Inference for reclassification statistics under nested and non-nested models for biomarker evaluation

Abstract

The Net Reclassification Improvement (NRI) and the Integrated Discrimination Improvement (IDI) are used to evaluate the diagnostic accuracy improvement for biomarkers in a wide range of applications. Most applications for these reclassification metrics are confined to nested model comparison. We emphasize the important extensions of these metrics to the non-nested comparison. Non-nested models are important in practice, in particular, in high-dimensional data analysis and in sophisticated semiparametric modeling. We demonstrate that the assessment of accuracy improvement may follow the familiar NRI and IDI evaluation. While the statistical properties of the estimators for NRI and IDI have been well studied in the nested setting, one cannot always rely on these asymptotic results to implement the inference procedure for practical data, especially for testing the null hypothesis of no improvement, and these properties have not been established for the non-nested setting. We propose a generic bootstrap re-sampling procedure for the construction of confidence intervals and hypothesis tests. Extensive simulations and real biomedical data examples illustrate the applicability of the proposed inference methods for both nested and non-nested models.     

Development and validation of two nomograms for predicting overall survival and cancer-specific survival in gastric cancer patients with liver metastases: A retrospective cohort study from SEER database

BackgroundGastric cancer is heterogeneous and aggressive, especially with liver metastasis. This study aims to develop two nomograms to predict the overall survival (OS) and cancer-specific survival (CSS) of gastric cancer with liver metastasis (GCLM) patients.MethodsFrom January 2000 to December 2018, a total of 1936 GCLM patients were selected from the Surveillance, Epidemiology, and End Results Program (SEER) database. They were further divided into a training cohort and a validation cohort, with the OS and CSS serving as the study's endpoints. The correlation analyses were used to determine the relationship between the variables. The univariate and multivariate Cox analyses were used to confirm the independent prognostic factors. To discriminate and calibrate the nomogram, calibration curves and the area under the time-dependent receiver operating characteristic curve (time-dependent AUC) were used. DCA curves were used to examine the accuracy and clinical benefits. The clinical utility of the nomogram and the AJCC Stage System was compared using net reclassification improvement (NRI) and integrated differentiation improvement (IDI) (IDI). Finally, the nomogram and the AJCC Stage System risk stratifications were compared.ResultsThere was no collinearity among the variables that were screened. The results of multivariate Cox regression analysis showed that six variables (bone metastasis, lung metastasis, surgery, chemotherapy, grade, age) and five variables (lung metastasis, surgery, chemotherapy, grade, N stage) were identified to establish the nomogram for OS and CSS, respectively. The calibration curves, time-dependent AUC curves, and DCA revealed that both nomograms had pleasant predictive power. Furthermore, NRI and IDI confirmed that the nomogram outperformed the AJCC Stage System.ConclusionBoth nomograms had satisfactory accuracy and were validated to assist clinicians in evaluating the prognosis of GCLM patients.     

Developing Genetic Epidemiological Models to Predict Risk for Nasopharyngeal Carcinoma in High-Risk Population of China

To date, the only established model for assessing risk for nasopharyngeal carcinoma (NPC) relies on the sero-status of the Epstein-Barr virus (EBV). By contrast, the risk assessment models proposed here include environmental risk factors, family history of NPC, and information on genetic variants. The models were developed using epidemiological and genetic data from a large case-control study, which included 1,387 subjects with NPC and 1,459 controls of Cantonese origin. The predictive accuracy of the models were then assessed by calculating the area under the receiver-operating characteristic curves (AUC). To compare the discriminatory improvement of models with and without genetic information, we estimated the net reclassification improvement (NRI) and integrated discrimination index (IDI). Well-established environmental risk factors for NPC include consumption of salted fish and preserved vegetables and cigarette smoking (in pack years). The environmental model alone shows modest discriminatory ability (AUC = 0.68; 95% CI: 0.66, 0.70), which is only slightly increased by the addition of data on family history of NPC (AUC = 0.70; 95% CI: 0.68, 0.72). With the addition of data on genetic variants, however, our model’s discriminatory ability rises to 0.74 (95% CI: 0.72, 0.76). The improvements in NRI and IDI also suggest the potential usefulness of considering genetic variants when screening for NPC in endemic areas. If these findings are confirmed in larger cohort and population-based case-control studies, use of the new models to analyse data from NPC-endemic areas could well lead to earlier detection of NPC.     

