How to Analyze Censored Concentration Data Using Modern Statistical Methods of Survival Analysis: Background and Nonparametric Methods

Abstract

Quantitative analytical methods for measuring concentrations of chemical substances in aquatic systems typically have acceptable accuracy and precision only for an intermediate range of analyte concentrations. Outside this range, the uncertainty of concentration estimates is too high to justify reporting them as valid measurements for use in statistical analyses. Therefore, concentration estimates falling below the lower reporting limit (LRL) are typically reported as the LRL, along with a code indicating that the measured values fell below the LRL. Such data are called left-censored data. Similarly, concentration estimates falling above the upper reporting limit (URL) are typically reported as the URL, along with a code indicating that the measured values exceeded the URL. Such data are known as right-censored data. Censored data violate assumptions underlying most traditional statistical methods, such as t-tests, regression analysis, and analysis of variance. We briefly review various statistical methods that have been employed for the analysis of censored concentration data, then review in greater detail some modern statistical survival-analysis methods that have become available in standard software within the last 10 years and can be applied to concentration data with both left- and right-censored values. Methods are illustrated with real data.

1. Introduction

Field studies designed to assess water quality commonly measure concentrations of various nutrients, organic and inorganic chemical contaminants, and pathogenic or fecal indicator microorganisms. The purpose of these studies is often to characterize spatial or temporal trends in the concentrations of specific analytes, assess spatial patterns of contaminant concentrations in the vicinity of a known source, or assess compliance with water-quality standards. All of these study goals require high-quality quantitative data on concentrations of the target analytes, and most require rigorous statistical methods that are appropriate for the types of data produced.

A common nuisance in such studies is that a significant proportion of the concentration estimates fall below the lower reporting limit (LRL) for one or more of the analytical methods being used to estimate analyte concentrations, as previous authors have pointed out for diverse sources of freshwater and marine surface water, groundwater, and wastewater, as well as a wide variety of analytes (e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]). Because they fall below the relevant LRL, these estimates are, by definition, too uncertain to be treated as valid concentration estimates in statistical analyses; one can only be confident that the actual concentrations to which they correspond are less than the LRL. For that reason, their numerical values typically are not reported or used in statistical analyses. They are, instead, reported only as being less than the LRL, along with the numerical value of the LRL. In statistics, such values are referred to as left-censored data.

Some of the analytical methods that are commonly used in field studies of aquatic systems also have an upper reporting limit (URL) that is low enough to be encountered in practice. An example is the Colilert-18^® enzyme–substrate method for quantifying Escherichia coli (E. coli) and total coliform bacteria. Like concentration measurements that fall below the LRL, those that exceed the URL are, by definition, too uncertain to be treated as valid concentration estimates in statistical analyses; one can only be confident that the corresponding actual concentrations are greater than the URL. Such measurements are typically reported only as being greater than the URL, along with the numerical value of the URL. In statistics, such values are referred to as right-censored data.

Datasets that include a mix of valid and censored concentration estimates are common in studies of aquatic systems, with left-censoring being particularly prevalent. To illustrate this point, we briefly describe three examples of studies conducted in different types of water bodies where the percentage of censored concentration estimates was reported. The prevalence of such data in studies of aquatic systems is the main reason it is important for researchers to be familiar with rigorous statistical methods that can be applied to censored concentration data.

The first example is an extensive river survey conducted to assess potential impacts of effluent from an industrial facility on water quality and biological communities of the Sabine River in the state of Texas (USA) [18]. Part of the study involved analysis of 18 different polycyclic aromatic hydrocarbons (PAHs) and 13 different chlorinated organic compounds known to be present in the effluent in sediment samples from several river reaches located upstream, adjacent to, and downstream from the facility’s effluent outfalls. For PAHs, 7.4% of the concentration estimates were below the respective detection limits (hence, left-censored), with marked differences between analytes in the percentage of censored values. For chlorinated organic compounds, 37.4% of the analyte concentration estimates were below the respective detection limits, again with marked differences between analytes in the percentage of left-censored values.

The second example is a study of the concentrations of three different SARS-CoV-2 genetic markers in wastewater from various municipal and institutional facilities in southeastern Michigan estimated using ddPCR (droplet digital polymerase chain reaction) analysis [19]. Monitoring concentrations of pathogenic viruses in wastewater has become an important epidemiological tool for detecting outbreaks of COVID-19 and other viral diseases around the world [20,21,22]. The twofold purpose of the Michigan study was to assess and compare the prevalence of the SARS-CoV-2 virus in wastewater originating from different types of facilities (and, hence, the prevalence of infected individuals living in or using those facilities) and to assess the relative utilities of the three genetic markers for detecting and quantifying the virus. The percentage of concentration estimates for different facilities that fell below the detection limit (i.e., left-censored estimates) ranged from 4% to 50% and was 29% for the combined dataset.

The third example is one that we will use in subsequent sections of this paper to illustrate the use of the main statistical methods we review. Real-time quantitative polymerase chain reaction (qPCR) analysis is increasingly being used to assess levels of E. coli contamination at recreational beaches because its turnaround time for producing results is much shorter than that for Colilert-18 and colony-count methods [23,24,25,26]. However, qPCR also tends to produce much higher percentages of censored data than Colilert-18 does. McNair et al. [27] used data comprising paired Colilert-18 and qPCR estimates of E. coli concentrations in split samples from bathing beaches in the state of Michigan (USA) to compare the two types of data with respect to their usefulness for characterizing E. coli concentration distributions and detecting differences between such distributions for beaches on inland lakes, rivers, and the Laurentian Great Lakes. At one of the beaches (Wolverine Lake), 2.4% of the Colilert-18 concentrations in 2019 were right-censored and none were left-censored, while 32.8% of the qPCR concentrations were left-censored and none were right-censored. At another beach (Ross Lake), 14.8% of the Colilert-18 concentrations in 2019 were right-censored and none were left-censored, while 25.9% of the qPCR concentrations were left-censored and none were right-censored. In the combined 2019 and 2020 data for all beaches and sampling dates (3205 pairs of observations), 2.8% of the Colilert-18 concentrations were left-censored and 0.3% were right-censored, while 52.2% of the qPCR concentrations were left-censored and none were right-censored.

In sharp contrast to the prevalence of censored concentration data in field studies, standard parametric statistical methods such as t-tests, least-squares regression, and analysis of variance require that the data to be analyzed consist entirely of valid concentration measurements. These are the statistical methods that would normally be used to assess potential differences between sampling sites or dates or to compare concentrations with water quality standards if all the data are valid measurements, but they are not appropriate for datasets that include a significant proportion of censored values. How, then, can such datasets be rigorously analyzed?

Historically, two different ways of approaching this problem have been popular with environmental scientists and engineers. The most common approach is to replace censored concentrations with fictitious values fabricated in some convenient way (e.g., by arbitrarily replacing each <LRL value with LRL/2), then proceed with statistical analysis as if all the data are valid measurements. The other approach is simply to treat concentration estimates below the LRL or above the URL as if they are valid measurements. The rationale for the latter approach is that, while values outside the reporting limits do, indeed, have unacceptably high uncertainty, one can argue that they are nevertheless likely to be closer to the true concentrations than are the fictitious values fabricated by the former approach.

Both of these traditional approaches to handling censored concentration data are clearly unsatisfactory and scientifically indefensible. Beginning in the 1980s, various authors drew attention to this fact. Helsel [4] provides a good summary of evidence that the two historical approaches to handling censored concentration data just mentioned (and several others) can lead to incorrect or misleading conclusions, in addition to providing a good survey of alternative statistical methods for handling such data that were available in standard statistical software in the early 2000s.

The main purpose of the present paper is to provide an updated survey of modern statistical methods of survival analysis that are appropriate for analyzing concentration data that may include a mix of valid, left-censored, and right-censored values and to provide examples of how to implement these methods using state-of-the-art R ([28], version 4.5.2) and SAS ([29], version 9.4M9) statistical software designed primarily for application to time-to-event data in medical research. Though certain classical statistical methods can be applied to data that include censored values (e.g., by partitioning observed concentrations into two or more discrete classes, with <LRL and >URL values included in separate classes), we focus mainly on methods from the specialized statistical discipline of survival analysis that are specifically designed for datasets that may include multiple types of censored values.

Various authors have previously advocated for the use of traditional methods of survival analysis (e.g., the Kaplan–Meier estimator of the survival curve, estimates of its point-wise confidence limits, and estimates of distribution quantiles and their confidence intervals) that were originally developed for medical studies where the only type of censoring is right censoring. In contrast, most of the older environmental studies that produced censored concentration data used detection limits as the sole basis for censoring (a practice that we discourage; see Section 2), so left censoring was the only form of censoring that was possible. Use of traditional methods of survival analysis for such data requires the reversal of the concentration scale to transform left-censored values into right-censored values, which is a kluge introduced by Ware and Demets in 1976 [30] (Helsel [4] calls this transformation “flipping”; Gillespie et al. [31] call the Kaplan–Meier estimator for left-censored data the “reverse Kaplan–Meier estimator”). But reversing the concentration scale cannot resolve the problem of both left-censored and right-censored values being present, as will sometimes be the case if proper methods of quantitative analytical chemistry are used to establish both lower and upper reporting limits.

In this paper, we focus on modern survival-analysis methods that do not require reversal of the concentration scale and that can be applied to datasets that include any mix of valid, left-censored, and right-censored values. The methods we review permit one to estimate the probability distribution function and its pointwise confidence intervals for the concentration of a given analyte for an individual sampling site or date, estimate concentration quantiles and their 95% confidence intervals, perform one-sided and two-sided pairwise comparisons of samples from different sites or dates, test the null hypothesis of homogeneity for samples for multiple sites or dates, and perform tests for monotonic trends across samples for multiple sites or dates.

The present review restricts attention to nonparametric methods. These methods are widely used in medical studies involving time-to-event data, can accommodate data that include a mix of valid and different types of censored data, and are available in standard statistical software. The other class of statistical methods that is widely used in medical studies involving time-to-event data consists of semiparametric methods. These include regression-like methods that allow one to incorporate one or more covariates (explanatory variables) in statistical analyses, where the dependence on covariates is parametric but all other aspects of the statistical models are nonparametric. Examples include the Cox proportional hazards model and the accelerated failure-time model. However, versions of these methods that are currently available in standard statistical software cannot accommodate data that include both left-censored and right-censored values and can only accommodate data that include left-censored values if the data are flipped. We therefore exclude them from this review.

