Statistical considerations when using a composite endpoint for comparing treatment groups.

When comparing two treatment groups in a time-to-event analysis, it is common to use a composite event consisting of two or more distinct outcomes. The goal of this paper is to develop a statistical methodology to derive efficiency guidelines for deciding whether to expand a study primary endpoint from E1 (for example, non-fatal myocardial infarction and cardiovascular death) to the composite of E1 and E2 (for example, non-fatal myocardial infarction, cardiovascular death or revascularisation). We investigate this problem by considering the asymptotic relative efficiency of a log-rank test for comparing treatment groups with respect to a primary relevant endpoint E1 versus the composite primary endpoint, say E, of E1 and E2, where E2 is some additional endpoint.

A likelihood estimation of HIV incidence incorporating information on past prevalence.

SUMMARY: The prevalence and incidence of an epidemic are basic characteristics that are essential for monitoring its impact, determining public health priorities, assessing the effect of interventions, and for planning purposes. A direct approach for estimating incidence is to undertake a longitudinal cohort study where a representative sample of disease free individuals are followed for a specified period of time and new cases of infection are observed and recorded. This approach is expensive, time consuming and prone to bias due to loss-to-follow-up. An alternative approach is to estimate incidence from cross sectional surveys using biomarkers to identify persons recently infected as in (Brookmeyer and Quinn, 1995; Janssen et al., 1998). This paper builds on the work of Janssen et al. (1998) and extends the theoretical framework proposed by Balasubramanian and Lagakos (2010) by incorporating information on past prevalence and deriving maximum likelihood estimators of incidence. The performance of the proposed method is evaluated through a simulation study, and its use is illustrated using data from the Botswana AIDS Impact (BAIS) III survey of 2008.

Testing and interval estimation for two-sample survival comparisons with small sample sizes and unequal censoring.

While the commonly used log-rank test for survival times between 2 groups enjoys many desirable properties, sometimes the log-rank test and its related linear rank tests perform poorly when sample sizes are small. Similar concerns apply to interval estimates for treatment differences in this setting, though their properties are less well known. Standard permutation tests are one option, but these are not in general valid when the underlying censoring distributions in the comparison groups are unequal. We develop 2 methods for testing and interval estimation, for use with small samples and possibly unequal censoring, based on first imputing survival and censoring times and then applying permutation methods. One provides a heuristic justification for the approach proposed recently by Heinze and others (2003, Exact log-rank tests for unequal follow-up. Biometrics 59, 1151-1157). Simulation studies show that the proposed methods have good Type I error and power properties. For accelerated failure time models, compared to the asymptotic methods of Jin and others (2003, Rank-based inference for the accelerated failure time model. Biometrika 90, 341-353), the proposed methods yield confidence intervals with better coverage probabilities in small-sample settings and similar efficiency when sample sizes are large. The proposed methods are illustrated with data from a cancer study and an AIDS clinical trial.

HIV-1 Subtype C Phylodynamics in the Global Epidemic.

The diversity of HIV-1 and its propensity to generate escape mutants present fundamental challenges to control efforts, including HIV vaccine design. Intra-host diversification of HIV is determined by immune responses elicited by an HIV-infected individual over the course of the infection. Complex and dynamic patterns of transmission of HIV lead to an even more complex population viral diversity over time, thus presenting enormous challenges to vaccine development. To address inter-patient viral evolution over time, a set of 653 unique HIV-1 subtype C gag sequences were retrieved from the LANL HIV Database, grouped by sampling year as <2000, 2000, 2001-2002, 2003, and 2004-2006, and analyzed for the site-specific frequency of translated amino acid residues. Phylogenetic analysis revealed that a total of 289 out of 653 (44.3%) analyzed sequences were found within 16 clusters defined by aLRT of more than 0.90. Median (IQR) inter-sample diversity of analyzed gag sequences was 8.7% (7.7%; 9.8%). Despite the heterogeneous origins of analyzed sequences, the gamut and frequency of amino acid residues in wild-type Gag were remarkably stable over the last decade of the HIV-1 subtype C epidemic. The vast majority of amino acid residues demonstrated minor frequency fluctuation over time, consistent with the conservative nature of the HIV-1 Gag protein. Only 4.0% (20 out of 500; HXB2 numbering) amino acid residues across Gag displayed both statistically significant (p<0.05 by both a trend test and heterogeneity test) changes in amino acid frequency over time as well as a range of at least 10% in the frequency of the major amino acid. A total of 59.2% of amino acid residues with changing frequency of 10%+ were found within previously identified CTL epitopes. The time of the most recent common ancestor of the HIV-1 subtype C was dated to around 1950 (95% HPD from 1928 to 1962). This study provides evidence for the overall stability of HIV-1 subtype C Gag among viruses circulating in the epidemic over the last decade. However selected sites across HIV-1C Gag with changing amino acid frequency are likely to be under selection pressure at the population level.

