Estimating Decision-Relevant Comparative Effects Using Instrumental Variables

Instrumental variables methods (IV) are widely used in the health economics literature to adjust for hidden selection biases in observational studies when estimating treatment effects. Less attention has been paid in the applied literature to the proper use of IVs if treatment effects are heterogeneous across subjects. Such a heterogeneity in effects becomes an issue for IV estimators when individuals’ self-selected choices of treatments are correlated with expected idiosyncratic gains or losses from treatments. We present an overview of the challenges that arise with IV estimators in the presence of effect heterogeneity and self-selection and compare conventional IV analysis with alternative approaches that use IVs to directly address these challenges. Using a Medicare sample of clinically localized breast cancer patients, we study the impact of breast-conserving surgery and radiation with mastectomy on 3-year survival rates. Our results reveal the traditional IV results may have masked important heterogeneity in treatment effects. In the context of these results, we discuss the advantages and limitations of conventional and alternative IV methods in estimating mean treatment-effect parameters, the role of heterogeneity in comparative effectiveness research and the implications for diffusion of technology.     

A new u-statistic with superior design sensitivity in matched observational studies

Summary In an observational or nonrandomized study of treatment effects, a sensitivity analysis indicates the magnitude of bias from unmeasured covariates that would need to be present to alter the conclusions of a naïve analysis that presumes adjustments for observed covariates suffice to remove all bias. The power of sensitivity analysis is the probability that it will reject a false hypothesis about treatment effects allowing for a departure from random assignment of a specified magnitude; in particular, if this specified magnitude is “no departure” then this is the same as the power of a randomization test in a randomized experiment. A new family of u‐statistics is proposed that includes Wilcoxon's signed rank statistic but also includes other statistics with substantially higher power when a sensitivity analysis is performed in an observational study. Wilcoxon's statistic has high power to detect small effects in large randomized experiments—that is, it often has good Pitman efficiency—but small effects are invariably sensitive to small unobserved biases. Members of this family of u‐statistics that emphasize medium to large effects can have substantially higher power in a sensitivity analysis. For example, in one situation with 250 pair differences that are Normal with expectation 1/2 and variance 1, the power of a sensitivity analysis that uses Wilcoxon's statistic is 0.08 while the power of another member of the family of u‐statistics is 0.66. The topic is examined by performing a sensitivity analysis in three observational studies, using an asymptotic measure called the design sensitivity, and by simulating power in finite samples. The three examples are drawn from epidemiology, clinical medicine, and genetic toxicology.     

Sensitivity analysis for m-estimates, tests, and confidence intervals in matched observational studies

Huber's m-estimates use an estimating equation in which observations are permitted a controlled level of influence. The family of m-estimates includes least squares and maximum likelihood, but typical applications give extreme observations limited weight. Maritz proposed methods of exact and approximate permutation inference for m-tests, confidence intervals, and estimators, which can be derived from random assignment of paired subjects to treatment or control. In contrast, in observational studies, where treatments are not randomly assigned, subjects matched for observed covariates may differ in terms of unobserved covariates, so differing outcomes may not be treatment effects. In observational studies, a method of sensitivity analysis is developed for m-tests, m-intervals, and m-estimates: it shows the extent to which inferences would be altered by biases of various magnitudes due to nonrandom treatment assignment. The method is developed for both matched pairs, with one treated subject matched to one control, and for matched sets, with one treated subject matched to one or more controls. The method is illustrated using two studies: (i) a paired study of damage to DNA from exposure to chromium and nickel and (ii) a study with one or two matched controls comparing side effects of two drug regimes to treat tuberculosis. The approach yields sensitivity analyses for: (i) m-tests with Huber's weight function and other robust weight functions, (ii) the permutational t-test which uses the observations directly, and (iii) various other procedures such as the sign test, Noether's test, and the permutation distribution of the efficient score test for a location family of distributions. Permutation inference with covariance adjustment is briefly discussed.     