The remainder of this paper is organized as follows. Section 2 reviews some background information from quantitative analytical chemistry that explains how censored concentration data arise. Section 3 provides a brief overview of various methods that have been used in the past for analysis of censored data. Section 4 reviews some important concepts and terminology from statistical survival analysis. Section 5 presents an overview of the main nonparametric methods of survival analysis that can be applied to censored concentration data for one, two, or multiple sites or dates. Section 5 also includes examples where each statistical method is applied to real concentration data using R software (examples using SAS software are included in the online Supporting Information). We conclude with a general discussion in Section 6.

2. How Do Censored Concentration Data Arise?

Methods of quantitative analytical chemistry typically have a working range of concentrations of the target analyte within which the reliability of concentration estimates is adequate and outside of which it is not. The lower and upper boundaries of this range serve as reporting limits for the method, meaning that only concentrations lying between the two limits are sufficiently reliable to justify reporting them as numerical concentrations and using them in statistical analyses. In laboratory studies, one can often dilute or concentrate samples that lie outside the reporting limits and re-analyze them. But in large-scale field studies and monitoring programs, this is usually either not necessary (e.g., if the goal is to assess compliance with a regulatory standard and the standard lies well within the reporting limits), not feasible (e.g., if funding levels do not permit re-analysis), or not possible (e.g., if the sample analysis time exceeds the sample holding time, as is often the case when monitoring E. coli levels with the Colilert-18 enzyme–substrate method). In such cases, values lying below the lower reporting limit (LRL) are reported simply as less than the LRL, along with the numerical value of the LRL. Similarly, values lying above the upper reporting limit (URL) are reported simply as greater than the URL, along with the numerical value of the URL. This common data-reporting practice ensures that any dataset that includes unreliable measurements lying outside the reporting limits will contain censored observations.

The specific estimation methods and numerical criteria for the LRL and URL vary for different types of analytical methods and often vary among different authors and standards organizations for the same analytical method. But because of the importance of these limits as the source of censoring for concentration data, it will be useful to briefly consider an example that illustrates some of the main reasons they are necessary while largely avoiding the specific numerical criteria that vary among authors and standards organizations (see [32,33,34] for additional details). For this purpose, we will consider the large class of analytical methods that estimate concentrations using a calibration curve. Examples of such methods that are commonly used to analyze samples from aquatic systems include molecular absorption spectroscopy, gas chromatography–mass spectrometry (GC-MS), liquid chromatography–tandem mass spectrometry (LC-MS/MS), inductively coupled plasma mass spectrometry (ICP-MS), and qPCR. In all these cases, the calibration curve relates instrument response (signal intensity, peak area, ion counts, and threshold cycle number) to known standard concentrations. The curve is typically constructed using multiple standards across a defined concentration range.

For the many methods that rely on them, calibration curves play a central role in defining the statistical boundaries of environmental measurement data. Before any statistical treatment of censored values occurs, analytical chemistry has already imposed quantitative boundaries through the calibration model. These constraints establish the lower and upper limits within which measured concentrations are considered quantitatively reliable. Thus, the calibration curve is not merely a laboratory artifact; it is the mechanism that defines the measurable domain of a dataset. For this reason, the development and validation of the calibration curve should be viewed as the first statistical decision in any environmental measurement program that requires one because it defines the quantitative limits within which environmental data can be interpreted defensibly.

In mathematical language, a calibration curve is a strictly monotonic function, i.e., $y=S\left(x\right)$ (in most cases, increasing with analyte concentration—but in some cases, decreasing), that maps the concentration (x) of a target analyte to the intensity (y) of a signal from an analytical instrument (Figure 1A, blue curve). Once fitted to data from a series of known analyte concentrations, the calibration curve can be inverted to yield a strictly monotonic function, i.e., $x=S^{-1}\left(y\right)$ (Figure 1B, blue curve), that can be used to estimate unknown concentrations of the analyte in field samples by measuring the intensity of the instrument signal.

A common example is the degree to which light of a specific wavelength is absorbed by a dissolved analyte (as measured with a spectrophotometer) over a fixed path length as a function of analyte concentration. According to Beer’s law, the proportion of transmitted light (i.e., transmittance) should decline exponentially with increasing analyte concentration, so the negative logarithm of transmittance (i.e., absorbance) should be directly proportional to the analyte concentration. But as Skoog et al. ([35], p. 306) note, systematic deviations from direct proportionality commonly occur, especially at high concentrations, although sometimes at low concentrations as well. Thus, the relationship between analyte concentration and absorbance—the calibration curve—is reliably linear only over an intermediate range of concentrations.

As an example, Fritz and Schenk ([36], p. 350) make the following specific recommendation: “Spectrophotometric measurement of absorbance (or transmittance) is very inaccurate at both very low and very high readings. For this reason, the concentration of absorbing substance should always be adjusted until the absorbance is in the range 0.10–1.00 (or to 1.50 for some precision spectrophotometers)”. But as noted above, when conducting large-scale field studies with limited funding, dilution or concentration of samples is often either not necessary (e.g., if the purpose of the study is to assess compliance with a numerical regulatory criterion that is well within the limits of quantification without dilution), not feasible, or not possible. In such cases, left-censored, right-censored, or doubly censored data may result, depending on the range of field concentrations.

An analytical method used to create a calibration curve is typically adequately sensitive to analyte concentration only over an intermediate range of concentrations, with sensitivity (slope of the calibration curve, Figure 1A) markedly declining at concentrations below and above this range. The insensitive portions of the curve appear as “shoulders” at low and high concentrations (Figure 1A, blue curve). The sensitive part of the calibration curve is typically well approximated by a straight line (Figure 1A, dashed red line), possibly after transforming the analyte concentration or instrument signal intensity—or both. Linearity at intermediate concentrations is often supported by a mechanistic physicochemical theory (e.g., Beer’s law), which breaks down at low and high concentrations where the calibration curve becomes distinctly nonlinear. Also, to the best of our knowledge, the rigorous statistical theory of calibration (including estimation of confidence intervals for predicted concentrations) has been adequately developed only for linear calibration curves (([37], Section 4.6), [38], ([39], Section 15.3)). For these reasons, the LRL must be no smaller than the lower limit of the concentration interval where the calibration curve is well approximated by a straight line, and the URL must be no larger than the upper limit of that interval (Figure 1A, gray arrows).

An additional reason for using only the linear portion of the calibration curve is that when the curve is inverted for use in estimating analyte concentrations from measured values of instrument signal intensity, the shoulders at low and high instrument signal intensities become steeply sloped regions where concentration estimates are highly sensitive to variation in instrument signal intensity (Figure 1B, blue curve). When signal intensity measurements are converted to estimated concentrations, these steep, nonlinear portions of the inverted curve greatly magnify the uncertainty of concentration estimates, as indicated by the relative standard deviation (RSD, called the coefficient of variation in statistics) (Figure 1B, orange curve). The degree of magnification is approximately proportional to the local slope of the inverted curve: the steeper the slope is, the more sensitive the inverted curve is to error in the signal intensity and, hence, the greater the error in predicted analyte concentration. The result is unacceptably high uncertainty of concentration estimates for signal intensities in the nonlinear parts of the inverted curve, with the acceptable limit of uncertainty commonly being chosen as RSD = 0.1 (Figure 1B, solid gray horizontal line). The concentrations corresponding to the lower and upper limits of this interval of acceptable measurement uncertainty are the lower and upper limits of quantification (LLOQ and ULOQ) (Figure 1B, gray arrows).

As noted above, the linear part of the calibration curve is fitted to calibration data consisting of measured values of instrument signal intensity for replicates of a series of standards with known concentrations of the target analyte using an appropriate statistical regression method (Figure 1C). This procedure yields estimates of the slope and intercept parameters of the linear regression model. The inverse function is then easily determined, yielding a function that allows one to estimate unknown analyte concentrations in samples based on measured values of the instrument signal intensity (Figure 1D). The associated statistical theory for linear calibration (e.g., [38]) also allows one to estimate 95% confidence intervals for estimated concentrations. But because the fitted calibration curve and its inverse function are only valid for the employed range of standard concentrations, the minimum and maximum standard concentrations and their associated minimum and maximum instrument signal stengths impose another set of limits on how small the LRL can be and how large the URL can be (Figure 1C, gray arrows).

Taking all of the above considerations into account, the LRL should be chosen as the maximum of the lower limit of linearity, the lowest standard concentration, and the LLOQ, while the URL should be chosen as the minimum of the upper limit of linearity, the highest standard concentration, and the ULOQ (Figure 1D). Only measured concentrations (x) such that LRL $⩽x⩽$ URL are reported as numerical concentrations. Values below the LRL are reported simply as the LRL, along with a code indicating that the measured value was below the LRL. Similarly, values above the URL are reported as the URL, along with a code indicating that the measured value was above the URL. The exact format for recording this information in a spreadsheet or database depends jointly on which statistical methods and which statistical software (e.g., R or SAS) will be used to analyze the data (see Section 4.2). But the key point is that because sample concentration estimates less than the LRL or greater than the URL are not reported as numerical concentrations, datasets containing such values will be left-censored by the LRL and right-censored by the URL. The entire process of sample collection, sample preparation, sample analysis, estimation of analyte concentrations with an inverse calibration curve, data reporting subject to reporting limits, and statistical analysis of the resulting censored data is summarized schematically in Figure 2 (adapted from Figure 1 of [32]).

Finally, we note that older studies of contamination in aquatic systems often ignore the quantification limits of concentration measurements and focus only on the limit of detection (LOD), which is then used as the sole reporting limit. Definitions of the LOD vary, but it is essentially a measure of the lowest instrument signal intensity (and corresponding estimate of analyte concentration) that one can be reasonably confident is too high to have been produced by a sample that does not contain the target analyte [32,33,34]. An important fact is that the LLOQ is typically greater than the LOD. For example, Eurachem ([34], pp. 24–25) notes that, as a rule of thumb, the LOD is roughly 3 times the standard deviation of the blanks, while the LLOQ is roughly 10 times the standard deviation of the blanks and, hence, about 3.3 times the LOD. The LOD is useful for providing evidence that an analyte is or is not present in a sample at a concentration high enough for the analytical method to reliably detect it, but it does not provide evidence that the concentration can be measured with acceptable uncertainty. That evidence is provided by the LLOQ, ULOQ, and (for methods that employ a calibration curve) the limits of linearity of the calibration curve and the maximum and minimum calibration standards. It follows that the LOD is typically irrelevant in determining reporting limits. The relationships between the LOD, LRL, and URL are shown schematically in Figure 3.

