Related Articles

We computationally investigate two approaches for uncertainty quantification in inverse problems for nonlinear parameter dependent dynamical systems. We compare the bootstrapping and asymptotic theory approaches for problems involving data with several noise forms and levels. We consider both constant variance absolute error data and relative error which produces non-constant variance data in our parameter estimation formulations. We compare and contrast parameter estimates, standard errors, confidence intervals, and computational times for both bootstrapping and asymptotic theory methods.

Many applications aim to learn a high dimensional parameter of a data generating distribution based on a sample of independent and identically distributed observations. For example, the goal might be to estimate the conditional mean of an outcome given a list of input variables. In this prediction context, bootstrap aggregating (bagging) has been introduced as a method to reduce the variance of a given estimator at little cost to bias. Bagging involves applying an estimator to multiple bootstrap samples, and averaging the result across bootstrap samples. In order to address the curse of dimensionality, a common practice has been to apply bagging to estimators which themselves use cross-validation, thereby using cross-validation within a bootstrap sample to select fine-tuning parameters trading off bias and variance of the bootstrap sample-specific candidate estimators. In this article we point out that in order to achieve the correct bias variance trade-off for the parameter of interest, one should apply the cross-validation selector externally to candidate bagged estimators indexed by these fine-tuning parameters. We use three simulations to compare the new cross-validated bagging method with bagging of cross-validated estimators and bagging of non-cross-validated estimators.

Background

Scoring systems are a very attractive family of clinical predictive models, because the patient score can be calculated without using any data processing system. Their weakness lies in the difficulty of associating a reliable prognostic probability with each score. In this study a bootstrap approach for estimating confidence intervals of outcome probabilities is described and applied to design and optimize the performance of a scoring system for morbidity in intensive care units after heart surgery.

Methods

The bias-corrected and accelerated bootstrap method was used to estimate the 95% confidence intervals of outcome probabilities associated with a scoring system. These confidence intervals were calculated for each score and each step of the scoring-system design by means of one thousand bootstrapped samples. 1090 consecutive adult patients who underwent coronary artery bypass graft were assigned at random to two groups of equal size, so as to define random training and testing sets with equal percentage morbidities. A collection of 78 preoperative, intraoperative and postoperative variables were considered as likely morbidity predictors.

Results

Several competing scoring systems were compared on the basis of discrimination, generalization and uncertainty associated with the prognostic probabilities. The results showed that confidence intervals corresponding to different scores often overlapped, making it convenient to unite and thus reduce the score classes. After uniting two adjacent classes, a model with six score groups not only gave a satisfactory trade-off between discrimination and generalization, but also enabled patients to be allocated to classes, most of which were characterized by well separated confidence intervals of prognostic probabilities.

Conclusions

Scoring systems are often designed solely on the basis of discrimination and generalization characteristics, to the detriment of prediction of a trustworthy outcome probability. The present example demonstrates that using a bootstrap method for the estimation of outcome-probability confidence intervals provides useful additional information about score-class statistics, guiding physicians towards the most convenient model for predicting morbidity outcomes in their clinical context.

Random error (misclassification) in exposure measurements usually biases a relative risk, regression coefficient, or other effect measure towards the null value (no association). The most important exception is Berkson type error, which causes little or no bias. Berkson type error arises, in particular, due to use of group average exposure in place of individual values. Random error in exposure measurements, Berkson or otherwise, reduces the power of a study, making it more likely that real associations are not detected. Random error in confounding variables compromises the control of their effect, leaving residual confounding. Random error in a variable that modifies the effect of exposure on health--for example, an indicator of susceptibility--tends to diminish the observed modification of effect, but error in the exposure can create a supurious appearance of modification. Methods are available to correct for bias (but not generally power loss) due to measurement error, if information on the magnitude and type of error is available. These methods can be complicated to use, however, and should be used cautiously as "correction" can magnify confounding if it is present.

 

Summary

A bootstrap based methodology is introduced for analyzing repeated measures/longitudinal microarray gene expression data over ordered categories. The proposed non-parametric procedure uses order-restricted inference to compare gene expressions among ordered experimental conditions. The null distribution for determining significance is derived by suitably bootstrapping the residuals. The procedure addresses two potential sources of correlation in the data, namely, (a) correlations among genes within a chip (“intra-chip” correlation), and (b) correlation within subject due to repeated/longitudinal measurements (“temporal” correlation). To make the procedure computationally efficient, the adaptive bootstrap methodology of Guo and Peddada (2008) is implemented such that the resulting procedure controls the false discovery rate (FDR) at the desired nominal level.

