Missing at random: a stochastic process perspective.

We offer a natural and extensible measure-theoretic treatment of missingness at random. Within the standard missing-data framework, we give a novel characterization of the observed data as a stopping-set sigma algebra. We demonstrate that the usual missingness-at-random conditions are equivalent to requiring particular stochastic processes to be adapted to a set-indexed filtration. These measurability conditions ensure the usual factorization of likelihood ratios. We illustrate how the theory can be extended easily to incorporate explanatory variables, to describe longitudinal data in continuous time, and to admit more general coarsening of observations.

Familial effects on the clinical course of multiple sclerosis.

BACKGROUND: Familial factors influence susceptibility to multiple sclerosis (MS) but it is unknown whether there are additional effects on the natural history of the disease. METHOD: We evaluated 1,083 families with > or =2 first-degree relatives with MS for concordance of age at onset, clinical course, and disease severity and investigated transmission patterns of these clinical features in affected parent-child pairs. RESULTS: There is concordance for age at onset for all families (correlation coefficient 0.14; p < 0.001), as well as for affected siblings (correlation coefficient 0.15; p < 0.001), and affected parent-child pairs (correlation coefficient 0.12; p = 0.03) when each is evaluated separately. Concordance for year of onset is present among affected siblings (correlation coefficient 0.18; p < 0.001) but not the parent-child group (correlation coefficient 0.08; p = 0.15). The clinical course is similar between siblings (kappa 0.12; p < 0.001) but not affected parents and their children (kappa -0.04; p = 0.09). This influence on the natural history is present in all clinical subgroups of relapsing-remitting, and primary and secondary progressive MS, reflecting a familial effect on episodic and progressive phases of the disease. There is no concordance for disease severity within any of the considered family groups (correlation coefficients: all families analyzed together, 0.02, p = 0.53; affected sibling group, 0.02, p = 0.61; affected parent-child group, 0.02, p = 0.69). Furthermore, there are no apparent transmission patterns of any of the investigated clinical features in affected parent-child pairs and no evidence for anticipation or effects of genetic loading. CONCLUSION: Familial factors do not significantly affect eventual disease severity. However, they increase the probability of a progressive clinical course, either from onset or after a phase of relapsing remitting disease. The familial effect is more likely to reflect genetic than environmental conditions. The results are relevant for counseling patients and have implications for the design of studies seeking to identify factors that influence the natural history of the disease.

Effect of genotyping error on type-I error rate of affected sib pair studies with genotyped parents.

OBJECTIVE: In affected sib pair studies without genotyped parents the effect of genotyping error is generally to reduce the type I error rate and power of tests for linkage. The effect of genotyping error when parents have been genotyped is unknown. We investigated the type I error rate of the single-point Mean test for studies in which genotypes of both parents are available. METHODS: Datasets were simulated assuming no linkage and one of five models for genotyping error. In each dataset, Mendelian-inconsistent families were either excluded or regenotyped, and then the Mean test applied. RESULTS: We found that genotyping errors lead to an inflated type I error rate when inconsistent families are excluded. Depending on the genotyping-error model assumed, regenotyping inconsistent families has one of several effects. It may produce the same type I error rate as if inconsistent families are excluded; it may reduce the type I error, but still leave an anti-conservative test; or it may give a conservative test. Departures of the type I error rate from its nominal level increase with both the genotyping error rate and sample size. CONCLUSION: We recommend that markers with high error rates either be excluded from the analysis or be regenotyped in all families.

Multiple Sclerosis Severity Score: using disability and disease duration to rate disease severity.

BACKGROUND: There is no consensus method for determining progression of disability in patients with multiple sclerosis (MS) when each patient has had only a single assessment in the course of the disease. METHODS: Using data from two large longitudinal databases, the authors tested whether cross-sectional disability assessments are representative of disease severity as a whole. An algorithm, the Multiple Sclerosis Severity Score (MSSS), which relates scores on the Expanded Disability Status Scale (EDSS) to the distribution of disability in patients with comparable disease durations, was devised and then applied to a collection of 9,892 patients from 11 countries to create the Global MSSS. In order to compare different methods of detecting such effects the authors simulated the effects of a genetic factor on disability. RESULTS: Cross-sectional EDSS measurements made after the first year were representative of overall disease severity. The MSSS was more powerful than the other methods the authors tested for detecting different rates of disease progression. CONCLUSION: The Multiple Sclerosis Severity Score (MSSS) is a powerful method for comparing disease progression using single assessment data. The Global MSSS can be used as a reference table for future disability comparisons. While useful for comparing groups of patients, disease fluctuation precludes its use as a predictor of future disability in an individual.

