Characterizing nonconstant instrumental variance in emerging miniaturized analytical techniques

Abstract

Measurement variance is a crucial aspect of quantitative chemical analysis. Variance directly affects important analytical figures of merit, including detection limit, quantitation limit, and confidence intervals. Most reported analyses for emerging analytical techniques implicitly assume constant variance (homoskedasticity) by using unweighted regression calibrations. Despite the assumption of constant variance, it is known that most instruments exhibit heteroskedasticity, where variance changes with signal intensity. Ignoring nonconstant variance results in suboptimal calibrations, invalid uncertainty estimates, and incorrect detection limits. Three techniques where homoskedasticity is often assumed were covered in this work to evaluate if heteroskedasticity had a significant quantitative impact-naked-eye, distance-based detection using paper-based analytical devices (PADs), cathodic stripping voltammetry (CSV) with disposable carbon-ink electrode devices, and microchip electrophoresis (MCE) with conductivity detection. Despite these techniques representing a wide range of chemistries and precision, heteroskedastic behavior was confirmed for each. The general variance forms were analyzed, and recommendations for accounting for nonconstant variance discussed. Monte Carlo simulations of instrument responses were performed to quantify the benefits of weighted regression, and the sensitivity to uncertainty in the variance function was tested. Results show that heteroskedasticity should be considered during development of new techniques; even moderate uncertainty (30%) in the variance function still results in weighted regression outperforming unweighted regressions. We recommend utilizing the power model of variance because it is easy to apply, requires little additional experimentation, and produces higher-precision results and more reliable uncertainty estimates than assuming homoskedasticity.

Uncertainty Analysis of Fluorescence-Based Oil-In-Water Monitors for Oil and Gas Produced Water

“…The default identification from the manufacturer uses ordinary least square (OLS) regression to estimate the best-fit to a first- or second-order polynomial, and thereby implicitly assume homoskedastic random error ( ) by using unweighted regression calibrations. For OLS to be the best linear unbiased estimator (BLUE), five assumptions must be assumed according to the Gauss–Markov theorem: the true data trend is linear and that for all values of x the ’s are independent, normally distributed with a mean of zero and have the same standard deviation throughout the region of interest (homoscedasticity) [ 40 , 41 ]. Although the first assumptions simplify the reality, proper calibration of the instrument might still achieve a close approximation to the true value.…”

“…Although the first assumptions simplify the reality, proper calibration of the instrument might still achieve a close approximation to the true value. However, if heteroskedasticity is present, OLS loses its efficiency and is not BLUE anymore [ 41 ]. By looking at the box plot of the calibration data for all four OiW monitors in Figure 2 , the combined 280 samples, 70 samples for each OiW monitor, clearly indicates that exhibit heteroskedasticity.…”

“…Ignoring the fact that nonconstant variance might result in suboptimal calibration and invalid uncertainty estimates [ 41 , 42 ], instead, weighted least square (WLS) could be used as it may not be reasonable to assume that every calibration concentration should be treated equally. One disadvantage of using WLS is the weighting is often unknown, but an appropriate weighting factor ( ) for WLS may well be the reciprocal of the variance ( ) for each calibration concentration.…”

“…As the estimation of the parameters for a linear regression a and b uses two degrees of freedom; . The confidence interval (CI) and prediction interval (PI) are calculated, respectively, where the subscript “0” indicates the set of values which falls within the CI and PI [ 40 , 41 ]: and …”

Electrochemical dithiothreitol assay for large-scale particulate matter studies

“…The end-point assay principle can be applied to both the electrochemical and UV/vis assays due to the linear reactivity of DTT used in the DTT assay. With the linear reactivity of DTT and having a proper calibration curve for accurate and precise measurements (Noblitt et al 2016), the DTT rate can be accurately calculated from two time points. Replicates at each time point are still needed to decrease the uncertainty.…”