Augmented cross-sectional prevalence testing for estimating HIV incidence.

Estimation of an HIV incidence rate based on a cross-sectional sample of individuals evaluated with both a sensitive and less-sensitive diagnostic test offers important advantages to incidence estimation based on a longitudinal cohort study. However, the reliability of the cross-sectional approach has been called into question because of two major concerns. One is the difficulty in obtaining a reliable external approximation for the mean "window period" between detectability of HIV infection with the sensitive and less-sensitive test, which is used in the cross-sectional estimation procedure. The other is how to handle false negative results with the less-sensitive diagnostic test; that is, subjects who may test negative-implying a recent infection-long after they are infected. We propose and investigate an augmented design for cross-sectional incidence estimation studies in which subjects found in the recent infection state are followed for transition to the nonrecent infection state. Inference is based on likelihood methods that account for the length-biased nature of the window periods of subjects found in the recent infection state, and relate the distribution of their forward recurrence times to the population distribution of the window period. The approach performs well in simulation studies and eliminates the need for external approximations of the mean window period and, where applicable, the false negative rate.

Nonparametric inference and uniqueness for periodically observed progressive disease models.

In many studies examining the progression of HIV and other chronic diseases, subjects are periodically monitored to assess their progression through disease states. This gives rise to a specific type of panel data which have been termed "chain-of-events data"; e.g. data that result from periodic observation of a progressive disease process whose states occur in a prescribed order and where state transitions are not observable. Using a discrete time semi-Markov model, we develop an algorithm for nonparametric estimation of the distribution functions of sojourn times in a J state progressive disease model. Issues of uniqueness for chain-of-events data are not well-understood. Thus, a main goal of this paper is to determine the uniqueness of the nonparametric estimators of the distribution functions of sojourn times within states. We develop sufficient conditions for uniqueness of the nonparametric maximum likelihood estimator, including situations where some but not all of its components are unique. We illustrate the methods with three examples.

Estimating HIV incidence based on combined prevalence testing.

Knowledge of incidence rates of HIV and other infectious diseases is important in evaluating the state of an epidemic as well as for designing interventional studies. Estimation of disease incidence from longitudinal studies can be expensive and time consuming. Alternatively, Janssen et al. (1998, Journal of the American Medical Association 280, 42-48) proposed the estimation of HIV incidence at a single point in time based on the combined use of a standard and "detuned" antibody assay. This article frames the problem from a longitudinal perspective, from which the maximum likelihood estimator of incidence is determined and compared with the Janssen estimator. The formulation also allows estimation for general situations, including different batteries of tests among subjects, inclusion of covariates, and a comparative evaluation of different test batteries to help guide study design. The methods are illustrated with data from an HIV interventional trial and a seroprevalence survey recently conducted in Botswana.

Timing constraints of in vivo gag mutations during primary HIV-1 subtype C infection.