In many environmental epidemiology studies, the locations and/or times of exposure measurements and health assessments do not match. In such settings, health effects analyses often use the predictions from an exposure model as a covariate in a regression model. Such exposure predictions contain some measurement error as the predicted values do not equal the true exposures. We provide a framework for spatial measurement error modeling, showing that smoothing induces a Berkson-type measurement error with nondiagonal error structure. From this viewpoint, we review the existing approaches to estimation in a linear regression health model, including direct use of the spatial predictions and exposure simulation, and explore some modified approaches, including Bayesian models and out-of-sample regression calibration, motivated by measurement error principles. We then extend this work to the generalized linear model framework for health outcomes. Based on analytical considerations and simulation results, we compare the performance of all these approaches under several spatial models for exposure. Our comparisons underscore several important points. First, exposure simulation can perform very poorly under certain realistic scenarios. Second, the relative performance of the different methods depends on the nature of the underlying exposure surface. Third, traditional measurement error concepts can help to explain the relative practical performance of the different methods. We apply the methods to data on the association between levels of particulate matter and birth weight in the greater Boston area.

We consider asymptotic properties of the maximum likelihood and related estimators in a clustered logistic joinpoint model with an unknown joinpoint. Sufficient conditions are given for the consistency of confidence bounds produced by the parametric bootstrap; one of the conditions required is that the true location of the joinpoint is not at one of the observation times. A simulation study is presented to illustrate the lack of consistency of the bootstrap confidence bounds when the joinpoint is an observation time. A removal algorithm is presented which corrects this problem, but at the price of an increased mean square error. Finally, the methods are applied to data on yearly cancer mortality in the United States for individuals age 65 and over.

We propose a computationally intensive method, the random lasso method, for variable selection in linear models. The method consists of two major steps. In step 1, the lasso method is applied to many bootstrap samples, each using a set of randomly selected covariates. A measure of importance is yielded from this step for each covariate. In step 2, a similar procedure to the first step is implemented with the exception that for each bootstrap sample, a subset of covariates is randomly selected with unequal selection probabilities determined by the covariates’ importance. Adaptive lasso may be used in the second step with weights determined by the importance measures. The final set of covariates and their coefficients are determined by averaging bootstrap results obtained from step 2. The proposed method alleviates some of the limitations of lasso, elastic-net and related methods noted especially in the context of microarray data analysis: it tends to remove highly correlated variables altogether or select them all, and maintains maximal flexibility in estimating their coefficients, particularly with different signs; the number of selected variables is no longer limited by the sample size; and the resulting prediction accuracy is competitive or superior compared to the alternatives. We illustrate the proposed method by extensive simulation studies. The proposed method is also applied to a Glioblastoma microarray data analysis.

Summary

Measurement error data or errors-in-variable data are often collected in many studies. Natural criterion functions are often unavailable for general functional measurement error models due to the lack of information on the distribution of the unobservable covariates. Typically, the parameter estimation is via solving estimating equations. In addition, the construction of such estimating equations routinely requires solving integral equations, hence the computation is often much more intensive compared with ordinary regression models. Because of these difficulties, traditional best subset variable selection procedures are not applicable, and in the measurement error model context, variable selection remains an unsolved issue. In this paper, we develop a framework for variable selection in measurement error models via penalized estimating equations. We first propose a class of selection procedures for general parametric measurement error models and for general semiparametric measurement error models, and study the asymptotic properties of the proposed procedures. Then, under certain regularity conditions and with a properly chosen regularization parameter, we demonstrate that the proposed procedure performs as well as an oracle procedure. We assess the finite sample performance via Monte Carlo simulation studies and illustrate the proposed methodology through the empirical analysis of a familiar data set.

Background

In prognostic studies model instability and missing data can be troubling factors. Proposed methods for handling these situations are bootstrapping (B) and Multiple imputation (MI). The authors examined the influence of these methods on model composition.

Methods

Models were constructed using a cohort of 587 patients consulting between January 2001 and January 2003 with a shoulder problem in general practice in the Netherlands (the Dutch Shoulder Study). Outcome measures were persistent shoulder disability and persistent shoulder pain. Potential predictors included socio-demographic variables, characteristics of the pain problem, physical activity and psychosocial factors. Model composition and performance (calibration and discrimination) were assessed for models using a complete case analysis, MI, bootstrapping or both MI and bootstrapping.

