Linear calibrations in chromatography: the incorrect use of ordinary least squares for determinations at low levels

Ordinary least squares is widely applied as the standard regression method for analytical calibrations, and it is usually accepted that this regression method can be used for quantification starting at the limit of quantification. However, it requires calibration being homoscedastic and this is not common. Different calibrations have been evaluated to assess whether ordinary least squares is adequate to quantify estimates at low levels. All calibrations evaluated were linear and heteroscedastic. Despite acceptable values for precision at limit of quantification levels were obtained, ordinary least squares fitting resulted in significant and unacceptable bias at low levels. When weighted least squares regression was applied, bias at low levels were solved and accurate estimates were obtained. With heteroscedastic calibrations, limit values determined by conventional methods are only appropriate if weighted least squares is used. A “practical limit of quantification” can be determined with ordinary least squares in heteroscedastic calibrations, which should be fixed at a minimum of 20 times the value calculated with conventional methods. Biases obtained above this “practical limit” were acceptable applying ordinary least squares and no significant differences were obtained between the estimates measured using weighted and ordinary least squares when analyzing real-world samples.


Introduction
Analytical methods used in laboratories must be evaluated and tested to ensure they produce valid results that are suitable for their intended purpose. It should be taken into account that the main objective in quantitative analysis is to provide an estimate of the content of an analyte with acceptable accuracy (i.e., trueness and precision [1]). Therefore, the most important factor should be the quality of the inverse predictions (i.e., backcalculated values) rather than the quality of fit [2,3].
Analytical calibrations are required to find a function that can describe the relationship between the instrumental response and the concentration of the target analyte. It is common to assume that the simplest model adequately describing this concentrationresponse relationship should be used [4]. In chemical and biological analysis many instruments show linear detector responses over several orders of magnitude; therefore, linear regression models, mainly ordinary least squares (OLS), are extensively used in practical applications; which are more intuitive and easier to fit than non-linear ones, and estimators are simpler to determine [5,6]. Despite mathematical simplicity being desirable, this is not a significant limitation as today's software programs can fit complex models without specialized knowledge, eliminating the need for a mathematical background to calculate parameters.
In any case, it must be understood that regression models are based on the fulfillment of certain preliminary conditions, which were adopted in formulating the model, and the failure to meet some of these conditions can lead to significant biases and imprecisions in the concentration estimates [7,8]. The most extensive fitting model used in laboratory calibrations is linear OLS, which requires variance of the dependent variable to be constant www.jss-journal.com Page 4 Journal of Separation Science This article is protected by copyright. All rights reserved.
at all values of the independent variable (homoscedasticity). However, in practice analytical and biological methods yield non-constant variance over the working range [7,[9][10][11][12][13][14][15][16][17][18][19], unless this range is particularly narrow (usually up to one order of magnitude). Previous studies [16,[19][20][21][22][23][24][25][26][27][28][29] have already demonstrated that the use of OLS with heteroscedastic data may result in significant bias and underestimation of the precision at concentrations that are close to the limit of detection (LOD), due to the overestimation of the high concentration standards. Despite this, it is still very common to see researchers applying OLS regression in calibrations without evaluating whether the calibration presents homoscedasticity or heteroscedasticity. Unfortunately, "a great number of people using least squares have just enough training to be able to apply it, but not enough training to see why it often shouldn't be applied" [30].
From a quantitative point of view, a basic parameter to be determined in the validation of an analytical method is the limit of quantification (LOQ), which has been defined as the lowest amount or concentration of an analyte at which performance is acceptable for a typical application [31], or that can be estimated with acceptable reliability [32] or precision [33]. Once this parameter has been determined, it is always assumed that the regression model used can be applied starting from the LOQ to obtain accurate estimates.
Unfortunately, the common methodologies used for the determination of LOQ values are only based on analyte response at a single concentration, making it inconsistent in situations of non-constant variance [11]. Usually the determination of the LOQ is simply based on an extrapolation of the IUPAC limit of detection or determination (LOD), which is only based on instrumental repeatability [4,33,34]. Sometimes the precision chosen at the LOQ level is defined as 10% relative standard deviation (RSD), as was suggested by Currie [35]. In other cases the LOQ is taken as being a fixed multiple of the LOD or the www.jss-journal.com

