Bayesian statistics in oncology: a guide for the clinical investigator

The rise of evidence-based medicine as well as important progress in statistical methods and computational power have led to a second birth of the >200-year-old Bayesian framework. The use of Bayesian techniques, in particular in the design and interpretation of clinical trials, offers several substantial advantages over the classical statistical approach. First, in contrast to classical statistics, Bayesian analysis allows a direct statement regarding the probability that a treatment was beneficial. Second, Bayesian statistics allow the researcher to incorporate any prior information in the analysis of the experimental results. Third, Bayesian methods can efficiently handle complex statistical models, which are suited for advanced clinical trial designs. Finally, Bayesian statistics encourage a thorough consideration and presentation of the assumptions underlying an analysis, which enables the reader to fully appraise the authors' conclusions. Both Bayesian and classical statistics have their respective strengths and limitations and should be viewed as being complementary to each other; we do not attempt to make a head-to-head comparison, as this is beyond the scope of the present review. Rather, the objective of the present article is to provide a nonmathematical, reader-friendly overview of the current practice of Bayesian statistics coupled with numerous intuitive examples from the field of oncology. It is hoped that this educational review will be a useful resource to the oncologist and result in a better understanding of the scope, strengths, and limitations of the Bayesian approach.

Statistical evaluation of alternative models of human evolution

An appropriate model of recent human evolution is not only important to understand our own history, but it is necessary to disentangle the effects of demography and selection on genome diversity. Although most genetic data support the view that our species originated recently in Africa, it is still unclear if it completely replaced former members of the Homo genus, or if some interbreeding occurred during its range expansion. Several scenarios of modern human evolution have been proposed on the basis of molecular and paleontological data, but their likelihood has never been statistically assessed. Using DNA data from 50 nuclear loci sequenced in African, Asian and Native American samples, we show here by extensive simulations that a simple African replacement model with exponential growth has a higher probability (78%) as compared with alternative multiregional evolution or assimilation scenarios. A Bayesian analysis of the data under this best supported model points to an origin of our species approximately 141 thousand years ago (Kya), an exit out-of-Africa approximately 51 Kya, and a recent colonization of the Americas approximately 10.5 Kya. We also find that the African replacement model explains not only the shallow ancestry of mtDNA or Y-chromosomes but also the occurrence of deep lineages at some autosomal loci, which has been formerly interpreted as a sign of interbreeding with Homo erectus.

Can we constrain the interior structure of rocky exoplanets from mass and radius measurements?

Aims. We present an inversion method based on Bayesian analysis to constrain the interior structure of terrestrial exoplanets, in the form of chemical composition of the mantle and core size. Specifically, we identify what parts of the interior structure of terrestrial exoplanets can be determined from observations of mass, radius, and stellar elemental abundances. Methods. We perform a full probabilistic inverse analysis to formally account for observational and model uncertainties and obtain confidence regions of interior structure models. This enables us to characterize how model variability depends on data and associated uncertainties. Results. We test our method on terrestrial solar system planets and find that our model predictions are consistent with independent estimates. Furthermore, we apply our method to synthetic exoplanets up to 10 Earth masses and up to 1.7 Earth radii, and to exoplanet Kepler-36b. Importantly, the inversion strategy proposed here provides a framework for understanding the level of precision required to characterize the interior of exoplanets. Conclusions. Our main conclusions are (1) observations of mass and radius are sufficient to constrain core size; (2) stellar elemental abundances (Fe, Si, Mg) are principal constraints to reduce degeneracy in interior structure models and to constrain mantle composition; (3) the inherent degeneracy in determining interior structure from mass and radius observations does not only depend on measurement accuracies, but also on the actual size and density of the exoplanet. We argue that precise observations of stellar elemental abundances are central in order to place constraints on planetary bulk composition and to reduce model degeneracy. We provide a general methodology of analyzing interior structures of exoplanets that may help to understand how interior models are distributed among star systems. The methodology we propose is sufficiently general to allow its future extension to more complex internal structures including hydrogen- and water-rich exoplanets.

