Optimal matching with minimal deviation from fine balance in a study of obesity and surgical outcomes

In multivariate matching, fine balance constrains the marginal distributions of a nominal variable in treated and matched control groups to be identical without constraining who is matched to whom. In this way, a fine balance constraint can balance a nominal variable with many levels while focusing efforts on other more important variables when pairing individuals to minimize the total covariate distance within pairs. Fine balance is not always possible; that is, it is a constraint on an optimization problem, but the constraint is not always feasible. We propose a new algorithm that returns a minimum distance finely balanced match when one is feasible, and otherwise minimizes the total distance among all matched samples that minimize the deviation from fine balance. Perhaps we can come very close to fine balance when fine balance is not attainable; moreover, in any event, because our algorithm is guaranteed to come as close as possible to fine balance, the investigator may perform one match, and on that basis judge whether the best attainable balance is adequate or not. We also show how to incorporate an additional constraint. The algorithm is implemented in two similar ways, first as an optimal assignment problem with an augmented distance matrix, second as a minimum cost flow problem in a network. The case of knee surgery in the Obesity and Surgical Outcomes Study motivated the development of this algorithm and is used as an illustration. In that example, 2 of 47 hospitals had too few nonobese patients to permit fine balance for the nominal variable with 47 levels representing the hospital, but our new algorithm came very close to fine balance. Moreover, in that example, there was a shortage of nonobese diabetic patients, and incorporation of an additional constraint forced the match to include all of these nonobese diabetic patients, thereby coming as close as possible to balance for this important but recalcitrant covariate.     

Directional penalties for optimal matching in observational studies

Multivariate matching in observational studies tends to view covariate differences symmetrically: a difference in age of 10 years is thought equally problematic whether the treated subject is older or younger than the matched control. If matching is correcting an imbalance in age, such that treated subjects are typically older than controls, then the situation in need of correction is asymmetric: a matched pair with a difference in age of 10 years is much more likely to have an older treated subject and a younger control than the opposite. Correcting the bias may be easier if matching tries to avoid the typical case that creates the bias. We describe several easily used, asymmetric, directional penalties and illustrate how they can improve covariate balance in a matched sample. The investigator starts with a matched sample built in a conventional way, then diagnoses residual covariate imbalances in need of reduction, and achieves the needed reduction by slightly altering the distance matrix with directional penalties, creating a new matched sample. Unlike penalties commonly used in matching, a directional penalty can go too far, reversing the direction of the bias rather than reducing the bias, so the magnitude of the directional penalty matters and may need adjustment. Our experience is that two or three adjustments, guided by balance diagnostics, can substantially improve covariate balance, perhaps requiring fifteen minutes effort sitting at the computer. We also explore the connection between directional penalties and a widely used technique in integer programming, namely Lagrangian relaxation of problematic linear side constraints in a minimum cost flow problem. In effect, many directional penalties are Lagrange multipliers, pushing a matched sample in the direction of satisfying a linear constraint that would not be satisfied without penalization. The method and example are in an R package DiPs at CRAN .     

Propensity score matching with time-dependent covariates

In observational studies with a time-dependent treatment and time-dependent covariates, it is desirable to balance the distribution of the covariates at every time point. A time-dependent propensity score based on the Cox proportional hazards model is proposed and used in risk set matching. Matching on this propensity score is shown to achieve a balanced distribution of the covariates in both treated and control groups. Optimal matching with various designs is conducted and compared in a study of a surgical treatment, cystoscopy and hydrodistention, given in response to a chronic bladder disease, interstitial cystitis. Simulation studies also suggest that the statistical analysis after matching outperforms the analysis without matching in terms of both point and interval estimations.     

