Augmented cross-sectional studies with abbreviated follow-up for estimating HIV incidence

Cross-sectional HIV incidence estimation based on a sensitive and less-sensitive test offers great advantages over the traditional cohort study. However, its use has been limited due to concerns about the false negative rate of the less-sensitive test, reflecting the phenomenon that some subjects may remain negative permanently on the less-sensitive test. Wang and Lagakos (2010, Biometrics 66, 864-874) propose an augmented cross-sectional design that provides one way to estimate the size of the infected population who remain negative permanently and subsequently incorporate this information in the cross-sectional incidence estimator. In an augmented cross-sectional study, subjects who test negative on the less-sensitive test in the cross-sectional survey are followed forward for transition into the nonrecent state, at which time they would test positive on the less-sensitive test. However, considerable uncertainty exists regarding the appropriate length of follow-up and the size of the infected population who remain nonreactive permanently to the less-sensitive test. In this article, we assess the impact of varying follow-up time on the resulting incidence estimators from an augmented cross-sectional study, evaluate the robustness of cross-sectional estimators to assumptions about the existence and the size of the subpopulation who will remain negative permanently, and propose a new estimator based on abbreviated follow-up time (AF). Compared to the original estimator from an augmented cross-sectional study, the AF estimator allows shorter follow-up time and does not require estimation of the mean window period, defined as the average time between detectability of HIV infection with the sensitive and less-sensitive tests. It is shown to perform well in a wide range of settings. We discuss when the AF estimator would be expected to perform well and offer design considerations for an augmented cross-sectional study with abbreviated follow-up.

Estimation of the timing of perinatal transmission of HIV

Knowledge of the timing of perinatal transmission of HIV would be valuable for the determination and evaluation of preventive treatments and would shed light on the mechanism of transmission. Estimation of the distribution of the time of perinatal transmission is difficult, however, because tests of infection status can only be undertaken after birth. DNA and RNA polymerase chain reaction (PCR) assays and HIV culture have been the most commonly used diagnostic tests for perinatal HIV infection. Such tests have high sensitivity and specificity, except when they are given shortly after infection. In this paper we use the time-dependent sensitivity of these diagnostic tests to make nonparametric and semiparametric inferences about the distribution of the time of perinatal HIV transmission as well as the cumulative probability of perinatal transmission. The methods are illustrated with data from a clinical trial conducted by the AIDS Clinical Trials group.

Efficiency comparisons of rank and permutation tests based on summary statistics computed from repeated measures data

A popular method of using repeated measures data to compare treatment groups in a clinical trial is to summarize each individual's outcomes with a scalar summary statistic, and then to perform a two-group comparison of the resulting statistics using a rank or permutation test. Many different types of summary statistics are used in practice, including discrete and continuous functions of the underlying repeated measures data. When the repeated measures processes of the comparison groups differ by a location shift at each time point, the asymptotic relative efficiency of (continuous) summary statistics that are linear functions of the repeated measures has been determined and used to compare tests in this class. However, little is known about the non-null behaviour of discrete summary statistics, about continuous summary statistics when the groups differ in more complex ways than location shifts or where the summary statistics are not linear functions of the repeated measures. Indeed, even simple distributional structures on the repeated measures variables can lead to complex differences between the distribution of common summary statistics of the comparison groups. The presence of left censoring of the repeated measures, which can arise when these are laboratory markers with lower limits of detection, further complicates the distribution of, and hence the ability to compare, summary statistics. This paper uses recent theoretical results for the non-null behaviour of rank and permutation tests to examine the asymptotic relative efficiencies of several popular summary statistics, both discrete and continuous, under a variety of common settings. We assume a flexible linear growth curve model to describe the repeated measures responses and focus on the types of settings that commonly arise in HIV/AIDS and other diseases.

Randomized study of saquinavir with ritonavir or nelfinavir together with delavirdine, adefovir, or both in human immunodeficiency virus-infected adults with virologic failure on indinavir: AIDS Clinical Trials Group Study 359

This study compared antiretroviral activity among 6 "salvage" therapy regimens. The study was a prospective, randomized, 2x3 factorial, multicenter study of the AIDS Clinical Trials Group. The study enrolled 277 human immunodeficiency virus (HIV)-infected patients naive to nonnucleoside analogues who had taken indinavir >6 months. The patients had 2000-200,000 HIV RNA copies/mL. Patients received saquinavir with ritonavir or nelfinavir together with delavirdine and/or adefovir and were followed for safety and antiretroviral response between baseline and week 16. At week 16, 30% (77/254) of patients had </=500 HIV RNA copies/mL. Virologic response did not differ significantly between pooled ritonavir and nelfinavir groups (28% vs. 33%; P=.50) or between pooled delavirdine and delavirdine/adefovir groups (40% vs. 33%; P=.42). Pooled delavirdine groups had a greater virologic response rate than did adefovir groups (40% vs. 18%; P=.002). Overall, one-third of patients who experienced virologic failure on an indinavir-containing regimen suppressed virus load levels while they were taking a new salvage regimen.