Growth hormone receptor polymorphism and growth hormone therapy response in children: a bayesian meta-analysis

Recombinant human growth hormone (rhGH) therapy is used in the long-term treatment of children with growth disorders, but there is considerable treatment response variability. The exon 3-deleted growth hormone receptor polymorphism (GHR(d3)) may account for some of this variability. The authors performed a systematic review (to April 2011), including investigator-only data, to quantify the effects of the GHR(fl-d3) and GHR(d3-d3) genotypes on rhGH therapy response and used a recently established Bayesian inheritance model-free approach to meta-analyze the data. The primary outcome was the 1-year change-in-height standard-deviation score for the 2 genotypes. Eighteen data sets from 12 studies (1,527 children) were included. After several prior assumptions were tested, the most appropriate inheritance model was codominant (posterior probability = 0.93). Compared with noncarriers, carriers had median differences in 1-year change-in-height standard-deviation score of 0.09 (95% credible interval (CrI): 0.01, 0.17) for GHR(fl-d3) and of 0.14 (95% CrI: 0.02, 0.26) for GHR(d3-d3). However, the between-study standard deviation of 0.18 (95% CrI: 0.10, 0.33) was considerable. The authors tested by meta-regression for potential modifiers and found no substantial influence. They conclude that 1) the GHR(d3) polymorphism inheritance is codominant, contrasting with previous reports; 2) GHR(d3) genotypes account for modest increases in rhGH effects in children; and 3) considerable unexplained variability in responsiveness remains.

Antiplatelets versus Anticoagulants for the Treatment of Cervical Artery Dissection: Bayesian Meta-Analysis

OBJECTIVE To compare the effects of antiplatelets and anticoagulants on stroke and death in patients with acute cervical artery dissection. DESIGN Systematic review with Bayesian meta-analysis. DATA SOURCES The reviewers searched MEDLINE and EMBASE from inception to November 2012, checked reference lists, and contacted authors. STUDY SELECTION Studies were eligible if they were randomised, quasi-randomised or observational comparisons of antiplatelets and anticoagulants in patients with cervical artery dissection. DATA EXTRACTION Data were extracted by one reviewer and checked by another. Bayesian techniques were used to appropriately account for studies with scarce event data and imbalances in the size of comparison groups. DATA SYNTHESIS Thirty-seven studies (1991 patients) were included. We found no randomised trial. The primary analysis revealed a large treatment effect in favour of antiplatelets for preventing the primary composite outcome of ischaemic stroke, intracranial haemorrhage or death within the first 3 months after treatment initiation (relative risk 0.32, 95% credibility interval 0.12 to 0.63), while the degree of between-study heterogeneity was moderate (τ(2) = 0.18). In an analysis restricted to studies of higher methodological quality, the possible advantage of antiplatelets over anticoagulants was less obvious than in the main analysis (relative risk 0.73, 95% credibility interval 0.17 to 2.30). CONCLUSION In view of these results and the safety advantages, easier usage and lower cost of antiplatelets, we conclude that antiplatelets should be given precedence over anticoagulants as a first line treatment in patients with cervical artery dissection unless results of an adequately powered randomised trial suggest the opposite.

Introduction to Bayesian Data Analysis

A workshop providing an introduction to Bayesian data analysis and hypothesis testing using R, Jags and the BayesFactor package.

Immune reconstitution inflammatory syndrome in patients starting antiretroviral therapy for HIV infection: a systematic review and meta-analysis

