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Methods for the Analysis of Continuous Biomarker Assay Data With Increased Sensitivity

Lau, Bryan; Gange, Stephen J.

doi: 10.1097/01.ede.0000142154.86749.e4
Original Article
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Prospective studies must be able to adapt to improved technology and adopt new assays with increased limits of detection. Our objective is to describe methods for incorporating new technologies in which the lower limits of quantification of a biomarker are enhanced. One may conduct an analysis of data with new and old sensitivity levels using a variety of methods, including retesting a sample of stored specimens from which multiple imputation may be applied, and a parametric approach that accounts for the changing limit of detection. We compare these methods in terms of their statistical bias and efficiency and identify the conditions under which the various methods perform well and then demonstrate our methods by evaluating differences in HIV RNA levels obtained from 2 prospective cohort studies.

From the Department of Epidemiology, Johns Hopkins Bloomberg School of Public Health, Baltimore, Maryland.

Submitted 5 March 2003; final version accepted 25 June 2004.

The Women's Interagency HIV Study and Multicenter AIDS Cohort Study are funded by the National Institutes of Health. This research was supported by cooperative agreements AI-42590 and AI-34043.

Correspondence: Stephen J. Gange, Department of Epidemiology, Johns Hopkins Bloomberg School of Public Health, 615 N. Wolfe Street, Room E7638, Baltimore, MD 21205. E-mail: sgange@jhsph.edu

The new era of “molecular epidemiology”1 has been driven by advances in technology that measure increasingly precise and relevant biologic markers of exposures and intermediate disease outcomes. These improving measures, when instituted within well-designed epidemiologic studies, offer great potential for an increased understanding of disease etiology. Because of rapid technological advances, prospective studies often face the challenge of adopting newer, more precise methods to replace older, less precise techniques.2 Assays for measuring quantitative plasma HIV-1 RNA (“viral load”), which have been shown to be an important risk factor for disease and mortality,3 provide an excellent example of this evolution. Initial quantitative data from serial dilutions of cultures provided relatively crude measures of viral concentration.4,5 During the past several years, measurement of plasma HIV-1 RNA by nucleic acid amplification assays has been used with increasing accuracy and sensitivity. Prospective cohort studies of HIV/AIDS have adopted these newer technologies, driven by studies demonstrating the importance of low values of viral load for defining the risk of disease progression.6

Improvements in assay methods resulting in lower limits of quantification of the biomarker are not restricted to HIV research. Another recent example is analysis of C-reactive protein markers. Levels of C-reactive protein greater than 10 mg/L are an important indicator of inflammation that is attributable to infection, and standard assays with a lower limit of 3–5 mg/L have been adequate to measure this marker. However, the risk with cardiovascular disease occurs at far lower levels7,8 and requires the use of high-sensitivity assays.7

Analyses of biomarker data from assays with improved precision present 2 challenges. First, one needs to evaluate whether the 2 assays are comparable in the range where both assays are able to quantify the biomarker. If systematic differences occur, methods need to be used to standardize the results of the assays. For example, conversion equations have been derived for a variety of different HIV-1 RNA-quantification methods.9–12 This presupposes that the assays are measuring the same fundamental quantity (ie, copies of HIV RNA per ml or mg of C-reactive protein per liter). If the assays are measuring targets that have different composition, then a more comprehensive evaluation of the relationship between the 2 assays needs to be undertaken.13,14 Presumably, the need to combine assays in such a situation rests on the existence of some association between the 2 assays. In this article, we are interested in methods for situations in which the difference between the new and old assays is solely a change in the limit of quantification of the biomarker, with both assays producing the same measurements in the range covered by both. It is common that new assays in molecular epidemiology evolve from older assays and have the same underlying basis but a lower limit of detection.