Meta‐analysis of genome‐wide association from genomic prediction models

Genome‐wide association (GWA) studies based on GBLUP models are a common practice in animal breeding. However, effect sizes of GWA tests are small, requiring larger sample sizes to enhance power of detection of rare variants. Because of difficulties in increasing sample size in animal populations, one alternative is to implement a meta‐analysis (MA), combining information and results from independent GWA studies. Although this methodology has been used widely in human genetics, implementation in animal breeding has been limited. Thus, we present methods to implement a MA of GWA, describing the proper approach to compute weights derived from multiple genomic evaluations based on animal‐centric GBLUP models. Application to real datasets shows that MA increases power of detection of associations in comparison with population‐level GWA, allowing for population structure and heterogeneity of variance components across populations to be accounted for. Another advantage of MA is that it does not require access to genotype data that is required for a joint analysis. Scripts related to the implementation of this approach, which consider the strength of association as well as the sign, are distributed and thus account for heterogeneity in association phase between QTL and SNPs. Thus, MA of GWA is an attractive alternative to summarizing results from multiple genomic studies, avoiding restrictions with genotype data sharing, definition of fixed effects and different scales of measurement of evaluated traits.     

Adaptive treatment allocation for comparative clinical studies with recurrent events data

In long-term clinical studies, recurrent event data are sometimes collected and used to contrast the efficacies of two different treatments. The event reoccurrence rates can be compared using the popular negative binomial model, which incorporates information related to patient heterogeneity into a data analysis. For treatment allocation, a balanced approach in which equal sample sizes are obtained for both treatments is predominately adopted. However, if one treatment is superior, then it may be desirable to allocate fewer subjects to the less-effective treatment. To accommodate this objective, a sequential response-adaptive treatment allocation procedure is derived based on the doubly adaptive biased coin design. Our proposed treatment allocation schemes have been shown to be capable of reducing the number of subjects receiving the inferior treatment while simultaneously retaining a test power level that is comparable to that of a balanced design. The redesign of a clinical study illustrates the advantages of using our procedure.     

Assessment of the diagnostic potential of Immmunocapture-PCR and Immuno-PCR for Citrus Variegated Chlorosis

Xylella fastidiosa causes significant losses in many economically important crops. An efficient pathogen detection system is critical for epidemiology studies, particularly when large sample size is involved. In this study we report the development of immunomolecular assays like Immmunocapture-PCR and Immuno-PCR for direct detection of X. fastidiosa without DNA isolation. Whereas the reactivity of ELISA and PCR ranged from 10(6) to 10(4) bacterial cells, the IC-PCR sensitivity was up to 10(3) and the detection limit of I-PCR was up to 10(1) bacterial cells. These methods can use either plant sample extracts or cultivated media, and show no cross reaction for any other endophytic citrus-bacteria. Therefore, IC-PCR and I-PCR assays provide an alternative for quick and very sensitive methods to screening X. fastidiosa, with the advantage of not requiring any concentration or DNA purification steps while still allowing an accurate diagnosis of CVC.     

Good-genes effects in sexual selection

The magnitude of the effect of good genes as a viability benefit accruing to choosy females remains a controversial theoretical and empirical issue. We collected all available data from the literature to estimate the magnitude of good-genes viability effects, while adjusting for sample size. The average correlation coefficient between male traits and offspring survival in 22 studies was 0.122, which differed highly significantly from zero. This implies that male characters chosen by females reveal on average 1.5% of the variance in viability. The studies demonstrated considerable heterogeneity in effect size; some of this heterogeneity could be accounted for by differences among taxa (birds demonstrating stronger effects), and by differences in the degree of mating skew in the species (high skew reflecting stronger effects). Although these results suggest that viability-based sexual selection is widespread across taxa, they indicate that the effect is relatively minor. Finally, there was also an effect of publication year in that the more recent studies reported reduced effects. This may reflect publication biases during paradigm shifts of this debated issue, but it should also be recalled that the studies have only partly estimated the full fitness consequences of mate choice for offspring.     

Aspirin in the Treatment of Cancer: Reductions in Metastatic Spread and in Mortality: A Systematic Review and Meta-Analyses of Published Studies