Covariate analysis of viral eradication studies

An important issue arising in therapeutic studies of hepatitis C and HIV is the identification of and adjustment for covariates associated with viral eradication and resistance. Analyses of such data are complicated by the fact that eradication is an occult event that is not directly observable, resulting in unique types of censored observations that do not arise in other competing risks settings. This paper proposes a semiparametric regression model to assess the association between multiple covariates and the eradication/resistance processes. The proposed methods are based on a piecewise proportional hazards model that allows parameters to vary between observation times. We illustrate the methods with data from recent hepatitis C clinical trials.

The one-sample problem from eradication studies of chronic viral infections

In studies of chronic viral infections, the objective is to estimate probabilities of developing viral eradication and resistance. Complications arise as the laboratory methods used to assess eradication status result in unusual types of censored observations. This paper proposes nonparametric methods for the one-sample analysis of viral eradication/resistance data. We show that the unconstrained nonparametric maximum likelihood estimator of the subdistributions of eradication and resistance are obtainable in closed form. In small samples, these estimators may be inadmissible; thus, we also present an algorithm for obtaining the constrained MLEs based on an isotonic regression of the unconstrained MLEs. Estimators of several functionals of the eradication and resistance subdistributions are also developed and discussed. The methods are illustrated with results from recent hepatitis C clinical trials.

Should data and safety monitoring boards share confidential interim data?

Consider the following situation: Two clinical trials are underway, closely related in terms of the interventions being compared and the target populations. In preparing for a planned interim analysis, the statistician for trial 1 finds that the results support a recommendation to stop the trial early. Should the statistician ask the investigators for trial 2 to make interim results of their trial available to the data and safety monitoring board (DSMB) for trial 1? If so, in what form? Would the answers change if the trial 1 results showed a strong but not convincing trend? What is the obligation of the trial 2 investigators to respond to such a request? What role do the two DSMBs have, either in initiating a request or in agreeing to respond to it? In this article, we examine this situation in some detail, having faced it occasionally in our own experience with clinical trials and DSMBs. The chief argument in favor of sharing data is that data from trial 2 are obviously relevant to the question being addressed by trial 1 and therefore ought to be available to those who must interpret the results from that trial. On the other hand, there are several reasons for not sharing interim data. For example, sharing is incompatible with the independence of the trials; the time for synthesizing evidence from both trials is after the two teams of investigators have presented the full analysis and interpretation of their separate trials. For this and other conceptual and practical reasons we conclude that it is better, in most cases, for DSMBs to consider only information that has already been made public in some form.

Group sequential analyses for the mean function of a repeated measure process

In many clinical trials, treatment efficacy is based upon response to a biological marker that is measured repeatedly during the course of follow-up. However, in some of these trials it is not clear, a priori, how treatment effects on the marker may manifest themselves or what kinds of effects are clinically meaningful and/or acceptable. It is, therefore, desirable to allow flexibility in design and monitoring process by not prespecifying a stopping rule or even the parameter on which inferences will be based. Using the more general results in Hu and Lagakos, this paper extends the idea of the repeated confidence intervals for a parameter (Jennison and Turnbull) to repeated confidence bands for the mean function of a repeated measure process. We illustrate the approach and some considerations in its application with the results of a recent AIDS clinical trial.

Use of the Wei-Lin-Weissfeld method for the analysis of a recurring and a terminating event

We consider application of the Wei-Lin-Weissfeld (WLW) method for multiple failure time data when analysing a disease process consisting of a recurring outcome, such as clinical progression, and a terminating outcome, such as death. In order to adapt WLW for this situation, 'events' must be specified that define multiple failure times and whether these are censored. Various choices of events are possible, and each corresponds to inferences about a different aspect of the underlying disease process. Definitions which regard the terminating outcome as a censor of the recurring outcome focus on specific cause-specific hazard functions, while event definitions which make no distinction between a recurring and terminating outcome focus on hazard functions of the induced failure times. Some event definitions require strong statistical assumptions to yield valid inferences and are not recommended. The application of WLW for recurring/terminating processes is illustrated with the results of two recently conducted clinical trials in persons with HIV.

