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The Richards model and its generalized version are deterministic models that are often implemented to fit and forecast the cumulative number of infective cases in an epidemic outbreak. In this paper we employ a generalized Richards model to predict the cumulative number of COVID-19 cases in Spain and Italy, based on available epidemiological data. To quantify uncertainty in the parameter estimation, we use a parametric bootstrapping approach to construct a 95% confidence interval estimation for the parameter model. Here we assume that the time series data follow a Poisson distribution. It is found that the 95% confidence interval of each parameter becomes narrow with the increasing number of data. All in all, the model predicts daily new cases of COVID-19 reasonably well during calibration periods. However, the model fails to produce good forecasts when the amount of data used for parameter estimations is not sufficient. Based on our parameter estimates, it is found that the early stages of COVID-19 epidemic, both in Spain and in Italy, followed an almost exponentially growth. The epidemic peak in Spain and Italy is respectively on 2 April 2020 and 28 March 2020. The final sizes of cumulative number of COVID-19 cases in Spain and Italy are forecasted to be at 293220 and 237010, respectively.

There are known limitations in methods of detecting positive selection. Common methods do not enable differentiation between positive selection and compensatory covariation, a major limitation. Further, the traditional method of calculating the ratio of nonsynonymous to synonymous substitutions (dN/dS) does not take into account the 3D structure of biomacromolecules nor differences between amino acids. It also does not account for saturation of synonymous mutations (dS) over long evolutionary time that renders codon-based methods ineffective for older divergences. This work aims to address these shortcomings for detecting positive selection through the development of a statistical model that examines clusters of substitutions in clusters of variable radii. Additionally, it uses a parametric bootstrapping approach to differentiate positive selection from compensatory processes. A previously reported case of positive selection in the leptin protein of primates was reexamined using this methodology.

Understanding links between thermal performance and environmental variation is necessary to predict organismal responses to climate change, and remains an ongoing challenge for ectotherms with complex life cycles. Distinct life stages can differ in thermal sensitivity, experience different environmental conditions as development unfolds, and, because stages are by nature interdependent, environmental effects can carry over from one stage to affect performance at others. Thermal performance may therefore respond to carryover effects of prior thermal environments, yet detailed insights into the nature, strength, and direction of those responses are still lacking. Here, in an aquatic ectotherm whose early planktonic stages (gametes, embryos, and larvae) govern adult abundances and dynamics, we explore the effects of prior thermal environments at fertilization and embryogenesis on thermal performance curves at the end of planktonic development. We factorially manipulate temperatures at fertilization and embryogenesis, then, for each combination of prior temperatures, measure thermal performance curves for survival of planktonic development (end of the larval stage) throughout the performance range. By combining generalized linear mixed modeling with parametric bootstrapping, we formally estimate and compare curve descriptors (thermal optima, limits, and breadth) among prior environments, and reveal carryover effects of temperature at embryogenesis, but not fertilization, on thermal optima at completion of development. Specifically, thermal optima shifted to track temperature during embryogenesis, while thermal limits and breadth remained unchanged. Our results argue that key aspects of thermal performance are shaped by prior thermal environment in early life, warranting further investigation of the possible mechanisms underpinning that response, and closer consideration of thermal carryover effects when predicting organismal responses to climate change.

This paper proposes a method for constructing a predictive estimator for logistic regression. We make a provisional assumption that the predictive estimator is given by multiplying the maximum likelihood estimators by constants, which are estimated using a parametric bootstrap method. The relative merits of the maximum likelihood estimator and the predictive estimator produced by this method are determined by cross-validation. The results show that the predictiveestimators derived by this method lead to a smaller deviance than that obtained by the maximum likelihood estimator in many instances.

BackgroundEnsemble modeling aims to boost the forecasting performance by systematically integrating the predictive accuracy across individual models. Here we introduce a simple-yet-powerful ensemble methodology for forecasting the trajectory of dynamic growth processes that are defined by a system of non-linear differential equations with applications to infectious disease spread.MethodsWe propose and assess the performance of two ensemble modeling schemes with different parametric bootstrapping procedures for trajectory forecasting and uncertainty quantification. Specifically, we conduct sequential probabilistic forecasts to evaluate their forecasting performance using simple dynamical growth models with good track records including the Richards model, the generalized-logistic growth model, and the Gompertz model. We first test and verify the functionality of the method using simulated data from phenomenological models and a mechanistic transmission model. Next, the performance of the method is demonstrated using a diversity of epidemic datasets including scenario outbreak data of theEbola Forecasting Challengeand real-world epidemic data outbreaks of including influenza, plague, Zika, and COVID-19.ResultsWe found that the ensemble method that randomly selects a model from the set of individual models for each time point of the trajectory of the epidemic frequently outcompeted the individual models as well as an alternative ensemble method based on the weighted combination of the individual models and yields broader and more realistic uncertainty bounds for the trajectory envelope, achieving not only better coverage rate of the 95% prediction interval but also improved mean interval scores across a diversity of epidemic datasets.ConclusionOur new methodology for ensemble forecasting outcompete component models and an alternative ensemble model that differ in how the variance is evaluated for the generation of the prediction intervals of the forecasts.

