables the construction of valid and feasible uniform confidence bands for the unknown density function. A crucial feature of dyadic distributions is that they may be "degenerate" at certain points in the support of the data, a property making our analysis somewhat delicate. Nonetheless our methods for uniform inference remain robust to the potential presence of such points. For implementation purposes, we discuss procedures based on positive semi-definite covariance estimators, mean squared error optimal bandwidth selectors and robust bias-correction techniques. We illustrate the empirical finite-sample performance of our methods both in simulations and with real-world data. Our technical results concerning strong approximations and maximal inequalities are of potential independent interest.
The asymptotic distribution of a wide class of V- and U-statistics with estimated parameters is derived in the case when the kernel is not necessarily differentiable along the parameter. The results have their application in goodness-of-fit problems.
Many research questions concern treatment effects on outcomes that can recur several times in the same individual. For example, medical researchers are interested in treatment effects on hospitalizations in heart failure patients and sports injuries in athletes. Competing events, such as death, complicate causal inference in studies of recurrent events because once a competing event occurs, an individual cannot have more recurrent events. Several statistical estimands have been studied in recurrent event settings, with and without competing events. However, the causal interpretations of these estimands, and the conditions that are required to identify these estimands from observed data, have yet to be formalized. Here we use a counterfactual framework for causal inference to formulate several causal estimands in recurrent event settings, with and without competing events. When competing events exist, we clarify when commonly used classical statistical estimands can be interpreted as causal quantities from the causal mediation literature, such as (controlled) direct effects and total effects. Furthermore, we show that recent results on interventionist mediation estimands allow us to define new causal estimands with recurrent and competing events that may be of particular clinical relevance in many subject matter settings. We use causal directed acyclic graphs and single world intervention graphs to illustrate how to reason about identification conditions for the various causal estimands using subject matter knowledge. Furthermore, using results on counting processes, we show how our causal estimands and their identification conditions, which are articulated in discrete time, converge to classical continuous-time counterparts in the limit of fine discretizations of time. Finally, we propose several estimators and establish their consistency for the various identifying functionals.
This paper analyzes the working or default assumptions researchers in the formal, statistical, and case study traditions typically hold regarding the sources of unexplained variance, the meaning of outliers, parameter values, human motivation, functional forms, time, and external validity. We argue that these working assumptions are often not essential to each method, and that these assumptions can be relaxed in ways that allow multimethod work to proceed. We then analyze the comparative advantages of different combinations of formal, statistical, and case study methods for various theory-building and theory-testing research objectives. We illustrate these advantages and offer methodological advice on how to combine different methods, through analysis and critique of prominent examples of multimethod research.
Consider a situation where a new patient arrives in the Intensive Care Unit (ICU) and is monitored by multiple sensors. We wish to assess relevant unmeasured physiological variables (e.g., cardiac contractility and output and vascular resistance) that have a strong effect on the patients diagnosis and treatment. We do not have any information about this specific patient, but, extensive offline information is available about previous patients, that may only be partially related to the present patient (a case of dataset shift). This information constitutes our prior knowledge, and is both partial and approximate. The basic question is how to best use this prior knowledge, combined with online patient data, to assist in diagnosing the current patient most effectively. Our proposed approach consists of three stages: (i) Use the abundant offline data in order to create both a non-causal and a causal estimator for the relevant unmeasured physiological variables. (ii) Based on the non-causal estimator constructed, and a set of measurements from a new group of patients, we construct a causal filter that provides higher accuracy in the prediction of the hidden physiological variables for this new set of patients. (iii) For any new patient arriving in the ICU, we use the constructed filter in order to predict relevant internal variables. Overall, this strategy allows us to make use of the abundantly available offline data in order to enhance causal estimation for newly arriving patients. We demonstrate the effectiveness of this methodology on a (non-medical) real-world task, in situations where the offline data is only partially related to the new observations. We provide a mathematical analysis of the merits of the approach in a linear setting of Kalman filtering and smoothing, demonstrating its utility.
Large margin nearest neighbor (LMNN) is a metric learner which optimizes the performance of the popular $k$NN classifier. However, its resulting metric relies on pre-selected target neighbors. In this paper, we address the feasibility of LMNN's optimization constraints regarding these target points, and introduce a mathematical measure to evaluate the size of the feasible region of the optimization problem. We enhance the optimization framework of LMNN by a weighting scheme which prefers data triplets which yield a larger feasible region. This increases the chances to obtain a good metric as the solution of LMNN's problem. We evaluate the performance of the resulting feasibility-based LMNN algorithm using synthetic and real datasets. The empirical results show an improved accuracy for different types of datasets in comparison to regular LMNN.
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