Latent variable models for gene–environment interactions in longitudinal studies with multiple correlated exposures

Many existing cohort studies designed to investigate health effects of environmental exposures also collect data on genetic markers. The Early Life Exposures in Mexico to Environmental Toxicants project, for instance, has been genotyping single nucleotide polymorphisms on candidate genes involved in mental and nutrient metabolism and also in potentially shared metabolic pathways with the environmental exposures. Given the longitudinal nature of these cohort studies, rich exposure and outcome data are available to address novel questions regarding gene–environment interaction (G × E). Latent variable (LV) models have been effectively used for dimension reduction, helping with multiple testing and multicollinearity issues in the presence of correlated multivariate exposures and outcomes. In this paper, we first propose a modeling strategy, based on LV models, to examine the association between repeated outcome measures (e.g., child weight) and a set of correlated exposure biomarkers (e.g., prenatal lead exposure). We then construct novel tests for G × E effects within the LV framework to examine effect modification of outcome–exposure association by genetic factors (e.g., the hemochromatosis gene). We consider two scenarios: one allowing dependence of the LV models on genes and the other assuming independence between the LV models and genes. We combine the two sets of estimates by shrinkage estimation to trade off bias and efficiency in a data‐adaptive way. Using simulations, we evaluate the properties of the shrinkage estimates, and in particular, we demonstrate the need for this data‐adaptive shrinkage given repeated outcome measures, exposure measures possibly repeated and time‐varying gene–environment association. Copyright © 2014 John Wiley & Sons, Ltd.     

Using Bayes model averaging to leverage both gene main effects and G × E interactions to identify genomic regions in genome-wide association studies

Genome-wide association studies typically search for marginal associations between a single-nucleotide polymorphism (SNP) and a disease trait while gene-environment (G × E) interactions remain generally unexplored. More powerful methods beyond the simple case–control (CC) approach leverage either marginal effects or CC ascertainment to increase power. However, these potential gains depend on assumptions whose aptness is often unclear a priori. Here, we review G × E methods and use simulations to highlight performance as a function of main and interaction effects and the association of the two factors in the source population. Substantial variation in performance between methods leads to uncertainty as to which approach is most appropriate for any given analysis. We present a framework that (a) balances the robustness of a CC approach with the power of the case-only (CO) approach; (b) incorporates main SNP effects; (c) allows for incorporation of prior information; and (d) allows the data to determine the most appropriate model. Our framework is based on Bayes model averaging, which provides a principled statistical method for incorporating model uncertainty. We average over inclusion of parameters corresponding to the main and G × E interaction effects and the G–E association in controls. The resulting method exploits the joint evidence for main and interaction effects while gaining power from a CO equivalent analysis. Through simulations, we demonstrate that our approach detects SNPs within a wide range of scenarios with increased power over current methods. We illustrate the approach on a gene-environment scan in the USC Children's Health Study.     

Genome‐wide association scans for secondary traits using case‐control samples

Genome‐wide association studies (GWAS) require considerable investment, so researchers often study multiple traits collected on the same set of subjects to maximize return. However, many GWAS have adopted a case‐control design; improperly accounting for case‐control ascertainment can lead to biased estimates of association between markers and secondary traits. We show that under the null hypothesis of no marker‐secondary trait association, naïve analyses that ignore ascertainment or stratify on case‐control status have proper Type I error rates except when both the marker and secondary trait are independently associated with disease risk. Under the alternative hypothesis, these methods are unbiased when the secondary trait is not associated with disease risk. We also show that inverse‐probability‐of‐sampling‐weighted (IPW) regression provides unbiased estimates of marker‐secondary trait association. We use simulation to quantify the Type I error, power and bias of naïve and IPW methods. IPW regression has appropriate Type I error in all situations we consider, but has lower power than naïve analyses. The bias for naïve analyses is small provided the marker is independent of disease risk. Considering the majority of tested markers in a GWAS are not associated with disease risk, naïve analyses provide valid tests of and nearly unbiased estimates of marker‐secondary trait association. Care must be taken when there is evidence that both the secondary trait and tested marker are associated with the primary disease, a situation we illustrate using an analysis of the relationship between a marker in FGFR2 and mammographic density in a breast cancer case‐control sample. Genet. Epidemiol. 33:717–728, 2009. © 2009 Wiley‐Liss, Inc.     