Non-parametric inference of a failure time distribution when the failure times are estimated

We consider the estimation of a failure time distribution F when, instead of N i.i.d. realizations T1, T2, ..., TN from F, the observations consist of estimates of the Ti. If the Ti could be observed, a natural non-parametric estimator of F would be the Kaplan-Meier estimator. Thus, we examine the properties of the Kaplan-Meier estimator based on the estimates of the Ti. We also consider a weighted Kaplan-Meier estimator which gives more emphasis to those estimated times based on more information. We evaluate the small sample bias and precision of F when the estimated failure times arise from additive or multiplicative error structures. Because this problem has particular application in the study of non-compliance of subjects in clinical trials, we also investigate the bias and precision of the estimators of the distribution function based on a complex error structure that would arise in the non-compliance setting. Here Ti denotes the unobserved time to non-compliance for the ith subject, and is estimated using repeated observations from a laboratory marker whose behaviour is affected by non-compliance. The techniques are illustrated with the results of a recent AIDS clinical trial.

A comparison of immediate with deferred zidovudine therapy for asymptomatic HIV-infected adults with CD4 cell counts of 500 or more per cubic millimeter. AIDS Clinical Trials Group

BACKGROUND

The clinical benefits of zidovudine remain unproved in patients with asymptomatic human immunodeficiency virus (HIV) infection when CD4 cell counts exceed 500 per cubic millimeter. We compared zidovudine therapy given immediately with deferred therapy in such subjects.

METHODS

Beginning in 1987, subjects with asymptomatic HIV infection and 500 or more CD4 cells per cubic millimeter were randomly assigned to receive placebo or zidovudine (either 500 or 1500 mg per day, starting immediately). In 1989, the study was modified so that open-label treatment with 500 mg of zidovudine per day (deferred therapy) was offered when CD4 cell counts fell below 500 per cubic millimeter. The study end points included overall survival, survival free of the acquired immunodeficiency syndrome (AIDS), toxic effects, and changes in CD4 cell counts.

RESULTS

There were 1637 subjects who could be evaluated: 547 in the deferred-therapy group, 549 in the group receiving 500 mg of zidovudine immediately, and 541 in the 1500-mg group. The subjects were followed for up to 6.5 years (group medians, 4.8, 4.8, and 4.9, respectively). There was no significant difference in AIDS-free survival in the deferred-therapy group as compared with the low-dose or high-dose groups (81 cases of progression to AIDS or death vs. 81 and 74, respectively; P = 0.95 and P = 0.13) or in overall survival (51 deaths vs. 47 and 46; P = 0.25 and P = 0.16). The decline in CD4 cells was slower in both immediate-therapy groups than in the deferred-therapy group (P < 0.001 for both). Adverse effects were uncommon, and before the study modification their incidence was similar among the treatment groups, but severe anemia and granulocytopenia were more frequent in the 1500-mg group than in the deferred-therapy group (P < 0.001).

CONCLUSIONS

In asymptomatic, HIV-infected adults with 500 or more CD4 cells per cubic millimeter, treatment with zidovudine slows the decline in the CD4 cell count but does not significantly prolong either AIDS-free or overall survival. These results do not encourage the routine use of zidovudine monotherapy in this population.

Assessing drug compliance using longitudinal marker data, with application to AIDS

The assessment of non-compliance to a study medication is an important issue in the evaluation of clinical trials of self-administered drugs. Traditional methods for evaluating the compliance of subjects include self-reported questionnaires and pharmacologic assays of drug levels in randomly-drawn blood samples, but each of these has important limitations. This paper adapts and extends changepoint methods to assess compliance from longitudinal data on laboratory markers that are affected by the drug. The maximum likelihood estimators for two models are developed and examined. The effect of the drug on the marker process, as well as the spacing of the observations of the marker process relative to the time of non-compliance determine which model parameters are estimable. For the situations examined, the method of maximum likelihood is found to perform well in most cases. However, when non-compliance begins shortly before the last observation of the marker process, these (as well as any other) estimators cannot reliably distinguish non-compliance from compliance. The methods are illustrated with an example from a recent clinical trial of persons infected with HIV.

The duration of zidovudine benefit in persons with asymptomatic HIV infection. Prolonged evaluation of protocol 019 of the AIDS Clinical Trials Group

OBJECTIVE

To determine the durability of zidovudine-induced delay in clinical progression of asymptomatic human immunodeficiency virus (HIV) disease and to assess the relationship between this effect and the entry CD4+ cell count.