Development and validation of a nomogram to predict overall survival of T1 esophageal squamous cell carcinoma patients with lymph node metastasis

PurposeTo develop a nomogram for predicting the prognosis of T1 esophageal squamous cell carcinoma (ESCC) patients with positive lymph node.MethodsT1 ESCC patients with lymph node metastasis diagnosed between 2010 and 2015 were selected from the Surveillance, Epidemiology, and Final Results (SEER) database. The entire cohort was randomly divided in the ratio of 7:3 into a training group (n=457) and validation group (n=192), respectively. Prognostic factors were identified by univariate and multivariate Cox regression models. Harrell''s concordance index (C-index), receiver operating characteristic (ROC) curve, and calibration curve were used to evaluate the discrimination and calibration of the nomogram. The accuracy and clinical net benefit of the nomogram compared with the 7^th AJCC staging system were evaluated using net reclassification improvement (NRI), integrated discrimination improvement (IDI), and decision curve analysis (DCA).ResultsThe nomogram consisted of eight factors: insurance, T stage, summary stage, primary site, radiation code, chemotherapy, surgery, and radiation sequence with surgery. In the training and validation cohorts, the AUCs exceeded 0.700, and the C-index scores were 0.749 and 0.751, respectively, indicating that the nomogram had good discrimination. The consistency between the survival probability predicted by the nomogram and the actual observed probability was indicated by the calibration curve in the training and validation cohorts. For NRI>0 and IDI>0, the predictive power of the nomogram was more accurate than that of the 7^th AJCC staging system. Furthermore, the DCA curve indicated that the nomogram achieved better clinical utility than the traditional system.ConclusionsUnlike the 7^th AJCC staging system, the developed and validated nomogram can help clinical staff to more accurately, personally and comprehensively predict the 1-year and 3-year OS probability of T1 ESCC patients with lymph node metastasis.     

Exact tests for the analysis of case-control studies of genetic markers

OBJECTIVES: The association of a candidate gene with disease can be evaluated by a case-control study in which the genotype distribution is compared for diseased cases and unaffected controls. Usually, the data are analyzed with Armitage's test using the asymptotic null distribution of the test statistic. Since this test does not generally guarantee a type I error rate less than or equal to the significance level alpha, tests based on exact null distributions have been investigated. METHODS: An algorithm to generate the exact null distribution for both Armitage's test statistic and a recently proposed modification of the Baumgartner-Weiss-Schindler statistic is presented. I have compared the tests in a simulation study. RESULTS: The asymptotic Armitage test is slightly anticonservative whereas the exact tests control the type I error rate. The exact Armitage test is very conservative, but the exact test based on the modification of the Baumgartner-Weiss-Schindler statistic has a type I error rate close to alpha. The exact Armitage test is the least powerful test; the difference in power between the other two tests is often small and the comparison does not show a clear winner. CONCLUSION: Simulation results indicate that an exact test based on the modification of the Baumgartner-Weiss-Schindler statistic is preferable for the analysis of case-control studies of genetic markers.     

Adiponectin Provides Additional Information to Conventional Cardiovascular Risk Factors for Assessing the Risk of Atherosclerosis in Both Genders

Background

This study evaluated the relation between adiponectin and atherosclerosis in both genders, and investigated whether adiponectin provides useful additional information for assessing the risk of atherosclerosis.

Methods

We measured serum adiponectin levels and other cardiovascular risk factors in 1033 subjects (454 men, 579 women) from the Korean Genomic Rural Cohort study. Carotid intima–media-thickness (CIMT) was used as measure of atherosclerosis. Odds ratios (ORs) with 95% confidence intervals (95% CI) were calculated using multiple logistic regression, and receiver operating characteristic curves (ROC), the category-free net reclassification improvement (NRI) and integrated discrimination improvement (IDI) were calculated.