Finding Novel Genes by Testing G × E Interactions in a Genome‐Wide Association Study

In a genome‐wide association study (GWAS), investigators typically focus their primary analysis on the direct (marginal) associations of each single nucleotide polymorphism (SNP) with the trait. Some SNPs that are truly associated with the trait may not be identified in this scan if they have a weak marginal effect and thus low power to be detected. However, these SNPs may be quite important in subgroups of the population defined by an environmental or personal factor, and may be detectable if such a factor is carefully considered in a gene–environment (G × E) interaction analysis. We address the question “Using a genome wide interaction scan (GWIS), can we find new genes that were not found in the primary GWAS scan?” We review commonly used approaches for conducting a GWIS in case‐control studies, and propose a new two‐step screening and testing method (EDG×E) that is optimized to find genes with a weak marginal effect. We simulate several scenarios in which our two‐step method provides 70–80% power to detect a disease locus while a marginal scan provides less than 5% power. We also provide simulations demonstrating that the EDG×E method outperforms other GWIS approaches (including case only and previously proposed two‐step methods) for finding genes with a weak marginal effect. Application of this method to a G × Sex scan for childhood asthma reveals two potentially interesting SNPs that were not identified in the marginal‐association scan. We distribute a new software program (G×Escan, available at http://biostats.usc.edu/software ) that implements this new method as well as several other GWIS approaches.     

Meta‐analysis of gene‐environment interaction exploiting gene‐environment independence across multiple case‐control studies

Multiple papers have studied the use of gene‐environment (G‐E) independence to enhance power for testing gene‐environment interaction in case‐control studies. However, studies that evaluate the role of G‐E independence in a meta‐analysis framework are limited. In this paper, we extend the single‐study empirical Bayes type shrinkage estimators proposed by Mukherjee and Chatterjee (2008) to a meta‐analysis setting that adjusts for uncertainty regarding the assumption of G‐E independence across studies. We use the retrospective likelihood framework to derive an adaptive combination of estimators obtained under the constrained model (assuming G‐E independence) and unconstrained model (without assumptions of G‐E independence) with weights determined by measures of G‐E association derived from multiple studies. Our simulation studies indicate that this newly proposed estimator has improved average performance across different simulation scenarios than the standard alternative of using inverse variance (covariance) weighted estimators that combines study‐specific constrained, unconstrained, or empirical Bayes estimators. The results are illustrated by meta‐analyzing 6 different studies of type 2 diabetes investigating interactions between genetic markers on the obesity related FTO gene and environmental factors body mass index and age.     

A simple approximation to the bias of gene–environment interactions in case–control studies with silent disease

One of the most important research areas in case–control Genome-Wide Association Studies is to determine how the effect of a genotype varies across the environment or to measure the gene–environment interaction (G × E). We consider the scenario when some of the “healthy” controls actually have the disease and when the frequency of these latent cases varies by the environmental variable of interest. In this scenario, performing logistic regression with the clinically diagnosed disease status as an outcome variable and will result in biased estimates of G × E interaction. Here, we derive a general theoretical approximation to the bias in the estimates of the G × E interaction and show, through extensive simulation, that this approximation is accurate in finite samples. Moreover, we apply this approximation to evaluate the bias in the effect estimates of the genetic variants related to mitochondrial proteins a large-scale prostate cancer study.     

A Genome‐Wide Association Study for Serum Bilirubin Levels and Gene‐Environment Interaction in a Chinese Population

Bilirubin is an effective antioxidant and is influenced by both genetic and environmental factors. Recent genome‐wide association studies (GWAS) have identified multiple loci affecting serum total bilirubin levels. However, most of the studies were conducted in European populations and little attention has been devoted either to genetic variants associated with direct and indirect bilirubin levels or to the gene‐environment interactions on bilirubin levels. In this study, a two‐stage GWAS was performed to identify genetic variants associated with all types of bilirubin levels in 10,282 Han Chinese individuals. Gene‐environment interactions were further examined. Briefly, two previously reported loci, UGT1A1 on 2q37 (rs6742078 and rs4148323, combined P = 1.44 × 10^?89 and P = 5.05 × 10^?69, respectively) and SLCO1B3 on 12p12 (rs2417940, combined P = 6.93 × 10^?19) were successfully replicated. The two loci explained 9.2% and 0.9% of the total variations of total bilirubin levels, respectively. Ethnic genetic differences were observed between Chinese and European populations. More importantly, a significant interaction was found between rs2417940 in SLCO1B3 gene and smoking on total bilirubin levels (P = 1.99 × 10^?3). Single nucleotide polymorphism (SNP) rs2417940 had stronger effects on total bilirubin levels in nonsmokers than in smokers, suggesting that the effects of SLCO1B3 genotype on bilirubin levels were partly dependent on smoking status. Consistent associations and interactions were observed for serum direct and indirect bilirubin levels.     