Rapid simulation of P values for product methods and multiple-testing adjustment in association studies.

A major aim of association studies is the identification of polymorphisms (usually SNPs) associated with a trait. Tests of association may be based on individual SNPs or on sets of neighboring SNPs, by use of (for example) a product P value method or Hotelling's T test. Linkage disequilibrium, the nonindependence of SNPs in physical proximity, causes problems for all these tests. First, multiple-testing correction for individual-SNP tests or for multilocus tests either leads to conservative P values (if Bonferroni correction is used) or is computationally expensive (if permutation is used). Second, calculation of product P values usually requires permutation. Here, we present the direct simulation approach (DSA), a method that accurately approximates P values obtained by permutation but is much faster. It may be used whenever tests are based on score statistics--for example, with Armitage's trend test or its multivariate analogue. The DSA can be used with binary, continuous, or count traits and allows adjustment for covariates. We demonstrate the accuracy of the DSA on real and simulated data and illustrate how it might be used in the analysis of a whole-genome association study.

A Bayesian partition model for case-control studies on highly polymorphic candidate genes.

We present a new statistical model for the analysis of case-control or cohort studies examining a highly polymorphic candidate disease susceptibility gene. Many genotypes are possible for such a gene. Consequently, the average number of subjects having each genotype will be modest. If analyzed separately, the risks associated with most genotypes will be estimated imprecisely. Our Bayesian partition model clusters genotypes according to risk, only allowing partitions that satisfy a particular assumption about the joint effect of the two alleles making up a genotype. This assumption is genetically plausible, imposes structure on the set of genotype risks, and still leaves a highly flexible model. By Bayesian model averaging over partitions, the model becomes, in effect, a semiparametric model for genotype risk. It allows borrowing of strength, i.e., estimates of risk for one genotype are informed by the risk estimates of all the genotypes. We present the results of fitting the model to two datasets, one simulated and one genuine case-control study of the NAT1 gene and lung cancer, and compare it in a simulation study with a haplotype relative risk model. The partition model enables genotype risks to be estimated more accurately and the alleles to be ranked according to risk.

Proportional hazards model for interval-censored failure times and time-dependent covariates: application to hazard of HIV infection of injecting drug users in prison.

Interval-censored survival data are data in which the failure times are not known precisely, but are known to lie within an interval. Such data can be analysed using a proportional hazards model with piecewise-exponential baseline hazard, a model which can be fitted by an EM algorithm easily programmed in standard statistical software. In this paper we extend the model to allow for time-dependent covariates and left-truncation, and demonstrate its use by assessing the effect of imprisonment on hazard of HIV infection in a cohort of injecting drug users from Edinburgh. No conclusive effect of incarceration on hazard of HIV infection was found, but there was a suggestion that imprisonment might have been a significant relative risk factor for infection in the later period, when risk behaviour among drug users in the community was reduced.

Historical HIV prevalence in Edinburgh Prison: a database-linkage study.

BACKGROUND: The prevalence of HIV in prisons is often higher than in the surrounding community, because prisons contain a high proportion of injecting drug users (IDUs). Reliable estimation of HIV prevalence in UK prisons only began in the 1990s. Edinburgh, Scotland, experienced a major IDU-related HIV epidemic which began in 1983. We sought retrospectively to estimate HIV prevalence in Edinburgh Prison over the period 1983-94. METHODS: Prison records of all 477 male HIV-positive patients (332 IDUs) in the Edinburgh City Hospital Cohort (believed to include three-quarters of HIV-positive Edinburgh IDUs) were abstracted from Edinburgh Prison. Using this information and the seroconversion intervals of the patients, the number of person-years spent inside the prison by these individuals while HIV-positive was estimated for each calendar month. From this, HIV prevalence was inferred. RESULTS: HIV prevalence in the prison rose from January 1983, as prevalence among Edinburgh IDUs increased, reaching a peak of 8% in December 1984. Prevalence during 1985-86 was 5-6% and then gradually declined, as the surviving HIV-infected IDUs spent less time in the prison. DISCUSSION: These figures are probably underestimates, as some HIV-positive prisoners are not in the cohort. However, the degree of underestimation should not be great and trends over time are reliable. Our estimate for August 1991, 4.1%, compares favourably with the estimate 4.5%, from an anonymous unlinked survey conducted in the prison that month. Prevalence estimates from other UK prisons are reviewed and suggestions made for other uses of database linkage in HIV and IDU epidemiology.