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concentration that produces a signal of k times (usually k=10) the standard deviation of the blank (s bl ) [31,36]. The most significant limitation of these approaches is that they are only based on the characterization of the target parameter without an assessment of uncertainty and there is a lack of bias accountability for this limit [4]. Therefore, the risk of accepting an unsuitable assay and rejecting a suitable assay is unknown and uncontrolled [14,37].
Moreover, when some statistical test is applied, it only computes type I error (i.e., probability of false positives), which involves that methods giving imprecise results can be more easily validated than more precise ones [37].
Some recent validation guidelines require the evaluation of the trueness in the determination of the LOQ, and indicate that it is the lowest amount of an analyte that can be determined with acceptable precision and trueness [4,38]. Moreover, different studies [14,23,24] have found that the selection of the regression model may have a significant effect on the estimation of LOQ values when trueness and precision are taken into account.
Scientists should be aware that a linear relationship between the dependent variable and the concentration does not guarantee method trueness when a bias is present [39], and this may easily occur at low concentration levels with heteroscedastic calibrations [21].

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The aim of this study is to compare the two most common linear regression fittings, OLS and weighted least squares (WLS), and to assess whether the LOQ values determined by conventional methods based on precision at blank level genuinely give a limit where the precision and the trueness of estimates are adequate for quantification purposes with both regression models. The accuracy profile methodology has been used for assessing the accuracy of the results obtained. This article is protected by copyright. All rights reserved.

Materials and Methods
when p<0.05. The weighting factor applied in WLS regressions was determined as the inverse of the experimental variance (w i =1/s i 2 ). However, it was also evaluated the use of other factors, such as w i =1/x i 2 and w i =1/y i 2 (where x i corresponds to the concentration of the i-standard and y i to its analytical response).

Accuracy profile approach
As indicated by Feinberg [37], the basic idea behind this concept is to translate the fitnessfor-purpose objective into the acceptability criterion (λ): where X is the analytical result, Z the true value and the limit λ is not arbitrary and depends on the goal of the analytical procedure.
This methodology is based on the calculation of a tolerance interval at each concentration level (TI j ). Briefly, it requires the determination of the bias for the back-calculated concentrations at each concentration level and the calculation of a β expectation tolerance interval (eq. 2). Full equations and detailed description of the methodology are described in the literature [2,16,39].
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To draw the accuracy profile plot, the relative error (%) for the back-calculated concentration at each concentration level is plotted against the corresponding standard concentration, together with the corresponding upper and lower tolerance limits. Two lines are drawn, one connecting the lower tolerance limits obtained at each concentration level and another connecting the upper tolerance limits, which allow showing specified acceptance limits in the graph.
When the whole accuracy profile, including the tolerance intervals, is within in the acceptance limits, the analytical method is expected to provide accurate results for its intended purpose. If some point of the profile steps outside the limits, the method should not be considered for that concentration level. In the present study, the acceptance limits at LOQ level have been set at ±20%, which is considered acceptable by the US-FDA for bioanalytical methods [4].

Evaluation of linearity and heteroscedasticity of calibrations
The linearity of all calibrations in the working ranges evaluated was assessed graphically by checking the residual plots, and statistically by applying the lack-of-fit (LOF) [

Accuracy profile plots
The accuracy profile plots obtained for the calibrations evaluated can be seen in Figure 1 and Supplementary Materials. In all calibrations where the first standard was clearly above www.jss-journal.com

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the LOQ, the accuracy profiles for OLS and WLS regression models gave results that were equivalent (Figures 1a and 1b) and the tolerance intervals were inside the acceptance limits for the whole calibration range. This indicates that OLS can be considered a correct regression method from a practical point of view when measuring at levels well above the LOQ. It is already known that, in linear heteroscedastic calibrations, OLS yield poorly and inaccurate assessment of the intercept but does not introduce significant changes in the slope [19,21], which means that discrepancies in estimates obtained between OLS and WLS can be found at the lowest levels of concentrations and can be very important.
When the first standard was at a level equivalent to the LOQ (Figures 1c and 1d), the accuracy profile for the OLS model clearly showed that this regression gave excessive bias and incorrect results at LOQ level, whereas the WLS model gave adequate tolerance limits in the whole range. Finally, when the first standard was set at a concentration between LOD and LOQ (Figures 1e and 1f), despite WLS regressions always giving a bias inside a ±20% limit, tolerance intervals were beyond the acceptance limits.