DESIGN AND INTERVENTIONS

Extended follow-up data from subjects participating in protocol 019 of the AIDS [acquired immunodeficiency syndrome] Clinical Trials Group were examined. Subjects were offered a total daily dose of 500 mg of open-label zidovudine after the unblinding of the original randomized trial in 1989. Original treatment groups included placebo, 500 mg of zidovudine, or 1500 mg of zidovudine daily in divided doses. Three distinct analyses were conducted to assess the duration of zidovudine's effect on progression to AIDS or death: (1) analysis of all follow-up information from all subjects, (2) analysis of all subjects but with follow-up of original placebo-assigned subjects censored at the time open-label zidovudine was initiated, and (3) analysis of the effect of initiating zidovudine in subjects initially assigned to receive placebo.

SETTING

University-based and university-affiliated AIDS research clinics participating in AIDS Clinical Trials Group protocol 019.

PATIENTS

A total of 1565 asymptomatic HIV-infected subjects with entry CD4+ cell counts less than 0.50 x 10(9)/L (500/microL).

MAIN OUTCOME MEASURE

Time to progression to AIDS or death.

RESULTS

During follow-up of up to 4.5 years (mean, 2.6 years), 232 subjects progressed to AIDS or died. In each of the three analyses described herein, zidovudine was associated with a significant (P = .008, .004, .007) decrease in the risk of such progression. However, each of these analyses also indicated a decreasing placebo:zidovudine relative risk with duration of use (P = .002, .08, .04), suggesting a nonpermanent effect. The duration of benefit appeared to be related to entry CD4+ cell count, with greater benefit in those with higher counts at entry. No significant differences in survival were found between those originally randomized to zidovudine or placebo.

CONCLUSIONS

Zidovudine at 500 mg/d caused a significant delay in progression to AIDS or death, but its earlier use in asymptomatic disease was not associated with an additional prolongation of survival compared with delayed initiation. The delay in progression diminished over time especially in subjects with entry CD4+ cell counts less than 0.30 x 10(9)/L (300/microL). Treatment strategies that alter drug regimens before the loss of zidovudine benefit should be explored.

Estimation of the infection time and latency distribution of AIDS with doubly censored data

We consider the nonparametric estimation of the time to infection with HIV and the latency period between infection and the onset of AIDS in data where both the events of infection and AIDS are not directly observed. The methods use self-consistency equations that are more easily and quickly solvable than the nonparametric estimators proposed by De Gruttola and Lagakos (1989, Biometrics 45, 1-11). The techniques are illustrated with data on hemophiliacs who became infected through contamination of the blood factor they were given to control their hemophilia.

Size and power of two-sample tests of repeated measures data

One method of using repeated measures data to compare treatment groups in a clinical trial is to summarize each subject's outcomes with a single summary statistic, and then perform a distribution-free comparison based on the resulting statistics. We examine extensions of this approach and conditions under which they retain proper size in the presence of missing data. The asymptotic relative efficiencies of several summary statistic tests are calculated to show which perform best in a variety of situations. The techniques are illustrated using data from an AIDS clinical trial.

CD4+ lymphocytes are an incomplete surrogate marker for clinical progression in persons with asymptomatic HIV infection taking zidovudine

OBJECTIVE

To determine the extent to which lymphocytes, particularly those with the CD4 surface antigen, are a surrogate marker for the development of the acquired immunodeficiency syndrome (AIDS) in persons with asymptomatic human immunodeficiency virus (HIV) infection.

DESIGN

Analysis of data from the AIDS Clinical Trials Group Protocol 019, a placebo-controlled, double-blind, randomized trial.

SETTING

University-based referral centers.

PATIENTS

Asymptomatic HIV-infected patients with 500 or fewer CD4+ cells/mm3 at baseline who were given placebo (350 patients) or one of two daily doses of zidovudine (725 patients).

MEASUREMENTS

Baseline and interim measurements of CD4+ and other leukocytes were assessed. Patients were followed for progression to AIDS.

RESULTS

Patients' lymphocyte levels were correlated with progression to AIDS (P < 0.001; relative risk for each depletion of 50 CD4+ cells/mm3, 1.75; 95% CI, 1.53 to 2.01); however, only a small portion (0% to 37%) of the effect of zidovudine on this progression was statistically explained by its effect on CD4+ lymphocyte levels. A substantial portion of zidovudine's effect on delaying progression to AIDS that was independent of the levels of these markers occurred within the first 16 weeks of therapy. In patients who had not progressed to AIDS by week 16, most of the subsequent zidovudine effect in reducing the risk for progression could be explained by its effect on net CD4+ percent (percentage of CD4+ lymphocytes among all leukocytes) for the first 16 weeks of therapy.

CONCLUSION

Levels of CD4+ lymphocytes are an incomplete surrogate marker for progression to AIDS, and the association is especially weak during the first 16 weeks of zidovudine therapy.