3. Examples of Methods for Analyzing Censored Concentration Data

Helsel [4] and Shoari [10] review various rigorous and not-so-rigorous methods that have been used in the past by researchers in environmental science and engineering to analyze censored concentration data. Both of these reviews include numerous examples of studies from the older literature that involve left-censored concentration data, to which we refer interested readers. In this section, we outline a few of the most commonly used methods to illustrate the variety of alternative approaches.

3.1. Deleting Censored Concentrations

One approach to handling censored data is to delete them or, slightly less flagrantly, to delete all data from sampling locations where some of the data are censored. This approach eliminates the problem of deciding how to extract valid information from the censored data and also mitigates the problem of violating assumptions of standard statistical methods such as ordinary least-squares regression and analysis of variance (these statistical problems are briefly discussed in Section 3.2). Though it is not difficult to find examples of published studies that use this approach, it should never even be considered. For example, left-censored values will be the lowest concentrations in a dataset if there is a single LRL and will be among the lowest if the data include subsets with different LRLs. It should be obvious that deleting these values will bias any inferences about the mean or median concentration for a given sampling site or time and, therefore, will invalidate statistical comparisons between different sites or times.

3.2. Data-Fabrication Methods

The most common approach to handling censored concentration data in the past has been to replace censored observations with fabricated values, then analyze the edited dataset using standard statistical methods for uncensored data such as t-tests, least-squares regression models, or analysis of variance. For purposes of illustration, we discuss two examples of the various types of data-fabrication methods: replacing each <LRL value with LRL/2 and replacing all <LRL values with fabricated values derived from a regression model based on quantiles or order statistics of an assumed parametric probability distribution.

3.2.1. One-Half LRL

Replacement of censored values with one-half the LRL is, in our experience, by far, the most common method of handling left-censored concentration data in applied environmental studies. If, for example, there is a single LRL and 20% of the concentrations in a dataset comprising 100 observations fall below it, then after replacement, 20% of the data will share exactly the same numerical value (=LRL/2), and this value will have no objective support in the data. Aside from the completely arbitrary nature of this method, the resulting large number of ties in the revised dataset clearly is not consistent with a normal distribution or with any other continuous distribution. Therefore, parametric statistical methods that assume the data come from a normal distribution, as well as nonparametric methods that simply assume the data come from a continuous distribution, will fail any valid assessment of their assumptions. The same will be true if observations less than the LRL are replaced with a value other than LRL/2, such as zero, the LRL, or LRL/$\sqrt{2}$, all of which have been employed in the past (e.g., [4,14,17,40,41,42]). Because any single replacement value for left-censored data will be both fabricated and arbitrary and because replacement of all left-censored data with the same value will produce a large number of ties when the proportion of left-censored data is significant, we strongly discourage the use of this method.

3.2.2. Methods Based on Quantiles or Order Statistics

An obvious and serious problem with data-fabrication methods that substitute the same value for every left-censored observation is that any significant number of censored observations will result in a sufficient number of ties to invalidate most standard statistical methods. This fact stimulated interest in data-fabrication methods that spread the fictitious concentrations over the interval between 0 (or $-\infty$ for log-transformed data) and the LRL instead of concentrating them on a single value (Figure 4). Here, we mention two methods of this type: one based on quantiles and the other on order statistics. These methods are more complicated than the LRL/2 method just described, so to conserve space, we describe them only briefly here (Shumway et al. [43] provide a good, concise overview). Importantly, both methods depend critically on the choice of a specific parametric probability distribution (typically, a normal or lognormal distribution) as the distribution from which the data were sampled. The basic idea behind the methods is to fit the chosen probability distribution to the uncensored data, then assign fabricated values to the censored data based on the theoretical quantiles or expected values of the order statistics for the chosen distribution. These methods avoid the problem of assigning the same value to all censored data, but they suffer from two serious shortcomings: the values assigned to the censored data are fabricated, and—especially when a nontrivial proportion of the data are censored—the missing portion of the concentration distribution precludes an adequate assessment of the appropriateness of the chosen probability distribution.

3.3. Ignoring the Reporting Limits

The simplest approach to dealing with data that are below the LRL or above the URL is to ignore the problem and use all the estimated concentrations as if they were valid concentration measurements, including readings that are outside the reporting limits and therefore known to have unacceptably high levels of measurement error. As mentioned in the Introduction, the original rationale for this approach is that, while concentration measurements outside the range of quantification admittedly have unacceptably high imprecision and/or bias, they may nevertheless be closer to the actual concentrations than the purely fictitious values created by data-fabrication methods. In our experience, this approach is not nearly as common as replacing censored values with half the LRL.

Helsel ([4], Section 3.3) discusses the problems inherent in this method. Gilbert ([44], p. 178) states that this method produces biased estimates of the mean and variance of the analyte concentration. Antwiler and Taylor [40] found that it performed poorly in estimating the mean, quartiles, and standard deviation compared to various data-fabrication methods, parametric maximum-likelihood estimators, and nonparametric survival-analysis methods. However, based on the results of a simulation study, George et al. [42] concluded that ignoring the reporting limits is the best approach. But because their simulation study did not include a plausible representation of measurement error, the excessive levels of measurement bias and imprecision that characterize data outside the reporting limits in real datasets—the two main problems that reporting limits are designed to mitigate—are absent from the simulated data. Reporting limits were therefore prevented from playing their intended role in filtering out highly biased and imprecise data (because no such data were present), and it is therefore not surprising that statistical analyses were not degraded by treating data outside the reporting limits as valid measurements.

Our own view is that concentration estimates that lie outside the reporting limits—and that are therefore known to be contaminated with unacceptably high levels of measurement error—should not be used in statistical analyses, even if their numerical values are available. A guiding principle of rigorous statistical analysis of censored data is to use only information about the data that is known with high confidence. Survival-analysis methods applied to properly censored concentration data are fully consistent with this principle. They use numerical measurements only for data whose bias and imprecision are known to be acceptably low. Censored values are included in analyses, but only information about them that is known with high confidence is employed—namely, that the true concentration corresponding to each censored concentration estimate lies below a particular numerical value (the LRL) or above a particular numerical value (the URL). These methods are statistically rigorous and have an extensive and proven record of successful application in medical research involving censored time-to-event data.

3.4. Partitioning Concentrations into Discrete Classes

An obvious way to handle data that include censored values is to partition the concentration scale into two or more discrete classes, one of which contains all concentrations less than the LRL as a subset and another that contains all concentrations greater than the URL as a subset. In our experience, the most common example of this approach occurs in monitoring or assessment studies where the goal is to determine whether the concentration of a particular analyte exceeds a threshold value (e.g., a water-quality standard or some other management target). Typically, it is possible to choose or adapt an analytical method such that the LRL and URL for every analyzed sample bracket the threshold value (if the URL is greater than any observed concentration, it can be treated as infinite). Every analyzed sample can then be classified as either exceeding the threshold (“success”) or not (“failure”), and standard nonparametric statistical methods for binomial success probabilities or proportions can be used to rigorously analyze the data. The binomial parameter here represents the probability that the concentration measured in a randomly chosen sample will exceed the threshold value, but for most purposes, it can be interpreted as the theoretical proportion of a large number of samples that would exceed the threshold.

Several kinds of statistical analyses can be performed using such data, with the appropriate methods depending on sample size and, in some cases, on whether the estimated success probability is very close to 0 or 1. Examples include estimating the success probability and its 95% confidence interval, estimating the difference between success probabilities and their 95% confidence intervals for two sites or dates, and testing the null hypothesis that the success probabilities for two sites or dates are the same versus the two-sided or appropriate one-sided alternative hypothesis. Particularly good estimators for confidence intervals of a binomial success probability are the Wilson interval and the Agresti–Coull interval [45,46]. For large samples, good estimators for confidence intervals of the difference between two success probabilities are the Newcombe hybrid score interval and the Agresti–Caffo interval [47,48,49]; for small samples, exact intervals can also be computed (e.g., [50,51]). For testing of the null hypothesis of no difference between the success probabilities for two sites or dates against one-sided or two-sided alternative hypotheses, classic large-sample tests are available [48,52], but small-sample tests are often preferable, with the best methods being the mid-P version of the exact conditional binomial test and various versions and modifications of Barnard’s exact unconditional binomial test [48,52,53,54].

3.5. Survival-Analysis Methods

The fact that the statistical methods of survival analysis are specifically designed to accommodate censored data distinguishes them from most other statistical methods, but this is not their only distinguishing feature. Another fundamental difference is that survival analysis focuses mainly on characterizing and comparing entire probability distributions rather than measures of location (central tendency). As discussed in the next section, probability distributions can be characterized in several mathematically equivalent ways, including using the probability distribution function, complementary probability distribution function (survivor function), and probability density function. Rigorous methods are also available for estimating distribution quantiles and their confidence limits and for assessing effects of covariates, but the main emphasis is always on characterizing and comparing entire distributions.

Three main types of statistical methods for characterizing and comparing probability distributions are available in survival analysis: nonparametric, semiparametric, and fully parametric methods. We briefly review all three of these approaches in the present section. The remainder of this paper, however, focuses entirely on nonparametric methods.

Nonparametric methods of survival analysis avoid assuming any specific probability distribution from which observations are sampled. The main tools employed for concentration data that include a mix of valid, left-censored, right-censored, and/or interval-censored observations are the Turnbull estimator of the complementary distribution function, various nonparametric tests of homogeneity and pairwise differences such as the log-rank test and its various generalizations, and nonparametric tests for monotonic trends. We discuss these tools in more detail in Section 5. In the Supplementary Materials, we provide examples of R and SAS code to illustrate applications of these methods to data.

Semiparametric methods also make no assumption about the mathematical form of the probability distribution from which observations are assumed to be sampled, but they do assume a parametric form for the dependence of this distribution on categorical or quantitative covariates. The best-known semiparametric tools are the Cox proportional hazards model, the accelerated failure-time model, and various extensions of both. Modern forms of these methods that can be applied to the full range of types of censored concentration data commonly encountered in studies of aquatic systems (i.e., left-censored, right-censored, doubly-censored, and interval-censored data) are still being actively developed by survival-analysis statisticians. As mentioned in the Introduction, we do not address semiparametric methods in this review because versions of these methods that can be applied to datasets that include both left-censored and right-censored data are not yet available in standard statistical software.