Second, one needs to consider the impact of differences in the assays’ precision upon the analysis.15 If the study has created a repository of specimens, an obvious solution is to retest the specimens, or perhaps a portion of the specimens, with the newer assay. This assumes that the effect of freezing is minimal.16 An important alternative to retesting specimens is to use statistical methods that appropriately account for the different censoring limits. The taxonomy of methods for handling censored data for this problem is broad and includes mixture models,17 statistical imputation,18 and incorporating censoring as part of the model.15,18

The purpose of this work is 2-fold. First, we demonstrate how to apply 2 methods for handling data with increased level of sensitivity: an imputation approach that is able to complete the data by retesting a representative sample of specimens and an analytical approach that incorporates the censored information into the model. We demonstrate our methods with HIV RNA data from 2 prospective cohort studies: the Women's Interagency HIV Study and the Multicenter AIDS Cohort Study. Second, we compare the bias and efficiency of these 2 approaches through simulations. The efficiency is an important consideration: although retesting specimens and multiple imputation complete the dataset and eliminate differential censoring, they may incur a large cost, particularly if the number of specimens to be retested is large and precious. Therefore, we investigated whether the parametric model can provide unbiased estimates that are close to those obtained with retesting, thereby evaluating the cost-effectiveness of these methods in lieu of expending resources.

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Prologue: Plasma HIV RNA and Sex of Subject

In 1998, Farzadegan et al19 described data from the AIDS Link to Intravenous Experience study that showed a difference in HIV RNA levels in men and women despite similar CD4+ cell counts. A strength of their study was the common protocol and laboratory methods used in the 2 groups. However, their study had relatively few women participants, all of whom had a history of injection drug use (IDU) and most of whom were African-American. Our goal was to investigate the differences in HIV RNA among a larger group and to further investigate the differences by IDU history and ethnicity.20 Data for this analysis came from 2 of the largest prospective cohort studies in the United States: the Women's Interagency HIV Study and the Multicenter AIDS Cohort Study.

The Women's Interagency HIV Study, which was established in 1993, is a long-term prospective cohort for investigating the impact of HIV infection among women in the United States. The recruitment and data management methods have been described elsewhere.21 Briefly, the study enrolled 2058 HIV-seropositive women and 569 seronegative women between October 1994 and November 1995 at 6 clinical consortia. Participants return to the clinics semiannually to undergo physical examination, an interviewer-administered questionnaire, and collection of specimens for laboratory analysis. The demographic and HIV exposure risk characteristics of the seropositive population in the cohort are comparable with the characteristics of the nationally reported AIDS cases in U.S. women.21 Plasma HIV-1 RNA quantification was initially determined using the Nucleic Acid Sequence-Based Amplification (NASBA) assay with a lower limit of quantification of 4000 cps/ml, an assay that is now outdated.

The Multicenter AIDS Cohort Study, established in 1984, is a long-term prospective cohort study of homosexual men. The methods of recruitment, management of data, and handing of laboratory specimens have been described previously.22 The cohort initially enrolled 4954 men without AIDS who were older than 18 years of age from 4 U.S. cities. Similar to Women's Study, participants returned to the clinics semiannually to undergo a physical examination, to provide specimens for laboratory analyses, and to complete data forms and an interviewer-administered questionnaire. Their plasma HIV-1 RNA quantity was determined through the use of a branched-DNA assay on reposited samples with a lower limit of quantification of 500 cps/ml.

We compared the data on women from 1994 and 1995 with men's specimens that were collected in 1985 and 1986 and then tested for HIV levels in 1995. To adjust for temporal changes in HIV treatment, for this study we selected only women who were not on therapy (no therapy existed before 1987). With these criteria, 1256 women had baseline HIV RNA viral load and CD4+ lymphocyte measurements available. The men we selected for this analysis were 1603 HIV-seropositive individuals for whom HIV RNA and CD4+ lymphocyte measurements were available.20

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STATISTICAL METHODS FOR CENSORED BIOMARKER DATA

Notation

Let random variable Yi be the biomarker of interest for individual i = 1,..., n and ci be an indicator for whether the outcome is measured with a test with lower limit of l1 (ci = 1) or l2 (ci = 2). Let Xi be a vector of the covariates to be studied in relationship to Yi. Let yi be the true value of Yi and yi1 and yi2 be the observable Yi such that

where 11 > 12.

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Parametric Censored Regression Approach

A parametric censored regression approach to this problem proceeds by establishing a statistical model that acknowledges the different contributions of censored and uncensored observations. Uncensored data, which are above the limit of quantification (δi = 1), contribute to the likelihood function in the “usual” fashion via the density function f(yi;β). Censored data, which are below the limit of quantification (δi = 0), contribute to the likelihood function through the cumulative density function F(yi;β). The log likelihood (LL) can be constructed as the sum of 2 terms:

This partitioning of the likelihood is analogous to the partitioning in parametric models for time-to-event/survival data, which consist of a mix of uncensored events, with known times, and censored observations that represent the lower limit of the time for the occurrence of the event. Maximizing LL will provide maximum likelihood estimates of the regression parameters.