Analyzing doubly censored data with covariates, with application to AIDS

This paper proposes a method for incorporating covariate information in the analysis of survival data when both the time of the originating event and the failure event can be right- or interval-censored. This method generalizes the one-sample estimation results of De Gruttola and Lagakos (1989, Biometrics 45, 1-11) by allowing the distribution of time between the two events to be a function of covariates under a proportional hazards model. Estimates for the model coefficients, as well as the underlying distributions, are obtained by an iterative fitting procedure based on Turnbull's (1976, Journal of the Royal Statistical Society, Series B 38, 290-295) self-consistency algorithm in combination with the Newton-Raphson algorithm. The method is illustrated with data from a study of hemophiliacs infected with the human immunodeficiency virus.

Surrogate markers in AIDS clinical trials: conceptual basis, validation, and uncertainties

The identification of surrogate markers for clinical end points has important implications for the conduct of AIDS clinical trials, the approval of drugs for the treatment of infection with human immunodeficiency virus (HIV), and the management of HIV infection. In this paper, the concept of a surrogate marker and the properties of an ideal marker are discussed. The steps required for the empirical verification of a potential marker are then addressed, and current information on surrogate markers for AIDS is reviewed. Studies conducted to date indicate that the effects of a new drug on numbers of CD4 lymphocytes account only partly for its ultimate impact on the clinical progression of HIV infection. Consequently, the potential benefits of early approval of a drug based on its effect on CD4 lymphocytes must be weighed against the uncertainty about its ability to actually delay clinical progression.

Toxic effects of zidovudine in asymptomatic human immunodeficiency virus-infected individuals with CD4+ cell counts of 0.50 x 10(9)/L or less. Detailed and updated results from protocol 019 of the AIDS Clinical Trials Group

BACKGROUND

Protocol 019 of the AIDS Clinical Trials Group is a multicenter, double-blind, placebo-controlled trial of zidovudine (3'-azido-3'-deoxythymidine; formerly AZT) in human immunodeficiency virus-infected asymptomatic individuals. The initial results in the stratum of subjects entering with CD4+ cell counts of 0.50 x 10(9)/L or less have been reported, but without a detailed analysis of toxic effects.

METHODS

This detailed and updated report analyzes the toxic effects that occurred in 1567 subjects (91% men; 89% white) in this stratum of protocol 019 who received placebo (494 subjects), a 500-mg daily dose of zidovudine (544 subjects), or a 1500-mg daily dose of zidovudine (529 subjects). Hematologic, hepatic, and renal effects and patient-reported symptoms and clinical signs were monitored.

RESULTS

Severe anemia (hemoglobin level, < 80 g/L) was associated with both the 500-mg zidovudine group and the 1500-mg group compared with placebo. The estimated 18-month risks of severe anemia were 0.4%, 2.0%, and 9.7% for the placebo, 500-mg zidovudine, and 1500-mg zidovudine groups, respectively. Predictive baseline measures were lower hemoglobin level in the 1500-mg group and the two zidovudine groups combined and lower platelet count in the 500-mg zidovudine group. The risk of a first severe anemia developing was greatest in months 3 through 8 of treatment. Of the 44 subjects with severe anemia in the zidovudine groups, 18 (41%) progressed from mild anemia (hemoglobin level, 95 to 109 g/L) to severe anemia on consecutive visits (usually 2 to 4 weeks apart). Severe neutropenia (absolute neutrophil count, < 750 x 10(6)/L) did not occur significantly more often in the 500-mg zidovudine group but did in the 1500-mg zidovudine group. Moderate neutropenia (absolute neutrophil count, < 1300 x 10(6)/L) did develop significantly more often in the 500-mg zidovudine group (165 subjects) than in the placebo group (71 subjects). Mild (or worse) elevations of bilirubin levels were uncommon but occurred more often with zidovudine. Severe nausea (and/or vomiting) was rare (2.8% of subjects) but was associated with zidovudine. Milder patient-reported events were common, and a number were associated with zidovudine.

CONCLUSION

Zidovudine at the 500-mg/d dosage appears to be tolerable in many patients with asymptomatic human immunodeficiency virus infection and CD4+ cell counts of 0.50 x 10(9)/L or less. Increased clinical surveillance for anemia may be warranted in certain patients.

A controlled trial comparing continued zidovudine with didanosine in human immunodeficiency virus infection. The NIAID AIDS Clinical Trials Group

BACKGROUND

Although zidovudine is effective in patients with human immunodeficiency virus (HIV) infection, its efficacy may decline with prolonged use. Didanosine is another inhibitor of HIV reverse transcriptase. We evaluated the effectiveness of changing anti-HIV treatment from zidovudine to didanosine.