In patients with HIV-1 infection who are starting combination antiretroviral therapy (ART), the incidence of immune reconstitution inflammatory syndrome (IRIS) is not well defined. We did a meta-analysis to establish the incidence and lethality of the syndrome in patients with a range of previously diagnosed opportunistic infections, and examined the relation between occurrence and the degree of immunodeficiency. Systematic review identified 54 cohort studies of 13 103 patients starting ART, of whom 1699 developed IRIS. We calculated pooled cumulative incidences with 95% credibility intervals (CrI) by Bayesian methods and did a random-effects metaregression to analyse the relation between CD4 cell count and incidence of IRIS. In patients with previously diagnosed AIDS-defining illnesses, IRIS developed in 37.7% (95% CrI 26.6-49.4) of those with cytomegalovirus retinitis, 19.5% (6.7-44.8) of those with cryptococcal meningitis, 15.7% (9.7-24.5) of those with tuberculosis, 16.7% (2.3-50.7) of those with progressive multifocal leukoencephalopathy, and 6.4% (1.2-24.7) of those with Kaposi's sarcoma, and 12.2% (6.8-19.6) of those with herpes zoster. 16.1% (11.1-22.9) of unselected patients starting ART developed any type of IRIS. 4.5% (2.1-8.6) of patients with any type of IRIS died, 3.2% (0.7-9.2) of those with tuberculosis-associated IRIS died, and 20.8% (5.0-52.7) of those with cryptococcal meningitis died. Metaregression analyses showed that the risk of IRIS is associated with CD4 cell count at the start of ART, with a high risk in patients with fewer than 50 cells per microL. Occurrence of IRIS might therefore be reduced by initiation of ART before immunodeficiency becomes advanced.

Effects of glucosamine, chondroitin, or placebo in patients with osteoarthritis of hip or knee: network meta-analysis

OBJECTIVE: To determine the effect of glucosamine, chondroitin, or the two in combination on joint pain and on radiological progression of disease in osteoarthritis of the hip or knee. Design Network meta-analysis. Direct comparisons within trials were combined with indirect evidence from other trials by using a Bayesian model that allowed the synthesis of multiple time points. MAIN OUTCOME MEASURE: Pain intensity. Secondary outcome was change in minimal width of joint space. The minimal clinically important difference between preparations and placebo was prespecified at -0.9 cm on a 10 cm visual analogue scale. DATA SOURCES: Electronic databases and conference proceedings from inception to June 2009, expert contact, relevant websites. Eligibility criteria for selecting studies Large scale randomised controlled trials in more than 200 patients with osteoarthritis of the knee or hip that compared glucosamine, chondroitin, or their combination with placebo or head to head. Results 10 trials in 3803 patients were included. On a 10 cm visual analogue scale the overall difference in pain intensity compared with placebo was -0.4 cm (95% credible interval -0.7 to -0.1 cm) for glucosamine, -0.3 cm (-0.7 to 0.0 cm) for chondroitin, and -0.5 cm (-0.9 to 0.0 cm) for the combination. For none of the estimates did the 95% credible intervals cross the boundary of the minimal clinically important difference. Industry independent trials showed smaller effects than commercially funded trials (P=0.02 for interaction). The differences in changes in minimal width of joint space were all minute, with 95% credible intervals overlapping zero. Conclusions Compared with placebo, glucosamine, chondroitin, and their combination do not reduce joint pain or have an impact on narrowing of joint space. Health authorities and health insurers should not cover the costs of these preparations, and new prescriptions to patients who have not received treatment should be discouraged.

Sacral neuromodulation for neurogenic lower urinary tract dysfunction: systematic review and meta-analysis

Context Treatment of neurogenic lower urinary tract dysfunction (LUTD) is a challenge, because conventional therapies often fail. Sacral neuromodulation (SNM) has become a well-established therapy for refractory non-neurogenic LUTD, but its value in patients with a neurologic cause is unclear. Objective To assess the efficacy and safety of SNM for neurogenic LUTD. Evidence acquisition Studies were identified by electronic search of PubMed, EMBASE, and ScienceDirect (on 15 April 2010) and hand search of reference lists and review articles. SNM articles were included if they reported on efficacy and/or safety of tested and/or permanently implanted patients suffering from neurogenic LUTD. Two reviewers independently selected studies and extracted data. Study estimates were pooled using Bayesian random-effects meta-analysis. Evidence synthesis Of the 26 independent studies (357 patients) included, the evidence level ranged from 2b to 4 according to the Oxford Centre for Evidence-Based Medicine. Half (n = 13) of the included studies reported data on both test phase and permanent SNM; the remaining studies were confined to test phase (n = 4) or permanent SNM (n = 9). The pooled success rate was 68% for the test phase (95% credibility interval [CrI], 50–87) and 92% (95% CrI, 81–98%) for permanent SNM, with a mean follow-up of 26 mo. The pooled adverse event rate was 0% (95% CrI, 0–2%) for the test phase and 24% (95% CrI, 6–48%) for permanent SNM. Conclusions There is evidence indicating that SNM may be effective and safe for the treatment of patients with neurogenic LUTD. However, the number of investigated patients is low with high between-study heterogeneity, and there is a lack of randomised, controlled trials. Thus, well-designed, adequately powered studies are urgently needed before more widespread use of SNM for neurogenic LUTD can be recommended.