BACKGROUND: Aiming to answer the broad question "When does mutation occur?" this study examined the time of appearance, dominance, and completeness of in vivo Gag mutations in primary HIV-1 subtype C infection. METHODS: A primary HIV-1C infection cohort comprised of 8 acutely and 34 recently infected subjects were followed frequently up to 500 days post-seroconversion (p/s). Gag mutations were analyzed by employing single-genome amplification and direct sequencing. Gag mutations were determined in relation to the estimated time of seroconversion. Time of appearance, dominance, and completeness was compared for different types of in vivo Gag mutations. RESULTS: Reverse mutations to the wild type appeared at a median (IQR) of 62 (44;139) days p/s, while escape mutations from the wild type appeared at 234 (169;326) days p/s (p<0.001). Within the subset of mutations that became dominant, reverse and escape mutations appeared at 54 (30;78) days p/s and 104 (47;198) days p/s, respectively (p<0.001). Among the mutations that reached completeness, reverse and escape mutations appeared at 54 (30;78) days p/s and 90 (44;196) days p/s, respectively (p=0.006). Time of dominance for reverse mutations to and escape mutations from the wild type was 58 (44;105) days p/s and 219 (90;326) days p/s, respectively (p<0.001). Time of completeness for reverse and escape mutations was 152 (100;176) days p/s and 243 (101;370) days p/s, respectively (p=0.001). Fitting a Cox proportional hazards model with frailties confirmed a significantly earlier time of appearance (hazard ratio (HR): 2.6; 95% CI: 2.3-3.0), dominance (4.8 (3.4-6.8)), and completeness (3.6 (2.3-5.5)) of reverse mutations to the wild type Gag than escape mutations from the wild type. Some complex mutational pathways in Gag included sequential series of reversions and escapes. CONCLUSIONS: The study identified the timing of different types of in vivo Gag mutations in primary HIV-1 subtype C infection in relation to the estimated time of seroconversion. Overall, the in vivo reverse mutations to the wild type occurred significantly earlier than escape mutations from the wild type. This shorter time to incidence of reverse mutations remained in the subsets of in vivo Gag mutations that reached dominance or completeness.

On the use of adjusted cross-sectional estimators of HIV incidence.

OBJECTIVE: To elucidate when and how cross-sectional estimators of HIV incidence rates based on a sensitive and less sensitive diagnostic test should be adjusted. METHODS: Evaluate the statistical properties of unadjusted and adjusted cross-sectional estimators of HIV incidence, including the adjusted estimators considered by McDougal et al, for the 2 settings where (a) all infected subjects eventually become reactive to the less sensitive test, and (b) a subset of infected subjects indefinitely remain nonreactive to the less sensitive test. Derive the maximum likelihood estimator of incidence for the latter setting and use analytical results and simulation studies to compare the performance of the various estimators. RESULTS: When every infected subject would eventually become reactive to the less sensitive test, the McDougal adjusted estimator is uniformly less precise than the unadjusted estimator and more susceptible to bias. When a subset of the infected population would indefinitely remain nonreactive to the less sensitive test, the McDougal adjusted estimator is less precise than the maximum likelihood estimator, which coincides with an estimator developed by McWalter and Welte using a mathematical modeling approach. When the assumed model is incorrect, the unadjusted estimator overestimates incidence, whereas the maximum likelihood estimator can be biased in either direction. CONCLUSIONS: The standard unadjusted cross-sectional estimator of HIV incidence should be used when all infected individuals would eventually become reactive to the less sensitive test. When a subset of individuals would indefinitely remain nonreactive to the less sensitive test, the maximum likelihood estimator for this setting should be used. Characterizing the proportion of individuals who would indefinitely remain nonreactive is crucial for accurate estimation of HIV incidence.

Better control of early viral replication is associated with slower rate of elicited antiviral antibodies in the detuned enzyme immunoassay during primary HIV-1C infection.

BACKGROUND: Estimation of HIV incidence rates is important for timing interventions, planning prevention studies, and monitoring the epidemic. This requires accurate estimation of the "recency period" (also known as the "window period") between seroconversion and achievement of specific detectable levels of anti-HIV antibody titers, such as the standardized optical density (SOD) in the early phase of HIV-1 infection. METHODS: To obtain a better understanding of interpatient variation of the recency period, prospective measurements of antiviral antibody titers in the early phase of HIV-1 subtype C infection were quantified by Vironostika-LS. Time of seroconversion was estimated by Fiebig staging. RESULTS: The profiles of SOD values during the first year of infection commonly showed slow initial increase followed by a more rapid increase, although in some patients, SOD values increased rapidly soon after seroconversion. Using an SOD cutoff of 1.0, the average duration of the recency period in subtype C infection in the local epidemic in Botswana was estimated to be 151 days (95% confidence interval: 130 to 172 days) post seroconversion. The recency period was significantly associated (P = 0.007) with the level of viral replication during the first 2-3 months post seroconversion. Reduction of SOD values after initiation of antiretroviral therapy (ART) was a dominant pattern in antiretroviral drug (ARV)-treated subjects. CONCLUSIONS: Our data suggest that HIV incidence estimation based on sensitive/less sensitive enzyme immunoassay cross-sectional testing could be potentially improved by incorporation of viral load levels at the time of detection of a recent infection.