BackgroundLow-dose aspirin has been shown to reduce the incidence of cancer, but its role in the treatment of cancer is uncertain.ObjectivesWe conducted a systematic search of the scientific literature on aspirin taken by patients following a diagnosis of cancer, together with appropriate meta-analyses.MethodsSearches were completed in Medline and Embase in December 2015 using a pre-defined search strategy. References and abstracts of all the selected papers were scanned and expert colleagues were contacted for additional studies. Two reviewers applied pre-determined eligibility criteria (cross-sectional, cohort and controlled studies, and aspirin taken after a diagnosis of cancer), assessed study quality and extracted data on cancer cause-specific deaths, overall mortality and incidence of metastases. Random effects meta-analyses and planned sub-group analyses were completed separately for observational and experimental studies. Heterogeneity and publication bias were assessed in sensitivity analyses and appropriate omissions made. Papers were examined for any reference to bleeding and authors of the papers were contacted and questioned.ResultsFive reports of randomised trials were identified, together with forty two observational studies: sixteen on colorectal cancer, ten on breast and ten on prostate cancer mortality. Pooling of eleven observational reports of the effect of aspirin on cause-specific mortality from colon cancer, after the omission of one report identified on the basis of sensitivity analyses, gave a hazard ratio (HR) of 0.76 (95% CI 0.66, 0.88) with reduced heterogeneity (P = 0.04). The cause specific mortality in five reports of patients with breast cancer showed significant heterogeneity (P<0.0005) but the omission of one outlying study reduced heterogeneity (P = 0.19) and led to an HR = 0.87 (95% CI 0.69, 1.09). Heterogeneity between nine studies of prostate cancer was significant, but again, the omission of one study led to acceptable homogeneity (P = 0.26) and an overall HR = 0.89 (95% CI 0.79–0.99). Six single studies of other cancers suggested reductions in cause specific mortality by aspirin, and in five the effect is statistically significant. There were no significant differences between the pooled HRs for the three main cancers and after the omission of three reports already identified in sensitivity analyses heterogeneity was removed and revealed an overall HR of 0.83 (95% CI 0.76–0.90). A mutation of PIK3CA was present in about 20% of patients, and appeared to explain most of the reduction in colon cancer mortality by aspirin. Data were not adequate to examine the importance of this or any other marker in the effect of aspirin in the other cancers. On bleeding attributable to aspirin two reports stated that there had been no side effect or bleeding attributable to aspirin. Authors on the other reports were written to and 21 replied stating that no data on bleeding were available.

Conclusions and Implications

The study highlights the need for randomised trials of aspirin treatment in a variety of cancers. While these are awaited there is an urgent need for evidence from observational studies of aspirin and the less common cancers, and for more evidence of the relevance of possible bio-markers of the aspirin effect on a wide variety of cancers. In the meantime it is urged that patients in whom a cancer is diagnosed should be given details of this research, together with its limitations, to enable each to make an informed decision as to whether or not to take low-dose aspirin.

Systematic Review Protocol Number

CRD42015014145     

Semiparametric additive time-varying coefficients model for longitudinal data with censored time origin

Statistical analysis of longitudinal data often involves modeling treatment effects on clinically relevant longitudinal biomarkers since an initial event (the time origin). In some studies including preventive HIV vaccine efficacy trials, some participants have biomarkers measured starting at the time origin, whereas others have biomarkers measured starting later with the time origin unknown. The semiparametric additive time-varying coefficient model is investigated where the effects of some covariates vary nonparametrically with time while the effects of others remain constant. Weighted profile least squares estimators coupled with kernel smoothing are developed. The method uses the expectation maximization approach to deal with the censored time origin. The Kaplan–Meier estimator and other failure time regression models such as the Cox model can be utilized to estimate the distribution and the conditional distribution of left censored event time related to the censored time origin. Asymptotic properties of the parametric and nonparametric estimators and consistent asymptotic variance estimators are derived. A two-stage estimation procedure for choosing weight is proposed to improve estimation efficiency. Numerical simulations are conducted to examine finite sample properties of the proposed estimators. The simulation results show that the theory and methods work well. The efficiency gain of the two-stage estimation procedure depends on the distribution of the longitudinal error processes. The method is applied to analyze data from the Merck 023/HVTN 502 Step HIV vaccine study.     

Improving trial generalizability using observational studies

Complementary features of randomized controlled trials (RCTs) and observational studies (OSs) can be used jointly to estimate the average treatment effect of a target population. We propose a calibration weighting estimator that enforces the covariate balance between the RCT and OS, therefore improving the trial-based estimator's generalizability. Exploiting semiparametric efficiency theory, we propose a doubly robust augmented calibration weighting estimator that achieves the efficiency bound derived under the identification assumptions. A nonparametric sieve method is provided as an alternative to the parametric approach, which enables the robust approximation of the nuisance functions and data-adaptive selection of outcome predictors for calibration. We establish asymptotic results and confirm the finite sample performances of the proposed estimators by simulation experiments and an application on the estimation of the treatment effect of adjuvant chemotherapy for early-stage non-small-cell lung patients after surgery.     