Results

After adjustment for conventional cardiovascular risk factors, such as age, waist circumference, smoking history, low-density and high-density lipoprotein cholesterol, triglycerides, systolic blood pressure and insulin resistance, the ORs (95%CI) of the third tertile adiponectin group were 0.42 (0.25–0.72) in men and 0.47 (0.29–0.75) in women. The area under the curve (AUC) on the ROC analysis increased significantly by 0.025 in men and 0.022 in women when adiponectin was added to the logistic model of conventional cardiovascular risk factors (AUC in men: 0.655 to 0.680, p = 0.038; AUC in women: 0.654 to 0.676, p = 0.041). The NRI was 0.32 (95%CI: 0.13–0.50, p<0.001), and the IDI was 0.03 (95%CI: 0.01–0.04, p<0.001) for men. For women, the category-free NRI was 0.18 (95%CI: 0.02–0.34, p = 0.031) and the IDI was 0.003 (95%CI: −0.002–0.008, p = 0.189).

Conclusion

Adiponectin and atherosclerosis were significantly related in both genders, and these relationships were independent of conventional cardiovascular risk factors. Furthermore, adiponectin provided additional information to conventional cardiovascular risk factors regarding the risk of atherosclerosis.     

Comparative application of the coefficient of variation and range-based statistics for assessing the taxonomic composition of fossil samples

Dental variation has been used commonly to assess taxonomic composition in morphologically homogeneous fossil samples. While the coefficient of variation (CV) has been used traditionally, range-based measures of variation, such as the range as a percentage of the mean (R%) and the maximum/minimum index (I_max/min) have recently become popular alternatives. The current study compares the performance of these statistics when applied to single- and pooled-species dental samples of extant Cercopithecus species. A common methodology for such problems of species discrimination has been to simply compare the maximum value of a variation statistic observed in extant samples with that observed in the fossil sample. However, regardless of what statistic is used, this approach has an unknowable Type I error rate, and usually has low power to detect multiple species. A more appropriate method involves a formal hypothesis test. The null hypothesis is that the level of variation in the fossil sample does not exceed what might be expected in a sample drawn randomly from a reference population, taking into account sampling error and the size of the fossil sample. Previous research using this method with the CV has indicated that it offers considerable power at an acceptable Type I error rate. In the current study, the data of primary interest were posterior dental dimensions for single- and pooled species samples from extant Cercopithecus species. In addition, the study also investigated the relative performance of variation statistics when applied to highly dimorphic canine dimensions, since much recent work has employed sexually dimorphic dental dimensions for assessing single-species hypotheses. The results indicate that the CV consistently out-performed the range-based statistics when using posterior dental dimensions to test a single-species hypothesis. Regardless of which statistic was used, tests on sexually dimorphic dimensions offered minimal power. In consideration of these results and the problem of studywise Type I error rates, we recommend against the use of multiple measures of variation to test for multiple species composition, and advocate the CV as the statistic of choice when using the method of Cope & Lacy (1992). For similar reasons, we argue for careful selection of dental variables for inclusion in such analyses, and in particular recommend against including sexually dimorphic dimensions when testing for multiple species composition.     

Thirty-One Novel Biomarkers as Predictors for Clinically Incident Diabetes

Background

The prevalence of diabetes is increasing in all industrialized countries and its prevention has become a public health priority. However, the predictors of diabetes risk are insufficiently understood. We evaluated, whether 31 novel biomarkers could help to predict the risk of incident diabetes.