METHODS

This multicenter, double-blind study involved 913 patients who had tolerated zidovudine for at least 16 weeks. The patients had the acquired immunodeficiency syndrome (AIDS), AIDS-related complex with less than or equal to 300 CD4 cells per cubic milliliter, or asymptomatic HIV infection with less than or equal to 200 CD4 cells per cubic milliliter. They were randomly assigned to receive 600 mg per day of zidovudine, 750 mg per day of didanosine, or 500 mg per day of didanosine.

RESULTS

There were significantly fewer new AIDS-defining events and deaths among the 298 subjects assigned to 500 mg per day of didanosine than among the subjects who continued to receive zidovudine (relative risk, 1.39; 95 percent confidence interval, 1.06 to 1.82; P = 0.015). With 750 mg of didanosine, there was no clear benefit over zidovudine (relative risk, 1.10; 95 percent confidence interval, 0.86 to 1.42). The efficacy of didanosine was unrelated to the duration of previous zidovudine treatment. In the two didanosine groups, there were improvements in the number of CD4 cells (P less than 0.001 for both groups) and in p24 antigen levels (P = 0.03 in the 500-mg group; P = 0.005 in the 750-mg group), as compared with the zidovudine group.

CONCLUSIONS

Changing treatment from zidovudine to 500 mg per day of didanosine appears to slow the progression of HIV disease.

Cage allocation designs for rodent carcinogenicity experiments

Cage allocation designs for rodent carcinogenicity experiments are discussed and presented with the goal of avoiding dosage group biases related to cage location. Considerations in selecting a cage design are first discussed in general terms. Specific designs are presented for use in experiments involving three, four, and five dose groups and with one, four, and five rodents per cage. Priorities for balancing treatment groups include horizontal position on shelf and shelf of rack, nearest neighbor balance, and male-female balance. It is proposed that these balance criteria be considered together with practical issues, such as the ability to accurately conform to a design and to determine a sensible and efficient design for each experiment.

Estimating the infectivity of HIV from partner studies

The main sources of data for estimating infectivity of the human immunodeficiency virus (HIV) come from studies of the susceptible partners of individuals known to be infected with HIV. Estimation of infectivity from these studies is complicated by limitations imposed by the sampling designs. This paper reviews three sampling schemes that have been used in partner studies and, for each, proposes methods for estimating various theoretic models for infectivity. The methods, as well as procedures for checking goodness-of-fit, are illustrated using data from a study by Peterman and colleagues.

Adjusting for early treatment termination in comparative clinical trials

In clinical trials of long-term therapies, patients often terminate their treatments earlier than planned. When analysing time-to-failure data, one approach to account for early treatment termination censors failure at the time of termination of therapy. In general, however, this does not produce valid inferences about the distribution of time to failure that would have occurred had treatment not been terminated. In contrast, intent-to-treat analyses, which are based on time to failure regardless of whether and when treatment is terminated, always produce valid inferences about the unconditional distribution of time to failure. Early treatment termination does not distort the size (type I error rate) of intent-to-treat tests but can cause a loss in power. Modifications to ordinary logrank tests can be used to recover some of the lost power without affecting test size, and can be most useful when the proportion of at-risk patients still taking their treatment changes substantially during periods when failures are observed. Extensions of the modified test to include strata are straightforward, although important design questions require further research.

Zidovudine in asymptomatic human immunodeficiency virus infection. A controlled trial in persons with fewer than 500 CD4-positive cells per cubic millimeter. The AIDS Clinical Trials Group of the National Institute of Allergy and Infectious Diseases

Zidovudine (AZT) is a potent inhibitor of the replication of the human immunodeficiency virus (HIV), and it has been shown to improve survival in advanced HIV disease. We conducted a randomized, double-blind trial in adults with asymptomatic HIV infection who had CD4+ cell counts of fewer than 500 per cubic millimeter on entry into the study. The subjects (92 percent male) were randomly assigned to one of three treatment groups: placebo (428 subjects); zidovudine, 500 mg per day (453); or zidovudine, 1500 mg per day (457). After a mean follow-up of 55 weeks (range, 19 to 107), 33 of the subjects assigned to placebo had the acquired immunodeficiency syndrome (AIDS), as compared with 11 of those assigned to receive 500 mg of zidovudine (P = 0.002; relative risk, 2.8; 95 percent confidence interval, 1.4 to 5.6) and 14 of those assigned to receive 1500 mg of zidovudine (P = 0.05; relative risk, 1.9; 95 percent confidence interval, 1.0 to 3.5). In the three treatment groups, the rates of progression (per 100 person-years) to either AIDS or advanced AIDS-related complex were 7.6, 3.6, and 4.3, respectively. As compared with those assigned to placebo, the subjects in the zidovudine groups had significant increases in the number of CD4+ cells and significant declines in p24 antigen levels. In the 1500-mg zidovudine group, severe hematologic toxicity (anemia or neutropenia) was more frequent than in the other groups (P less than 0.0001). In the 500-mg zidovudine group, nausea was the only toxicity that was significantly more frequent (in 3.3 percent) than in the placebo group (P = 0.001). We conclude that zidovudine is safe and effective in persons with asymptomatic HIV infection and fewer than 500 CD4+ cells per cubic millimeter. Additional study will be required to determine whether such treatment will ultimately improve survival for persons infected with HIV.