ABCtoolbox: a versatile toolkit for approximate Bayesian computations

Background The estimation of demographic parameters from genetic data often requires the computation of likelihoods. However, the likelihood function is computationally intractable for many realistic evolutionary models, and the use of Bayesian inference has therefore been limited to very simple models. The situation changed recently with the advent of Approximate Bayesian Computation (ABC) algorithms allowing one to obtain parameter posterior distributions based on simulations not requiring likelihood computations. Results Here we present ABCtoolbox, a series of open source programs to perform Approximate Bayesian Computations (ABC). It implements various ABC algorithms including rejection sampling, MCMC without likelihood, a Particle-based sampler and ABC-GLM. ABCtoolbox is bundled with, but not limited to, a program that allows parameter inference in a population genetics context and the simultaneous use of different types of markers with different ploidy levels. In addition, ABCtoolbox can also interact with most simulation and summary statistics computation programs. The usability of the ABCtoolbox is demonstrated by inferring the evolutionary history of two evolutionary lineages of Microtus arvalis. Using nuclear microsatellites and mitochondrial sequence data in the same estimation procedure enabled us to infer sex-specific population sizes and migration rates and to find that males show smaller population sizes but much higher levels of migration than females. Conclusion ABCtoolbox allows a user to perform all the necessary steps of a full ABC analysis, from parameter sampling from prior distributions, data simulations, computation of summary statistics, estimation of posterior distributions, model choice, validation of the estimation procedure, and visualization of the results.

Evaluation of a Bayesian MCMC Random Effects Inference Methodology for Capture-Mark-Recapture Data

Monte Carlo simulation was used to evaluate properties of a simple Bayesian MCMC analysis of the random effects model for single group Cormack-Jolly-Seber capture-recapture data. The MCMC method is applied to the model via a logit link, so parameters p, S are on a logit scale, where logit(S) is assumed to have, and is generated from, a normal distribution with mean μ and variance σ2 . Marginal prior distributions on logit(p) and μ were independent normal with mean zero and standard deviation 1.75 for logit(p) and 100 for μ ; hence minimally informative. Marginal prior distribution on σ2 was placed on τ2=1/σ2 as a gamma distribution with α=β=0.001 . The study design has 432 points spread over 5 factors: occasions (t) , new releases per occasion (u), p, μ , and σ . At each design point 100 independent trials were completed (hence 43,200 trials in total), each with sample size n=10,000 from the parameter posterior distribution. At 128 of these design points comparisons are made to previously reported results from a method of moments procedure. We looked at properties of point and interval inference on μ , and σ based on the posterior mean, median, and mode and equal-tailed 95% credibility interval. Bayesian inference did very well for the parameter μ , but under the conditions used here, MCMC inference performance for σ was mixed: poor for sparse data (i.e., only 7 occasions) or σ=0 , but good when there were sufficient data and not small σ .