Methods and Findings

The biomarkers were evaluated primarily in the FINRISK97 cohort (n = 7,827; 417 cases of clinically incident diabetes during the follow-up). The findings were replicated in the Health 2000 cohort (n = 4,977; 179 cases of clinically incident diabetes during the follow-up). We used Cox proportional hazards models to calculate the relative risk of diabetes, after adjusting for the classic risk factors, separately for each biomarker. Next, we assessed the discriminatory ability of single biomarkers using receiver operating characteristic curves and C-statistics, integrated discrimination improvement (IDI) and net reclassification improvement (NRI). Finally, we derived a biomarker score in the FINRISK97 cohort and validated it in the Health 2000 cohort. A score consisting of adiponectin, apolipoprotein B, C-reactive protein and ferritin almost doubled the relative risk of diabetes in the validation cohort (HR per one standard deviation increase 1.88, p = 2.8 e-5). It also improved discrimination of the model (IDI = 0.0149, p<0.0001) and reclassification of diabetes risk (NRI = 11.8%, p = 0.006). Gender-specific analyses suggested that the best score differed between men and women. Among men, the best results were obtained with the score of four biomarkers: adiponectin, apolipoprotein B, ferritin and interleukin-1 receptor antagonist, which gave an NRI of 25.4% (p<0.0001). Among women, the best score included adiponectin, apolipoprotein B, C-reactive protein and insulin. It gave an NRI of 13.6% (p = 0.041).

Conclusions

We identified novel biomarkers that were associated with the risk of clinically incident diabetes over and above the classic risk factors. This gives new insights into the pathogenesis of diabetes and may help with targeting prevention and treatment.     

Notes on Testing Equality in Dichotomous Data with Matched Pairs

In attempting to improve the efficiency of McNemar's test statistic, we develop two test procedures that account for the information on both the discordant and concordant pairs for testing equality between two comparison groups in dichotomous data with matched pairs. Furthermore, we derive a test procedure derived from one of the most commonly‐used interval estimators for odds ratio. We compare these procedures with those using McNemar's test, McNemar's test with the continuity correction, and the exact test with respect to type I error and power in a variety of situations. We note that the test procedures using McNemar's test with the continuity correction and the exact test can be quite conservative and hence lose much efficiency, while the test procedure using McNemar's test can actually perform well even when the expected number of discordant pairs is small. We also find that the two test procedures, which incorporate the information on all matched pairs into hypothesis testing, may slightly improve the power of using McNemar's test without essentially losing the precision of type I error. On the other hand, the test procedure derived from an interval estimator of adds ratio with use of the logarithmic transformation may have type I error much larger than the nominal α‐level when the expected number of discordant pairs is not large and therefore, is not recommended for general use.     

Tailoring the Implementation of New Biomarkers Based on Their Added Predictive Value in Subgroups of Individuals

Background

The value of new biomarkers or imaging tests, when added to a prediction model, is currently evaluated using reclassification measures, such as the net reclassification improvement (NRI). However, these measures only provide an estimate of improved reclassification at population level. We present a straightforward approach to characterize subgroups of reclassified individuals in order to tailor implementation of a new prediction model to individuals expected to benefit from it.

Methods

In a large Dutch population cohort (n = 21,992) we classified individuals to low (<5%) and high (≥5%) fatal cardiovascular disease risk by the Framingham risk score (FRS) and reclassified them based on the systematic coronary risk evaluation (SCORE). Subsequently, we characterized the reclassified individuals and, in case of heterogeneity, applied cluster analysis to identify and characterize subgroups. These characterizations were used to select individuals expected to benefit from implementation of SCORE.

Results

Reclassification after applying SCORE in all individuals resulted in an NRI of 5.00% (95% CI [-0.53%; 11.50%]) within the events, 0.06% (95% CI [-0.08%; 0.22%]) within the nonevents, and a total NRI of 0.051 (95% CI [-0.004; 0.116]). Among the correctly downward reclassified individuals cluster analysis identified three subgroups. Using the characterizations of the typically correctly reclassified individuals, implementing SCORE only in individuals expected to benefit (n = 2,707,12.3%) improved the NRI to 5.32% (95% CI [-0.13%; 12.06%]) within the events, 0.24% (95% CI [0.10%; 0.36%]) within the nonevents, and a total NRI of 0.055 (95% CI [0.001; 0.123]). Overall, the risk levels for individuals reclassified by tailored implementation of SCORE were more accurate.