A bound for publication bias based on the fraction of unpublished studies

Publication bias in meta-analysis is usually modeled in terms of an accept/reject selection procedure in which the selected studies are the "published" studies and the rejected studies are the "unpublished" studies. One possible selection mechanism is to suppose that only studies that report an estimated treatment effect exceeding (or falling short of) some threshold are accepted. We show that, with appropriate choice of thresholds, this attains the maximum bias among all selection mechanisms in which the probability of selection increases with study size. It is impossible to estimate the selection mechanism from the observed studies alone: this result leads to a "worst-case" sensitivity analysis for publication bias, which is remarkably easy to implement in practice. The method is illustrated using data on the effectiveness of prophylactic corticosteroids.     

Generalized linear mixed models for mapping multiple quantitative trait loci

Many biological traits are discretely distributed in phenotype but continuously distributed in genetics because they are controlled by multiple genes and environmental variants. Due to the quantitative nature of the genetic background, these multiple genes are called quantitative trait loci (QTL). When the QTL effects are treated as random, they can be estimated in a single generalized linear mixed model (GLMM), even if the number of QTL may be larger than the sample size. The GLMM in its original form cannot be applied to QTL mapping for discrete traits if there are missing genotypes. We examined two alternative missing genotype-handling methods: the expectation method and the overdispersion method. Simulation studies show that the two methods are efficient for multiple QTL mapping (MQM) under the GLMM framework. The overdispersion method showed slight advantages over the expectation method in terms of smaller mean-squared errors of the estimated QTL effects. The two methods of GLMM were applied to MQM for the female fertility trait of wheat. Multiple QTL were detected to control the variation of the number of seeded spikelets.     

Risk Sensitivity in Foraging Jack Dempsey Cichlids (Cichlasoma octofasciatum)

Studies have documented that animals with positive energy budgets tend to prefer feeding sites offering constant amounts of food over those offering variable amounts of food. The present study tested whether Jack Dempsey cichlids (Cichlasoma octofasciatum) in a two-patch environment preferred the patch offering a constant amount of food over the one offering a variable amount of food. The study also examined (1) whether fish displayed shifts in risk sensitivity when they were allowed only one choice per trial (discrete-choice treatment), as opposed to when they could freely sample both alternatives (free-choice treatment), and (2) whether or not fish displayed shifts in risk sensitivity between the first trial of the day, when they were hungrier, and the final trial of the day, when they were more satiated. Two of six fish significantly preferred the constant alternative and one showed a numerical preference for the constant alternative that approached significance. The mean proportion of first choices to the constant side did not differ significantly between the discrete-choice treatment and the free-choice treatment. In the discrete-choice treatment, however, fish were significantly more risk averse in the first trial of daily sessions than in the final trial — a shift in risk sensitivity opposite to that seen in a number of other studies. This suggests that animals may pass through more than one threshold in risk sensitivity on different points along an energy budget continuum.     

Type I error rates, coverage of confidence intervals, and variance estimation in propensity-score matched analyses

Propensity-score matching is frequently used in the medical literature to reduce or eliminate the effect of treatment selection bias when estimating the effect of treatments or exposures on outcomes using observational data. In propensity-score matching, pairs of treated and untreated subjects with similar propensity scores are formed. Recent systematic reviews of the use of propensity-score matching found that the large majority of researchers ignore the matched nature of the propensity-score matched sample when estimating the statistical significance of the treatment effect. We conducted a series of Monte Carlo simulations to examine the impact of ignoring the matched nature of the propensity-score matched sample on Type I error rates, coverage of confidence intervals, and variance estimation of the treatment effect. We examined estimating differences in means, relative risks, odds ratios, rate ratios from Poisson models, and hazard ratios from Cox regression models. We demonstrated that accounting for the matched nature of the propensity-score matched sample tended to result in type I error rates that were closer to the advertised level compared to when matching was not incorporated into the analyses. Similarly, accounting for the matched nature of the sample tended to result in confidence intervals with coverage rates that were closer to the nominal level, compared to when matching was not taken into account. Finally, accounting for the matched nature of the sample resulted in estimates of standard error that more closely reflected the sampling variability of the treatment effect compared to when matching was not taken into account.     