Fully parametric methods of survival analysis require one to assume a specific mathematical form for the probability distribution from which observations are assumed to be sampled. Methods that make use of covariates also require one to assume a specific parametric form for the dependence of the assumed distribution on the covariates. Applications of these methods to censored time-to-event data in medical research have shown that the statistical results often depend strongly on the chosen probability distribution, while the data typically do not provide sufficient evidence to adequately justify any particular choice [55]. For that reason, modern statistical survival analysis in medical research almost always uses nonparametric and semiparametric methods, and we suggest that these are likely to be the most useful methods for environmental concentration data as well.

4. Basic Concepts and Terminology

4.1. Functions for Specifying Probability Distributions

The specialized statistical discipline of survival analysis deals with time-to-event data where some of the event times are censored in the sense that one only knows they are greater than some value (e.g., the final observation time in an experiment), less than some value (e.g., the first observation time in an experiment), or between two values (e.g., successive observation times). In biological and medical applications, where the term “survival analysis” originated, the event of interest might be (for example) the time until an individual organism dies, a seed germinates, or an embryo transitions from one developmental stage to the next. In engineering applications, the event of interest might be the break-down or failure of a machine or electrical system, and in this context, the same statistical discipline is often referred to as reliability analysis or failure-time analysis. The broader term, time-to-event analysis, emphasizes the conceptual unity of these applications and reduces the terminological confusion created by applying methods with names referring to survival or failure times to, for example, seed germination times.

In all of these traditional applications of survival analysis, the focus is on the time until an event occurs. But if one takes a step back and considers more generally the basis of these statistical methods, it becomes clear that the underlying concepts are not restricted to processes unfolding in time. In reality, the fundamental concepts simply involve properties of probability distributions, and the discipline of survival analysis deals with methods for characterizing and comparing those distributions when some of the data are censored. Thus, there is no reason why survival analysis cannot be applied to data comprising concentrations of chemical compounds or microorganisms where some of the observations are censored (e.g., below the LRL). In our experience, however, the traditional terminology of survival analysis acts as an impediment to acceptance of these methods for applications involving concentrations instead of event times because the picturesque terms typically used (e.g., survivor function, failure-time distribution function, and hazard function) only make intuitive sense if the data are generated by a process that acts over time and terminates the life of an organism or a machine. Therefore, we suggest alternative terms that, while less vivid, have the advantage of not suggesting associations that are incompatible with applications to chemical or microbial concentration data.

We assume that concentration data from any given sampling site or date are sampled from a probability distribution and that the true values of all concentrations (not the estimates generated by an analytical instrument) lie in the interval of $[0,\infty )$. For technical reasons, we assume the probability is zero that the true value of the analyte concentration is identically zero, since we wish to exclude probability distributions that have an “atom” of probability at a concentration of zero. Implicitly, then, we assume that “everything is everywhere” so that any contaminant of interest will be present not only at pristine sites but also at concentrations that might be so low as to be undetectable by available chemical or microbiological analytical methods.

We use capital letters (e.g., X) to denote random variables representing concentrations and lower-case letters (e.g., x) to denote observed or measured values of random variables. Four functions related to continuous random variables are important in survival analysis, with their traditional names being the probability distribution function, survivor function, probability density function, and hazard function (

Table 1. Names of the four main probability functions used in continuous-time survival analysis. Left column: Standard names used in the literature for time-to-event data. Two of these (survivor function and hazard function) are likely to create confusion when presenting results for concentration data. Right column: Suggested generic names that may prevent this confusion when presenting results, along with abbreviations used in this paper.

Name for Time-to-Event Data	Suggested Name for Concentration Data
Probability distribution function	Probability distribution function (PDF)
Survivor function	Complementary probability distribution function (cPDF)
Probability density function	Probability density function (pdf)
Hazard function	Attenuation function (AF)

). In order to understand the statistical methods of survival analysis, it is necessary to be familiar with these functions, which we now briefly discuss.

The probability distribution function (PDF) of concentration X, denoted as $F\left(x\right)$, is a dimensionless function that tells us the probability that the value of random variable X is less than or equal to x. That is,

$$F\left(x\right)=Pr\{X⩽x\},$$

(1)

where x typically has dimensions of Mass · Volume⁻¹ and its possible values lie in the interval of $[0,\infty )$. It is sometimes desirable to analyze concentration data after log-transforming them, in which case possible values of x lie in the interval of $(-\infty ,\infty )$; to simplify our exposition, we assume concentration data have not been log-transformed and, therefore, lie in the interval of $[0,\infty )$). We restrict attention to distribution functions ($F\left(x\right)$) that are differentiable and for which the limit of $F\left(x\right)$ as x approaches zero from above is zero, i.e., ${lim}_{x\downarrow 0}F\left(x\right)=0$.

Be aware that, whereas “probability distribution function” is the term used for $F\left(x\right)$ in probability theory (with the more general term “distribution function” being used in measure theory for functions that have nothing to do with probability), it is common in applied statistics to refer to $F\left(x\right)$ as the cumulative probability distribution function. But as probability theorists dutifully point out (e.g., [56], p. 179), the qualifier “cumulative” is redundant because a distribution function is cumulative by definition.

The complementary probability distribution function (cPDF) of concentration X, denoted as $\mathcal{F}\left(x\right)$, is a dimensionless function with the same domain as $F\left(x\right)$ that represents the probability that $X>x$. It is simply the complement of $F\left(x\right)$ and is given by

$$\mathcal{F}\left(x\right)=1-F\left(x\right).$$

(2)

In survival analysis, $\mathcal{F}\left(x\right)$ is called the survivor function (or survival function), but we find this term confusing when dealing with concentration data and therefore usually employ the generic term from probability theory.

The probability density function (pdf), denoted as $f\left(x\right)$, is the derivative of the PDF with respect to its argument (x). In symbols,

$$f\left(x\right)=F^{\prime }\left(x\right)=-{\mathcal{F}}^{\prime }\left(x\right),$$

(3)

where the prime (′) denotes differentiation with respect to x. It follows from Equation (3) that $F\left(x\right)$ can be expressed in terms of $f\left(x\right)$ as

$$F\left(x\right)={\int }_0^{x}f\left(\xi \right)\mathrm{d}\xi .$$

(4)

Note that both $F\left(x\right)$ and $\mathcal{F}\left(x\right)$ are dimensionless, while $f\left(x\right)$ has the same dimensions as $1/x$.

The final function related to random variable X that is important to be familiar with is usually called the hazard function in survival analysis and reliability analysis and denoted by $h\left(x\right)$. However, we feel the term “hazard function” is confusing when dealing with concentration data, so we propose an alternative. We have been unable to think of a similarly picturesque term for applications to concentration data, so we propose an alternative that, like probability density function, probability distribution function, and complementary probability distribution function, does not have a strong connection to any particular application and, therefore, should at least not be confusing.

Before stating the alternative term we propose, it will be helpful to provide a generic definition of the function ($h\left(x\right)$) that is called the hazard function in survival analysis. For probability distributions whose distribution function is differentiable, we may define $h\left(x\right)$ as

$$h\left(x\right):={\displaystyle \frac{f\left(x\right)}{\mathcal{F}\left(x\right)}}=-{\displaystyle \frac{{\mathcal{F}}^{\prime }\left(x\right)}{\mathcal{F}\left(x\right)}}.$$

(5)

Because $\mathcal{F}\left(x\right)$ is dimensionless, it is clear from Equation (5) that $h\left(x\right)$ has the same dimensions as $f\left(x\right)$, which are those of $1/x$.

It follows from Equation (5) that function $\mathcal{F}$ obeys the differential equation, i.e.,

$${\displaystyle \frac{\mathrm{d}\mathcal{F}}{\mathrm{d}x}}=-h\left(x\right)\mathcal{F}\left(x\right),x\ge 0.$$

(6)

The solution, subject to an initial condition of $\mathcal{F}\left(0\right)=1$, is

$$\mathcal{F}\left(x\right)=exp\left(-{\int }_0^{x}h\left(\xi \right)\mathrm{d}\xi \right).$$

(7)

This result and Equations (2) through (5) show that functions $F\left(x\right)$, $\mathcal{F}\left(x\right)$, $f\left(x\right)$, and $h\left(x\right)$ are mathematically equivalent in the sense that knowledge of one of them implies the other three. However, each function has a different use in survival analysis, with the cPDF or survivor function ($\mathcal{F}\left(x\right)$) and hazard function ($h\left(x\right)$) playing particularly important roles in the statistical theory.

Equation (7) shows that $h\left(x\right)$ determines the rate at which $\mathcal{F}\left(x\right)$ decreases with an increasing x. In survival applications, this rate of decrease is the per capita mortality rate, while in reliability analysis, it is the per machine failure rate. In both of these time-to-event applications, $h\left(x\right)$ is often referred to by the less application-specific but still picturesque term “hazard function”, as noted above. While picturesque terms like mortality rate, failure rate, and hazard function are useful as mnemonic devices and in stimulating intuition in survival and reliability applications, they become confusing in applications to other types of data, such as seed germination times (where seed survival and germination are biologically distinct phenomena) or chemical concentrations. Therefore, bearing in mind that $h\left(x\right)$ determines the rate at which $\mathcal{F}\left(x\right)$ decreases with an increasing x, we suggest attenuation function (AF) as a reasonable application-independent term.

Examples illustrating the role of $h\left(x\right)$ in attenuating $\mathcal{F}\left(x\right)$ as x increases are shown in Figure 5. The figure shows plots of $h\left(x\right)$ (left panel) and $\mathcal{F}\left(x\right)$ (right panel) for the Weibull distribution. The AF and cPDF for this distribution are given by

$$h\left(x\right)=rs{\left(rx\right)}^{s-1},\mathcal{F}\left(x\right)=e^{-{\left(rx\right)}^s},x\ge 0,$$

(8)

where $s>0$ is the shape parameter and $r>0$ is the rate parameter. Three examples of each function are shown in Figure 5. All examples have a shape parameter of $s=3$, but values of the rate parameter differ ($r=1.0,1.5,2.0$). Note that as r increases, the $h\left(x\right)$ curves increase faster with increasing x values, causing the $\mathcal{F}\left(x\right)$ curves to attenuate faster with increasing x values.