A critical decision is the choice of the density function f(yi;β), which is used to model the entire distribution of outcomes (eg, viral load). This distribution will determine how the censored observations contribute. The strength of this approach is the gain in efficiency by fitting a parametric model, whereby all data are used to estimate the small number of parameters. A weakness of this approach, however, is the reliance on this assumption since the actual distribution may not be appropriately modeled.

An obvious choice for f(yi;β) is a standard unimodal distribution, which enables capitalizing upon routines in standard statistical software (eg, SAS; SAS Institute, Cary, NC), although the method is applicable to more general distribution functions. For example, an appropriate distribution may be the lognormal distribution, with E[log(Y) | X] = α + βX. Then the LL for the expected outcome Y can be written as:

for which the estimates of α and β can be determined through the maximum likelihood. In research on HIV, this parametric approach has been suggested for analyses of the magnitude of change in HIV-1 RNA levels as end points in clinical trials.23 Another likelihood approach has been described for left-censored biomarker data using a combination of the simplex algorithm and Marquardt algorithm for the maximization of the likelihood.24

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Multiple Imputation Approach

A multiple imputation approach to the problem involves filling in the undetectable biomarker observations tested by the older assay with the higher limit of quantification (l1) and then analyzing the completed dataset. This method requires a distribution of appropriate values from which the missing data may be imputed. These values must be representative of individuals who fall between the limits of quantification for the old (l1) and new (l2) assays. Let set R of m specimens represent this distribution, which may be derived through several different approaches. The most natural approach, if feasible, is to actually retest a portion of specimens to determine R. Other approaches would be to use external data sources for R or to use the data between the limits of quantification of the (l1) and (l2) when the new assay is adopted. These approaches may be inappropriate, however, because of temporal trends or the representativeness of those that fell beneath the old assay's limit of quantification.

Once the distribution R is determined, the process of imputing values for the missing data and performing the analysis is repeated multiple times. The results of the analysis performed on each of the imputations are summarized and the corresponding variance is adjusted for the fact that the imputed values are a substitute for the unknown true values.25–27

Algorithmically, repeat the following T times:

  • Replace any values where yi1 = l1 with rq, a value randomly selected with replacement from R, such that the resulting data for Y will be:
  • Run a regression of the model of interest:

Perform this regression incorporating the imputed data in the model and recording the β(t) and the corresponding variance, var(t) (β) for t = 1,2,...,T.

  • After the T imputations, summarize the estimates of the relationship between X and Y through the following formula:

with the variance calculated by the incorporation of the variability from the imputed values as:25–27

for the g = 1,...,k covariates.

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Hybrid Approach

Use of the parametric censored model regression and multiple imputation approach is not mutually exclusive. Specifically, although a new assay might lower the detection limit (l1 to l2), there may remain data that fall below the new lower limit l2. A hybrid approach that uses both the parametric model and the multiple imputation approach could be formulated by replacing the likelihood function in the multiple imputation with the censored likelihood model in Eq 1. This approach would continue to take advantage of the detectable and retested specimens, but acknowledge the data below the new limit of quantification (l2).

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EXAMPLE: PLASMA HIV RNA AND SEX OF SUBJECT

Methods

The WIHS used the NASBA assay (l1 = 4000 copies/ml) and MACS used the branched-DNA assay (l2 = 500 copies/ml branched-DNA scale). To eliminate the problem of bias between the HIV RNA assays in the detectable ranges, viral load data for both cohorts was transformed from the scale of the respective assays of the cohorts to a scale of a third method of quantification, reverse-transcription PCR (RT-PCR). This was done using published comparison data of the assays in which duplicate samples were tested by the RT-PCR assay and either the NASBA or branched-DNA assay.3,10,11 Values converted from NASBA to RT-PCR have been found to be statistically equivalent in testing done by WIHS laboratories.11 Therefore, analysis was performed without the need to convert NASBA to RT-PCR. The MACS had a lower limit of 1466 HIV RNA cps/ml (l2 RT-PCR scale) copies/ml after transformation using a published conversion formula:3 (RT-PCR) = 5.364 * {b − DNA}0.9029.