The impact of stopping rules on heterogeneity of results in overviews of clinical trials.

This paper explores the extent to which application of statistical stopping rules in clinical trials can create an artificial heterogeneity of treatment effects in overviews (meta-analyses) of related trials. For illustration, we concentrate on overviews of identically designed group sequential trials, using either fixed nominal or O'Brien and Fleming two-sided boundaries. Some analytic results are obtained for two-group designs and simulation studies are otherwise used, with the following overall findings. The use of stopping rules leads to biased estimates of treatment effect so that the assessment of heterogeneity of results in an overview of trials, some of which have used stopping rules, is confounded by this bias. If the true treatment effect being studied is small, as is often the case, then artificial heterogeneity is introduced, thus increasing the Type I error rate in the test of homogeneity. This could lead to erroneous use of a random effects model, producing exaggerated estimates and confidence intervals. However, if the true mean effect is large, then between-trial heterogeneity may be underestimated. When undertaking or interpreting overviews, one should ascertain whether stopping rules have been used (either formally or informally) and should consider whether their use might account for any heterogeneity found.     

Detection of significant antiviral drug effects on COVID-19 with reasonable sample sizes in randomized controlled trials: A modeling study

BackgroundDevelopment of an effective antiviral drug for Coronavirus Disease 2019 (COVID-19) is a global health priority. Although several candidate drugs have been identified through in vitro and in vivo models, consistent and compelling evidence from clinical studies is limited. The lack of evidence from clinical trials may stem in part from the imperfect design of the trials. We investigated how clinical trials for antivirals need to be designed, especially focusing on the sample size in randomized controlled trials.Methods and findingsA modeling study was conducted to help understand the reasons behind inconsistent clinical trial findings and to design better clinical trials. We first analyzed longitudinal viral load data for Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) without antiviral treatment by use of a within-host virus dynamics model. The fitted viral load was categorized into 3 different groups by a clustering approach. Comparison of the estimated parameters showed that the 3 distinct groups were characterized by different virus decay rates (p-value < 0.001). The mean decay rates were 1.17 d⁻¹ (95% CI: 1.06 to 1.27 d⁻¹), 0.777 d⁻¹ (0.716 to 0.838 d⁻¹), and 0.450 d⁻¹ (0.378 to 0.522 d⁻¹) for the 3 groups, respectively. Such heterogeneity in virus dynamics could be a confounding variable if it is associated with treatment allocation in compassionate use programs (i.e., observational studies).Subsequently, we mimicked randomized controlled trials of antivirals by simulation. An antiviral effect causing a 95% to 99% reduction in viral replication was added to the model. To be realistic, we assumed that randomization and treatment are initiated with some time lag after symptom onset. Using the duration of virus shedding as an outcome, the sample size to detect a statistically significant mean difference between the treatment and placebo groups (1:1 allocation) was 13,603 and 11,670 (when the antiviral effect was 95% and 99%, respectively) per group if all patients are enrolled regardless of timing of randomization. The sample size was reduced to 584 and 458 (when the antiviral effect was 95% and 99%, respectively) if only patients who are treated within 1 day of symptom onset are enrolled. We confirmed the sample size was similarly reduced when using cumulative viral load in log scale as an outcome.We used a conventional virus dynamics model, which may not fully reflect the detailed mechanisms of viral dynamics of SARS-CoV-2. The model needs to be calibrated in terms of both parameter settings and model structure, which would yield more reliable sample size calculation.ConclusionsIn this study, we found that estimated association in observational studies can be biased due to large heterogeneity in viral dynamics among infected individuals, and statistically significant effect in randomized controlled trials may be difficult to be detected due to small sample size. The sample size can be dramatically reduced by recruiting patients immediately after developing symptoms. We believe this is the first study investigated the study design of clinical trials for antiviral treatment using the viral dynamics model.

Using a viral dynamics model, Shingo Iwami and colleagues investigate the sample sizes required to detect significant antiviral drug effects on COVID-19 in randomized controlled trials.     