Optimal multivariate matching before randomization

Although blocking or pairing before randomization is a basic principle of experimental design, the principle is almost invariably applied to at most one or two blocking variables. Here, we discuss the use of optimal multivariate matching prior to randomization to improve covariate balance for many variables at the same time, presenting an algorithm and a case-study of its performance. The method is useful when all subjects, or large groups of subjects, are randomized at the same time. Optimal matching divides a single group of 2n subjects into n pairs to minimize covariate differences within pairs-the so-called nonbipartite matching problem-then one subject in each pair is picked at random for treatment, the other being assigned to control. Using the baseline covariate data for 132 patients from an actual, unmatched, randomized experiment, we construct 66 pairs matching for 14 covariates. We then create 10000 unmatched and 10000 matched randomized experiments by repeatedly randomizing the 132 patients, and compare the covariate balance with and without matching. By every measure, every one of the 14 covariates was substantially better balanced when randomization was performed within matched pairs. Even after covariance adjustment for chance imbalances in the 14 covariates, matched randomizations provided more accurate estimates than unmatched randomizations, the increase in accuracy being equivalent to, on average, a 7% increase in sample size. In randomization tests of no treatment effect, matched randomizations using the signed rank test had substantially higher power than unmatched randomizations using the rank sum test, even when only 2 of 14 covariates were relevant to a simulated response. Unmatched randomizations experienced rare disasters which were consistently avoided by matched randomizations.     

Propensity score methods for time-dependent cluster confounding

In observational studies, subjects are often nested within clusters. In medical studies, patients are often treated by doctors and therefore patients are regarded as nested or clustered within doctors. A concern that arises with clustered data is that cluster-level characteristics (e.g., characteristics of the doctor) are associated with both treatment selection and patient outcomes, resulting in cluster-level confounding. Measuring and modeling cluster attributes can be difficult and statistical methods exist to control for all unmeasured cluster characteristics. An assumption of these methods however is that characteristics of the cluster and the effects of those characteristics on the outcome (as well as probability of treatment assignment when using covariate balancing methods) are constant over time. In this paper, we consider methods that relax this assumption and allow for estimation of treatment effects in the presence of unmeasured time-dependent cluster confounding. The methods are based on matching with the propensity score and incorporate unmeasured time-specific cluster effects by performing matching within clusters or using fixed- or random-cluster effects in the propensity score model. The methods are illustrated using data to compare the effectiveness of two total hip devices with respect to survival of the device and a simulation study is performed that compares the proposed methods. One method that was found to perform well is matching within surgeon clusters partitioned by time. Considerations in implementing the proposed methods are discussed.     

Substantial gains in bias reduction from matching with a variable number of controls

In observational studies that match several controls to each treated subject, substantially greater bias reduction is possible if the number of controls is not fixed but rather is allowed to vary from one matched set to another. In certain cases, matching with a fixed number of controls may remove only 50% of the bias in a covariate, whereas matching with a variable number of controls may remove 90% of the bias, even though both control groups have the same number of controls in total. An example of matching in a study of surgical mortality is discussed in detail.     

A comparison of machine learning algorithms and covariate balance measures for propensity score matching and weighting

Propensity score matching (PSM) and propensity score weighting (PSW) are popular tools to estimate causal effects in observational studies. We address two open issues: how to estimate propensity scores and assess covariate balance. Using simulations, we compare the performance of PSM and PSW based on logistic regression and machine learning algorithms (CART; Bagging; Boosting; Random Forest; Neural Networks; naive Bayes). Additionally, we consider several measures of covariate balance (Absolute Standardized Average Mean (ASAM) with and without interactions; measures based on the quantile‐quantile plots; ratio between variances of propensity scores; area under the curve (AUC)) and assess their ability in predicting the bias of PSM and PSW estimators. We also investigate the importance of tuning of machine learning parameters in the context of propensity score methods. Two simulation designs are employed. In the first, the generating processes are inspired to birth register data used to assess the effect of labor induction on the occurrence of caesarean section. The second exploits more general generating mechanisms. Overall, among the different techniques, random forests performed the best, especially in PSW. Logistic regression and neural networks also showed an excellent performance similar to that of random forests. As for covariate balance, the simplest and commonly used metric, the ASAM, showed a strong correlation with the bias of causal effects estimators. Our findings suggest that researchers should aim at obtaining an ASAM lower than 10% for as many variables as possible. In the empirical study we found that labor induction had a small and not statistically significant impact on caesarean section.     