Ultimately, to perform this analysis, the left censoring must be comparable between the 2 assays to resolve the large discrepancy in the lower limit of quantification of the assays on the RT-PCR scale (l1= 4000; l2 = 1466 copies/ml). Figure 1A shows the distribution of HIV RNA values of the women's samples before application of multiple imputation. Of the 1256 women included in the analysis of HIV RNA and sex, 418 individuals at the baseline visit had a viral load less than the lower limit of quantification of the NASBA assay. Figure 1B displays the distribution of men's viral load with 122 individuals censored at l2. Comparing Figure 1A and 1B, one can see the discrepancy between the lower limits of quantification of the 2 cohort studies and the differences in the viral load distribution. To apply multiple imputation, a random sample of 395 specimens undetectable by the NASBA assay in the Women's Interagency Health Study (of 1396 undetectable specimens from visits 1 and 3) had been retested by the Nuclisens assay (lower limit = 80 cps/ml), shown in Figure 1C; 49 remained below the lower limit of quantification of the newer assay.28 The data from the retested specimens provide an empirical distribution of viral load from which NASBA censored observations may be imputed. This lowers the viral load limit from 4000 cps/ml to 80 cps/ml (Fig. 1D), which we censored at the level of l2 to maintain a level comparable with the men’s data.

FIGURE 1.

FIGURE 1.

In summary, the primary exposures of interest were sex/cohort, history of IDU, and ethnicity. These were adjusted for CD4+ cell count and presence of clinical symptoms, which included fever for longer than 2 weeks, weight loss greater than 10% in the last 6 months, or diarrhea in the last 2 weeks. We used a hybrid lognormal regression model as described above since we continued to have a number of observations below l2.

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Results: Retested HIV RNA Distributions

The randomly selected subsample of undetectable individuals was similar to the entire undetectable cohort for age, race, marital status, education, sexual identity, and exposure risk category (data not shown).28Figure 1C shows the distribution of the retested specimens, which were initially undetectable by the NASBA assay. Figure 1D shows the distribution of the full HIV RNA data after one iteration of multiple imputation. The imputed values assume the distribution of the retested samples. Due to the random selection of the specimens that were tested by the Nuclisens assay, these empirical distributions of HIV RNA quantity can be assumed to describe those that were undetectable by the NASBA assay for the corresponding visit.

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Results of HIV RNA Analysis by Sex

The characteristics of the 1256 women and 1603 men in the 2 cohorts are shown in Table 1. A greater proportion of the women were either black or Hispanic compared with the male participants, and they were older. The women were more likely to report previous use of injected drugs, to havelower CD4+ cell counts, and to have clinical symptoms at the time the data were collected. Results in Table 2 compare the hybrid approach with the use of the parametric approach alone. We found similarities in the estimates between the 2 models for several variables: race, history of IDU, the presence of clinical symptoms, and the effect of an increase in CD4+ cell count. However there are important differences between the models with respect to inferences of sex. In the model without multiple imputation, there was little overall difference in HIV RNA levels between sex/cohort except for the category including CD4+ cell count levels between 350 and 500. This is in contrast to the model that included multiple imputation, in which a difference in HIV RNA between sex/cohort was found in all categories of CD4+ cell counts greater than 200 cells. It is at these CD4+ cell counts that imputation would have the greatest effect because that is where most NASBA-undetectable individuals would be as the result of the inverse relationship between CD4+ cell counts and HIV RNA levels. In a model without imputation and with an interaction term for CD4+ cell count level and sex, the independent effect of sex was a reduction in median viral load in women (−30.61%; 95% confidence interval = −44.16 to −13.34). Only when using multiple imputation is the interaction between categorical CD4+ cell count level and sex elucidated.

TABLE 1

TABLE 1

TABLE 2

TABLE 2

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COMPARISON OF PARAMETRIC AND MULTIPLE IMPUTATION APPROACHES

Motivation for the Simulation

The goal of the simulation was to examine how well the various methods compare in identifying a relationship between 2 variables. To accomplish this, data were simulated to represent the relationship between CD4+ cell counts and HIV RNA levels. To simulate the lower limit of quantification, values under several cutoffs were identified such that the parametric or multiple imputation methods presented above could be applied. Knowing the true relationship, we examined the relative bias, variance, and mean-squared error.