Background The hardware and software differences between MR vendors and individual sites influence the quantification of MR spectroscopy data. An analysis of a large data set may help to better understand sources of the total variance in quantified metabolite levels. Purpose To compare multisite quantitative brain MR spectroscopy data acquired in healthy participants at 26 sites by using the vendor-supplied single-voxel point-resolved spectroscopy (PRESS) sequence. Materials and Methods An MR spectroscopy protocol to acquire short-echo-time PRESS data from the midparietal region of the brain was disseminated to 26 research sites operating 3.0-T MR scanners from three different vendors. In this prospective study, healthy participants were scanned between July 2016 and December 2017. Data were analyzed by using software with simulated basis sets customized for each vendor implementation. The proportion of total variance attributed to vendor-, site-, and participant-related effects was estimated by using a linear mixed-effects model. P values were derived through parametric bootstrapping of the linear mixed-effects models (denoted Pboot). Results In total, 296 participants (mean age, 26 years ± 4.6; 155 women and 141 men) were scanned. Good-quality data were recorded from all sites, as evidenced by a consistent linewidth of N-acetylaspartate (range, 4.4-5.0 Hz), signal-to-noise ratio (range, 174-289), and low Cramér-Rao lower bounds (≤5%) for all of the major metabolites. Among the major metabolites, no vendor effects were found for levels of myo-inositol (Pboot > .90), N-acetylaspartate and N-acetylaspartylglutamate (Pboot = .13), or glutamate and glutamine (Pboot = .11). Among the smaller resonances, no vendor effects were found for ascorbate (Pboot = .08), aspartate (Pboot > .90), glutathione (Pboot > .90), or lactate (Pboot = .28). Conclusion Multisite multivendor single-voxel MR spectroscopy studies performed at 3.0 T can yield results that are coherent across vendors, provided that vendor differences in pulse sequence implementation are accounted for in data analysis. However, the site-related effects on variability were more profound and suggest the need for further standardization of spectroscopic protocols. © RSNA, 2020 Online supplemental material is available for this article. ispartof: RADIOLOGY vol:295 issue:1 pages:171-180 ispartof: location:United States status: published

A new robust pairwise statistic, the pairwise median scaled difference (MSD), is proposed for the detection of anomalous location/uncertainty pairs in heteroscedastic interlaboratory study data with associated uncertainties. The distribution for the IID case is presented and approximate critical values for routine use are provided. The determination of observation-specific quantiles and p-values for heteroscedastic data, using parametric bootstrapping, is demonstrated by example. It is shown that the statistic has good power for detecting anomalies compared to a previous pairwise statistic, and offers much greater resistance to multiple outlying values.

Digital games controlled by body movements (exergames) have been proposed as a way to improve postural control among older adults. Exergames are meant to be played at home in an unsupervised way. However, only few studies have investigated the effect of unsupervised home-exergaming on postural control. Moreover, suitable methods to dynamically assess postural control during exergaming are still scarce. Dynamic postural control (DPC) assessment could be used to provide both meaningful feedback and automatic adjustment of exergame difficulty. These features could potentially foster unsupervised exergaming at home and improve the effectiveness of exergames as tools to improve balance control. The main aim of this study is to investigate the effect of six weeks of unsupervised home-exergaming on DPC as assessed by a recently developed probabilistic model. High probability values suggest ‘deteriorated’ postural control, whereas low probability values suggest ‘good’ postural control. In a pilot study, ten healthy older adults (average 77.9, SD 7.2 years) played an ice-skating exergame at home half an hour per day, three times a week during six weeks. The intervention effect on DPC was assessed using exergaming trials recorded by Kinect at baseline and every other week. Visualization of the results suggests that the probabilistic model is suitable for real-time DPC assessment. Moreover, linear mixed model analysis and parametric bootstrapping suggest a significant intervention effect on DPC. In conclusion, these results suggest that unsupervised exergaming for improving DPC among older adults is indeed feasible and that probabilistic models could be a new approach to assess DPC.