Marginal analysis of panel counts through estimating functions.

We develop nonparametric estimation procedures for the marginal mean function of a counting process based on periodic observations, using two types of self-consistent estimating equations. The first is derived from the likelihood studied in Wellner & Zhang (2000), assuming a Poisson counting process, and gives a nondecreasing estimator, which is the same as the nonparametric maximum likelihood estimator of Wellner & Zhang and thus is consistent without the Poisson assumption. Motivated by the construction of parametric generalized estimating equations, the second type is a set of data-adaptive quasi-score functions, which are likelihood estimating functions under a mixed-Poisson assumption. We evaluate the procedures via simulation, and illustrate them with the data from a bladder cancer study.

Viral load and CD4+ T-cell dynamics in primary HIV-1 subtype C infection.

BACKGROUND: Most knowledge of primary HIV-1 infection is based on subtype B studies, whereas the evolution of viral parameters in the early phase of HIV-1 subtype C infection is not well characterized. METHODS: The kinetics of viral RNA, proviral DNA, CD4+ T-cell count, and subsets of CD4+ T cells expressing CCR5 or CXCR4 were characterized in 8 acute and 62 recent subtype C infections over the first year postseroconversion. RESULTS: The viral RNA peak was 6.25 +/- 0.92 log10 copies per milliliter. After seroconversion, heterogeneity among acute cases was evident by patterns of change in viral load and CD4+ T-cell count over time. The patterns were supported by the rate of viral RNA decline from peak (P = 0.022), viral RNA means (P = 0.005), CD4 levels (P < 0.001), and CD4 decline to 350 (P = 0.011) or 200 (P = 0.046). Proviral DNA had no apparent peak and its mean was 2.59 +/- 0.69 log10 per 106 peripheral blood mononuclear cell. In recent infections, viral RNA set point was 4.00 +/- 0.97 log10 and viral RNA correlated inversely with CD4+ T cells (P < 0.001) and directly with proviral DNA (P < 0.001). CONCLUSIONS: Distinct patterns of viral RNA evolution may exist shortly after seroconversion in HIV-1 subtype C infection. The study provides better understanding of the early phase of subtype C infection.

Evolution of proviral gp120 over the first year of HIV-1 subtype C infection.

The evolution of proviral gp120 during the first year after seroconversion in HIV-1 subtype C infection was addressed in a case series of eight subjects. Multiple viral variants were found in two out of eight cases. Slow rate of viral RNA decline and high early viral RNA set point were associated with a higher level of proviral diversity from 0 to 200 days after seroconversion. Proviral divergence from MRCA over the same period also differed between subjects with slow and fast decline of viral RNA, suggesting that evolution of proviral gp120 early in infection may be linked to the level of viral RNA replication. Changes in the length of variable loops were minimal, and length reduction was more common than length increase. Potential N-linked glycosylation sites ranged +/-one site, showing common fluctuations in the V4 and V5 loops. These results highlight the role of proviral gp120 diversity and diversification in the pathogenesis of acute HIV-1 subtype C infection.

Statistical monitoring of clinical trials with multivariate response and/or multiple arms: a flexible approach.

Randomized clinical trials with a multivariate response and/or multiple treatment arms are increasingly common, in part because of their efficiency and a greater concern about balancing risks with benefits. In some trials, the specific types and magnitudes of treatment group differences that would warrant early termination cannot easily be specified prior to the onset of the trial and/or could change as the trial progresses. This underscores the need for more flexible monitoring methods than traditional approaches. This paper extends the repeated confidence bands approach for interim monitoring to more general settings where there can be a multivariate response and/or multiple treatment arms and where the metrics for comparing treatment groups can change during the conduct of the trial. We illustrate the approach using the results of a recent AIDS clinical trial and examine its efficiency and robustness via simulation.

GENERALIZED LEAST SQUARES ESTIMATION OF THE MEAN FUNCTION OF A COUNTING PROCESS BASED ON PANEL COUNTS.