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Simulation Methods

Several datasets were created for the simulations. The first dataset was composed of 1000 individuals with 5 independent visits each, for whom the CD4+ cell counts were generated using an exponential distribution with a mean of 400 and log10(HIV RNA) = 9–0.21 (CD4+ cell count)1/2 + ε, where ε is an error term that is normally distributed around zero with a variance of 4. A second dataset simulated 1000 individuals each followed for 5 visits to simulate a cohort being followed over time. The covariance matrix for each individual was structured as Vi = ZiDZiT + σ2I where

and ZiT = [1 2 3 4 5] and σ2 = 0.5. Otherwise, the CD4+ cell counts were generated as in the first dataset with the same relationship to HIV RNA. To compare the efficiencies of the methods when the parametric approach is not appropriately applied, a t-distribution was used to generate a dataset such that the assumed distribution for the parametric method would be incorrect. Four approaches were used to analyze the simulated data:

  1. The parametric left-censored regression model approach, which was performed as described above using a lognormal distribution.
  2. A cold-deck approach, whereby any censored observation was assigned a value of three fourths the lower limit. This method was meant to represent a naive approach that acknowledges that the censored values are below the lower limit.
  3. A multiple imputation approach as described above with 2 different sampling schemes. To create the distribution from which the imputed values were to be selected for the censored observations, a subsample of the censored values must be selected and their true values obtained. Thus, for the multiple imputation analyses at each limit of quantification, the analyses were performed using 25% and 50% of the censored observations as the subsample from which the remaining censored values would be imputed. The imputation analyses followed the methods outlined above using log10(HIV RNA) = 9 - 0.21 (CD4+ cell count)1/2 as the model in the place of Eq 1.
  4. A weighted multiple imputation approach which combines the standard-multiple imputation approach with estimates obtained from an analysis based only on the detectable observations. This includes values that are above the lower limit of quantification and those observations that were selected for the imputation distribution. The overall estimate is

where wi is the weight that is the percent detectable observations either initially or when selected for retesting, βobs represents the estimate determined from the analysis based upon only the detectable observations, and the mean corresponds to the average estimate based on the multiple imputation (βimp). To determine the variance for the resulting estimates, a bootstrap method was applied. This approach was motivated by the fact that most of the data usually are not beneath the lower limit of quantification. Hence, it was thought that it might be more efficient to weight the results by the percent detectable since the relationship between the marker and outcome is truly described by these marker values and not imputed.

For the correlated data, a random effects model was used for approaches 2, 3, and 4 with a random intercept. The model was correctly specified by including only CD4+ as the independent variable. The amount of censoring was determined by changing the limit of quantification of HIV RNA. Each approach analyzed the datasets with 17.7, 41.2, 56.1, and 83.0% censoring. Efficiency of the methods was examined by assessing the relative bias, the variance, and the mean squared error of the estimates for the intercept and for the coefficient of (CD4+ cell count)1/2.

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Simulation Results

The results of the simulations for various censoring proportions are shown in Figure 2, which presents the relative bias, the variance and the mean squared error. For brevity, the results pertaining to the covariate (CD4+ cell count)1/2 for both the noncorrelated and correlated datasets are displayed in the figures.

FIGURE 2.

FIGURE 2.

Overall, the cold-deck approach was the most biased method and the parametric method the least biased, with levels of relative bias less than 3.0% (Figs. 2A, B). The bias generally increased with increasing proportion of censoring, although the biases for the weighted-multiple imputation and parametric approaches were low and relatively stable until reaching a high level of censoring. The standard and weighted multiple imputation approaches had similar bias at low levels of censoring within the correlated dataset, but the standard multiple imputation was slightly better at low levels of censoring in the noncorrelated dataset. Both multiple imputation approaches were less biased if a greater proportion of censored observations were selected to construct the imputation distribution.

When the levels of variance (Figs. 2C, D) and the mean-squared error (Figs. 2E, F) were examined, the cold-deck approach was the least efficient method, whereas the parametric approach was the most efficient. At low levels of censoring, the standard multiple imputation, weighted multiple imputation, and the parametric approaches were similar in level of mean squared error. With greater censoring, the parametric and weighted multiple imputation approaches showed continued efficiency, whereas the other approaches dramatically increased.