Post-stratified model-assisted (MA) and hybrid (HY) estimators are used with repeated airborne laser scanning (ALS) strip sampling and national forest inventory field data for stratum-wise and overall estimation of aboveground biomass (AGB) stock and change. The study area covered the southern portion of the Hedmark County in Norway. Both MA and HY estimation substantially reduced the uncertainty in AGB change when compared with estimation using the field survey only. Relative efficiencies (relative variance) of 4.15 (MA) and 3.36 (HY) for overall estimates were found. The results suggest the MA estimator for single-time estimation and the HY as more appropriate for change estimation by cover class. With the HY estimator, a nested post-stratification scheme is demonstrated, combining cover classes with change classes, which enables detailed reporting for change according to cause within each cover class, and has the potential to improve the estimation precision. Finally, parametric bootstrapping is demonstrated as an empirical alternative to estimate the model-error component in the HY estimator. The model error estimated with parametric bootstrapping converged to the analytically determined value of the HY estimator within 1000 bootstrap samples.

Understanding thermal performance at life stages that limit persistence is necessary to predict responses to climate change, especially for ectotherms whose fitness (survival and reproduction) depends on environmental temperature. Ectotherms often undergo stage-specific changes in size, complexity, and duration that are predicted to modify thermal performance. Yet performance is mostly explored for adults, while performance at earlier stages that typically limit persistence remains poorly understood. Here, we experimentally isolate thermal performance curves at fertilization, embryo development, and larval development in an aquatic ectotherm whose early planktonic stages (gametes, embryos, and larvae) govern adult abundances and dynamics. Unlike previous studies based on short-term exposures, responses with unclear links to fitness, or proxies in lieu of explicit curve descriptors (thermal optima, limits, and breadth), we measure performance as successful completion of each stage after exposure throughout, and at temperatures that explicitly capture curve descriptors at all stages. Formal comparisons of descriptors using a combination of generalized linear mixed modelling and parametric bootstrapping reveal important differences among life stages. Thermal performance differs significantly from fertilization to embryo development (with thermal optimum declining by ∼2 °C, thermal limits shifting inwards by ∼8–10 °C, and thermal breadth narrowing by ∼10 °C), while performance declines independently of temperature thereafter. Our comparisons show that thermal performance at one life stage can misrepresent performance at others, and point to gains in complexity during embryogenesis, rather than subsequent gains in size or duration of exposure, as a key driver of thermal sensitivity in early life.

Students pursuing baccalaureate degrees in electrical engineering and computer engineering are required to take a course in probability and statistics. While the course continues to be mostly conceptual, author started initiatives to introduce data analytics in this course with special emphasis on machine vision applications. Topics such as receiver operating characteristics curves and hypothesis testing are covered through examples and exercises with students having individual datasets. Continuing with this theme, bootstrapping and associated methodologies have now been introduced to facilitate interpretation of machine vision experiments. A demo created that illustrates simple, stratified, and parametric bootstrapping as a means to understand the statistics of a machine vision sensor is presented. It encompasses a number of conceptual topics such as random variables, densities, parameter estimation, chi square testing, etc. alongside data analytics offering a holistic picture of machine learning and machine vision to the undergraduate students.

Determining population trends is critical for evaluating management actions and prioritizing species protections. In this study, we used empirical data to produce an estimate of the population trend for sub-adult and adult white sharks in central California. We used the unique dorsal fin morphology to build a mark-recapture data set in a modified Jolly-Seber model (POPAN formulation) to estimate annual abundance and then investigate population growth rates using parametric bootstrapping methods for sub-adult and adult sharks (males and females). For all demographic groups combined, we found equivocal evidence for a positive regional population growth (λ = 1.07 (95% CI = 0.91 to 1.23)). However, sex- and size-specific population growth rate estimates provided some evidence of population increases for reproductively mature males (λ = 1.06 (95% CI = 0.99 to 1.13)) and females (λ = 1.06 (95% CI = 0.95 to 1.17)). For sub-adult male and female white sharks, point estimates of λ were positive but uncertainty prevents strong inference (λ = 1.07 (95% CI = 0.85 to 1.29)) and (λ = 1.08 (95% CI = 0.88 to 1.28)), respectively. Our findings of a potential increase in reproductive-aged white sharks in central California may be a result of regional fluxes in density or attributed in part to current protection efforts and subsequent increase in abundance of pinnipeds as well as reduced gill-net fisheries mortality of juveniles. A trend estimate for the entire northeastern Pacific will require obtaining similar data across known aggregation areas along the west coast of North America.