Discussion

In our empirical example the presented approach successfully characterized subgroups of reclassified individuals that could be used to improve reclassification and reduce implementation burden. In particular when newly added biomarkers or imaging tests are costly or burdensome such a tailored implementation strategy may save resources and improve (cost-)effectiveness.     

An evaluation of power and type I error of single-nucleotide polymorphism transmission/disequilibrium-based statistical methods under different family structures, missing parental data, and population stratification

Researchers conducting family-based association studies have a wide variety of transmission/disequilibrium (TD)-based methods to choose from, but few guidelines exist in the selection of a particular method to apply to available data. Using a simulation study design, we compared the power and type I error of eight popular TD-based methods under different family structures, frequencies of missing parental data, genetic models, and population stratifications. No method was uniformly most powerful under all conditions, but type I error was appropriate for nearly every test statistic under all conditions. Power varied widely across methods, with a 46.5% difference in power observed between the most powerful and the least powerful method when 50% of families consisted of an affected sib pair and one parent genotyped under an additive genetic model and a 35.2% difference when 50% of families consisted of a single affection-discordant sibling pair without parental genotypes available under an additive genetic model. Methods were generally robust to population stratification, although some slightly less so than others. The choice of a TD-based test statistic should be dependent on the predominant family structure ascertained, the frequency of missing parental genotypes, and the assumed genetic model.     

Simultaneous Non‐inferiority Test of Sensitivity and Specificity for Two Diagnostic Procedures in the Presence of a Gold Standard

Sensitivity and specificity have traditionally been used to assess the performance of a diagnostic procedure. Diagnostic procedures with both high sensitivity and high specificity are desirable, but these procedures are frequently too expensive, hazardous, and/or difficult to operate. A less sophisticated procedure may be preferred, if the loss of the sensitivity or specificity is determined to be clinically acceptable. This paper addresses the problem of simultaneous testing of sensitivity and specificity for an alternative test procedure with a reference test procedure when a gold standard is present. The hypothesis is formulated as a compound hypothesis of two non‐inferiority (one‐sided equivalence) tests. We present an asymptotic test statistic based on the restricted maximum likelihood estimate in the framework of comparing two correlated proportions under the prospective and retrospective sampling designs. The sample size and power of an asymptotic test statistic are derived. The actual type I error and power are calculated by enumerating the exact probabilities in the rejection region. For applications that require high sensitivity as well as high specificity, a large number of positive subjects and a large number of negative subjects are needed. We also propose a weighted sum statistic as an alternative test by comparing a combined measure of sensitivity and specificity of the two procedures. The sample size determination is independent of the sampling plan for the two tests.     

Graphical representations and summary indicators to assess the performance of risk predictors

The availability of novel biomarkers in several branches of medicine opens room for refining prognosis by adding factors on top of those having an established role. It is accepted that the impact of novel factors should not rely solely on regression coefficients and their significance but also on predictive power measures, such as Brier score and ROC‐based quantities. However, novel factors that are promising at the exploratory stage often result in disappointingly low impact in the predictive power. This motivated the proposal of the net reclassification improvement and the integrated discrimination improvement, as direct measures of predictive power gain due to additional factors based on the concept of reclassification tables. These measures became extremely popular in cardiovascular disease and cancer applications, given the apparently easy interpretation. However, recent contributions in the biostatistical literature enlightened the tendency to indicate as advantageous models obtained by adding unrelated factors. These measures should not be used in practice. A further measure proposed a decade ago, the net benefit, is becoming a standard in assessing the consequences in terms of costs and benefits when using a risk predictor in practice for classification. This work reviews the conceptual formulations and interpretations of the available graphical methods and summary measures for evaluating risk predictor models. The aim is to provide guidance in the evaluation process that from the model development brings the risk predictor to be used in clinical practice for binary decision rules.     