For the parametric simulation, the distribution of the data was assumed to be appropriate. However, the efficiency of the parametric approach relies on how appropriately the distribution represents the underlying distribution of the data. We also performed a simulation using a t-distribution with 2 degrees of freedom and a small sample size, to limit the impact of the central limit theorem. The results of this small simulation showed that, under these conditions, the standard multiple imputation with 50% sampling had a better relative bias compared with the parametric approach (up to 34 vs. 45% relative bias for 78% censoring). Furthermore, although the parametric model had slightly lower variance, the mean-squared error remained lower for the standard-multiple imputation method.

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DISCUSSION

Adopting appropriate methods to analyze data with various detection limits is an important consideration when more sensitive assays become available. The methods presented here should be applicable to any biomarker data that are continuous in nature. However, the results would not apply to assays measuring a marker by enzyme-linked immunosorbent assay resulting where data are recorded as either positive or negative. The methods are most applicable to data resulting from quantification of a marker in which there is a lower limit that is surpassed by a new assay with a lower limit.

With the exception of the cold-deck approach, each of the alternatives to the parametric method requires the estimation of the distribution R of the biomarker in the study population. Unless obtainable from external sources, estimation of this distribution requires retesting specimens. When stored specimens are available, our results put retesting into a proper perspective. Specifically, retesting a large number of specimens with a more sensitive assay will not result in estimates with a lower mean-squared error that than those from an appropriately applied statistical approach, that is, when the assumptions for the statistical method are not violated. This was demonstrated in our simulations, where the mean-squared errors of the parametric approach under a correctly specified model were lower than all the alternative methods for any level of censoring. The parametric method remained unbiased at even high levels of censoring, whereas the other methods tended to show increased bias with more censoring.

However, retesting is not without usefulness. The statistical methods for the parametric approach are more complex and, despite integration with standard software, might be disregarded in lieu of assigning a single value to those samples that are undetectable as per the cold-deck approach. Our results show that this method performs poorly with increasing bias almost linearly associated with greater censoring. Much can be gained by retesting even a modest portion of the samples to estimate R. Our results also demonstrate that the overall mean-squared error of the multiple imputation approach was nearly the same as the parametric approach when the amount of data below the limit of detection was low (eg, 20%).

Perhaps, more importantly, our results highlight the dependence of the parametric approach upon the validity of the underlying assumptions. It may be impossible to confirm that the assumptions are correct when analyzing real data, and the efficiency of the method as described in our simulations may be misleading. This problem was demonstrated in the t-distribution simulation: the assumptions of the parametric approach were violated, and the multiple imputation approach had greater efficiency. The multiple imputation approach by its nature makes minimal assumptions and uses the empirical distribution R. The value of the multiple imputation approach can also be seen in the HIV RNA example, whereby the interaction would not have been revealed using the parametric method alone. The use of the hybrid approach demonstrated that the multiple imputation and parametric methods are not necessarily disjoint, and they combine the best of 2 approaches: increased information about the distribution from a retested sample, and parametric modeling for those samples remaining below the new limit of detection. Further work evaluating this approach in the presence of different alternatives is warranted.