4.2. Censored Data

Survival analysis is unique among statistical disciplines in that it is specifically designed to handle censored data. In laboratory studies that produce time-to-event data (e.g., seed germination tests and bioassay tests with mortality as an endpoint), it is common to employ a fixed study duration that is long enough to ensure that most test subjects experience the event of interest (e.g., germination or mortality) but not long enough to ensure that all do. For test subjects that have not experienced the event by the end of the experiment, the actual event time (T) is unknown; all that is known is that T is greater than the duration (D) of the experiment. Thus, there is a “veil” covering the time axis to the right of D, preventing us from seeing the portion $(D,T]$ of the event time (T) that is beyond D. Such data are called right-censored data. Traditional statistical methods such as t-tests and analysis of variance—or even the usual maximum-likelihood estimator for the mean—cannot be applied to such data because they require a valid measurement for every observation.

With concentration data, censoring occurs most commonly when a measured concentration is less than the LRL of the analytical method being employed. Many analytical methods for environmental samples also have a URL that may be low enough so that field concentrations sometimes exceed it. Examples include the Colilert-18^® method for quantifying coliform bacteria and analytical methods that employ a calibration curve based on a fixed range of standards. For such methods, a concentration measurement will be censored if it is greater than the URL. LRLs and URLs are set by statistical methods and ensure that all reported concentration estimates have acceptably low bias and imprecision.

An important difference between censored survival times and censored concentrations is that, whereas no estimate of the true survival time is available for a censored event time, an estimate with unacceptably high uncertainty will be available for a censored concentration value if all concentration estimates, regardless of their quality, are reported. Thus, in statistical analyses of censored concentration data, a temptation exists to employ data of unacceptably high bias and imprecision that typically does not exist for time-to-event data. To avoid succumbing to this temptation, it is useful to bear in mind two fundamental principles of rigorous statistical analysis of censored data:

• Avoid fabricating data whenever possible.
• Use only information from available data that is known with high confidence.

All modern methods of survival analysis used in medical and reliability studies comply with both of these principles.

When a concentration estimate (x) is less than the LRL, we know that its numerical value is contaminated with too much measurement error and bias to justify treating the value as reliable and using it in statistical analyses; all we can be confident of is that $x<\mathrm{LRL}$. In this case, the concentration is said to be left-censored. Similarly, if a concentration estimate (x) exceeds the URL, all we can be confident of is that $x>\mathrm{URL}$, and the concentration is said to be right-censored. Datasets that include a mix of valid, left-censored, and right-censored concentrations are said to be doubly censored, though each censored observation is either left- or right-censored.

Another type of censoring that is important to know about is interval censoring, though it is rarely encountered with concentration data. An individual concentration estimate is interval-censored if we know only that it lies between two values of L and R, with $L<R$. This contrasts with double censoring, where each individual concentration is either left-censored or right-censored but both types of censored values are present in the dataset. In reality, all concentration data are either left-, right-, or interval-censored because concentrations can be measured to only a relatively small number of significant digits. But this complication is universally ignored in statistical analyses, and we know of no commonly used analytical method that produces concentration estimates that would be treated as strictly interval-censored in practice.

Nevertheless, there are two reasons why it is important to be aware of the term “interval-censored”. First, in the typical case where the LOD for an analytical method is less than the LLOQ, some authors (e.g., [4]) suggest treating concentrations between the LOD and the LLOQ as interval-censored and only values below the LOD as left-censored (we disagree with this approach because all concentration estimates below the LRL are too uncertain to treat as valid numerical measurements). The second and more important reason is that this term is commonly used in statistical software and the literature in a loose sense that includes valid, left-censored, right-censored, and strictly interval-censored data as special cases. Thus, statistical methods for interval-censored data are typically intended for application to datasets that are allowed but not required to include strictly interval-censored values, as well as valid and possibly left-censored and/or right-censored values.

In recording data from an empirical study where some of the concentrations are censored, it is very important to use a format for the data that is compatible with the statistical software that will be used to analyze the data and is also simple and intuitive for laboratory personnel to use when entering the data. The vast majority of time-to-event datasets include a mix of valid (usually called “exact” in the statistical literature) and right-censored data, and as a result, the proper way to record such data is highly standardized. In contrast, most environmental datasets consisting of measured concentrations include a mix of valid and left-censored data, and many include right-censored data as well. Formats for recording such data are not fully standardized in the statistical literature or software, so it is important to find out what format is compatible with the particular software one intends to use.

Based on our own experience, it is also important to consider that students and laboratory technicians who often produce and record the data find certain formats counter-intuitive and difficult to work with. A basic law of data entry is that the more logical and intuitive the format for entering data is, the less frequent recording errors will be. Data recorded in any of the common formats can be converted to any of the others by a few lines of computer code, so there is no need to sacrifice convenience and accuracy in recording data to achieve the data format required by a particular software program. As long as all required information is recorded, one can use different formats for data entry and data analysis.

Two rather different methods for characterizing doubly censored data are encountered in statistical software, both of which require that two values be recorded. One method characterizes each observation using a concentration (or time) and a status code, while the other characterizes each observation using the two endpoints of a concentration (or time) interval. We call these the value-plus-code and interval-endpoint approaches. With the value-plus-code approach (which R calls the interval format), the recorded concentration is either a valid concentration, the LRL (implying the concentration is left-censored at that value), or the URL (implying the concentration is right-censored at that value), and the status code tells us which. For example, with the value-plus-code data format in R, status codes 0, 1, and 2 mean that the recorded value is right-censored, valid, and left-censored. With the interval-endpoint approach (which R calls the interval2 format and which is the default format in SAS), each observation is viewed as an interval specified by its left and right endpoints. If the two endpoints are the same, the value of the shared endpoints is a valid concentration. If the left endpoint is missing (indicated by logical constant NA in R and by a blank field in SAS) and the right endpoint is numeric, then the concentration is left-censored and the right endpoint is the LRL. If the right endpoint is missing and the left endpoint is numeric, then the concentration is right-censored and the left endpoint is the URL. Figure 6 illustrates some alternative data coding schemes for the value-plus-code and interval-endpoint approaches to characterizing doubly censored data. As this figure suggests, datasets that are composites of data from different labs, analysis dates, technicians, and so on typically have a variety of different LRL and URL values, so these limits must be recorded for each analyzed sample.

The same two approaches can be used if strictly interval-censored data are present—but with a slight extension for the value-plus-code approach. Strictly interval-censored data require two concentration values (instead of only one) plus a status code when the value-plus-code approach is used. In R’s interval format, the first concentration value must be numeric and is used in statistical analysis regardless of the value of the status code; the second concentration value is ignored unless the status code is 3 (which means the observation is strictly interval-censored and lies between the first and second recorded concentrations), in which case it must be greater than the first concentration. With the interval-endpoint format, strictly interval-censored data are implied whenever both endpoints are numeric and the first is less than the second.

As a very simple illustration, the first several lines in a spreadsheet containing data from two stream sampling sites might be set up as shown in Figure 7. The left and middle examples show the two common ways to format censored data for R, while the right table shows standard SAS formatting. The site column in each example shows two hypothetical sampling sites (S1 and S2). With R’s interval formatting (left example), each entry in the conc column contains a numerical concentration (in mg L⁻¹, say), which represents either a valid concentration, the LRL for a left-censored concentration, or the URL for a right-censored concentration, as determined by the corresponding integer code in the status column (0: URL; 1: valid concentration; 2: LRL). With R’s interval2 formatting (middle example), each entry in the left and right columns contains either a concentration or R’s logical constant, i.e., NA (“not available”), for missing data, both of which are treated as numerical values by R and, therefore, can be included in the same vector or the same column of a data frame. SAS formatting is essentially the same as R’s interval2 format, except that spreadsheet cells for missing data are simply left blank. Note that these concentration data consist of one left-censored value (with LRL = 0.03 mg L⁻¹), one right-censored value (with URL = 2.50 mg L⁻¹), and four valid concentrations. Also shown in these examples is a column labeled turb, which represents turbidity (in nephelometric turbidity units, NTU). This is an example of a quantitative explanatory variable or covariate, which are often used in semiparametric regression models. If there were additional explanatory variables, their values for different samples would be included in additional columns, with one column for each variable.

An important censoring-related distinction in all types of survival analysis is whether the limits of quantification are fixed or are adjusted to ensure that a predefined proportion of the samples yields valid concentration measurements (or a predefined proportion of test subjects experiences the event of interest). For concentration data, it usually is appropriate to assume fixed censoring (known as Type I censoring in survival analysis), and we will do so throughout this paper. In studies that produce time-to-event data, it is sometimes desirable to run an experiment until a fixed number or proportion of test subjects experiences the event of interest instead of running it for a fixed total duration. Because the censoring time is random rather than fixed for this type of censoring (known in survival analysis as Type II censoring), somewhat different statistical methods are sometimes required for their analysis.

Another important distinction we must mention is between informative and non-informative censoring. A full explanation of this distinction requires a technical detour that is inappropriate for this paper (we refer the interested reader to the book by Kalbfleisch and Prentice ([58], Section 3.2)), but the basic idea is rather simple: For left-censored concentration data, the numerical value of the LRL or URL must not be correlated with the median or another location measure of the probability distribution from which concentrations in the dataset were sampled. This means that the numerical value of the LRL or URL must provide no information about whether the true median concentration at a sampling site is high or low or whether the true median concentration at one site is higher or lower than at another site. This condition would be violated if, for example, sampling sites with unusually high levels of a particular contaminant also had unusually high levels of a second contaminant that interfered with analysis of the first in a way that increased the LRL. In this paper, we restrict attention to non-informative censoring both because this is the most common type in applications and because most statistical methods of survival analysis currently available in software packages require it.

5. Nonparametric Survival-Analysis Methods

In this section, we review modern nonparametric methods of survival analysis that can accommodate the full range of censoring types likely to be encountered in concentration data from studies of aquatic systems and that are currently available in standard statistical software. But before we begin, it will be prudent to address a common misconception regarding nonparametric statistical methods in general.