Evaluating Propensity Score Methods in a Quasi-Experimental Study of the Impact of Menu-Labeling

Background

Quasi-experimental studies of menu labeling have found mixed results for improving diet. Differences between experimental groups can hinder interpretation. Propensity scores are an increasingly common method to improve covariate balance, but multiple methods exist and the improvements associated with each method have rarely been compared. In this re-analysis of the impact of menu labeling, we compare multiple propensity score methods to determine which methods optimize balance between experimental groups.

Methods

Study participants included adult customers who visited full-service restaurants with menu labeling (treatment) and without (control). We compared the balance between treatment groups obtained by four propensity score methods: 1) 1:1 nearest neighbor matching (NN), 2) augmented 1:1 NN (using caliper of 0.2 and an exact match on an imbalanced covariate), 3) full matching, and 4) inverse probability weighting (IPW). We then evaluated the treatment effect on differences in nutrients purchased across the different methods.

Results

1:1 NN resulted in worse balance than the original unmatched sample (average standardized absolute mean distance [ASAM]: 0.185 compared to 0.171). Augmented 1:1 NN improved balance (ASAM: 0.038) but resulted in a large reduction in sample size. Full matching and IPW improved balance over the unmatched sample without a reduction in sample size (ASAM: 0.049 and 0.031, respectively). Menu labeling was associated with decreased calories, fat, sodium and carbohydrates in the unmatched analysis. Results were qualitatively similar in the propensity score matched/weighted models.

Conclusions

While propensity scores offer an increasingly popular tool to improve causal inference, choosing the correct method can be challenging. Our results emphasize the benefit of examining multiple methods to ensure results are consistent, and considering approaches beyond the most popular method of 1:1 NN matching.     

Automatic variable selection for exposure-driven propensity score matching with unmeasured confounders

Multivariable model building for propensity score modeling approaches is challenging. A common propensity score approach is exposure-driven propensity score matching, where the best model selection strategy is still unclear. In particular, the situation may require variable selection, while it is still unclear if variables included in the propensity score should be associated with the exposure and the outcome, with either the exposure or the outcome, with at least the exposure or with at least the outcome. Unmeasured confounders, complex correlation structures, and non-normal covariate distributions further complicate matters. We consider the performance of different modeling strategies in a simulation design with a complex but realistic structure and effects on a binary outcome. We compare the strategies in terms of bias and variance in estimated marginal exposure effects. Considering the bias in estimated marginal exposure effects, the most reliable results for estimating the propensity score are obtained by selecting variables related to the exposure. On average this results in the least bias and does not greatly increase variances. Although our results cannot be generalized, this provides a counterexample to existing recommendations in the literature based on simple simulation settings. This highlights that recommendations obtained in simple simulation settings cannot always be generalized to more complex, but realistic settings and that more complex simulation studies are needed.     

The bias due to incomplete matching

Observational studies comparing groups of treated and control units are often used to estimate the effects caused by treatments. Matching is a method for sampling a large reservoir of potential controls to produce a control group of modest size that is ostensibly similar to the treated group. In practice, there is a trade-off between the desires to find matches for all treated units and to obtain matched treated-control pairs that are extremely similar to each other. We derive expressions for the bias in the average matched pair difference due to the failure to match all treated units--incomplete matching, and the failure to obtain exact matches--inexact matching. A practical example shows that the bias due to incomplete matching can be severe, and moreover, can be avoided entirely by using an appropriate multivariate nearest available matching algorithm, which, in the example, leaves only a small residual bias due to inexact matching.     