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REFERENCES

1.Schulte PA, Perera FP. Molecular Epidemiology Principles and Practices. San Diego: Academic Press, Inc.; 1993.
2.Muñoz A, Gange SJ. Methodological issues for biomarkers and intermediate outcomes in cohort studies. Am J Epidemiol. 1999;20:29–42.
3.Mellors JW, Muñoz A, Giorgi JV, et al. Plasma viral load and CD4+ lymphocytes as prognostic markers of HIV-1 infection. Ann Intern Med. 1997;126:946–954.
4.Ho DD, Moudgil T, Alam M. Quantitation of human immunodeficiency virus type 1 in the blood of infected persons. N Engl J Med. 1989;321:1621–1625.
5.Coombs RW, Collier AC, Allain J-P, et al. Plasma viremia in human immunodeficiency virus type 1 in the blood of infected persons. N Engl J Med 1989;321:1626–1632.
6.Raboud JM, Montaner JS, Conway B, et al. Suppression of plasma viral load below 20 copies/ml is required to achieve a long-term response to therapy. AIDS. 1998;12:1619–1624.
7.Ridker PM. High-sensitivity C-reactive protein potential adjunct for global risk assessment in the primary prevention of cardiovascular disease. Circulation. 2001;103:1813–1818.
8.Rifai N, Tracy RP, Ridker PM. Clinical efficacy of an automated high-sensitivity C-reactive protein assay. Clin Chem. 1999;45:2136–2141.
9.Mellors JW, Rinaldo CR, Gupta P, et al. Prognosis in HIV-1 infection predicted by the quantity of virus in plasma. Science. 1996;272:1167–1170.
10.Vandamme AM, Schmit JC, Van Dooren S, et al. Quantification of HIV-1 RNA in plasma: comparable results with the NASBA HIV-1 RNA QT and the AMPLICOR HIV monitor test. J Acquir Immune Defic Syndr Hum Retrovirol. 1996;13:127–139.
11.Lew J, Reichelderfer P, Fowler M, et al. Determinations of levels of human immunodeficiency virus type 1 RNA in plasma: reassessment of parameters affecting assay outcome. TUBE Meeting Workshop Attendees. Technology utilization for HIV-1 blood evaluation and standardization in pediatrics. J Clin Microbiol. 1998;36:1471–1479.
12.Kirstein LM, Mellors JW, Rinaldo CR, et al. Effects of anticoagulant, processing delay, and assay method (branched DNA versus reverse transcriptase PCR) on measurement of human immunodeficiency virus type 1 RNA levels in plasma. J Clin Microbiol. 1999;37:2428–2433.
13.Lang JR, Bolton S. A comprehensive method validation strategy for bioanalytical applications in the pharmaceutical industry–1. Experimental considerations. J Pharm Biomed Anal. 1991;9:357–361.
14.Lang JR, Bolton S. A comprehensive method validation strategy for bioanalytical applications in the pharmaceutical industry–2. Statistical analyses. J Pharm Biomed Anal. 1991;9:435–442.
15.Hughes MD. Analysis and design issues for studies using censored biomarker measurements with an example of viral load measurements in HIV clinical trials. Stat Med. 2000;19:3171–3191.
16.Kleeberger CA, Lyles RH, Margolick JB, et al. Viability and recovery of peripheral blood mononuclear cells cryopreserved for up to 12 years in a multicenter study. Clin Diagn Lab Immunol. 1999;6:14–19.
17.Moulton LH, Curriero FC, Barroso PF. Mixture models for quantitative HIV RNA data. Stat Methods Med Res. 2002;11:317–325.
18.Lynn HS. Maximum likelihood inference for left-censored HIV RNA data. Stat Med. 2001;20:33–45.
19.Farzadegan H, Hoover DR, Astemborski J, et al. Sex differences in HIV-1 viral load and progression to AIDS. Lancet. 1998;352:1510–1514.
20.Anastos K, Gange SJ, Lau B, et al. The association of race and gender with HIV-1 RNA levels and immunologic progression. JAIDS. 2000;24:218–226.
21.Barkan SE, Melnick SL, Preston-Martin S, et al. The Women's Interagency HIV Study. Epidemiology. 1998;9:117–125.
22.Kaslow RA, Ostrow DG, Detels R, et al. The Multicenter AIDS Cohort Study: rationale, organization, and selected characteristics of the participants. Am J Epidemiol. 1987;126:310–318.
23.Marschner IC, Betensky RA, DeGruttola V, et al. Clinical trials using HIV-1 RNA-based primary endpoints: statistical analysis and potential biases. J Acquir Immune Defic Syndr Hum Retrovirol. 1999;20:220–227.
24.Jacqmin-Gadda H, Thiebaut R, Chene G, et al. Analysis of left-censored longitudinal data with application to viral load in HIV infection. Biostatistics. 2000;1:355–368.
25.Little RJ, Rubin DB. Statistical Analysis with Missing Data. New York: Wiley; 1987.
26.Muñoz A, Carey V, Taylor JMG, et al. Estimation of time since exposure for a prevalent cohort. Statistics in Medicine. 1992;11:939–952.
27.Rubin DB. Multiple Imputation for Nonresponse in Surveys. New York: Wiley; 1987.
28.Lau B. Methods for Adapting Biomarker Assays with Increased Sensitivity in Prospective Cohort Studies: Application to HIV RNA in the Women’s Interagency HIV Study. [dissertation]. Baltimore, MD: Johns Hopkins School of Public Health; 2000.
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