Weighted least squares
It is clear that WLS should be the golden regression method for linear analytical and biological calibrations in laboratories due to the existence of heteroscedasticity. that, in routine analysis, WLS can be applied without the need for replicate measurements at each calibration level using empirical factors such as 1/x i 2 and 1/y i 2 to obtain non-biased results at low levels.

Estimation of a "practical LOQ" using OLS
The evaluation of the accuracy profile plots shows that the bias and tolerance limits obtained at levels ≥5 times the LOQ were always acceptable for both OLS and WLS models ( Figure 1 and Table 1). This suggests that the use of OLS regression does not introduce any significant error, from a practical point of view, with heteroscedastic calibrations provided that the first standard is set at least 5 times above the LOQ and no quantification measurements are done below this point. From these plots, a "practical LOQ" could be defined as at least 5·LOQ to obtain non-biased results with OLS regression.
It was decided to assess whether the use of a minimum concentration of 5·LOQ, determined from accuracy plots, can really be applied in the analysis of real-world samples using OLS.
Therefore, a set of experiments were performed to study the bias obtained with OLS when measuring at low levels (between LOQ and 20·LOQ). First, three different batches of different commercial teas were evaluated (one green tea and two pu-erh teas), and independent replicate samples of each batch were analyzed for theobromine content. The solution obtained after the extraction of the target analyte was measured directly and after being diluted at a level close to the LOQ. Each replicate sample was analyzed on different www.jss-journal.com

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days over an eight-week period, with new analytical calibrations being made each day. The content of the target analyte in tea leaves (in mg g -1 ) was calculated from each replicate (both from the diluted and undiluted solutions analyzed) applying OLS and WLS regressions ( Figure 3). In a first group of samples (Figure 3a, n=13, pu-erh tea), the solution was diluted until a level that was around 10·LOQ (a level where accuracy profiles plots did not yield differences between OLS and WLS); a second group (Figure 3b, n=12, green tea) was diluted until 5·LOQ; and a last group (n=28, pu-erh tea) was diluted to 0.5·LOQ. In all cases, the signals measured for the diluted samples agreed with the dilution factor applied, as with dilution to 0.5·LOQ, Figure 3c). These results also confirmed that the percentage of bias obtained applying OLS regression increased when the content of the solution measured decreased (-6.2% bias at 10·LOQ, -12.1% at 5·LOQ and -32.5% at 0.5·LOQ). These results agree with those obtained with the accuracy profile plots as the percentage of bias obtained at concentration levels between 5-10 times the LOQ were inside the prefixed acceptance limits at this level (±20%); however, the analysis of a large number of replicate samples showed that a systematic bias was still found applying OLS.  (Figure 4). It was found that the use of OLS regression yielded negative bias for all samples when the amount determined was <20·LOQ, which increased exponentially when the concentration detected from the diluted solutions was <10·LOQ (LOQ=0.06 mg L -1 , Figure 4a). In the case of WLS regression (Figure 4b), no systematic errors were observed (the results were randomly distributed around 0% bias) and although the percentage of bias tended to increase as the concentration of the diluted solution decreased, it was always inside ±10% until the LOQ level.
These results indicate that, despite the accuracy profile plots yielded acceptable accuracy using OLS regression for heteroscedastic calibrations when the first standard was set at a level of 5·LOQ, systematic errors were still found in estimations determined with OLS at This article is protected by copyright. All rights reserved.   (a) WLS regression applied with a weighting factor w i =1/s i 2 , (b) w i =1/x i 2 , and (c) w i =1/y i 2 .