Soluble Forms of Intercellular and Vascular Cell Adhesion Molecules Independently Predict Progression to Type 2 Diabetes in Mexican American Families

Objective

While the role of type 2 diabetes (T2D) in inducing endothelial dysfunction is fairly well-established the etiological role of endothelial dysfunction in the onset of T2D is still a matter of debate. In the light of conflicting evidence in this regard, we conducted a prospective study to determine the association of circulating levels of soluble intercellular adhesion molecule 1 (sICAM-1) and soluble vessel cell adhesion molecule 1 (sVCAM-1) with incident T2D.

Methods

Data from this study came from 1,269 Mexican Americans of whom 821 initially T2D-free individuals were longitudinally followed up in the San Antonio Family Heart Study. These individuals were followed for 9752.95 person-years for development of T2D. Prospective association of sICAM-1 and sVCAM-1 with incident T2D was studied using Kaplan-Meier survival plots and mixed effects Cox proportional hazards modeling to account for relatedness among study participants. Incremental value of adhesion molecule biomarkers was studied using integrated discrimination improvement (IDI) and net reclassification improvement (NRI) indexes.

Results

Decreasing median values for serum concentrations of sICAM-1 and sVCAM-1 were observed in the following groups in this order: individuals with T2D at baseline, individuals who developed T2D during follow-up, individuals with prediabetes at baseline and normal glucose tolerant (NGT) individuals who remained T2D-free during follow-up. Top quartiles for sICAM-1 and sVCAM-1 were strongly and significantly associated with homeostatic model of assessment—insulin resistance (HOMA-IR). Mixed effects Cox proportional hazards modeling revealed that after correcting for important clinical confounders, high sICAM-1 and sVCAM-1 concentrations were associated with 2.52 and 1.99 times faster progression to T2D as compared to low concentrations, respectively. Individuals with high concentrations for both sICAM-1 and sVCAM-1 progressed to T2D 3.42 times faster than those with low values for both sICAM-1 and sVCAM-1. The results were similar in women in reproductive age group and the remainder of the cohort. Inclusion of sICAM-1 and sVCAM-1 in predictive models significantly improved reclassification and discrimination. The majority of these results were seen even when the analyses were restricted to NGT individuals.

Conclusion

Serum concentrations of sICAM-1 and sVCAM-1 independently and additively predict future T2D and represent important candidate biomarkers of T2D.     

An entropy-based statistic for genomewide association studies

Efficient genotyping methods and the availability of a large collection of single-nucleotide polymorphisms provide valuable tools for genetic studies of human disease. The standard chi2 statistic for case-control studies, which uses a linear function of allele frequencies, has limited power when the number of marker loci is large. We introduce a novel test statistic for genetic association studies that uses Shannon entropy and a nonlinear function of allele frequencies to amplify the differences in allele and haplotype frequencies to maintain statistical power with large numbers of marker loci. We investigate the relationship between the entropy-based test statistic and the standard chi2 statistic and show that, in most cases, the power of the entropy-based statistic is greater than that of the standard chi2 statistic. The distribution of the entropy-based statistic and the type I error rates are validated using simulation studies. Finally, we apply the new entropy-based test statistic to two real data sets, one for the COMT gene and schizophrenia and one for the MMP-2 gene and esophageal carcinoma, to evaluate the performance of the new method for genetic association studies. The results show that the entropy-based statistic obtained smaller P values than did the standard chi2 statistic.     

Hypothesis Testing Procedures for a Series of Independent Fourfold Tables under Inverse Sampling

This paper considers four summary test statistics, including the one recently proposed by Bennett (1986, Biometrical Journal 28, 859–862), for hypothesis testing of association in a series of independent fourfold tables under inverse sampling. This paper provides a systematic and quantitative evaluation of the small-sample performance for these summary test statistics on the basis of a Monte Carlo simulation. This paper notes that the test statistic developed by Bennett (1986) can be conservative and thereby possibly lose the power when the underlying disease is not rare. This paper also finds that for given a fixed total number of cases in each table, the conditional test statistic is the best in controlling type I error among all test statistics considered here.     