The Case‐Only Test for Gene–Environment Interaction is Not Uniformly Powerful: An Empirical Example

The case‐only test has been proposed as a more powerful approach to detect gene–environment (G × E) interactions. This approach assumes that the genetic and environmental factors are independent. Although it is well known that Type I error rate will increase if this assumption is violated, it is less widely appreciated that G × E correlation can also lead to power loss. We illustrate this phenomenon by comparing the performance of the case‐only test to other approaches to detect G × E interactions in a genome‐wide association study (GWAS) of esophageal squamous‐cell carcinoma (ESCC) in Chinese populations. Some of these approaches do not use information on the correlation between exposure and genotype (standard logistic regression), whereas others seek to use this information in a robust fashion to boost power without increasing Type I error (two‐step, empirical Bayes, and cocktail methods). G × E interactions were identified involving drinking status and two regions containing genes in the alcohol metabolism pathway, 4q23 and 12q24. Although the case‐only test yielded the most significant tests of G × E interaction in the 4q23 region, the case‐only test failed to identify significant interactions in the 12q24 region which were readily identified using other approaches. The low power of the case‐only test in the 12q24 region is likely due to the strong inverse association between the single nucleotide polymorphism (SNPs) in this region and drinking status. This example underscores the need to consider multiple approaches to detect G × E interactions, as different tests are more or less sensitive to different alternative hypotheses and violations of the G × E independence assumption.     

Bayesian analysis of a matched case–control study with expert prior information on both the misclassification of exposure and the exposure–disease association

We propose a Bayesian adjustment for the misclassification of a binary exposure variable in a matched case–control study. The method admits a priori knowledge about both the misclassification parameters and the exposure–disease association. The standard Dirichlet prior distribution for a multinomial model is extended to allow separation of prior assertions about the exposure–disease association from assertions about other parameters. The method is applied to a study of occupational risk factors for new‐onset adult asthma. Copyright © 2009 John Wiley & Sons, Ltd.     

A genetic stochastic process model for genome‐wide joint analysis of biomarker dynamics and disease susceptibility with longitudinal data

Unraveling the underlying biological mechanisms or pathways behind the effects of genetic variations on complex diseases remains one of the major challenges in the post‐GWAS (where GWAS is genome‐wide association study) era. To further explore the relationship between genetic variations, biomarkers, and diseases for elucidating underlying pathological mechanism, a huge effort has been placed on examining pleiotropic and gene‐environmental interaction effects. We propose a novel genetic stochastic process model (GSPM) that can be applied to GWAS and jointly investigate the genetic effects on longitudinally measured biomarkers and risks of diseases. This model is characterized by more profound biological interpretation and takes into account the dynamics of biomarkers during follow‐up when investigating the hazards of a disease. We illustrate the rationale and evaluate the performance of the proposed model through two GWAS. One is to detect single nucleotide polymorphisms (SNPs) having interaction effects on type 2 diabetes (T2D) with body mass index (BMI) and the other is to detect SNPs affecting the optimal BMI level for protecting from T2D. We identified multiple SNPs that showed interaction effects with BMI on T2D, including a novel SNP rs11757677 in the CDKAL1 gene (P = 5.77 × 10^?7). We also found a SNP rs1551133 located on 2q14.2 that reversed the effect of BMI on T2D (P = 6.70 × 10^?7). In conclusion, the proposed GSPM provides a promising and useful tool in GWAS of longitudinal data for interrogating pleiotropic and interaction effects to gain more insights into the relationship between genes, quantitative biomarkers, and risks of complex diseases.     