In this paper we derive a general parametric bootstrapping approach to compute density forecasts for various types of mixed-data sampling (MIDAS) regressions. We consider both classical and unrestricted MIDAS regressions with and without an autoregressive component. First, we compare the forecasting performance of the dierent MIDAS models in Monte Carlo simulation experiments. We nd that the results in terms of point and density forecasts are coherent. Moreover, the results do not clearly indicate a superior performance of one of the models under scrutiny when the persistence of the low frequency variable is low. Some dierences are instead more evident when the persistence is high, for which the ARMIDAS and the AR-U-MIDAS produce better forecasts. Second, in an empirical exercise we evaluate density forecasts for quarterly US output growth, exploiting information from typical monthly series. We nd that MIDAS models provide accurate and timely density forecasts.

Mixed-effects models are being used ever more frequently in the analysis of experimental data. However, in the lme4 package in R the standards for evaluating significance of fixed effects in these models (i.e., obtaining p-values) are somewhat vague. There are good reasons for this, but as researchers who are using these models are required in many cases to report p-values, some method for evaluating the significance of the model output is needed. This paper reports the results of simulations showing that the two most common methods for evaluating significance, using likelihood ratio tests and applying the z distribution to the Wald t values from the model output (t-as-z), are somewhat anti-conservative, especially for smaller sample sizes. Other methods for evaluating significance, including parametric bootstrapping and the Kenward-Roger and Satterthwaite approximations for degrees of freedom, were also evaluated. The results of these simulations suggest that Type 1 error rates are closest to .05 when models are fitted using REML and p-values are derived using the Kenward-Roger or Satterthwaite approximations, as these approximations both produced acceptable Type 1 error rates even for smaller samples.

Slow-downs in lineage accumulation in phylogenies suggest that speciation rates decline as diversity increases. Likelihood methods have been developed to detect such diversity-dependence. However, a thorough test of whether such approaches correctly infer diversity-dependence is lacking.Here we simulate phylogenetic branching under linear negative diversity-dependent and diversity-independent models and estimate from the simulated phylogenies the maximum likelihood parameters for three different conditionings – on survival of the birth-death process given the crown age, on tree size (N), and on tree size given the crown age. We report the accuracy of recovering the simulation parameters and the reliability of the model selection based on the χ2 likelihood ratio test.Parameter estimate accuracyConditioning on survival given the crown age yields a severe bias of the carrying capacity K toward N, and an upward bias of the speciation rate, particularly in clades where diversity-dependent feedbacks are still weak (N « K). Conditioning on N yields an overestimate of K and an underestimate of speciation rate, particularly when saturation has been reached. Dual conditioning yields relatively unbiased parameter estimates on average, but the deviation from the true value for any single estimate may be large.Model selection reliabilityThe frequency of incorrectly rejecting a diversity-independent model when the simulation was diversity-independent (Type I error) differs substantially from the significance level α used in the likelihood ratio test, rendering the likelihood ratio test inappropriate. The frequency of correctly rejecting the diversity-independent model when the simulation was diversity-dependent (power) is larger when the clade is closer to equilibrium, and for conditioning on crown age.We conclude that conditioning on crown age has the best statistical properties overall, but caution that parameter estimates may be biased. To assess parameter uncertainty in future studies of diversity-dependence on real data, we recommend parametric bootstrapping, examination of the likelihood surface and comparison of estimates across the types of conditioning. To assess model selection reliability we discourage the use of the χ2 likelihood ratio test or AIC (which are equivalent in this case), but recommend a likelihood ratio test based on parametric bootstrap. We illustrate this method for the diversification of Dendroica warblers.

Data sets from ecological experiments can be difficult to analyze, due to lack of independence of experimental units and complex variance structures. In addition, information of interest may lie in complicated contrasts among treatments, rather than direct output from statistical tests. Here, we present a statistical framework for analyzing data sets containing non-independent experimental units and differences in variance among treatments (heteroscedasticity) and apply this framework to experimental data on interspecific competition among three tadpole species. Our framework involves three steps: (1) use a multilevel regression model to calculate coefficients of treatment effects on response variables; (2) combine coefficients to quantify the strength of competition (the target information of our experiment); and (3) use parametric bootstrapping to calculate significance of competition strengths. We repeated this framework using three multilevel regression models to analyze data at the level of individual tadpoles, at the replicate level, and at the replicate level accounting for heteroscedasticity. Comparing results shows the need to correctly specify the statistical model, with the model that accurately accounts for heteroscedasticity leading to different conclusions from the other two models. This approach gives a single, comprehensive analysis of experimental data that can be used to extract informative biological parameters in a statistically rigorous way. © 2013 by the Ecological Society of America.