A Transmission/Disequilibrium Test That Allows for Genotyping Errors in the Analysis of Single-Nucleotide Polymorphism Data

The present study assesses the effects of genotyping errors on the type I error rate of a particular transmission/disequilibrium test (TDT(std)), which assumes that data are errorless, and introduces a new transmission/disequilibrium test (TDT(ae)) that allows for random genotyping errors. We evaluate the type I error rate and power of the TDT(ae) under a variety of simulations and perform a power comparison between the TDT(std) and the TDT(ae), for errorless data. Both the TDT(std) and the TDT(ae) statistics are computed as two times a log-likelihood difference, and both are asymptotically distributed as chi(2) with 1 df. Genotype data for trios are simulated under a null hypothesis and under an alternative (power) hypothesis. For each simulation, errors are introduced randomly via a computer algorithm with different probabilities (called "allelic error rates"). The TDT(std) statistic is computed on all trios that show Mendelian consistency, whereas the TDT(ae) statistic is computed on all trios. The results indicate that TDT(std) shows a significant increase in type I error when applied to data in which inconsistent trios are removed. This type I error increases both with an increase in sample size and with an increase in the allelic error rates. TDT(ae) always maintains correct type I error rates for the simulations considered. Factors affecting the power of the TDT(ae) are discussed. Finally, the power of TDT(std) is at least that of TDT(ae) for simulations with errorless data. Because data are rarely error free, we recommend that researchers use methods, such as the TDT(ae), that allow for errors in genotype data.     

Monte Carlo pedigree disequilibrium test for markers on the X chromosome

Because of the need for fine mapping of disease loci and the availability of dense single-nucleotide-polymorphism markers, many forms of association tests have been developed. Most of them are applicable only to triads, whereas some are amenable to nuclear families (sibships). Although there are a number of methods that can deal with extended families (e.g., the pedigree disequilibrium test [PDT]), most of them cannot accommodate incomplete data. Furthermore, despite a large body of literature on association mapping, only a very limited number of publications are applicable to X-chromosomal markers. In this report, we first extend the PDT to markers on the X chromosome for testing linkage disequilibrium in the presence of linkage. This method is applicable to any pedigree structure and is termed "X-chromosomal pedigree disequilibrium test" (XPDT). We then further extend the XPDT to accommodate pedigrees with missing genotypes in some of the individuals, especially founders. Monte Carlo (MC) samples of the missing genotypes are generated and used to calculate the XMCPDT (X-chromosomal MC PDT) statistic, which is defined as the conditional expectation of the XPDT statistic given the incomplete (observed) data. This MC version of the XPDT remains a valid test for association under linkage with the assumption that the pedigrees and their associated affection patterns are drawn randomly from a population of pedigrees with at least one affected offspring. This set of methods was compared with existing approaches through simulation, and substantial power gains were observed in all settings considered, with type I error rates closely tracking their nominal values.     

A Bayesian Chi‐Squared Goodness‐of‐Fit Test for Censored Data Models

Summary We propose a Bayesian chi‐squared model diagnostic for analysis of data subject to censoring. The test statistic has the form of Pearson's chi‐squared test statistic and is easy to calculate from standard output of Markov chain Monte Carlo algorithms. The key innovation of this diagnostic is that it is based only on observed failure times. Because it does not rely on the imputation of failure times for observations that have been censored, we show that under heavy censoring it can have higher power for detecting model departures than a comparable test based on the complete data. In a simulation study, we show that tests based on this diagnostic exhibit comparable power and better nominal Type I error rates than a commonly used alternative test proposed by Akritas (1988, Journal of the American Statistical Association 83, 222–230). An important advantage of the proposed diagnostic is that it can be applied to a broad class of censored data models, including generalized linear models and other models with nonidentically distributed and nonadditive error structures. We illustrate the proposed model diagnostic for testing the adequacy of two parametric survival models for Space Shuttle main engine failures.