Results

Results showed that model composition varied between models as a result of how missing data was handled and that bootstrapping provided additional information on the stability of the selected prognostic model.

Conclusion

In prognostic modeling missing data needs to be handled by MI and bootstrap model selection is advised in order to provide information on model stability.

The relationship between exposure to environmental chemicals during pregnancy and early childhood development is an important issue which has a spatial risk component. In this context, we have examined mental retardation and developmental delay (MRDD) outcome measures for children in a Medicaid population in South Carolina and sampled measures of soil chemistry (e.g. As, Hg, etc.) on a network of sites which are misaligned to the outcome residential addresses during pregnancy. The true chemical concentration at the residential addresses is not observed directly and must be interpolated from soil samples. In this study, we have developed a Bayesian joint model which interpolates soil chemical fields and estimates the associated MRDD risk simultaneously. Having multiple spatial fields to interpolate, we have considered a low-rank Kriging method for the interpolation which requires less computation than Bayesian Kriging. We performed a sensitivity analysis for a bivariate smoothing, changing the number of knots and the smoothing parameter. These analyses show that a low-rank Kriging method can be used as an alternative to a full-rank Kriging, reducing computational burden. However, the number of knots for the low-rank Kriging model need to be selected with caution as a bivariate surface estimation can be sensitive to the choice of the number of knots.

In a cocaine dependence treatment study, we use linear and nonlinear regression models to model posttreatment cocaine craving scores and first cocaine relapse time. A subset of the covariates are summary statistics derived from baseline daily cocaine use trajectories, such as baseline cocaine use frequency and average daily use amount. These summary statistics are subject to estimation error and can therefore cause biased estimators for the regression coefficients. Unlike classical measurement error problems, the error we encounter here is heteroscedastic with an unknown distribution, and there are no replicates for the error-prone variables or instrumental variables. We propose two robust methods to correct for the bias: a computationally efficient method-of-moments-based method for linear regression models and a subsampling extrapolation method that is generally applicable to both linear and nonlinear regression models. Simulations and an application to the cocaine dependence treatment data are used to illustrate the efficacy of the proposed methods. Asymptotic theory and variance estimation for the proposed subsampling extrapolation method and some additional simulation results are described in the online supplementary material.

Background

Accurately modeling the sequence substitution process is required for the correct estimation of evolutionary parameters, be they phylogenetic relationships, substitution rates or ancestral states; it is also crucial to simulate realistic data sets. Such simulation procedures are needed to estimate the null-distribution of complex statistics, an approach referred to as parametric bootstrapping, and are also used to test the quality of phylogenetic reconstruction programs. It has often been observed that homologous sequences can vary widely in their nucleotide or amino-acid compositions, revealing that sequence evolution has changed importantly among lineages, and may therefore be most appropriately approached through non-homogeneous models. Several programs implementing such models have been developed, but they are limited in their possibilities: only a few particular models are available for likelihood optimization, and data sets cannot be easily generated using the resulting estimated parameters.

Results

We hereby present a general implementation of non-homogeneous models of substitutions. It is available as dedicated classes in the Bio++ libraries and can hence be used in any C++ program. Two programs that use these classes are also presented. The first one, Bio++ Maximum Likelihood (BppML), estimates parameters of any non-homogeneous model and the second one, Bio++ Sequence Generator (BppSeqGen), simulates the evolution of sequences from these models. These programs allow the user to describe non-homogeneous models through a property file with a simple yet powerful syntax, without any programming required.

Conclusion

We show that the general implementation introduced here can accommodate virtually any type of non-homogeneous models of sequence evolution, including heterotachous ones, while being computer efficient. We furthermore illustrate the use of such general models for parametric bootstrapping, using tests of non-homogeneity applied to an already published ribosomal RNA data set.

Background

Predictive microbiology develops mathematical models that can predict the growth rate of a microorganism population under a set of environmental conditions. Many primary growth models have been proposed. However, when primary models are applied to bacterial growth curves, the biological variability is reduced to a single curve defined by some kinetic parameters (lag time and growth rate), and sometimes the models give poor fits in some regions of the curve. The development of a prediction band (from a set of bacterial growth curves) using non-parametric and bootstrap methods permits to overcome that problem and include the biological variability of the microorganism into the modelling process.