We used parametric bootstrapping to assess the performance of maximum parsimony and maximum likelihood phylogenetic analyses of a 12‐locus seed plant data set. Evidence of biases in maximum parsimony analyses of single‐locus data sets may explain some of the locus‐specific variation among DNA‐based hypotheses of seed plant phylogeny. In particular, there is strong evidence of bias in maximum parsimony analyses, especially of plastid loci, that favors placing Gnetales sister to other seed plants. We concatenated simulated single‐locus data sets to examine biases in analyses of a 12‐locus data set in which each locus is simulated with different substitution parameters and branch lengths. Maximum parsimony analyses of the simulated 12‐locus data set also show evidence of biases in favor of recovering trees with Gnetales sister to other seed plants and against recovering anthophyte, gnepine, and gnetifer trees. These biases are most evident in analyses that include the fastest‐evolving characters. In the maxi...

The classic Kriging variance formula is widely used in geostatistics and in the design and analysis of computer experiments. This paper proves that this formula is wrong. Furthermore, it shows that the formula underestimates the Kriging variance in expectation. The paper develops parametric bootstrapping to estimate the Kriging variance. The new method is tested on several artificial examples and a real-life case study. These results demonstrate that the classic formula underestimates the true Kriging variance.

Motivation: The Dss statistic was proposed by (Mol. Biol. Evol., 14, 1125–1131, McGuire et al. (1997)) for scanning data sets for the presence of recombination, an important step in some phylogenetic analyses. The statistic, however, could not distinguish well between among-site rate variation and recombination, and had no statistical test for significant values. This paper addresses these shortfalls. Results: A modification to the Dss statistic is proposed which accounts for rate variation to a large extent. A statistical test, based on parametric bootstrapping, is also suggested. Availability: The TOPAL package (version 2) may be accessed from http://www.bioss.sari.ac.uk/~frank/Geneticsand by anonymous ftp from ftp://ftp.bioss.sari.ac.ukin the directory pub/phylogeny/topal. Contact: frank@bioss.sari.ac.uk

SUMMARY Recent papers have questioned the appropriateness of the circular confidence regions often drawn around group mean points in canonical variate analysis and proposed improved elliptical regions. This paper develops new theoretical and bootstrap regions of an alternative type, with those constructed using parametric bootstrapping being shown by simulation to perform consistently well.

SUMMARY A large deviation result is established for the bootstrap empirical distribution in a finite sample space, thereby validating both nonparametric and parametric bootstrapping in certain phylogenetic inference problems. The bias previously observed in the bootstrap distribution of the estimated tree topology is shown to stem from dispersion effects in the joint distribution of sample and bootstrap empirical distributions. Both results are examined for maximum likelihood estimation in a three-taxon model having particularly simple geometry. They are also applicable to discrete parameter problems outside of phylogenetic inference.

Bootstrapping has gained popularity in different tests of hypotheses as an alternative in using asymptotic distribution if one is not sure of the distribution of the test statistic under a null hypothesis. This method, in general, has two variants – the parametric and the nonparametric approaches. However, issues on reliability of this method always arise in many applications. This paper addresses the issue on reliability by establishing a reliability measure in terms of quantiles with respect to asymptotic distribution, when this is approximately correct. The test of hypotheses used is Ftest. The simulated results show that using nonparametric bootstrapping in F-test gives better reliability than parametric bootstrapping with relatively higher degrees of freedom., {"references":["S. Amiri, D. von Rosen, and S. Zwanzig, \"On the comparison of\nparametric and nonparametric bootstrap,\" Uppsala University\nDepartment of Mathematics Report 2008:15. Uppsala University,\nUppsala, Sweden, 2008, unpublished.","R. Barlow and F. Proschan, Mathematical Theory of Reliability, SIAM\nClassics edition, 1996, pp. 5-6.","A. Davison and D. Hinkley, Bootstrap Methods and their Applications.\nCambridge, United Kingdom: Cambridge University Press, 1997, ch. 1.","P. Good and J. Hardin, Common Errors in Statistics: How to Avoid\nThem. New Jersey: John Wiley & Sons, 2005.","J. MacKinnon, \"Bootstrap hypothesis testing,\" Queen-s Economics\nDepartment Working Paper No. 1127, Queen-s University, Ontario,\nCanada, 2007, unpublished.","J. MacKinnon and R. Davidson, \"Improving the reliability of bootstrap\ntests with the fast double bootstrap,\" Queen-s Economics Department\nWorking Paper No. 1044. Queen-s University, Ontario, Canada, 2006,\nunpublished.","B. Efron and R. Tibshirani, An Introduction to the Bootstrap, New York:\nChapman & Hall, 1993, pp. 31-32.","D. Politis, \"The impact of bootstrap methods in time series,\" Statistical\nScience, Vol. 18, No. 2, 2003, pp. 219-230."]}