Age correction in dementia--matching to a healthy brain

In recent research, many univariate and multivariate approaches have been proposed to improve automatic classification of various dementia syndromes using imaging data. Some of these methods do not provide the possibility to integrate possible confounding variables like age into the statistical evaluation. A similar problem sometimes exists in clinical studies, as it is not always possible to match different clinical groups to each other in all confounding variables, like for example, early-onset (age<65 years) and late-onset (age≥65) patients with Alzheimer's disease (AD). Here, we propose a simple method to control for possible effects of confounding variables such as age prior to statistical evaluation of magnetic resonance imaging (MRI) data using support vector machine classification (SVM) or voxel-based morphometry (VBM). We compare SVM results for the classification of 80 AD patients and 79 healthy control subjects based on MRI data with and without prior age correction. Additionally, we compare VBM results for the comparison of three different groups of AD patients differing in age with the same group of control subjects obtained without including age as covariate, with age as covariate or with prior age correction using the proposed method. SVM classification using the proposed method resulted in higher between-group classification accuracy compared to uncorrected data. Further, applying the proposed age correction substantially improved univariate detection of disease-related grey matter atrophy using VBM in AD patients differing in age from control subjects. The results suggest that the approach proposed in this work is generally suited to control for confounding variables such as age in SVM or VBM analyses. Accordingly, the approach might improve and extend the application of these methods in clinical neurosciences.     

Optimal Caliper Width for Propensity Score Matching of Three Treatment Groups: A Monte Carlo Study

Propensity score matching is a method to reduce bias in non-randomized and observational studies. Propensity score matching is mainly applied to two treatment groups rather than multiple treatment groups, because some key issues affecting its application to multiple treatment groups remain unsolved, such as the matching distance, the assessment of balance in baseline variables, and the choice of optimal caliper width. The primary objective of this study was to compare propensity score matching methods using different calipers and to choose the optimal caliper width for use with three treatment groups. The authors used caliper widths from 0.1 to 0.8 of the pooled standard deviation of the logit of the propensity score, in increments of 0.1. The balance in baseline variables was assessed by standardized difference. The matching ratio, relative bias, and mean squared error (MSE) of the estimate between groups in different propensity score-matched samples were also reported. The results of Monte Carlo simulations indicate that matching using a caliper width of 0.2 of the pooled standard deviation of the logit of the propensity score affords superior performance in the estimation of treatment effects. This study provides practical solutions for the application of propensity score matching of three treatment groups.     

Estimation with correlated censored survival data with missing covariates

Incomplete covariate data are a common occurrence in studies in which the outcome is survival time. Further, studies in the health sciences often give rise to correlated, possibly censored, survival data. With no missing covariate data, if the marginal distributions of the correlated survival times follow a given parametric model, then the estimates using the maximum likelihood estimating equations, naively treating the correlated survival times as independent, give consistent estimates of the relative risk parameters Lipsitz et al. 1994 50, 842-846. Now, suppose that some observations within a cluster have some missing covariates. We show in this paper that if one naively treats observations within a cluster as independent, that one can still use the maximum likelihood estimating equations to obtain consistent estimates of the relative risk parameters. This method requires the estimation of the parameters of the distribution of the covariates. We present results from a clinical trial Lipsitz and Ibrahim (1996b) 2, 5-14 with five covariates, four of which have some missing values. In the trial, the clusters are the hospitals in which the patients were treated.     

Predictive margins with survey data

In the analysis of covariance, the display of adjusted treatment means allows one to compare mean (treatment) group outcomes controlling for different covariate distributions in the groups. Predictive margins are a generalization of adjusted treatment means to nonlinear models. The predictive margin for group r represents the average predicted response if everyone in the sample had been in group r. This paper discusses the use of predictive margins with complex survey data, where an important consideration is the choice of covariate distribution used to standardize the predictive margin. It is suggested that the textbook formula for the standard error of an adjusted treatment mean from the analysis of covariance may be inappropriate for applications involving survey data. Applications are given using data from the 1992 National Health Interview Survey (NHIS) and the Epidemiologic Followup Study to the first National Health and Nutrition Examination Survey (NHANES I).     