Analysis of doubly-censored survival data, with application to AIDS

This paper proposes nonparametric and weakly structured parametric methods for analyzing survival data in which both the time origin and the failure event can be right- or interval-censored. Such data arise in clinical investigations of the human immunodeficiency virus (HIV) when the infection and clinical status of patients are observed only at several time points. The proposed methods generalize the self-consistency algorithm proposed by Turnbull (1976, Journal of the Royal Statistical Society, Series B 38, 290-295) for singly-censored univariate data, and are illustrated with the results from a study of hemophiliacs who were infected with HIV by contaminated blood factor.

The value of AIDS incidence data in assessing the spread of HIV infection

Changes over time in the cumulative number of cases of AIDS depend in a complex way on several features of the epidemic, including the distribution of the induction time between infection with the human immunodeficiency virus (HIV) and onset of symptoms of AIDS, heterogeneity in such behaviours as sexual practices, selection of partners, and IV drug use, and changes over time in these behaviours. Consequently, the observed increase in the doubling time in cumulative AIDS incidence from 5 to 13 months, since AIDS was first recognized in 1982, demonstrates neither that the epidemic has begun to 'run its course' nor that behavioural changes have had a major impact in reducing incidence. Even in a homogeneous population with known induction distribution, AIDS incidence data currently are of little value in determining the number of persons infected with HIV unless additional information is available about the shape of cumulative incidence curve of HIV infection.

Analysis of agreement among findings of pathologists in ED01 experiment

The ED01 experiment, which involved determination of the effective dose to produce a 1% tumor rate, was carried out by the National Center for Toxicological Research (NCTR). The study involved greater than 20,000 BALB/c female mice exposed to various doses of 2-acetylaminofluorene, a chemical known to produce bladder and liver carcinomas. After death, tissues from each animal were evaluated for the presence of several types of tumors by one of a team of NCTR pathologists. After the ED01 experiment was completed, the Society of Toxicology commissioned another pathologist to carry out an independent review of the bladder and liver specimens from a stratified sample of the mice. There were substantial differences in the diagnoses of both tumor types by the pathologists, but the implications for detection of a dose-response relationship are important only for liver carcinomas.

Effects of mismodelling and mismeasuring explanatory variables on tests of their association with a response variable

We consider three commonly-used statistical tests for assessing the association between an explanatory variable and a measured, binary, or survival-time, response variable, and investigate the loss in efficiency from mismodelling or mismeasuring the explanatory variable. With respect to mismodelling, we examine the consequences of using an incorrect dose metameter in a test for trend, of mismodelling a continuous explanatory variable, and of discretizing a continuous explanatory variable. We also examine the consequences of classification errors for a discrete explanatory variable and of measurement errors for a continuous explanatory variable. For all three statistical tests, the asymptotic relative efficiency (ARE) corresponding to each type of mis-specification equals the square of the correlation between the correct and fitted form of the explanatory variable. This result is evaluated numerically for the different types of mis-specification to provide insight into the selection of tests, the interpretation of results, and the design of studies where the 'correct' explanatory variable cannot be measured exactly.

Statistical analysis of disease onset and lifetime data from tumorigenicity experiments

We present and discuss several methods for analyzing rodent tumorigenicity experiments. Two approaches are based on the age and tumor status (present/absent) of each animal at the time of death, and assume either that the tumor type is nonlethal or instantly lethal. Two other approaches avoid such restrictive assumptions about tumor lethality by requiring additional types of data. One method assumes that animals are randomly sacrificed at various ages throughout the study. The second approach requires that each animal which develops the tumor be classified as dying either from the tumor or from other causes.

