Nonparametric inference for median costs with censored data

Summary Increasingly, estimations of health care costs are used to evaluate competing treatments or to assess the expected expenditures associated with certain diseases. In health policy and economics, the primary focus of these estimations has been on the mean cost, because the total cost can be derived directly from the mean cost, and because information about total resources utilized is highly relevant for policymakers. Yet, the median cost also could be important, both as an intuitive measure of central tendency in cost distribution and as a subject of interest to payers and consumers. In many prospective studies, cost data collection is sometimes incomplete for some subjects due to right censoring, which typically is caused by loss to follow-up or by limited study duration. Censoring poses a unique challenge for cost data analysis because of so-called induced informative censoring, in that traditional methods suited for survival data generally are invalid in censored cost estimation. In this article, we propose methods for estimating the median cost and its confidence interval (CI) when data are subject to right censoring. We also consider the estimation of the ratio and difference of two median costs and their CIs. These methods can be extended to the estimation of other quantiles and other informatively censored data. We conduct simulation and real data analysis in order to examine the performance of the proposed methods.     

Median Regression Model with Interval Censored Data

Quantile regression methods have been used to estimate upper and lower quantile reference curves as the function of several covariates. Especially, in survival analysis, median regression models to the right‐censored data are suggested with several assumptions. In this article, we consider a median regression model for interval‐censored data and construct an estimating equation based on weights derived from interval‐censored data. In a simulation study, the performances of the proposed method are evaluated for both symmetric and right‐skewed distributed failure times. A well‐known breast cancer data are analyzed to illustrate the proposed method.     

Estimating incremental cost-effectiveness ratios and their confidence intervals with differentially censored data

With medical costs escalating over recent years, cost analysis is being conducted more and more to assess economic impact of new treatment options. An incremental cost-effectiveness ratio (ICER) is a measure that assesses the additional cost for a new treatment for each additional unit of effectiveness, such as saving 1 year of life. In this article, we consider cost-effectiveness analysis for new treatments evaluated in a randomized clinical trial setting with staggered entries. In particular, the censoring times are different for cost and survival data. We propose a method for estimating the ICER and obtaining its confidence interval when differential censoring exists. Simulation experiments are conducted to evaluate our proposed method. We also apply our methods to a clinical trial example comparing the cost-effectiveness of implanted defibrillators with conventional therapy for individuals with reduced left ventricular function after myocardial infarction.     

The challenge of estimating the direct and indirect effects of COVID-19 interventions – Toward an integrated economic and epidemiological approach

Decisions on public health measures to contain a pandemic are often based on parameters such as expected disease burden and additional mortality due to the pandemic. Both pandemics and non-pharmaceutical interventions to fight pandemics, however, produce economic, social, and medical costs. The costs are, for example, caused by changes in access to healthcare, social distancing, and restrictions on economic activity. These factors indirectly influence health outcomes in the short- and long-term perspective. In a narrative review based on targeted literature searches, we develop a comprehensive perspective on the concepts available as well as the challenges of estimating the overall disease burden and the direct and indirect effects of COVID-19 interventions from both epidemiological and economic perspectives, particularly during the early part of a pandemic. We review the literature and discuss relevant components that need to be included when estimating the direct and indirect effects of the COVID-19 pandemic. The review presents data sources and different forms of death counts, and discusses empirical findings on direct and indirect effects of the pandemic and interventions on disease burden as well as the distribution of health risks.     

Linear regression analysis of censored medical costs

This paper deals with the problem of linear regression for medical cost data when some study subjects are not followed for the full duration of interest so that their total costs are unknown. Standard survival analysis techniques are ill-suited to this type of censoring. The familiar normal equations for the least-squares estimation are modified in several ways to properly account for the incompleteness of the data. The resulting estimators are shown to be consistent and asymptotically normal with easily estimated variance-covariance matrices. The proposed methodology can be used when the cost database contains only the total costs for those with complete follow-up. More efficient estimators are available when the cost data are recorded in multiple time intervals. A study on the medical cost for ovarian cancer is presented.     