In our experience, scientists and engineers often assume nonparametric statistical methods are inherently much less powerful than parametric methods and therefore should be avoided unless there is no defensible alternative. In contrast, statisticians who design and analyze medical research studies that produce censored time-to-event data almost always use distribution-free nonparametric or semiparametric methods, and the same is true of statisticians working on various other types of applications. For example, Conover ([59], p. 2) states, “Nonparametric methods have become essential tools in the workshop of the applied scientist who needs to do statistical analyses. When the price for making a wrong decision is high, applied scientists are very concerned that the statistical methods they are using are not based on assumptions that appear to be invalid, or are impossible to verify”. Hollander et al. ([52], p. $xiii$) go further and state that “the nonparametric approach is the preferred methodology for statisticians”— except, of course, for certain types of designed experiments that produce sufficiently large quantities of uncensored data with statistician-friendly properties, so the simple distributional assumptions of standard parametric methods are appropriate and can be convincingly verified.

Hollander et al. ([52], pp. 1–2) provide a list of 10 advantages of nonparametric statistical methods, including the following (which we paraphrase) that are relevant to survival-analysis methods that can be applied to concentration data:

• Nonparametric methods are much less sensitive to outliers than parametric methods.
• Nonparametric methods make fewer assumptions that must be verified regarding properties of the underlying populations from which the data are obtained. In particular, no specific probability distribution is assumed for concentration data. These methods are therefore applicable in many cases where parametric methods are invalid because their distributional assumptions are clearly untenable or where these assumptions cannot be convincingly assessed.
• In cases where the probability distribution assumed by a parametric method is consistent with properties of the sampled population (regardless of whether this consistency can be convincingly demonstrated) nonparametric methods often are nearly as efficient as parametric methods. This means that they require only slightly larger sample sizes to achieve the same statistical power.

Regarding the last two points, Lehmann ([60], p. $viii$) states, “The feature of nonparametric methods mainly responsible for their great popularity (and to which they owe their name) is the weak set of assumptions required for their validity. Although it was believed at first that a heavy price in loss of efficiency would have to be paid for this robustness, it turned out, rather surprisingly, that the efficiency of the Wilcoxon tests and other nonparametric procedures holds up quite well under the classical assumption of normality and that these procedures may have considerable advantages in efficiency (as well as validity) when the assumption of normality is not satisfied”.

Having hopefully dispelled the notion that nonparametric statistical methods are inherently much less powerful than parametric methods, we turn now to our review of nonparametric survival-analysis methods that can be applied to concentration data that may include left-censored, right-censored, and interval-censored values. Three main types of problems can be addressed with these methods:

1.

Characterizing the distribution of concentrations for an individual study site or date;

2.

Pairwise comparison of concentration distributions for different sites or dates;

3.

Testing homogeneity and detecting monotonic trends in concentration distributions for three or more sites or dates.

We discuss these three problems in turn in this section.

As background, we note that previous expositions of survival-analysis methods for censored concentration data that we are aware of focus mainly on data where the only type of censoring is left censoring, making it possible to use the older methods of survival analysis after flipping the data. However, both R and SAS now have functions and procedures that permit most of the key nonparametric methods of survival analysis to be applied to datasets containing any mixture of valid, left-censored, right-censored, and interval-censored data on the original concentration scale. We restrict attention to these newer methods.

5.1. Characterizing Concentration Distributions

Nonparametric survival-analysis methods yield quantitative estimates of the PDF, cPDF, and corresponding pointwise confidence intervals for an individual site or date without requiring one to assume a specific probability distribution for the analyte concentration. They also yield estimates of the median and other quantiles and their 95% (or other) confidence intervals. These methods can be viewed as extensions of the simple Kaplan–Meier estimator of the cPDF and are the only type of survival-analysis method that both accommodates all major types of censoring and has been available in standard statistical software for more than 10 years.

Regarding the median and other quantiles, we note that the mean and standard deviation are almost never estimated in medical research studies involving censored time-to-event data. There are two reasons for this, both of which apply to environmental concentration data. The first is that the classical estimators of the mean and standard deviation employed in parametric statistics cannot be employed when censored values are present (unless fabricated values are substituted for censored data, which we discourage) because they require valid numerical estimates of all observed concentrations. The second reason is that the data typically come from distributions with pronounced positive skew, and the median rather than the mean is the preferred measure of location for such distributions (e.g., [61], p. 30). Alternative measures of dispersion that are more appropriate than the standard deviation for censored data from positively skewed distributions include the interquartile range (the difference between the third and first quartiles, or the 75th and 25th percentiles) and the semi-interquartile range (half the interquartile range) (e.g., [61], p. 31). Thus, unless one is conducting an applied study in which statistical methods are rigidly imposed by a governmental regulatory agency, one should characterize the location and dispersion of the distribution underlying a set of censored concentration data for a single sampling site or date by using the median (and 95% confidence interval) and interquartile or semi-interquartile range.

Nonparametric estimates of the cPDF for doubly censored data are usually obtained using an iterative procedure due to Turnbull [62,63] or various more recent extensions, with data represented in the interval2 format (or its SAS equivalent) discussed above. Each observation is recorded in one of three forms $(\mathtt{NA},x_i)$, $[x_i,x_i]$, or $(x_i,\mathtt{NA})$, representing left-censored, valid, or right-censored observations, respectively. For valid observations, $x_i$ is a measured concentration; for left-censored observations, $x_i={\mathrm{LRL}}_i$; and for right-censored observations, $x_i={\mathrm{URL}}_i$. Thus, each observation (i) includes a single distinct numerical value ($x_i$) that specifies one or both interval endpoints, depending on whether the observation is or is not censored.

The $x_i$ values for different observations typically include repeats or ties, meaning that two or more observations share the same value of $x_i$ (usually an LRL or URL). Let n be the total number of observations and suppose there are $m\le n$ distinct values among the $x_i$. Let $x_{\left[j\right]}$ denote these distinct values for $j=1,2,\dots ,m$, ordered so that $x_{\left[1\right]}<x_{\left[2\right]}<\cdots <x_{\left[m\right]}$. Finally, let $V_{\left[j\right]}$ denote the number of valid concentrations with $x_i=x_{\left[j\right]}$, $L_{\left[j\right]}$ denote the number of left-censored concentrations with $x_i={\mathrm{LRL}}_i=x_{\left[j\right]}$, and $R_{\left[j\right]}$ denote the number of right-censored concentrations with $x_i={\mathrm{URL}}_i=x_{\left[j\right]}$. With this notation, the data may be represented by the set of m 4-tuples $(x_{\left[j\right]},V_{\left[j\right]},L_{\left[j\right]},R_{\left[j\right]})$ for $j=1,2,3,\dots ,m$. The kernel of the likelihood function, given the data, is then

$${\displaystyle \prod _{j=1}^m}f{\left(x_{\left[j\right]}\right)}^{V_{\left[j\right]}}F{\left(x_{\left[j\right]}\right)}^{L_{\left[j\right]}}\mathcal{F}{\left(x_{\left[j\right]}\right)}^{R_{\left[j\right]}},$$

(9)

where $f(·)$ is the pdf, $F(·)$ is the PDF, and $\mathcal{F}(·)$ is the cPDF of the unknown probability distribution from which the $x_i$ values are assumed to be sampled.

In parametric survival analysis, functional forms would be specified for the pdf, PDF, and cPDF, and their parameter values would be chosen to maximize the likelihood kernel. Nonparametric approaches avoid choosing any specific parametric probability distribution. As mentioned above in Section 3.5, the main reason for avoiding parametric distributions is that results for censored data often depend strongly on which distribution is chosen, while evidence supporting any particular choice is often inconclusive [55]. The goal then shifts to estimating the numerical values of the PDF or cPDF at distinct concentrations ($x_{\left[j\right]}$, $j=1,2,3,\dots ,m$), with the value of the function remaining unchanged between these points, as in the classic Kaplan–Meier survivor function for right-censored data.

Unlike the simple case of right censoring, there is no explicit formula for the nonparametric maximum likelihood estimator of the cPDF when both left and right censoring are allowed. However, Turnbull [62,63] proposed an iterative algorithm for obtaining numerical estimates of the values of the cPDF at distinct values of $x_{\left[j\right]}$ (an example of what would later be called the expectation-maximization or EM algorithm), yielding a decreasing step function similar to the Kaplan–Meier survival curve. Turnbull’s method and various modifications of it are employed in several statistical functions now available in R and SAS, making it simple for users to estimate the cPDF and PDF, confidence intervals, and quantiles for doubly censored datasets (technical details underlying Turnbull’s method are beyond the scope of this review; see Turnbull’s papers [62,63]).

5.1.1. R Example

In R, the cPDF, PDF, and their pointwise confidence intervals for doubly censored data can be estimated using a combination of the Surv() and survfit() functions from the survival package in essentially the same way these functions and intervals are estimated for traditional right-censored data encountered in time-to-event analysis. Quantiles can be estimated from a survfit object using the quantile() function from the survival package.

To illustrate the use of R software for statistical analysis of censored concentration data, all examples in the present paper utilize a set of 3205 estimates of the concentration of fecal coliform bacterium Escherichia coli (E. coli) at freshwater bathing beaches across the state of Michigan, USA, in 2019 and 2020. The data we use here were produced for county health departments by multiple laboratories throughout the state as part of Michigan’s annual beach monitoring program, using the Colilert-18^® enumeration method. Details on sampling sites, dates, and methods are presented by McNair et al. [27]. The LRL and URL for the data are 1 and 2420 MPN/100 mL (MPN: Most Probably Number) for all samples. Roughly 3% of the data are left-censored, and 0.3% are right-censored, with the percentages varying markedly among counties. Of particular importance for the examples presented in the present paper is the fact that every beach was assigned to one of three classes (inland lake, river, or coastal beach) according to whether the beach was located on an inland lake, a river, or one of the Laurentian Great Lakes. A question of interest is whether E. coli concentrations tend to differ among these three types of beach.

Figure 8 shows the estimated cPDF (left panel) and PDF (right panel) for coastal beaches, along with pointwise 95% confidence intervals. The quartiles correspond to concentrations at which the PDF crosses the dashed horizontal lines at 0.25, 0.50, and 0.75 on the vertical axis.

Table 2. Estimated quantiles for probabilities of 0.25, 0.50, and 0.75 and their 95% confidence limits for E. coli concentrations (MPN/100 mL) at coastal beaches in Michigan. The corresponding cPDFs and PDFs are plotted in Figure 8. LCL and UCL denote the estimated lower and upper 95% confidence limits for the quantiles. Quantiles and confidence limits represent E. coli concentrations in units of MPN/100 mL. Computations were performed with R functions Surv(), survfit(), and quantile().