Properties of proportional-hazards score tests under misspecified regression models

The effects are investigated of misspecifying a proportional-hazards regression model on the associated partial-likelihood score test for comparing two randomized treatments in the presence of covariates. The asymptotic efficiency of the proportional-hazards score test, relative to the optimal partial-likelihood test, declines slowly as the hazard functions for the two treatments deviate from proportionality; the efficiency can be very low when the hazard functions cross or differ only at large survival times. Misspecification of the functional form of the regression portion of a proportional-hazards model introduces a quantitative treatment-covariate interaction. In the situations that we examine, based on a binary covariate, this misspecification usually results in only a minor drop in efficiency. The omission of a covariate that is balanced across treatments has a negligible effect on the size of the score test, but can substantially reduce power when the covariate effect is strong. The loss of power from mismodeling a balanced covariate is usually small.

Adriamycin and Cytoxan in the treatment of inoperable lung cancer

The results of a clinical trial involving 599 patients with inoperable squamous cell, large cell anaplastic, and adenocarcinoma of the lung are summarized. Patients were randomized to initial therapy with Cytoxan (CTX) (cyclophosphamide), or to one of two schedules of Adriamycin (doxorubicin) 50, or 75 mg/m2 IV every three weeks, or to a combined regimen of ADR and CTX. Upon disease progression, CTX patients were randomized to one of the two ADR schedules, while ADR patients were randomly assigned to CTX alone, or in combination with Cisdiamminedichloroplatinum (Cis-Platinum) 15 mg/m2 IV every three weeks. No statistically significant response or survival differences were observed between the two dose schedules of Adriamycin for any of the cell types studied. The two dose levels did, however, differ with respect to toxicity. There were some response and survival differences among the various cell types in the comparison of low-dose Adriamycin and Cytoxan: (1) patients with adenocarcinoma treated with low-dose Adriamycin tended to survive longer (P = 0.04) than those treated with Cytoxan; and (2) patients with large cell carcinoma receiving Cytoxan experienced a greater tumor response rate than those receiving low dose Adriamycin (P = 0.03). Because of the difficulties involved in distinguishing these two cell types on pathologic examination, the evidence of apparent treatment differences should not be regarded as definitive. During the period when Adriamycin plus Cytoxan was open to patient entry 61 evaluable patients received that regimen, 21 received low-dose Adriamycin and 22 received Cytoxan. Because relatively few patients received the latter two regimens, comparisons of these treatments with Adriamycin plus Cytoxan lack statistical power. However, there is no suggestion in the available data that Adriamycin plus Cytoxan increased survival either in the overall population or in the subset of patients with squamous histology. Initial performance status, metastatic disease symptoms, primary disease symptoms, and weight loss were significantly correlated to survival time, and are recommended as stratification factors in future studies.

Nonparametric estimation of lifetime and disease onset distributions from incomplete observations

In this paper we derive and investigate nonparametric estimators of the distributions of lifetime and time until onset associated with an irreversible disease that is detectable only at death. The nonparametric maximum likelihood solution requires an iterative algorithm. An alternative though closely related pair of estimators for the lifetime and onset distributions exists in closed form. These estimators are the familiar Kaplan-Meier estimator and an isotonic regression estimator, respectively. First-order approximations provide variance estimators. The proposed methods generalize and shed additional light on the constrained estimators presented by Kodell, Shaw and Johnson (1982, Biometrics 38, 43-58). Data from an animal experiment illustrate the techniques.

Practical experience of randomization in cancer trials: an international survey

The results from an international survey of 15 major cancer centres have clarified how randomization is being implemented in cancer trials. As regards the mechanics of obtaining treatment assignment for each patient a system of telephone registration to a central randomization office was widely used. We also advise formal checks for patient eligibility immediately before treatment assignment, and subsequent written confirmation of randomization to the investigators. As regards statistical methods, stratification of randomization by one or two prognostic factors (and institution in multicentre trials) is commonplace. Most centres used the standard approach of random permuted blocks within strata though some others used "dynamic" institution-balancing or "minimization" methods instead. The value of stratified allocation is chiefly for the trial's credibility in having comparable treatment groups, rather than for statistical efficiency. One should avoid overstratification and use only the really important prognostic factors. One essential is that randomization should in practice work for every patient, so undue complexity is to be avoided.

General right censoring and its impact on the analysis of survival data

This paper concerns general right censoring and some of the difficulties it creates in the analysis of survival data. A general formulation of censored-survival processes leads to the partition of all models into those based on noninformative and informative censoring. Nearly all statistical methods for censored data assume that censoring is noninformative. Topics considered within this class include: the relationships between three models for noninformative censoring, the use of likelihood methods for inferences about the distribution of survival time, the effects of censoring on the K-sample problem, and the effects of censoring on model testing. Also considered are several topics which relate to informative censoring models. These include: problems of nonidentifiability that can be encountered when attempting to assess a set of data for the type of censoring in effect, the consequences of falsely assuming that censoring is noninformative, and classes of informative censoring models.