This paper considers nonparametric estimation of the mean function of a counting process based on periodic observations, i.e., panel counts. We present estimators derived through minimizing a class of generalized sums of squares subject to a monotonicity constraint. We establish consistency of the estimators and provide procedures to implement them with various weight functions. For specific weight functions, they reduce to the estimator given in Sun and Kalbfleisch (1995), and are closely related to the nonparametric maximum likelihood estimator studied in Wellner and Zhang (2000). With other weight functions, the proposed estimators provide alternatives that can have better efficiency in non-Poisson situations than previous approaches. Simulations are used to examine the finite-sample performance of the proposed estimators.

Inference after variable selection using restricted permutation methods.

When confronted with multiple covariates and a response variable, analysts sometimes apply a variable-selection algorithm to the covariate-response data to identify a subset of covariates potentially associated with the response, and then wish to make inferences about parameters in a model for the marginal association between the selected covariates and the response. If an independent data set were available, the parameters of interest could be estimated by using standard inference methods to fit the postulated marginal model to the independent data set. However, when applied to the same data set used by the variable selector, standard ("naive") methods can lead to distorted inferences. The authors develop testing and interval estimation methods for parameters reflecting the marginal association between the selected covariates and response variable, based on the same data set used for variable selection. They provide theoretical justification for the proposed methods, present results to guide their implementation, and use simulations to assess and compare their performance to a sample-splitting approach. The methods are illustrated with data from a recent AIDS study.

Analysis of time to a silent event whose occurrence is monitored with error, with application to mother-to-child HIV transmission.

Inferences about the distribution of time to HIV infection in infants are complicated because infection is a silent event and imperfect diagnostic tests are used to detect its occurrence, leading to false-positive and false-negative results. Nonparametric likelihood approaches are computationally hampered by a large number of parameters and a possibly nonconcave likelihood function. To overcome these difficulties, we develop one-sample and regression methods based on profile likelihood and Markov chain Monte Carlo techniques. The methods also provide a useful diagnostic for assessing the infection status of individual subjects, and are illustrated using results from a recent clinical trial for the prevention of mother-to-child HIV transmission.

Response to antiretroviral therapy after a single, peripartum dose of nevirapine.

BACKGROUND: A single dose of nevirapine during labor reduces perinatal transmission of human immunodeficiency virus type 1 (HIV-1) but often leads to viral nevirapine resistance mutations in mothers and infants. METHODS: We studied the response to nevirapine-based antiretroviral treatment among women and infants who had previously been randomly assigned to a single, peripartum dose of nevirapine or placebo in a trial in Botswana involving the prevention of the transmission of HIV-1 from mother to child. All women were treated with antenatal zidovudine. The primary end point for mothers and infants was virologic failure by the 6-month visit after initiation of antiretroviral treatment, estimated within groups by the Kaplan-Meier method. RESULTS: Of 218 women who started antiretroviral treatment, 112 had received a single dose of nevirapine and 106 had received placebo. By the 6-month visit after the initiation of antiretroviral treatment, 5.0% of the women who had received placebo had virologic failure, as compared with 18.4% of those who had received a single dose of nevirapine (P=0.002). Among 60 women starting antiretroviral treatment within 6 months after receiving placebo or a single dose of nevirapine, no women in the placebo group and 41.7% in the nevirapine group had virologic failure (P<0.001). In contrast, virologic failure rates did not differ significantly between the placebo group and the nevirapine group among 158 women starting antiretroviral treatment 6 months or more post partum (7.8% and 12.0%, respectively; P=0.39). Thirty infants also began antiretroviral treatment (15 in the placebo group and 15 in the nevirapine group). Virologic failure by the 6-month visit occurred in significantly more infants who had received a single dose of nevirapine than in infants who had received placebo (P<0.001). Maternal and infant findings did not change qualitatively by 12 and 24 months after the initiation of antiretroviral treatment. CONCLUSIONS: Women who received a single dose of nevirapine to prevent perinatal transmission of HIV-1 had higher rates of virologic failure with subsequent nevirapine-based antiretroviral therapy than did women without previous exposure to nevirapine. However, this applied only when nevirapine-based antiretroviral therapy was initiated within 6 months after receipt of a single, peripartum dose of nevirapine. (ClinicalTrials.gov number, NCT00197587 [ClinicalTrials.gov].).