Median regression models for longitudinal data with dropouts

Summary .  Recently, median regression models have received increasing attention. When continuous responses follow a distribution that is quite different from a normal distribution, usual mean regression models may fail to produce efficient estimators whereas median regression models may perform satisfactorily. In this article, we discuss using median regression models to deal with longitudinal data with dropouts. Weighted estimating equations are proposed to estimate the median regression parameters for incomplete longitudinal data, where the weights are determined by modeling the dropout process. Consistency and the asymptotic distribution of the resultant estimators are established. The proposed method is used to analyze a longitudinal data set arising from a controlled trial of HIV disease ( Volberding et al., 1990 , The New England Journal of Medicine 322, 941–949). Simulation studies are conducted to assess the performance of the proposed method under various situations. An extension to estimation of the association parameters is outlined.     

Cost Drivers for Voluntary Medical Male Circumcision Using Primary Source Data from Sub-Saharan Africa

Background

As voluntary medical male circumcision (VMMC) programs scale up, there is a pressing need for information about the important cost drivers, and potential efficiency gains. We examine those cost drivers here, and estimate the potential efficiency gains through an econometric model.

Methods and Findings

We examined the main cost drivers (i.e., personnel and consumables) associated with providing VMMC in sub-Saharan Africa along a number of dimensions, including facility type and service provider. Primary source facility level data from Kenya, Namibia, South Africa, Tanzania, Uganda, and Zambia were utilized throughout. We estimated the efficiency gains by econometrically estimating a cost function in order to calculate the impact of scale and other relevant factors. Personnel and consumables were estimated at 36% and 28%, respectively, of total costs across countries. Economies of scale (EOS) is estimated to be eight at the median volume of VMMCs performed, and EOS falls from 23 at the 25th percentile volume of VMMCs performed to 5.1 at the 75th percentile.

Conclusions

The analysis suggests that there is significant room for efficiency improvement as indicated by declining EOS as VMMC volume increases. The scale of the fall in EOS as VMMC volume increases suggests that we are still at the ascension phase of the scale-up of VMMC, where continuing to add new sites results in additional start-up costs as well. A key aspect of improving efficiency is task sharing VMMC procedures, due to the large percentage of overall costs associated with personnel costs. In addition, efficiency improvements in consumables are likely to occur over time as prices and distribution costs decrease.     

Regression analysis of mean quality-adjusted lifetime with censored data

In clinical trials of chronic diseases such as acquired immunodeficiency syndrome, cancer, or cardiovascular diseases, the concept of quality-adjusted lifetime (QAL) has received more and more attention. In this paper, we consider the problem of how the covariates affect the mean QAL when the data are subject to right censoring. We allow a very general form for the mean model as a function of covariates. Using the idea of inverse probability weighting, we first construct a simple weighted estimating equation for the parameters in our mean model. We then find the form of the most efficient estimating equation, which yields the most efficient estimator for the regression parameters. Since the most efficient estimator depends on the distribution of the health history processes, and thus cannot be estimated nonparametrically, we consider different approaches for improving the efficiency of the simple weighted estimating equation using observed data. The applicability of these methods is demonstrated by both simulation experiments and a data example from a breast cancer clinical trial study.     

Smooth quantile ratio estimation with regression: estimating medical expenditures for smoking-attributable diseases

The methodological development of this paper is motivated by a common problem in econometrics where we are interested in estimating the difference in the average expenditures between two populations, say with and without a disease, as a function of the covariates. For example, let Y(1) and Y(2) be two non-negative random variables denoting the health expenditures for cases and controls. Smooth Quantile Ratio Estimation (SQUARE) is a novel approach for estimating Delta=E[Y(1)] - E[Y(2)] by smoothing across percentiles the log-transformed ratio of the two quantile functions. Dominici et al. (2005) have shown that SQUARE defines a large class of estimators of Delta, is more efficient than common parametric and nonparametric estimators of Delta, and is consistent and asymptotically normal. However, in applications it is often desirable to estimate Delta(x)=E[Y(1)|x]--E[Y(2)|x], that is, the difference in means as a function of x. In this paper we extend SQUARE to a regression model and we introduce a two-part regression SQUARE for estimating Delta(x) as a function of x. We use the first part of the model to estimate the probability of incurring any costs and the second part of the model to estimate the mean difference in health expenditures, given that a nonzero cost is observed. In the second part of the model, we apply the basic definition of SQUARE for positive costs to compare expenditures for the cases and controls having 'similar' covariate profiles. We determine strata of cases and control with 'similar' covariate profiles by the use of propensity score matching. We then apply two-part regression SQUARE to the 1987 National Medicare Expenditure Survey to estimate the difference Delta(x) between persons suffering from smoking-attributable diseases and persons without these diseases as a function of the propensity of getting the disease. Using a simulation study, we compare frequentist properties of two-part regression SQUARE with maximum likelihood estimators for the log-transformed expenditures.     