Effect of initial treatment strategy on survival of patients with advanced-stage Hodgkin's lymphoma: a systematic review and network meta-analysis

BACKGROUND Several treatment strategies are available for adults with advanced-stage Hodgkin's lymphoma, but studies assessing two alternative standards of care-increased dose bleomycin, etoposide, doxorubicin, cyclophosphamide, vincristine, procarbazine, and prednisone (BEACOPPescalated), and doxorubicin, bleomycin, vinblastine, and dacarbazine (ABVD)-were not powered to test differences in overall survival. To guide treatment decisions in this population of patients, we did a systematic review and network meta-analysis to identify the best initial treatment strategy. METHODS We searched the Cochrane Library, Medline, and conference proceedings for randomised controlled trials published between January, 1980, and June, 2013, that assessed overall survival in patients with advanced-stage Hodgkin's lymphoma given BEACOPPbaseline, BEACOPPescalated, BEACOPP variants, ABVD, cyclophosphamide (mechlorethamine), vincristine, procarbazine, and prednisone (C[M]OPP), hybrid or alternating chemotherapy regimens with ABVD as the backbone (eg, COPP/ABVD, MOPP/ABVD), or doxorubicin, vinblastine, mechlorethamine, vincristine, bleomycin, etoposide, and prednisone combined with radiation therapy (the Stanford V regimen). We assessed studies for eligibility, extracted data, and assessed their quality. We then pooled the data and used a Bayesian random-effects model to combine direct comparisons with indirect evidence. We also reconstructed individual patient survival data from published Kaplan-Meier curves and did standard random-effects Poisson regression. Results are reported relative to ABVD. The primary outcome was overall survival. FINDINGS We screened 2055 records and identified 75 papers covering 14 eligible trials that assessed 11 different regimens in 9993 patients, providing 59 651 patient-years of follow-up. 1189 patients died, and the median follow-up was 5·9 years (IQR 4·9-6·7). Included studies were of high methodological quality, and between-trial heterogeneity was negligible (τ(2)=0·01). Overall survival was highest in patients who received six cycles of BEACOPPescalated (HR 0·38, 95% credibility interval [CrI] 0·20-0·75). Compared with a 5 year survival of 88% for ABVD, the survival benefit for six cycles of BEACOPPescalated is 7% (95% CrI 3-10)-ie, a 5 year survival of 95%. Reconstructed individual survival data showed that, at 5 years, BEACOPPescalated has a 10% (95% CI 3-15) advantage over ABVD in overall survival. INTERPRETATION Six cycles of BEACOPPescalated significantly improves overall survival compared with ABVD and other regimens, and thus we recommend this treatment strategy as standard of care for patients with access to the appropriate supportive care.

Comparative effectiveness of long term drug treatment strategies to prevent asthma exacerbations: network meta-analysis

Objective To determine the comparative effectiveness and safety of current maintenance strategies in preventing exacerbations of asthma. Design Systematic review and network meta-analysis using Bayesian statistics. Data sources Cochrane systematic reviews on chronic asthma, complemented by an updated search when appropriate. Eligibility criteria Trials of adults with asthma randomised to maintenance treatments of at least 24 weeks duration and that reported on asthma exacerbations in full text. Low dose inhaled corticosteroid treatment was the comparator strategy. The primary effectiveness outcome was the rate of severe exacerbations. The secondary outcome was the composite of moderate or severe exacerbations. The rate of withdrawal was analysed as a safety outcome. Results 64 trials with 59 622 patient years of follow-up comparing 15 strategies and placebo were included. For prevention of severe exacerbations, combined inhaled corticosteroids and long acting β agonists as maintenance and reliever treatment and combined inhaled corticosteroids and long acting β agonists in a fixed daily dose performed equally well and were ranked first for effectiveness. The rate ratios compared with low dose inhaled corticosteroids were 0.44 (95% credible interval 0.29 to 0.66) and 0.51 (0.35 to 0.77), respectively. Other combined strategies were not superior to inhaled corticosteroids and all single drug treatments were inferior to single low dose inhaled corticosteroids. Safety was best for conventional best (guideline based) practice and combined maintenance and reliever therapy. Conclusions Strategies with combined inhaled corticosteroids and long acting β agonists are most effective and safe in preventing severe exacerbations of asthma, although some heterogeneity was observed in this network meta-analysis of full text reports.