Probability	Quantile	LCL	UCL
0.25	9	8	11
0.50	30	26	34
0.75	84	75	93

shows the estimated quartiles and 95% confidence intervals produced by the quantile() function. Confidence intervals for the cPDF, PDF, and quartiles reflect the value of the conf.type argument of survfit() (actually, survfit.formula()), which was set to “log-log” for consistency with the SAS default.

5.1.2. SAS Example

In SAS, the cPDF, PDF, their pointwise confidence intervals, quantiles, and confidence intervals for the quantiles all can be estimated with the ICLIFETEST procedure. An example demonstrating the estimation of these quantities is included in the online Supplementary Materials.

5.2. Pairwise Comparison of Concentration Distributions

Nonparametric methods of survival analysis include methods that can be used to test hypotheses about potential differences in concentration distributions for different sampling sites or dates when the data contain a mix of valid concentrations and one or more types of censored data. The main tests of interest are tests for pairwise differences between sites, tests of homogeneity for $k⩾3$ sites, and tests for monotonic trends across $k⩾3$ sites. We discuss tests for pairwise differences in the present section; homogeneity tests and tests for monotonic trends are discussed in Section 5.3.

The null and alternative hypotheses for pairwise tests between sites are most naturally stated in terms of the cPDFs rather than the PDFs, since the ordering of sites by the cPDF is the same as their ordering by the median when the cPDFs do not cross decisively (see below). The alternative hypothesis can be two-sided or one-sided. Following Oller and Langohr [64], the null and alternative hypotheses (H₀ and H₁) for pairwise tests between two sites ($i\ne j$) can be stated as follows:

H₀:

${\mathcal{F}}_i\left(x\right)={\mathcal{F}}_j\left(x\right)$ for all x;

H₁:

${\mathcal{F}}_i\left(x\right)\ne {\mathcal{F}}_j\left(x\right)$ for some x (two-sided);

H₁:

${\mathcal{F}}_i\left(x\right)\ge {\mathcal{F}}_j\left(x\right)$ for all x, with “>” for some x (one-sided, “site i greater than site j”);

H₁:

${\mathcal{F}}_i\left(x\right)\le {\mathcal{F}}_j\left(x\right)$ for all x, with “<” for some x (one-sided, “site i less than site j”).

As usual, the appropriate one-sided alternative must be chosen before examining the data and should be based on objective and rational considerations such as the location of a known contaminant source relative to different sampling sites.

Both R and SAS provide functions or procedures for performing pairwise tests based on extensions of the Fleming–Harrington class of tests appropriate for doubly censored and interval-censored data. This type of test employs two parameters, traditionally denoted $\rho$ and $\lambda$, that allow one to weight low, moderate, and high concentrations uniformly or differentially in comparing sites. However, unless one has a compelling a priori reason for using differential weighting, we recommend following accepted practice in survival analysis of time-to-event data and employing equal weighting ($\rho =\lambda =0$), in which case the test is often referred to as a log-rank test.

All of the tests for comparing concentration distributions for discrete groups (e.g., different sampling sites or dates) utilize tests from the Fleming–Harrington family of tests, which are “geared to detect alternative hypotheses where the hazards [AFs] between groups differ but do not cross” ([64], p. 3). To the best of our knowledge, available software packages for handling doubly censored and interval-censored data do not produce estimates of AFs, so the “no crossing” condition for AFs cannot be directly assessed. A practical indirect way to determine whether there is strong evidence that two AFs ($h_i\left(x\right)$ and $h_j\left(x\right)$) for distinct groups (i and j), cross decisively is to overlay plots of the Turnbull estimates of the PDFs or cPDFs for the two groups and visually assess whether they cross decisively. If so, there is strong evidence that the AFs cross decisively; otherwise, not.

The rationale for this simple visual diagnostic is presented as follows. If AFs $h_i\left(x\right)$ and $h_j\left(x\right)$ ($i\ne j$) do not cross, then either $h_i\left(x\right)\ge h_j\left(x\right)$ for all x or $h_i\left(x\right)\le h_j\left(x\right)$ for all x. Let ${\mathcal{F}}_i\left(x\right)$ denote the cPDF for any group (i). A basic property of continuous probability distributions with support ($0\le x<\infty$) is that the survival and hazard functions are related by

$${\mathcal{F}}_i\left(x\right)=exp\left(-{\int }_0^x{h}_i\left(\xi \right)d\xi \right)$$

(10)

for all x. If $h_i\left(x\right)\le h_j\left(x\right)$ for all x, then ${\int }_0^x{h}_i\left(\xi \right)d\xi \le {\int }_0^x{h}_j\left(\xi \right)d\xi$ and Equation (10) implies ${\mathcal{F}}_i\left(x\right)\ge {\mathcal{F}}_j\left(x\right)$ for all x. Similarly, if $h_i\left(x\right)\ge h_j\left(x\right)$ for all x, then ${\mathcal{F}}_i\left(x\right)\le {\mathcal{F}}_j\left(x\right)$ for all x. This argument shows that if two hazard functions do not cross, then neither do the corresponding survival functions. By contraposition, if two survival functions do cross, then the corresponding hazard functions also must cross, implying that Fleming–Harrington tests are not appropriate. Thus, a simple diagnostic for determining whether Fleming–Harrington tests are appropriate for two groups is to plot the corresponding survival functions and visually determine whether they cross decisively (we say “decisively” because the survival functions being plotted are only estimates, and we want convincing evidence that the unknown true survival functions cross). If they do, we can be confident that the hazard functions also cross and, therefore, that Fleming–Harrington tests are not appropriate. On the other hand, if the survival functions do not cross, then the evidence is consistent with the hypothesis that the hazard functions do not cross decisively and, hence, that Fleming–Harrington tests are appropriate.

We note that the same problem arises with Fleming–Harrington tests for right-censored time-to-event data, where the Kaplan–Meier (instead of Turnbull) estimator of the cPDF is employed. For example, Hosmer et al. [65] state the following:

A problem can occur if the estimated survival functions cross one another. This means that, in some time intervals, one group will have a more favorable survival experience, while in other time intervals, the other group will have the more favorable experience. This situation is analogous to having interaction present when applying Mantel-Haenszel methods to a stratified contingency table… Fleming, Harrington, and O’Sullivan [66] proposed a method that addresses the problem by using, as a test statistic, the maximum observed difference between the two survival functions. This test has not been implemented in any software package… For the time being, our only check is via a visual examination of the plot of the Kaplan-Meier estimator for the groups being compared. If we see that the curves cross, then this “interaction” may be present.

Thus, Hosmer et al. [65] recommend the same visual diagnostic that we outlined above: overlay plots of the two estimated cPDFs (or PDFs) and visually determine whether they decisively cross. Kaplan–Meier estimates of the cPDFs are used with right-censored data, and Turnbull estimates are used with left-censored, doubly censored, or interval-censored data.

5.2.1. R Example

R’s standard survival package includes functions that allow one to perform the types of hypothesis tests discussed above for datasets where all censored values are right-censored. Datasets that include both left-censored and right-censored values cannot be handled by these functions, even if the concentration scale is reversed.

These limitations of the survival package are not a serious concern because R add-on packages are available that provide an extensive set of statistical tools for dealing with data that include any mix of valid, left-censored, right-censored, or interval-censored data. In our opinion, the most useful of these packages for the types of hypothesis tests addressed in this section is currently the FHtest package [64], which includes functions that permit comparison of two or more cPDFs using an extended version of the Fleming–Harrington class of tests. The two key functions for our present purposes are FHtesticp() and FHtestics(), which implement the extended Fleming–Harrington test using a permutation distribution and a score-vector distribution, respectively. We are not aware of any guidance on minimum sample sizes, so we suggest employing the exact=TRUE option when the FHtesticp() function is used unless the computational method fails because the sample sizes are too large. An example of PDFs and cPDFs for E. coli concentrations (MPN/100 mL) at inland-lake and coastal beaches in Michigan is shown in Figure 9. Note that the 25th, 50th, and 75th percentiles for inland-lake beaches are all less than the corresponding percentiles for coastal beaches (left panel) and that the cPDF (${\mathcal{F}}_{\mathrm{I}}\left(x\right)$) for inland-lake beaches is less than the cPDF (${\mathcal{F}}_{\mathrm{C}}\left(x\right)$) for coastal beaches for all concentrations of $x>0$ (right panel). In this sense, then, the ordering of concentration distributions by quartiles is the same as the ordering by cPDF. This correspondence between orderings by quartiles and orderings by cPDFs makes cPDFs a more natural choice than PDFs for stating null and alternative hypotheses regarding pairwise differences and monotonic trends in concentration distributions for different sites or dates.

As just noted, visual inspection of Figure 9 reveals that the estimated quartiles for inland-lake beaches are consistently less than those for coastal beaches. But is there statistically sound evidence that the apparent difference between the two concentration distributions is real? First, we are not aware of any plausible connection between processes that might be responsible for producing unusually high or low true concentrations in the field and processes in the laboratory that are responsible for determining the censoring levels (LRL and URL), so it is reasonable to assume that censoring is non-informative. Next, we note that the cPDF for inland-lake beaches is less than that for coastal beaches at all E. coli concentrations, so the two PDFs certainly do not cross decisively. In view of these two properties of the data, we used the FHtesticp() function with parameters of $\rho =0$ and $\lambda =0$ to test the null hypothesis of no difference against the two-sided alternative hypothesis. R returns a p-value of $1.5\times {10}^{-6}$, which provides strong evidence that the null hypothesis is false and, hence, that the two distributions are different. If we had a valid a priori reason to be interested only in the one-sided alternative with inland-lake beaches less than coastal beaches, the p-value would decrease to $7.5\times {10}^{-7}$, which, again, provides strong evidence that the null hypothesis is false but now supports the alternative that E. coli concentrations tend to be lower at inland-lake beaches than at coastal beaches.

5.2.2. SAS Example

Pairwise tests comparing cPDFs can be performed in SAS with the ICLIFETEST procedure. An example is presented in the online Supplementary Materials.