Estimating the Cost of Type 1 Diabetes in the U.S.: A Propensity Score Matching Method

Background

Diabetes costs represent a large burden to both patients and the health care system. However, few studies that examine the economic consequences of diabetes have distinguished between the two major forms, type 1 and type 2 diabetes, despite differences in underlying pathologies. Combining the two diseases implies that there is no difference between the costs of type 1 and type 2 diabetes to a patient. In this study, we examine the costs of type 1 diabetes, which is often overlooked due to the larger population of type 2 patients, and compare them to the estimated costs of diabetes reported in the literature.

Methodology/Principal Findings

Using a nationally representative dataset, we estimate yearly and lifetime medical and indirect costs of type 1 diabetes by implementing a matching method to compare a patient with type 1 diabetes to a similar individual without the disease. We find that each year type 1 diabetes costs this country $14.4 billion (11.5–17.3) in medical costs and lost income. In terms of lost income, type 1 patients incur a disproportionate share of type 1 and type 2 costs. Further, if the disease were eliminated by therapeutic intervention, an estimated $10.6 billion (7.2–14.0) incurred by a new cohort and $422.9 billion (327.2–519.4) incurred by the existing number of type 1 diabetic patients over their lifetime would be avoided.

Conclusions/Significance

We find that the costs attributed to type 1 diabetes are disproportionately higher than the number of type 1 patients compared with type 2 patients, suggesting that combining the two diseases when estimating costs is not appropriate. This study and another recent contribution provides a necessary first step in estimating the substantial costs of type 1 diabetes on the U.S.     

Monetary valuation of road noise. Residential property prices as an indicator of the acoustic climate quality

Soundscape as an inherent part of landscape provides ecosystem services, first of all spiritual and symbolic services as well as educational and esthetic ones. The value of these services depends on the ability to hear sounds of nature. However, more often people can hear only sounds generated by humans and those generated by organisms and the physical environment become very desirable. Reports of the European Commission confirm that the noise in the environment is a very serious threat to public health and that the noise exposure in Europe is increasing. It is estimated that the main threat to the acoustic climate is road noise, both in the cities and outside them. Although the soundscape is a non-market good, the attempts of its evaluation have been increasing, usually by estimating the economic costs arising from exposure to noise: lost productivity, medical expenses, decreases in revenues from tourism. The authors used the hedonic pricing method to estimate the decline in undeveloped property prices associated with road noise around the city of Poznan. To extract the effects of noise also other factors that contribute to the land price were considered. The model chosen by using multiple regression showed, that plots located in the zone with noise exceedance at night were about 57% cheaper than those located outside this zone. The results can be helpful in spatial planning, especially for estimating costs of road investments in environmental and economic impact assessments.     

Estimating Incidence Rates from Population‐Based Case‐Control Studies in the Presence of Nonrespondents

In population‐based case‐control studies, it is of great public‐health importance to estimate the disease incidence rates associated with different levels of risk factors. This estimation is complicated by the fact that in such studies the selection probabilities for the cases and controls are unequal. A further complication arises when the subjects who are selected into the study do not participate (i.e. become nonrespondents) and nonrespondents differ systematically from respondents. In this paper, we show how to account for unequal selection probabilities as well as differential nonresponses in the incidence estimation. We use two logistic models, one relating the disease incidence rate to the risk factors, and one modelling the predictors that affect the nonresponse probability. After estimating the regression parameters in the nonresponse model, we estimate the regression parameters in the disease incidence model by a weighted estimating function that weights a respondent's contribution to the likelihood score function by the inverse of the product of his/her selection probability and his/her model‐predicted response probability. The resulting estimators of the regression parameters and the corresponding estimators of the incidence rates are shown to be consistent and asymptotically normal with easily estimated variances. Simulation results demonstrate that the asymptotic approximations are adequate for practical use and that failure to adjust for nonresponses could result in severe biases. An illustration with data from a cardiovascular study that motivated this work is presented.     