Gene‐Gene Interaction Among WNT Genes for Oral Cleft in Trios

Genome‐wide association studies (GWAS) for nonsyndromic cleft lip with or without cleft palate (CL/P) have identified multiple genes as important in the etiology of this common birth defect. We performed a candidate gene/pathway analysis explicitly considering gene‐gene (G × G) interaction to further explore the etiology of CL/P. Animal models have shown the WNT signaling pathway plays an important role in mid‐facial development, and various genes in this pathway have been associated with nonsyndromic CL/P in previous studies. We propose a combined approach to search for possible G × G interactions using machine learning and regression‐based methods to test for interactions between genes in the WNT family, and between these genes and other genes identified by GWAS in case‐parent trios. Using this combined approach of regression‐based and machine learning methods in CL/P case‐parent trios, we found robust evidence of G × G interaction between markers in WNT5B and MAFB (empiric P‐values = 0.0076 among Asian trios and P‐values = 0.018 among European trios). Additional evidence for epistatic interaction between markers in WNT5A, IRF6, and C1orf107 was seen among Asian trios, and markers in the 8q24 region and WNT5B among European trios.     

Efficient Generalized Least Squares Method for Mixed Population and Family‐based Samples in Genome‐wide Association Studies

Genome‐wide association studies (GWAS) that draw samples from multiple studies with a mixture of relationship structures are becoming more common. Analytical methods exist for using mixed‐sample data, but few methods have been proposed for the analysis of genotype‐by‐environment (G×E) interactions. Using GWAS data from a study of sarcoidosis susceptibility genes in related and unrelated African Americans, we explored the current analytic options for genotype association testing in studies using both unrelated and family‐based designs. We propose a novel method—generalized least squares (GLX)—to estimate both SNP and G×E interaction effects for categorical environmental covariates and compared this method to generalized estimating equations (GEE), logistic regression, the Cochran–Armitage trend test, and the W_QLS and M_QLS methods. We used simulation to demonstrate that the GLX method reduces type I error under a variety of pedigree structures. We also demonstrate its superior power to detect SNP effects while offering computational advantages and comparable power to detect G×E interactions versus GEE. Using this method, we found two novel SNPs that demonstrate a significant genome‐wide interaction with insecticide exposure—rs10499003 and rs7745248, located in the intronic and 3' UTR regions of the FUT9 gene on chromosome 6q16.1.     

Empirical Hierarchical Bayes Approach to Gene‐Environment Interactions: Development and Application to Genome‐Wide Association Studies of Lung Cancer in TRICL

The analysis of gene‐environment (G × E) interactions remains one of the greatest challenges in the postgenome‐wide association studies (GWASs) era. Recent methods constitute a compromise between the robust but underpowered case‐control and powerful case‐only methods. Inferences of the latter are biased when the assumption of gene‐environment (G‐E) independence in controls fails. We propose a novel empirical hierarchical Bayes approach to G × E interaction (EHB‐GE), which benefits from greater rank power while accounting for population‐based G‐E correlation. Building on Lewinger et al.'s ([2007] Genet Epidemiol 31:871–882) hierarchical Bayes prioritization approach, the method first obtains posterior G‐E correlation estimates in controls for each marker, borrowing strength from G‐E information across the genome. These posterior estimates are then subtracted from the corresponding case‐only G × E estimates. We compared EHB‐GE with rival methods using simulation. EHB‐GE has similar or greater rank power to detect G × E interactions in the presence of large numbers of G‐E correlations with weak to strong effects or only a low number of such correlations with large effect. When there are no or only a few weak G‐E correlations, Murcray et al.'s method ([2009] Am J Epidemiol 169:219–226) identifies markers with low G × E interaction effects better. We applied EHB‐GE and competing methods to four lung cancer case‐control GWAS from the Interdisciplinary Research in Cancer of the Lung/International Lung Cancer Consortium with smoking as environmental factor. A number of genes worth investigating were identified by the EHB‐GE approach.     

Investigation of a Candidate Gene,Environment, and G×E Interaction Using Case‐Control and Case‐Parent Study Designs

We investigated the independent contributions of a candidate gene and an environmental factor, and the presence of gene×environment (G×E) interaction, in the etiology of a disease in the Genetic Analysis Workshop (GAW) 12 problem 2 simulated data using a two‐stage approach utilizing both case‐control and case‐parent study designs. Using the case‐control design, several SNPs within candidate gene 1 (CG1) and environmental factor 1 (dichotomized using the 75^th percentile as a cut‐off) (EXP) were independently associated with disease status, in models adjusted for age and sex. We found evidence of gene×environment (G×E) interaction between EXP and two single‐nucleotide polymorphisms (SNPs) within CGI using the case‐control design. Using the case‐parent study design in the same population, we detected association between SNPs within CG1 and disease, but no G×E interaction was detected. © 2001 Wiley‐Liss, Inc.     