Exploiting collider bias to apply two-sample summary data Mendelian randomization methods to one-sample individual level data

Over the last decade the availability of SNP-trait associations from genome-wide association studies has led to an array of methods for performing Mendelian randomization studies using only summary statistics. A common feature of these methods, besides their intuitive simplicity, is the ability to combine data from several sources, incorporate multiple variants and account for biases due to weak instruments and pleiotropy. With the advent of large and accessible fully-genotyped cohorts such as UK Biobank, there is now increasing interest in understanding how best to apply these well developed summary data methods to individual level data, and to explore the use of more sophisticated causal methods allowing for non-linearity and effect modification.In this paper we describe a general procedure for optimally applying any two sample summary data method using one sample data. Our procedure first performs a meta-analysis of summary data estimates that are intentionally contaminated by collider bias between the genetic instruments and unmeasured confounders, due to conditioning on the observed exposure. These estimates are then used to correct the standard observational association between an exposure and outcome. Simulations are conducted to demonstrate the method’s performance against naive applications of two sample summary data MR. We apply the approach to the UK Biobank cohort to investigate the causal role of sleep disturbance on HbA1c levels, an important determinant of diabetes.Our approach can be viewed as a generalization of Dudbridge et al. (Nat. Comm. 10: 1561), who developed a technique to adjust for index event bias when uncovering genetic predictors of disease progression based on case-only data. Our work serves to clarify that in any one sample MR analysis, it can be advantageous to estimate causal relationships by artificially inducing and then correcting for collider bias.     

Combining parametric and nonparametric models to estimate treatment effects in observational studies

Performing causal inference in observational studies requires we assume confounding variables are correctly adjusted for. In settings with few discrete-valued confounders, standard models can be employed. However, as the number of confounders increases these models become less feasible as there are fewer observations available for each unique combination of confounding variables. In this paper, we propose a new model for estimating treatment effects in observational studies that incorporates both parametric and nonparametric outcome models. By conceptually splitting the data, we can combine these models while maintaining a conjugate framework, allowing us to avoid the use of Markov chain Monte Carlo (MCMC) methods. Approximations using the central limit theorem and random sampling allow our method to be scaled to high-dimensional confounders. Through simulation studies we show our method can be competitive with benchmark models while maintaining efficient computation, and illustrate the method on a large epidemiological health survey.     

Heterogeneous treatment effects in stratified clinical trials with time‐to‐event endpoints

When analyzing clinical trials with a stratified population, homogeneity of treatment effects is a common assumption in survival analysis. However, in the context of recent developments in clinical trial design, which aim to test multiple targeted therapies in corresponding subpopulations simultaneously, the assumption that there is no treatment‐by‐stratum interaction seems inappropriate. It becomes an issue if the expected sample size of the strata makes it unfeasible to analyze the trial arms individually. Alternatively, one might choose as primary aim to prove efficacy of the overall (targeted) treatment strategy. When testing for the overall treatment effect, a violation of the no‐interaction assumption renders it necessary to deviate from standard methods that rely on this assumption. We investigate the performance of different methods for sample size calculation and data analysis under heterogeneous treatment effects. The commonly used sample size formula by Schoenfeld is compared to another formula by Lachin and Foulkes, and to an extension of Schoenfeld's formula allowing for stratification. Beyond the widely used (stratified) Cox model, we explore the lognormal shared frailty model, and a two‐step analysis approach as potential alternatives that attempt to adjust for interstrata heterogeneity. We carry out a simulation study for a trial with three strata and violations of the no‐interaction assumption. The extension of Schoenfeld's formula to heterogeneous strata effects provides the most reliable sample size with respect to desired versus actual power. The two‐step analysis and frailty model prove to be more robust against loss of power caused by heterogeneous treatment effects than the stratified Cox model and should be preferred in such situations.     

Sample size for individually matched case-control studies

The standard formulas used to calculate sample size for an individually matched case-control study assume a constant probability of exposure throughout the pool of possible controls. We propose new formulas that allow for heterogeneity in the probability of exposure among controls in different matched sets. Since matching factors are suspected of being confounders, they are expected to divide the total population into subgroups with different proportions exposed. Thus, the assumption of homogeneity of exposure among controls, made by the currently used formulas, is inconsistent with the assumptions used to design a matched study. The proposed formulas avoid this inconsistency. We present an example to illustrate how heterogeneity can affect the required sample size.