ROC-based utility function maximization for feature selection and classification with applications to high-dimensional protease data

SUMMARY: In medical diagnosis, the diseased and nondiseased classes are usually unbalanced and one class may be more important than the other depending on the diagnosis purpose. Most standard classification methods, however, are designed to maximize the overall accuracy and cannot incorporate different costs to different classes explicitly. In this article, we propose a novel nonparametric method to directly maximize the weighted specificity and sensitivity of the receiver operating characteristic curve. Combining advances in machine learning, optimization theory, and statistics, the proposed method has excellent generalization property and assigns different error costs to different classes explicitly. We present experiments that compare the proposed algorithms with support vector machines and regularized logistic regression using data from a study on HIV-1 protease as well as six public available datasets. Our main conclusion is that the performance of proposed algorithm is significantly better in most cases than the other classifiers tested. Software package in MATLAB is available upon request.     

Semiparametric estimation of the covariate-specific ROC curve in presence of ignorable verification bias

Covariate-specific receiver operating characteristic (ROC) curves are often used to evaluate the classification accuracy of a medical diagnostic test or a biomarker, when the accuracy of the test is associated with certain covariates. In many large-scale screening tests, the gold standard is subject to missingness due to high cost or harmfulness to the patient. In this article, we propose a semiparametric estimation of the covariate-specific ROC curves with a partial missing gold standard. A location-scale model is constructed for the test result to model the covariates' effect, but the residual distributions are left unspecified. Thus the baseline and link functions of the ROC curve both have flexible shapes. With the gold standard missing at random (MAR) assumption, we consider weighted estimating equations for the location-scale parameters, and weighted kernel estimating equations for the residual distributions. Three ROC curve estimators are proposed and compared, namely, imputation-based, inverse probability weighted, and doubly robust estimators. We derive the asymptotic normality of the estimated ROC curve, as well as the analytical form of the standard error estimator. The proposed method is motivated and applied to the data in an Alzheimer's disease research.     

Estimating multilevel logistic regression models when the number of clusters is low: a comparison of different statistical software procedures

Multilevel logistic regression models are increasingly being used to analyze clustered data in medical, public health, epidemiological, and educational research. Procedures for estimating the parameters of such models are available in many statistical software packages. There is currently little evidence on the minimum number of clusters necessary to reliably fit multilevel regression models. We conducted a Monte Carlo study to compare the performance of different statistical software procedures for estimating multilevel logistic regression models when the number of clusters was low. We examined procedures available in BUGS, HLM, R, SAS, and Stata. We found that there were qualitative differences in the performance of different software procedures for estimating multilevel logistic models when the number of clusters was low. Among the likelihood-based procedures, estimation methods based on adaptive Gauss-Hermite approximations to the likelihood (glmer in R and xtlogit in Stata) or adaptive Gaussian quadrature (Proc NLMIXED in SAS) tended to have superior performance for estimating variance components when the number of clusters was small, compared to software procedures based on penalized quasi-likelihood. However, only Bayesian estimation with BUGS allowed for accurate estimation of variance components when there were fewer than 10 clusters. For all statistical software procedures, estimation of variance components tended to be poor when there were only five subjects per cluster, regardless of the number of clusters.     

Estimating costs and outcomes of invasive American mink (<Emphasis Type="Italic">Neovison vison</Emphasis>) management in continental areas: a framework for evidence based control and eradication

Introduced predators are a major threat to biodiversity. While in island ecosystems the favoured management option is species eradication, in continental areas most managers tend to control-orientated options, assuming that eradication is an impossible goal. However, many management actions are conducted without precise or quantifiable goals, and their output is difficult to assess due to the lack of experimental approaches and scientific evidence. Here, we analyse the results of a typical small-scale short-term management action consisting of live-trapping and culling invasive American mink. We estimate the American mink population size at the beginning of the study assuming three different scenarios, assess the output of the management action in each scenario and model the results of further trapping efforts. Based on the results we estimate the effort and economic costs required for culling different population percentages per unit area, as well as the costs and feasibility of eradication. Our results provide a basis for planning invasive predator management, setting realistic management goals and estimating funding required, as well as a framework for managers to evaluate on-going management actions. In addition, our results suggest that American mink eradication from some continental areas would be feasible with current techniques at a moderate-low cost. We suggest that invasive predator management in continental areas should move towards eradication when feasible, regarding control as a second option.     