Results

Absorbance data from Listeria monocytogenes cultured at 22, 26, 38, and 42°C were selected under different environmental conditions of pH (4.5, 5.5, 6.5, and 7.4) and percentage of NaCl (2.5, 3.5, 4.5, and 5.5). Transformation of absorbance data to viable count data was carried out. A random effect multiplicative heteroscedastic model was considered to explain the dynamics of bacterial growth. The concept of a prediction band for microbial growth is proposed. The bootstrap method was used to obtain resamples from this model. An iterative procedure is proposed to overcome the computer intensive task of calculating simultaneous prediction intervals, along time, for bacterial growth. The bands were narrower below the inflection point (0-8 h at 22°C, and 0-5.5 h at 42°C), and wider to the right of it (from 9 h onwards at 22°C, and from 7 h onwards at 42°C). A wider band was observed at 42°C than at 22°C when the curves reach their upper asymptote. Similar bands have been obtained for 26 and 38°C.

Conclusions

The combination of nonparametric models and bootstrap techniques results in a good procedure to obtain reliable prediction bands in this context. Moreover, the new iterative algorithm proposed in this paper allows one to achieve exactly the prefixed coverage probability for the prediction band. The microbial growth bands reflect the influence of the different environmental conditions on the microorganism behaviour, helping in the interpretation of the biological meaning of the growth curves obtained experimentally.

Background

Non-parametric bootstrapping is a widely-used statistical procedure for assessing confidence of model parameters based on the empirical distribution of the observed data [1] and, as such, it has become a common method for assessing tree confidence in phylogenetics [2]. Traditional non-parametric bootstrapping does not weigh each tree inferred from resampled (i.e., pseudo-replicated) sequences. Hence, the quality of these trees is not taken into account when computing bootstrap scores associated with the clades of the original phylogeny. As a consequence, traditionally, the trees with different bootstrap support or those providing a different fit to the corresponding pseudo-replicated sequences (the fit quality can be expressed through the LS, ML or parsimony score) contribute in the same way to the computation of the bootstrap support of the original phylogeny.

Results

In this article, we discuss the idea of applying weighted bootstrapping to phylogenetic reconstruction by weighting each phylogeny inferred from resampled sequences. Tree weights can be based either on the least-squares (LS) tree estimate or on the average secondary bootstrap score (SBS) associated with each resampled tree. Secondary bootstrapping consists of the estimation of bootstrap scores of the trees inferred from resampled data. The LS and SBS-based bootstrapping procedures were designed to take into account the quality of each "pseudo-replicated" phylogeny in the final tree estimation. A simulation study was carried out to evaluate the performances of the five weighting strategies which are as follows: LS and SBS-based bootstrapping, LS and SBS-based bootstrapping with data normalization and the traditional unweighted bootstrapping.

Conclusions

The simulations conducted with two real data sets and the five weighting strategies suggest that the SBS-based bootstrapping with the data normalization usually exhibits larger bootstrap scores and a higher robustness compared to the four other competing strategies, including the traditional bootstrapping. The high robustness of the normalized SBS could be particularly useful in situations where observed sequences have been affected by noise or have undergone massive insertion or deletion events. The results provided by the four other strategies were very similar regardless the noise level, thus also demonstrating the stability of the traditional bootstrapping method.

Summary

Estimation of a regression function is a well-known problem in the context of errors in variables, where the explanatory variable is observed with random noise. This noise can be of two types, which are known as classical or Berkson, and it is common to assume that the error is purely of one of these two types. In practice, however, there are many situations where the explanatory variable is contaminated by a mixture of the two errors. In such instances, the Berkson component typically arises because the variable of interest is not directly available and can only be assessed through a proxy, whereas the inaccuracy that is related to the observation of the latter causes an error of classical type. We propose a non-parametric estimator of a regression function from data that are contaminated by a mixture of the two errors. We prove consistency of our estimator, derive rates of convergence and suggest a data-driven implementation. Finite sample performance is illustrated via simulated and real data examples.

This paper considers the problem of estimation in a general semiparametric regression model when error-prone covariates are modeled parametrically while covariates measured without error are modeled nonparametrically. To account for the effects of measurement error, we apply a correction to a criterion function. The specific form of the correction proposed allows Monte Carlo simulations in problems for which the direct calculation of a corrected criterion is difficult. Therefore, in contrast to methods that require solving integral equations of possibly multiple dimensions, as in the case of multiple error-prone covariates, we propose methodology which offers a simple implementation. The resulting methods are functional, they make no assumptions about the distribution of the mismeasured covariates. We utilize profile kernel and backfitting estimation methods and derive the asymptotic distribution of the resulting estimators. Through numerical studies we demonstrate the applicability of proposed methods to Poisson, logistic and multivariate Gaussian partially linear models. We show that the performance of our methods is similar to a computationally demanding alternative. Finally, we demonstrate the practical value of our methods when applied to Nevada Test Site (NTS) Thyroid Disease Study data.