Mortality from overdose among injecting drug users recently released from prison: database linkage study.

OBJECTIVE: To assess whether injecting drug users have a higher than usual risk of death from overdose in the 2 weeks after release from prison. DESIGN: Soundex coding of surnames and information on date of birth were used to link entry and release dates from the local prison between 1983 and 1994 with clinical data from Edinburgh City Hospital's cohort of male injecting drug users who are infected with HIV. SETTING: Edinburgh City Hospital and Edinburgh Prison. SUBJECTS: 316/332 male injecting drug users infected with HIV in the City Hospital HIV cohort; 16 were excluded because they were enrolled after developing AIDS or because their precise date of death was not available. MAIN OUTCOME MEASURE: Relative risk of dying from overdose before developing AIDS and relative risk of dying of all causes before developing AIDS during the 2 weeks after release from prison; this was compared with relative risks of death during other time at liberty. RESULTS: 238/316 (75%) injecting drug users served time in the prison between 1983 and 1994. 33 out of 316 injecting drug users who were infected with HIV died before developing AIDS during 517,177 days at risk. 20 of these men died of an overdose; 6 of these deaths occurred within 2 weeks of release during 5903 days at risk. Death rates from overdose before the development of AIDS were 1.02/1000 days during the 2 weeks after release (recently released) and 0.029/1000 days during other times of liberty. The relative risk of death from overdose became 7.7 (1.5 to 39.1) after temporal matching (when the comparison was limited to the first 2 weeks after release v the next 10 weeks). The crude relative risk in an analysis combining stratified prison term and the 2 weeks after release was 4.5 (1.7 to 11.7) for death from overdose. After temporal matching these risks became 1.8 (0.4 to 9.2). CONCLUSION: Prisons should evaluate interventions to reduce the risk of death from overdose after release.

Pre-AIDS mortality in the Edinburgh City Hospital HIV cohort.

In this paper, we look at the incidence and predictive factors of pre-AIDS mortality among HIV-infected individuals, and injecting drug users (IDUs) in particular, and compare IDUs with non-IDUs. 627 patients (73 per cent IDUs) of the Edinburgh City Hospital HIV Cohort were enrolled pre-AIDS and followed up until September 1994. Analyses were performed using cumulative hazard and cumulative incidence estimators for a competing risks model, the Cox proportional hazards model and the non-parametric hazard estimator of Fusaro et al. (1993). The effects of age and CD4 T-lymphocyte cell count, progressively depleted during HIV progression, were investigated. 60 deaths occurred in AIDS-free patients during follow-up; 25 were drug-related deaths in IDUs. Pre-AIDS mortality was higher among IDUs than non-IDUs (p = 0.07). The cumulative incidences of pre-AIDS death after five years from enrollment were 11 per cent in IDUs and 6 per cent in non-IDUs; the cumulative AIDS incidences were, respectively, 19 per cent and 32 per cent. After eight years, cumulative pre-AIDS death incidence was 15 per cent among IDUs; cumulative AIDS incidence among IDUs was 35 per cent. Both groups had similar risks of medically-related (non-AIDS)-MRNA-death. Age and CD4 count were both individually predictive of MRNA death (relative risks (RRs); 2.1 per decade of life, p < 0.01; and 1.9 for each 100 cells per 100 microliters lost, p < 0.0001), although when used together age was less significant (RR 1.6, p = 0.07). Neither was statistically significant for drug-related mortality, although hazard may be lower in older individuals and may increase with falling CD4 count. The drug-related mortality was 1.1 per cent: 2.3 per cent in the first two years after enrollment, and 0.4 per cent thereafter. We conclude that older HIV-infected individuals are at greater risk of medically-related death before AIDS. This risk increases as CD4 count declines. Drug-related hazard may be greater in younger individuals and may increase as CD4 counts fall, but neither effect was formally significant.