Modified Gaussian estimation for correlated binary data

In this paper, we develop a Gaussian estimation (GE) procedure to estimate the parameters of a regression model for correlated (longitudinal) binary response data using a working correlation matrix. A two‐step iterative procedure is proposed for estimating the regression parameters by the GE method and the correlation parameters by the method of moments. Consistency properties of the estimators are discussed. A simulation study was conducted to compare 11 estimators of the regression parameters, namely, four versions of the GE, five versions of the generalized estimating equations (GEEs), and two versions of the weighted GEE. Simulations show that (i) the Gaussian estimates have the smallest mean square error and best coverage probability if the working correlation structure is correctly specified and (ii) when the working correlation structure is correctly specified, the GE and the GEE with exchangeable correlation structure perform best as opposed to when the correlation structure is misspecified.     

Isolating cost drivers in interstitial lung disease treatment using nonparametric Bayesian methods

Mixture modeling is a popular approach to accommodate overdispersion, skewness, and multimodality features that are very common for health care utilization data. However, mixture modeling tends to rely on subjective judgment regarding the appropriate number of mixture components or some hypothesis about how to cluster the data. In this work, we adopt a nonparametric, variational Bayesian approach to allow the model to select the number of components while estimating their parameters. Our model allows for a probabilistic classification of observations into clusters and simultaneous estimation of a Gaussian regression model within each cluster. When we apply this approach to data on patients with interstitial lung disease, we find distinct subgroups of patients with differences in means and variances of health care costs, health and treatment covariates, and relationships between covariates and costs. The subgroups identified are readily interpretable, suggesting that this nonparametric variational approach to inference can discover valid insights into the factors driving treatment costs. Moreover, the learning algorithm we employed is very fast and scalable, which should make the technique accessible for a broad range of applications.     

Cox regression methods for two-stage randomization designs

Two-stage randomization designs (TSRD) are becoming increasingly common in oncology and AIDS clinical trials as they make more efficient use of study participants to examine therapeutic regimens. In these designs patients are initially randomized to an induction treatment, followed by randomization to a maintenance treatment conditional on their induction response and consent to further study treatment. Broader acceptance of TSRDs in drug development may hinge on the ability to make appropriate intent-to-treat type inference within this design framework as to whether an experimental induction regimen is better than a standard induction regimen when maintenance treatment is fixed. Recently Lunceford, Davidian, and Tsiatis (2002, Biometrics 58, 48-57) introduced an inverse probability weighting based analytical framework for estimating survival distributions and mean restricted survival times, as well as for comparing treatment policies at landmarks in the TSRD setting. In practice Cox regression is widely used and in this article we extend the analytical framework of Lunceford et al. (2002) to derive a consistent estimator for the log hazard in the Cox model and a robust score test to compare treatment policies. Large sample properties of these methods are derived, illustrated via a simulation study, and applied to a TSRD clinical trial.     

Median percent change: A robust alternative for assessing temporal trends

A typical summary statistic for temporal trends is the average percent change (APC). The APC is estimated by using a generalized linear model, usually under the assumption of linearity on the logarithmic scale. A serious limitation of least-squares type estimators is their sensitivity to outliers. The goal of this study is twofold: firstly, we propose a robust and easy-to-compute measure of the temporal trend based on the median of the rates (median percent change – MPC), rather than their mean, under the hypothesis of constant relative change; secondly, we investigate the performance of several models for estimating the rate of change when some of the most common model assumptions are violated. We provide some guidance on the practices of the estimation of temporal trends when using different models under different circumstances. The robustness property of the median is assessed in a simulation study, which shows that the MPC provides strong reductions in estimation bias and variance in presence of outliers. We also demonstrate how a mathematical property of the median helps addressing the issue of zero counts when estimating trends on the log-scale. Finally, we analyzed an English cancer registration dataset to illustrate the proposed method. We believe that, as a good practice, both APC and MPC should be presented when sensitivity issues arise.