Efficient Two‐Step Testing of Gene‐Gene Interactions in Genome‐Wide Association Studies

Exhaustive testing of all possible SNP pairs in a genome‐wide association study (GWAS) generally yields low power to detect gene‐gene (G × G) interactions because of small effect sizes and stringent requirements for multiple‐testing correction. We introduce a new two‐step procedure for testing G × G interactions in case‐control GWAS to detect interacting single nucleotide polymorphisms (SNPs) regardless of their marginal effects. In an initial screening step, all SNP pairs are tested for gene‐gene association in the combined sample of cases and controls. In the second step, the pairs that pass the screening are followed up with a traditional test for G × G interaction. We show that the two‐step method is substantially more powerful to detect G × G interactions than the exhaustive testing approach. For example, with 2,000 cases and 2,000 controls, the two‐step method can have more than 90% power to detect an interaction odds ratio of 2.0 compared to less than 50% power for the exhaustive testing approach. Moreover, we show that a hybrid two‐step approach that combines our newly proposed two‐step test and the two‐step test that screens for marginal effects retains the best power properties of both. The two‐step procedures we introduce have the potential to uncover genetic signals that have not been previously identified in an initial single‐SNP GWAS. We demonstrate the computational feasibility of the two‐step G × G procedure by performing a G × G scan in the asthma GWAS of the University of Southern California Children's Health Study.     

Informed Genome‐Wide Association Analysis With Family History As a Secondary Phenotype Identifies Novel Loci of Lung Cancer

Lung cancer is the leading cause of cancer death worldwide. Although several genetic variants associated with lung cancer have been identified in the past, stringent selection criteria of genome‐wide association studies (GWAS) can lead to missed variants. The objective of this study was to uncover missed variants by using the known association between lung cancer and first‐degree family history of lung cancer to enrich the variant prioritization for lung cancer susceptibility regions. In this two‐stage GWAS study, we first selected a list of variants associated with both lung cancer and family history of lung cancer in four GWAS (3,953 cases, 4,730 controls), then replicated our findings for 30 variants in a meta‐analysis of four additional studies (7,510 cases, 7,476 controls). The top ranked genetic variant rs12415204 in chr10q23.33 encoding FFAR4 in the Discovery set was validated in the Replication set with an overall OR of 1.09 (95% CI = 1.04, 1.14, P = 1.63 × 10^?4). When combining the two stages of the study, the strongest association was found in rs1158970 at Ch4p15.2 encoding KCNIP4 with an OR of 0.89 (95% CI = 0.85, 0.94, P = 9.64 × 10^?6). We performed a stratified analysis of rs12415204 and rs1158970 across all eight studies by age, gender, smoking status, and histology, and found consistent results across strata. Four of the 30 replicated variants act as expression quantitative trait loci (eQTL) sites in 1,111 nontumor lung tissues and meet the genome‐wide 10% FDR threshold.     

Exposure enriched outcome dependent designs for longitudinal studies of gene–environment interaction

Joint effects of genetic and environmental factors have been increasingly recognized in the development of many complex human diseases. Despite the popularity of case‐control and case‐only designs, longitudinal cohort studies that can capture time‐varying outcome and exposure information have long been recommended for gene–environment (G × E) interactions. To date, literature on sampling designs for longitudinal studies of G × E interaction is quite limited. We therefore consider designs that can prioritize a subsample of the existing cohort for retrospective genotyping on the basis of currently available outcome, exposure, and covariate data. In this work, we propose stratified sampling based on summaries of individual exposures and outcome trajectories and develop a full conditional likelihood approach for estimation that adjusts for the biased sample. We compare the performance of our proposed design and analysis with combinations of different sampling designs and estimation approaches via simulation. We observe that the full conditional likelihood provides improved estimates for the G × E interaction and joint exposure effects over uncorrected complete‐case analysis, and the exposure enriched outcome trajectory dependent design outperforms other designs in terms of estimation efficiency and power for detection of the G × E interaction. We also illustrate our design and analysis using data from the Normative Aging Study, an ongoing longitudinal cohort study initiated by the Veterans Administration in 1963. Copyright © 2017 John Wiley & Sons, Ltd.     