Motivation: Although widely accepted that high-throughput biological data are typically highly noisy, the effects that this uncertainty has upon the conclusions we draw from these data are often overlooked. However, in order to assign any degree of confidence to our conclusions, we must quantify these effects. Bootstrap resampling is one method by which this may be achieved. Here, we present a parametric bootstrapping approach for time-course data, in which Gaussian process regression (GPR) is used to fit a probabilistic model from which replicates may then be drawn. This approach implicitly allows the time dependence of the data to be taken into account, and is applicable to a wide range of problems.

Results: We apply GPR bootstrapping to two datasets from the literature. In the first example, we show how the approach may be used to investigate the effects of data uncertainty upon the estimation of parameters in an ordinary differential equations (ODE) model of a cell signalling pathway. Although we find that the parameter estimates inferred from the original dataset are relatively robust to data uncertainty, we also identify a distinct second set of estimates. In the second example, we use our method to show that the topology of networks constructed from time-course gene expression data appears to be sensitive to data uncertainty, although there may be individual edges in the network that are robust in light of present data.

Availability: Matlab code for performing GPR bootstrapping is available from our web site: http://www3.imperial.ac.uk/theoreticalsystemsbiology/data-software/

Contact: paul.kirk@imperial.ac.uk, m.stumpf@imperial.ac.uk

Supplementary information:Supplementary data are available at Bioinformatics online.

It is a common practice to use resampling methods such as the bootstrap for calculating the p-value for each test when performing large scale multiple testing. The precision of the bootstrap p-values and that of the false discovery rate (FDR) relies on the number of bootstraps used for testing each hypothesis. Clearly, the larger the number of bootstraps the better the precision. However, the required number of bootstraps can be computationally burdensome, and it multiplies the number of tests to be performed. Further adding to the computational challenge is that in some applications the calculation of the test statistic itself may require considerable computation time. As technology improves one can expect the dimension of the problem to increase as well. For instance, during the early days of microarray technology, the number of probes on a cDNA chip was less than 10,000. Now the Affymetrix chips come with over 50,000 probes per chip. Motivated by this important need, we developed a simple adaptive bootstrap methodology for large scale multiple testing, which reduces the total number of bootstrap calculations while ensuring the control of the FDR. The proposed algorithm results in a substantial reduction in the number of bootstrap samples. Based on a simulation study we found that, relative to the number of bootstraps required for the Benjamini-Hochberg (BH) procedure, the standard FDR methodology which was the proposed methodology achieved a very substantial reduction in the number of bootstraps. In some cases the new algorithm required as little as 1/6th the number of bootstraps as the conventional BH procedure. Thus, if the conventional BH procedure used 1,000 bootstraps, then the proposed method required only 160 bootstraps. This methodology has been implemented for time-course/dose-response data in our software, ORIOGEN, which is available from the authors upon request.

In-motion alignment of Strapdown Inertial Navigation Systems (SINS) without any geodetic-frame observations is one of the toughest challenges for Autonomous Underwater Vehicles (AUV). This paper presents a novel scheme for Doppler Velocity Log (DVL) aided SINS alignment using Unscented Kalman Filter (UKF) which allows large initial misalignments. With the proposed mechanism, a nonlinear SINS error model is presented and the measurement model is derived under the assumption that large misalignments may exist. Since a priori knowledge of the measurement noise covariance is of great importance to robustness of the UKF, the covariance-matching methods widely used in the Adaptive KF (AKF) are extended for use in Adaptive UKF (AUKF). Experimental results show that the proposed DVL-aided alignment model is effective with any initial heading errors. The performances of the adaptive filtering methods are evaluated with regards to their parameter estimation stability. Furthermore, it is clearly shown that the measurement noise covariance can be estimated reliably by the adaptive UKF methods and hence improve the performance of the alignment.

