Steven G. Self

Abstract Large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described. Specifically, the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived. These results generalize the work of Moran (1971), Chant (1974), and Chernoff (1954). Some of Chant's results are shown to be incorrect. The approach used in deriving these results follows from comments made by Moran and Chant. The problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate Gaussian distribution from a sample of size 1. In this representation the Gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator. Thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the Gaussian random variable onto the region of admissible values for the mean. A variety of examples is provided for which the limiting distributions of likelihood ratio statistics are mixtures of chi-squared distributions. One example is provided with a nuisance parameter on the boundary for which the asymptotic distribution is not a mixture of chi-squared distributions.

Characterization of the immune responses induced in the initial stages of human immunodeficiency virus type 1 (HIV-1) infection is of critical importance for an understanding of early viral pathogenesis and prophylactic vaccine design. Here, we used sequential plasma samples collected during the eclipse and exponential viral expansion phases from subjects acquiring HIV-1 (or, for comparison, hepatitis B virus [HBV]or hepatitis C virus [HCV]) to determine the nature and kinetics of the earliest systemic elevations in cytokine and chemokine levels in each infection. Plasma viremia was quantitated over time, and levels of 30 cytokines and chemokines were measured using Luminex-based multiplex assays and enzyme-linked immunosorbent assays. The increase in plasma viremia in acute HIV-1 infection was found to be associated with elevations in plasma levels of multiple cytokines and chemokines, including rapid and transient elevations in alpha interferon (IFN-alpha) and interleukin-15 (IL-15) levels; a large increase in inducible protein 10 (IP-10) levels; rapid and more-sustained increases in tumor necrosis factor alpha and monocyte chemotactic protein 1 levels; more slowly initiated elevations in levels of additional proinflammatory factors including IL-6, IL-8, IL-18, and IFN-gamma; and a late-peaking increase in levels of the immunoregulatory cytokine IL-10. Notably, there was comparatively little perturbation in plasma cytokine levels during the same phase of HBV infection and a delayed response of more intermediate magnitude in acute HCV infection, indicating that the rapid activation of a striking systemic cytokine cascade is not a prerequisite for viral clearance (which occurs in a majority of HBV-infected individuals). The intense early cytokine storm in acute HIV-1 infection may have immunopathological consequences, promoting immune activation, viral replication, and CD4(+) T-cell loss.

BACKGROUND: Case reports and the results of a recent case-control study have raised questions about the potential neoplastic effects of medications used as treatment for infertility. METHODS: We examined the risk of ovarian tumors in a cohort of 3837 women evaluated for infertility between 1974 and 1985 in Seattle. Computer linkage with a population-based tumor registry was used to identify women in whom tumors were diagnosed before January 1, 1992. Data on infertility testing and treatment were abstracted from the medical records of women who had ovarian cancer and those of a randomly selected comparison group. The risk of ovarian tumors associated with exposure to ovulation-inducing medications was assessed through an age-standardized comparison with the rate of ovarian tumors in the general population, and Cox regression analysis was used to compare the risk of cancer among women who received these medications with the risk among infertile women who did not receive them. RESULTS: There were 11 invasive or borderline malignant ovarian tumors, as compared with an expected number of 4.4 (standardized incidence ratio, 2.5; 95 percent confidence interval, 1.3 to 4.5). Nine of the women in whom ovarian tumors developed had taken clomiphene; the adjusted relative risk among these women, as compared with that among infertile women who had not taken this drug, was 2.3 (95 percent confidence interval, 0.5 to 11.4). Five of the nine women had taken the drug during 12 or more monthly cycles. This period of treatment was associated with an increased risk of ovarian tumors among both women with ovarian abnormalities and those without apparent abnormalities (relative risk, 11.1; 95 percent confidence interval, 1.5 to 82.3), whereas treatment with the drug for less than one year was not associated with an increased risk. CONCLUSIONS: Prolonged use of clomiphene may increase the risk of a borderline or invasive ovarian tumor.

BACKGROUND: Among women who practice breast self-examination (BSE), breast cancers may be detected when they are at an earlier stage and are smaller than in women who do not practice BSE. However, the efficacy of breast self-examination for decreasing breast cancer mortality is unproven. This study was conducted to determine whether an intensive program of BSE instruction will reduce the number of women dying of breast cancer. METHODS: From October 1989 through October 1991, 266,064 women associated with 519 factories in Shanghai were randomly assigned to a BSE instruction group (132,979 women) or a control group (133,085 women). Initial instruction in BSE was followed by reinforcement sessions 1 and 3 years later, by BSE practice under medical supervision at least every 6 months for 5 years, and by ongoing reminders to practice BSE monthly. The women were followed through December 2000 for mortality from breast cancer. Cumulative risk ratios of dying from breast cancer were estimated using Cox proportional hazards models. All statistical tests were two-sided. RESULTS: There were 135 (0.10%) breast cancer deaths in the instruction group and 131 (0.10%) in the control group. The cumulative breast cancer mortality rates through 10 to 11 years of follow-up were similar (cumulative risk ratio for women in the instruction group relative to that in the control group = 1.04, 95% confidence interval = 0.82 to 1.33; P =.72). However, more benign breast lesions were diagnosed in the instruction group than in the control group. CONCLUSIONS: Intensive instruction in BSE did not reduce mortality from breast cancer. Programs to encourage BSE in the absence of mammography would be unlikely to reduce mortality from breast cancer. Women who choose to practice BSE should be informed that its efficacy is unproven and that it may increase their chances of having a benign breast biopsy.