5.3. Tests of Homogeneity and Monotonic Trends in Multiple Concentration Distributions

The purpose of homogeneity tests is to determine if the inevitable numerical differences between observed concentration distributions for $k⩾3$ sites (or dates) can reasonably be attributed merely to chance variation. If not, there is strong evidence that at least two of the distributions exhibit a real difference. Following Oller and Langohr [64], the null and alternative hypotheses for k-site tests of homogeneity can be stated in terms of the cPDFs as follows:

H₀:

${\mathcal{F}}_1\left(x\right)={\mathcal{F}}_2\left(x\right)=\cdots ={\mathcal{F}}_k\left(x\right)$ for all x;

H₁:

${\mathcal{F}}_i\left(x\right)\ne {\mathcal{F}}_j\left(x\right)$ for some concentration (x) and pair of sites ($i,j\ne i$).

Note that the alternative hypothesis is necessarily two-sided. If the null hypothesis is rejected, pairwise comparisons, each with a two-sided alternative hypothesis, can be run (as described in Section 5.2) for all $k(k-1)/2$ distinct pairs of data groups (sites or dates) to determine which pairs show strong evidence of a difference after adjusting p-values to account for the number of comparisons made (typically using Holm’s method).

The purpose of tests for monotonic trends is to determine if there is strong evidence that the concentration distributions for $k⩾3$ sites exhibit a specified monotonic ordering. Again following Oller and Langohr [64], the null and alternative hypotheses can be stated in terms of the cPDFs as follows:

H₀:

${\mathcal{F}}_1\left(x\right)={\mathcal{F}}_2\left(x\right)=\cdots ={\mathcal{F}}_k\left(x\right)$ for all x;

H₁:

${\mathcal{F}}_1\left(x\right)\ge {\mathcal{F}}_2\left(x\right)\ge \cdots \ge {\mathcal{F}}_k\left(x\right)$ for all x, with “>” for some x and pair ($i<j$) (“decreasing trend”);

H₁:

${\mathcal{F}}_1\left(x\right)\le {\mathcal{F}}_2\left(x\right)\le \cdots \le {\mathcal{F}}_k\left(x\right)$ for all x, with “<” for some x and pair ($i<j$) (“increasing trend”).

The alternative hypothesis is necessarily one-sided but can be either increasing or decreasing. As usual with one-sided tests, the ordering of sites in the putative trend must be chosen before examining the data and should be based on objective and rational considerations.

As an aid to interpreting strictly monotonic trends in cPFDs, Figure 10 shows a hypothetical example with pdfs (top panel) and corresponding cPDFs (bottom panel) for concentrations at three sampling sites. True distributions are shown rather than empirical estimates, so there is no censoring. To further simplify the example, the pdfs have the same shape and are symmetric, so we may use means to characterize their locations. The means increase from site 1 to site 2 to site 3 and, hence, form an increasing trend, that is, ${\mu }_1<{\mu }_2<{\mu }_3$. The corresponding cPDFs have the property that ${\mathcal{F}}_1\left(x\right)<{\mathcal{F}}_2\left(x\right)<{\mathcal{F}}_3\left(x\right)$ for all $x>0$.

As in the case of pairwise tests, both R and SAS provide functions or procedures for performing tests of homogeneity and monotonicity for $k⩾3$ sites based on extensions of the Fleming–Harrington class of tests appropriate for doubly censored and interval-censored data.

5.3.1. R Example

A particularly useful R package for assessing the homogeneity and monotonicity of $k⩾3$ sites is the FHtest package [64]. For our present purposes, the two key functions this package provides are the sames ones we mentioned earlier in connection with pairwise tests: FHtesticp() and FHtestics(). These functions implement an extended version of the Fleming–Harrington test using a permutation distribution and a score-vector distribution, respectively.

Figure 11 shows PDFs (left panel) and cPDFs (right panel) for concentration distributions of E. coli concentration at inland-lake, river, and coastal beaches in Michigan. Visual inspection suggests that the distributions are not all the same. We used the FHtesticp() function to test the null hypothesis that all three cPDFs are the same against the alternative hypothesis that at least two of them differ, with equal weighting for all concentrations ($\rho =\lambda =0$). FHtesticp returned a p-value of $p=2.48\times {10}^{-11}$ for the test, which provides strong evidence that the null hypothesis is false and therefore at least two of the distributions differ.

Now, suppose there is a valid a priori reason for expecting that the cPDFs for the three types of beach either show no monotonic trend (the null hypothesis) or show a monotonically increasing trend in the order of inland lake → river → coastal. Assuming, for the purpose of this example, that there is such a reason, we used the FHtesticp() function to test the null hypothesis of no ordering against the one-sided alternative just stated. FHtesticp returned a p-value of $p=2.73\times {10}^{-12}$ for the test, providing strong evidence that the null hypothesis is false and, hence, that the monotonic increasing trend is real.

5.3.2. SAS Example

Tests of homogeneity and monotonicity for $k⩾3$ sites (or dates) can be performed in SAS using the ICLIFETEST procedure. An example is presented in the online Supplementary Materials.

6. Discussion

Concentration estimates produced by the methods of quantitative analytical chemistry are always uncertain to some degree. A key point is that the degree of uncertainty varies markedly with analyte concentration, typically in a U-shaped pattern where uncertainty is unacceptably high for sufficiently low and high concentrations but acceptably low for intermediate concentrations. Only concentration estimates in this intermediate range have measurement uncertainty that is low enough to justify reporting their numerical values and using these values in statistical analyses. The limits of this range are the lower reporting limit (LRL) and upper reporting limit (URL), while the range itself may conveniently be referred to as the reporting interval. For concentration measurements outside the reporting interval, all we know with confidence is that the true analyte concentrations are either below the LRL or above the URL. Datasets containing such values will be a mix of valid numerical concentrations (measurements lying within the reporting interval) and censored concentrations (measurements lying outside the reporting interval).

Historically, studies of aquatic systems involving the acquisition and statistical analysis of concentration data have often employed a statistically estimated limit of detection (LOD) as the sole reporting limit, ignoring the lower and upper limits of quantification and other factors that are important in defining the range of analyte concentrations for which measurement uncertainty is acceptably low and valid confidence intervals for predicted analyte concentrations can be estimated. If rigorous and informative statistical analyses are to be conducted that go beyond merely determining whether there is evidence that some of the measured concentrations exceed numerical water-quality standards, it is important to employ reporting limits based on all properties of the chemical measurement process that affect the reliability of the resulting concentration measurements. A properly defined LRL will typically lie well above the LOD, and a properly defined URL will lie well above the LRL.

Datasets that include censored concentration estimates pose statistical challenges for studies designed to assess potential differences between analyte concentrations for different sampling sites or dates; detect spatial or temporal trends; or identify relationships between concentration levels and explanatory variables such as water temperature, turbidity, or pH. However, most of the traditional statistical methods that one would likely employ in such studies cannot be validly applied to censored data. Empirical researchers in the past have therefore been rather creative in devising various alternative approaches for statistically analyzing censored data. The most common approaches are to either replace the censored values with fabricated ones or simply ignore the problem (i.e., treat all available data as valid, including data outside the reporting interval), then employ traditional statistical methods for uncensored data. Such approaches clearly are not scientifically defensible, but the importance of the questions researchers need to answer requires that some approach be chosen and used.

Shoari [10] summarizes the results of 18 published studies that assessed different statistical estimation methods for left-censored concentration data, including most of the methods we outlined in Section 3, as well as several others. She concludes, “The literature review reveals that previous studies reached different conclusions about appropriate analysis of left-censored data” (p. 29). Most of the published assessments she reviewed were based on simulation studies in which concentration distributions were sampled numerically from simple probability distributions (e.g., normal, lognormal, and gamma). Among the possible causes of the observed lack of consistency, Shoari mentions effects of different degrees of skewness, lack of robustness to departures from assumed probability distributions, and differences between methods in the confidence intervals for estimated parameters.

The approach we favor for most statistical analyses of censored concentration data is to use nonparametric and semiparametric methods originally developed for censored time-to-event data in medical and engineering research. These methods make no use of fabricated data and use only information about each available observation that is known with acceptably low uncertainty. For left-censored values, this information consists of the LRL and the fact that the measurement was less than the LRL; for right-censored values, it consists of the URL and the fact that the measurement was greater than the URL. These methods also avoid assuming any specific probability distribution for event times, making them more robust than fully parametric methods.

In the medical and engineering studies for which most methods of survival analysis were developed, the most common study design involves recording the times to occurrence of some meaningful event (e.g., death, cure, relapse of a medical condition, or failure of a machine or component) for subjects in one or more treatment groups. For subjects where the event of interest did not occur during the fixed duration (D) of the study, the numerical value of (D) is recorded as the event time, along with a code indicating that the actual time was greater than D. A consequence of this simple study design is that the only type of censoring produced is right censoring. This is why most of the older statistical methods of survival analysis were designed specifically for right-censored data.

The first survival-analysis methods that could accommodate both left-censored and right-censored data were nonparametric and were developed by Turnbull [62,63] in the mid-1970s. These and other nonparametric and semiparametric methods that were subsequently developed have only begun to be included in standard statistical software in the last 10 to 15 years. The main types of nonparametric methods are now available in both R and SAS, and these are the methods we have reviewed here.

Table 3. Summary of nonparametric survival-analysis methods discussed in this review.

Statistical Task	Section
Characterize concentration distributions	Section 5.1
• Estimate PDF, cPDF, and point-wise confidence intervals
• Estimate quantiles and their confidence intervals
Pairwise comparison of concentration distributions	Section 5.2
Tests of homogeneity and monotonic trends	Section 5.3

lists these methods, together with references to the sections of this paper where they are discussed.

Little progress has been made to date in terms of including semiparametric methods of survival analysis that can be applied to doubly censored and interval-censored data in standard statistical software, though we are aware of statisticians who are developing R packages to address this problem at the time of writing. Statisticians rarely use fully parametric methods of survival analysis in medical applications because statistical results usually depend strongly on which parametric probability distribution the data are assumed to be sampled from but it is rarely possible to convincingly determine which distribution is most appropriate for a particular set of data unless the dataset is unusually large ([59], p. 2; [55], pp. 2–3). We expect the same to be true of environmental concentration data.

In closing, we reiterate that two fundamental principles of rigorous statistical methods for censored data analysis are to avoid fabricating data whenever possible and to use only information from available data that is known with high confidence. The latest nonparametric methods of survival analysis provide statistical tools that are consistent with both of these principles and can be applied to datasets that include any combination of valid, left-uncensored, right-censored, or interval-censored values. Versions of semiparametric regression-like methods like the Cox proportional hazards model and the accelerated failure-time model that can properly handle these different types of censoring are not yet available in standard statistical software, but we expect them to become available in the next few years, most likely in an R package.