This article uses a sequentialized experimental design to select simulation input combinations for global optimization, based on Kriging (also called Gaussian process or spatial correlation modeling); this Kriging is used to analyze the input/output data of the simulation model (computer code). This design and analysis adapt the classic "expected improvement" (EI) in "efficient global optimization" (EGO) through the introduction of an improved estimator of the Kriging predictor variance; this estimator uses parametric bootstrapping. Classic EI and bootstrapped EI are compared through various test functions, including the six-hump camel-back and several Hartmann functions. These empirical results demonstrate that in some applications bootstrapped EI finds the global optimum faster than classic EI does; in general, however, the classic EI may be considered to be a robust global optimizer. We thank the anonymous referee for a very detailed report that lead to the additional experiments in Table 1 aimed at investigating the effects of different initial sample sizes, the example in the Appendix, a simpler notation, and many other minor improvements. We also thank Emmanuel Vazquez (SUPÉLEC) for bringing Abt (1999) and Müller and Pronzato (2009) to our attention.

This paper concerns the asymptotic validity of the bootstrap method in a non-regular model. Specifically, it is shown that the parametric bootstrap of the change-point parameter in the change-point hazard rate model works.

The problem of distinguishing density-independent (DI) from density-dependent (DD) demographic time series has been addressed in the past via hypothesis testing based on parametric bootstrapping (PBLR) and, in later works, by Information Criteria such as FPE or SIC. Here, we address the problem in a novel way using Structural Risk Minimization (SRM). DI and DD time series corrupted with noise are extensively simulated using a drift (DI) and a Ricker (DD) model; on each generated time series, both models are identified, and then one is selected by FPE, SIC and SRM. The probability of density-[in]dependence recognition is statistically assessed and compared with the results obtained via PBLR in a previous work.

In recent years, likelihood ratio tests (LRTs) based on DNA and protein sequence data have been proposed for testing various evolutionary hypotheses. Because conducting an LRT requires an evolutionary model of nucleotide or amino acid substitution, which is almost always unknown, it becomes important to investigate the robustness of LRTs to violations of assumptions of these evolutionary models. Computer simulation was used to examine performance of LRTs of the molecular clock, transition/transversion bias, and among-site rate variation under different substitution models. The results showed that when correct models are used, LRTs perform quite well even when the DNA sequences are as short as 300 nt. However, LRTs were found to be biased under incorrect models. The extent of bias varies considerably, depending on the hypotheses tested, the substitution models assumed, and the lengths of the sequences used, among other things. A preliminary simulation study also suggests that LRTs based on parametric bootstrapping may be more sensitive to substitution models than are standard LRTs. When an assumed substitution model is grossly wrong and a more realistic model is available, LRTs can often reject the wrong model; thus, the performance of LRTs may be improved by using a more appropriate model. On the other hand, many factors of molecular evolution have not been considered in any substitution models so far built, and the possibility of an influence of this negligence on LRTs is often overlooked. The dependence of LRTs on substitution models calls for caution in interpreting test results and highlights the importance of clarifying the substitution patterns of genes and proteins and building more realistic models.

The video lottery terminal dataset contains observations on the three windows of an electronic slot machine for 345 plays together with the prize paid out for each play. The prize payout distribution is so badly skewed that confidence intervals for expected payout based on the central limit theorem are not accurate. This dataset can be used at the graduate or upper undergraduate level to illustrate parametric bootstrapping. The dataset can also be used in a graduate course to illustrate tests of independence for two and three-way contingency tables involving random zeroes, or these tables may be collapsed and used as examples in an introductory course.