The polymerase chain reaction (PCR) is a molecular biology tool with diverse applications in the aquatic sciences. Classical PCR is a nonquantitative method that can be used to detect target DNA sequences that are characteristic of particular microbial taxa but cannot determine their concentrations in water samples. Various quantitative forms of PCR have been developed to remove this limitation. Of these, the two that currently are used most widely are real-time quantitative PCR (qPCR) and droplet digital PCR (ddPCR). Several outlines of the mathematical and statistical basis of these methods for estimating target sequence concentrations are available in the literature, but we are aware of no thorough and rigorous derivation of the theoretical underpinnings of either. The purpose of this review is to provide such derivations, and to identify and compare the main strengths and weaknesses of the two methods. We find that both estimation methods are sound, provided careful attention is paid to specific details that differ between the two. With qPCR, it is especially important to reduce any significant PCR inhibition by sample constituents and to properly fit the standard curve to heteroskedastic calibration data. With ddPCR, it is important to ensure that the value of the mean droplet volume used in calculating concentrations is correct for the particular combination of droplet generator and master mix used. The advantages of qPCR include lower instrument and per-sample costs, a shorter turnaround time for obtaining results, a higher upper limit of quantification, and a wider dynamic range. The advantages of ddPCR include freedom from dependence on a standard curve, an inherently lower sensitivity to PCR inhibitors, a lower limit of quantification, a simpler theoretical basis, and simpler data analysis. We suggest qPCR often will be preferable in laboratory studies where investigators have significant control over the range of target sequence concentrations in samples, concentrations are sufficiently high so proper calibration does not require standards with concentrations low enough to exhibit exaggerated variability in the threshold cycle, and no significant inhibition is present, or more generally, in studies where funding levels do not permit the higher cost of instrumentation and supplies required by ddPCR or where the shorter turnaround time for qPCR is essential. If sufficient funds are available, ddPCR often will be preferable when the ability to quantify low concentrations is important, especially if inhibitors are likely to be present at concentrations that are problematic for qPCR.

One particular recurrent events data scenario involves patients experiencing events according to a common intensity rate, and then a treatment may be applied. The treatment might be effective for a limited amount of time, so that the intensity rate would be expected to change abruptly when the effect of the treatment wears out. In particular, we allow models for the intensity rate, post-treatment, to be at first decreasing and then change to increasing (and vice versa). Two estimators of the location of this change are proposed.

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.

The polymerase chain reaction (PCR) is a molecular biology tool with diverse applications in the aquatic sciences. Classical PCR is a nonquantitative method that can be used to detect target DNA sequences that are characteristic of particular microbial taxa but cannot determine their concentrations in water samples. Various quantitative forms of PCR have been developed to remove this limitation. Of these, the two that currently are used most widely are real-time quantitative PCR (qPCR) and droplet digital PCR (ddPCR). Several outlines of the mathematical and statistical basis of these methods for estimating target sequence concentrations are available in the literature, but we are aware of no thorough and rigorous derivation of the theoretical underpinnings of either. The purpose of this review is to provide such derivations, and to identify and compare the main strengths and weaknesses of the two methods. We find that both estimation methods are sound, provided careful attention is paid to specific details that differ between the two. With qPCR, it is especially important to reduce any significant PCR inhibition by sample constituents and to properly fit the standard curve to heteroskedastic calibration data. With ddPCR, it is important to ensure that the value of the mean droplet volume used in calculating concentrations is correct for the particular combination of droplet generator and master mix used. Advantages of qPCR include lower instrument and per-sample costs, a shorter turnaround time for obtaining results, a higher upper limit of quantification, and a wider dynamic range. Advantages of ddPCR include freedom from dependence on a standard curve, inherently lower sensitivity to PCR inhibitors, a lower limit of quantification, a simpler theoretical basis, and simpler data analysis.

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) typically are 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) typically are 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 parametric statistical methods, such as t-tests, regression analysis, and analysis of variance. We briefly review various statistical methods that have been employed for analyzing 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.