Double Inverse‐Weighted Estimation of Cumulative Treatment Effects Under Nonproportional Hazards and Dependent Censoring

Summary In medical studies of time‐to‐event data, nonproportional hazards and dependent censoring are very common issues when estimating the treatment effect. A traditional method for dealing with time‐dependent treatment effects is to model the time‐dependence parametrically. Limitations of this approach include the difficulty to verify the correctness of the specified functional form and the fact that, in the presence of a treatment effect that varies over time, investigators are usually interested in the cumulative as opposed to instantaneous treatment effect. In many applications, censoring time is not independent of event time. Therefore, we propose methods for estimating the cumulative treatment effect in the presence of nonproportional hazards and dependent censoring. Three measures are proposed, including the ratio of cumulative hazards, relative risk, and difference in restricted mean lifetime. For each measure, we propose a double inverse‐weighted estimator, constructed by first using inverse probability of treatment weighting (IPTW) to balance the treatment‐specific covariate distributions, then using inverse probability of censoring weighting (IPCW) to overcome the dependent censoring. The proposed estimators are shown to be consistent and asymptotically normal. We study their finite‐sample properties through simulation. The proposed methods are used to compare kidney wait‐list mortality by race.     

Regression models for infant mortality data in Norwegian siblings, using a compound Poisson frailty distribution with random scale

The power variance function distributions, which include the gamma and compound Poisson (CP) distributions among others, are commonly used in frailty models for family data. In a previous paper, we presented a frailty model constructed by randomizing the scale parameter in a CP distribution. When combined with a parametric baseline hazard, this yields a model with heterogeneity on both the individual and the family level and a subgroup with zero frailty, corresponding to people not experiencing the event. In this paper, we discuss covariates in the model. Depending on where the covariates are inserted in the model, one may have proportional hazards at the individual level, the family level, and a larger group level (for covariates shared by many families, e.g. ethnic groups) or get accelerated failure times. Each of these alternatives gives a specific interpretation of the covariate effects. An application to data infant mortality in siblings from the Medical Birth Registry of Norway is included. We compare the results for some of the different covariate modeling options.     

On power and sample size calculations for likelihood ratio tests in generalized linear models

A direct extension of the approach described in Self, Mauritsen, and Ohara (1992, Biometrics 48, 31-39) for power and sample size calculations in generalized linear models is presented. The major feature of the proposed approach is that the modification accommodates both a finite and an infinite number of covariate configurations. Furthermore, for the approximation of the noncentrality of the noncentral chi-square distribution for the likelihood ratio statistic, a simplification is provided that not only reduces substantial computation but also maintains the accuracy. Simulation studies are conducted to assess the accuracy for various model configurations and covariate distributions.     

Augmented designs to assess immune response in vaccine trials

This article introduces methods for use in vaccine clinical trials to help determine whether the immune response to a vaccine is actually causing a reduction in the infection rate. This is not easy because immune response to the (say HIV) vaccine is only observed in the HIV vaccine arm. If we knew what the HIV-specific immune response in placebo recipients would have been, had they been vaccinated, this immune response could be treated essentially like a baseline covariate and an interaction with treatment could be evaluated. Relatedly, the rate of infection by this baseline covariate could be compared between the two groups and a causative role of immune response would be supported if infection risk decreased with increasing HIV immune response only in the vaccine group. We introduce two methods for inferring this HIV-specific immune response. The first involves vaccinating everyone before baseline with an irrelevant vaccine, for example, rabies. Randomization ensures that the relationship between the immune responses to the rabies and HIV vaccines observed in the vaccine group is the same as what would have been seen in the placebo group. We infer a placebo volunteer's response to the HIV vaccine using their rabies response and a prediction model from the vaccine group. The second method entails vaccinating all uninfected placebo patients at the closeout of the trial with the HIV vaccine and recording immune response. We pretend this immune response at closeout is what they would have had at baseline. We can then infer what the distribution of immune response among placebo infecteds would have been. Such designs may help elucidate the role of immune response in preventing infections. More pointedly, they could be helpful in the decision to improve or abandon an HIV vaccine with mediocre performance in a phase III trial.     

Defining the Study Population for an Observational Study to Ensure Sufficient Overlap: A Tree Approach

The internal validity of an observational study is enhanced by only comparing sets of treated and control subjects which have sufficient overlap in their covariate distributions. Methods have been developed for defining the study population using propensity scores to ensure sufficient overlap. However, a study population defined by propensity scores is difficult for other investigators to understand. We develop a method of defining a study population in terms of a tree which is easy to understand and display, and that has similar internal validity as that of the study population defined by propensity scores.