Time series studies of environmental exposures often involve comparing daily changes in a toxicant measured at a point in space with daily changes in an aggregate measure of health. Spatial misalignment of the exposure and response variables can bias the estimation of health risk, and the magnitude of this bias depends on the spatial variation of the exposure of interest. In air pollution epidemiology, there is an increasing focus on estimating the health effects of the chemical components of particulate matter (PM). One issue that is raised by this new focus is the spatial misalignment error introduced by the lack of spatial homogeneity in many of the PM components. Current approaches to estimating short-term health risks via time series modeling do not take into account the spatial properties of the chemical components and therefore could result in biased estimation of those risks. We present a spatial–temporal statistical model for quantifying spatial misalignment error and show how adjusted health risk estimates can be obtained using a regression calibration approach and a 2-stage Bayesian model. We apply our methods to a database containing information on hospital admissions, air pollution, and weather for 20 large urban counties in the United States.

We present a heuristic approach to the DNA assignment problem based on phylogenetic inferences using constrained neighbour joining and non-parametric bootstrapping. We show that this method performs as well as the more computationally intensive full Bayesian approach in an analysis of 500 insect DNA sequences obtained from GenBank. We also analyse a previously published dataset of environmental DNA sequences from soil from New Zealand and Siberia, and use these data to illustrate the fact that statistical approaches to the DNA assignment problem allow for more appropriate criteria for determining the taxonomic level at which a particular DNA sequence can be assigned.

Background

Large-scale statistical analyses have become hallmarks of post-genomic era biological research due to advances in high-throughput assays and the integration of large biological databases. One accompanying issue is the simultaneous estimation of p-values for a large number of hypothesis tests. In many applications, a parametric assumption in the null distribution such as normality may be unreasonable, and resampling-based p-values are the preferred procedure for establishing statistical significance. Using resampling-based procedures for multiple testing is computationally intensive and typically requires large numbers of resamples.

Results

We present a new approach to more efficiently assign resamples (such as bootstrap samples or permutations) within a nonparametric multiple testing framework. We formulated a Bayesian-inspired approach to this problem, and devised an algorithm that adapts the assignment of resamples iteratively with negligible space and running time overhead. In two experimental studies, a breast cancer microarray dataset and a genome wide association study dataset for Parkinson's disease, we demonstrated that our differential allocation procedure is substantially more accurate compared to the traditional uniform resample allocation.

Conclusion

Our experiments demonstrate that using a more sophisticated allocation strategy can improve our inference for hypothesis testing without a drastic increase in the amount of computation on randomized data. Moreover, we gain more improvement in efficiency when the number of tests is large. R code for our algorithm and the shortcut method are available at .

Background

This work has investigated under what conditions confidence intervals around the differences in mean costs from a cluster RCT are suitable for estimation using a commonly used cluster-adjusted bootstrap in preference to methods that utilise the Huber-White robust estimator of variance. The bootstrap's main advantage is in dealing with skewed data, which often characterise patient costs. However, it is insufficiently well recognised that one method of adjusting the bootstrap to deal with clustered data is only valid in large samples. In particular, the requirement that the number of clusters randomised should be large would not be satisfied in many cluster RCTs performed to date.

Methods

The performances of confidence intervals for simple differences in mean costs utilising a robust (cluster-adjusted) standard error and from two cluster-adjusted bootstrap procedures were compared in terms of confidence interval coverage in a large number of simulations. Parameters varied included the intracluster correlation coefficient, the sample size and the distributions used to generate the data.

Results

The bootstrap's advantage in dealing with skewed data was found to be outweighed by its poor confidence interval coverage when the number of clusters was at the level frequently found in cluster RCTs in practice. Simulations showed that confidence intervals based on robust methods of standard error estimation achieved coverage rates between 93.5% and 94.8% for a 95% nominal level whereas those for the bootstrap ranged between 86.4% and 93.8%.

Conclusion

In general, 24 clusters per treatment arm is probably the minimum number for which one would even begin to consider the bootstrap in preference to traditional robust methods, for the parameter combinations investigated here. At least this number of clusters and extremely skewed data would be necessary for the bootstrap to be considered in favour of the robust method. There is a need for further investigation of more complex bootstrap procedures if economic data from cluster RCTs are to be analysed appropriately.

Summary

The existing theory of the wild bootstrap has focused on linear estimators. In this note, we broaden its validity by providing a class of weight distributions that is asymptotically valid for quantile regression estimators. As most weight distributions in the literature lead to biased variance estimates for nonlinear estimators of linear regression, we propose a modification of the wild bootstrap that admits a broader class of weight distributions for quantile regression. A simulation study on median regression is carried out to compare various bootstrap methods. With a simple finite-sample correction, the wild bootstrap is shown to account for general forms of heteroscedasticity in a regression model with fixed design points.