Case‐only genome‐wide interaction study of disease risk,prognosis and treatment

Case‐control genome‐wide association (GWA) studies have facilitated the identification of susceptibility loci for many complex diseases; however, these studies are often not adequately powered to detect gene‐environment (G×E) and gene‐gene (G×G) interactions. Case‐only studies are more efficient than case‐control studies for detecting interactions and require no data on control subjects. In this article, we discuss the concept and utility of the case‐only genome‐wide interaction (COGWI) study, in which common genetic variants, measured genome‐wide, are screened for association with environmental exposures or genetic variants of interest. An observed G‐E (or G‐G) association, as measured by the case‐only odds ratio (OR), suggests interaction, but only if the interacting factors are unassociated in the population from which the cases were drawn. The case‐only OR is equivalent to the interaction risk ratio. In addition to risk‐related interactions, we discuss how the COGWI design can be used to efficiently detect G×G, G×E and pharmacogenetic interactions related to disease outcomes in the context of observational clinical studies or randomized clinical trials. Such studies can be conducted using only data on individuals experiencing an outcome of interest or individuals not experiencing the outcome of interest. Sharing data among GWA and COGWI studies of disease risk and outcome can further enhance efficiency. Sample size requirements for COGWI studies, as compared to case‐control GWA studies, are provided. In the current era of genome‐wide analyses, the COGWI design is an efficient and straightforward method for detecting G×G, G×E and pharmacogenetic interactions related to disease risk, prognosis and treatment response. Genet. Epidemiol. 34:7–15, 2010. © 2009 Wiley‐Liss, Inc.     

Gene–environment interactions related to blood pressure traits in two community-based Korean cohorts

Hypertension is a complex disorder caused by genetic and environmental risk factors. Recently, genome-wide association studies (GWASs) identified more than 100 genetic variants for blood pressure traits and hypertension. However, the interactions between these genetic variants and environmental factors have not been systematically investigated. Therefore, we examined the interaction between genetic and environmental risk factors in blood pressure traits using the genetic risk score (GRS). Two Korean community-based cohorts, Cohort I (KARE; N = 8,840) and Cohort II (CAVAS; N = 9,599), were used for this study, and GRSs were calculated from 42 GWAS single-nucleotide polymorphisms (SNPs) that were validated for their association in these cohorts. We calculated GRSs in both ways by considering the effect sizes of each SNP (weighted GRS) and not considering the effect sizes (unweighted GRS). The unweighted GRS was strongly associated with systolic blood pressure, diastolic blood pressure, and hypertension (p = 9.03 × 10 ^–47, p = 9.41 × 10 ^–48, and p = 3.22 × 10 ^–55 by meta-analysis, respectively) and the weighted GRS showed the similar results. The environmental factors of body mass index, waist circumference, and drinking status were significantly associated with blood pressure traits, and the interaction between these factors and GRSs were examined. However, no interactions were found with either the GRS or the individual SNPs considered for the GRS. Our findings show that it is challenging to find GRS–environment interactions regarding blood pressure traits.     

Statistical Inference for Cost–Effectiveness Ratios

Methods for statistical inference for cost–effectiveness (C/E) ratios for individual treatments and for incremental cost–effectiveness (ΔC/ΔE) ratios when two treatments are compared are presented. In a lemma, we relate the relative magnitude of two C/E ratios to the ΔC/ΔE ratio. We describe a statistical procedure to test for dominance, or admissibility, that can be used to eliminate an inferior treatment. The one-sided Bonferroni's confidence interval procedure is generalized to the two- sided case. The method requires only that two confidence intervals be available, one for cost and one for effectiveness. We describe Fieller-based confidence intervals and show them to be shorter than Bonferroni intervals. When distribution assumptions hold and variance and covariance estimates are available, Fieller intervals are preferable. However, Bonferroni intervals can be applied in more diverse situations and are easier to calculate. A simple Bonferroni based technique, and a likelihood ratio statistic given by Siegel, Laska and Meisner, for testing the null hypothesis that the C/E ratios of two treatments are equal is presented. The approaches are applied to the data from a phase II clinical trial of a new treatment for sepsis considered previously by others. © 1997 by John Wiley & Sons, Ltd.