A direct comparison of patient-based real-time quality control techniques: The importance of the analyte distribution

Abstract

Background

Patient-based real-time quality control (PBRTQC) techniques have been described in clinical chemistry for over 50 years. PBRTQC has a number of advantages over traditional quality control including commutability, cost and the opportunity for real-time monitoring. However, there are few systematic investigations assessing how different PBRTQC techniques perform head-to-head.

Methods

In this study, we compare moving averages with and without truncation and moving medians. For analytes with skewed distributions such as alanine aminotransferase and creatinine, we also investigate the effect of Box–Cox transformation of the data. We assess the ability of each technique to detect simulated analytical bias in real patient data for multiple analytes and to retrospectively detect a real analytical shift in a creatinine and urea assay.

Results

For analytes with symmetrical distributions, we show that error detection is similar for a moving average with and without four standard deviation truncation limits and for a moving median. In contrast to analytes with symmetrically distributed results, moving averages perform poorly for right skewed distributions such as alanine aminotransferase and creatinine and function only with a tight upper truncation limit. Box–Cox transformation of the data both improves the performance of moving averages and allows all data points to be used. This was also confirmed for retrospective detection of a real analytical shift in creatinine and urea.

Conclusions

Our study highlights the importance of careful assessment of the distribution of patient results for each analyte in a PBRTQC program with the optimal approaches dependent on whether the patient result distribution is symmetrical or skewed.

Keywords

Quality assurance and control statistics patient-based quality control

Introduction

Patient-based real-time quality control (PBRTQC) techniques have been described in the clinical laboratory for more than 50 years.¹ PBRTQC has a number of purported advantages over traditional quality control (QC) including commutability, cost and the opportunity for real-time monitoring.^1,2 All PBRTQC techniques suffer from what has been termed the ‘population problem,’¹ in which variations in the patient population by time of day, week or month due to phlebotomy schedules or outpatient clinics cannot be distinguished from shifts in analytical performance.¹ Censoring data from certain patient populations (e.g. hospital inpatients, intensive care, dialysis unit) can be used but may lead to a reduction in the number of data points in small hospital laboratories with a high number of inpatient samples, delaying the detection of bias.²

There have been various statistical attempts to reduce the impact of population variation in PBRTQC techniques. In the earliest use of the average of (patient) normals (AoN) PBRTQC technique in a clinical chemistry setting, Hoffman and Waid used the daily average of patient results truncated for results falling within the prespecified reference interval in order to reduce the effect of data points from diseased patients biasing the AoN metric.³ Reed modified the approach of Hoffman and Waid to specify that averages are calculated from the same number of tests (uniform block size) rather than daily and to specify that truncation limits were based on the distribution of the laboratories’ own results rather than the prespecified reference interval.⁴ Similarly, van Rossum and Kemperman⁵ and Ng et al.⁶ used truncated means in their moving average (MA) PBRTQC approaches, with van Rossum and Kemperman optimizing truncation limits based on a bias detection curve and Ng et al. optimizing truncation limits and other parameters (e.g. block size) iteratively using a simulated annealing algorithm.

In addition to the use of truncation (i.e. trimmed means), moving medians have been used in PBRTQC techniques.^7,8 It is also recognized that the distribution of patient results for some analytes is not normally distributed but skewed (for example transaminases). Transformations such as log or square root have been proposed to normalize skewed distributions to improve AoN or MA approaches.⁹ The median is a more robust indicator of central location less affected by the skewedness of the distribution of patient results and also does not require the censoring of any patient data. This allows a moving median to be calculated more frequently than an MA from truncated data, theoretically allowing shifts to be detected sooner.

For the derivation of reference intervals from patient data, the Box–Cox family of transformations has been used to transform skewed and non-Gaussian patient data with success.¹⁰ This approach has not yet been used to transform data in a PBRTQC program.

Overall, there are few systematic investigations assessing how different PBRTQC techniques perform head-to-head. In this study, we compare MAs with and without truncation and moving medians both with and without and Box–Cox transformation for analytes with skewed distributions. We assess the ability of each technique to detect analytical bias using the technique of Ng et al., simulating bias on historical patient data and calculating the average number of patient samples until the induced bias was detected.⁶

Methods

Patient data and setting

Raw de-identified patient results for a number of clinical chemistry analytes (full list presented in Table 1) were downloaded from an Abbott Architect (c1600) using the AMS middleware for the period of 26 December 2017 to 25 June 2018.

Table 1.

List of analyte and PBRTQC parameters used in this study.

Analyte	Distribution	RCPAQAP APS (%)	Low TL (wide)	High TL (wide)	Low TL (narrow)	High TL (narrow)	Lambda
Symmetrical distributions
Albumin	Symmetrical	6	5	57	35	50	NA
Calcium	Symmetrical	4	1.4	3	2.1	2.6	NA
Cholesterol	Symmetrical	6	0	9.4	2.5	6	NA
Chloride	Symmetrical	3	86	123	100	110	NA
Bicarbonate	Symmetrical	10	9.9	39	20	30	NA
Iron	Symmetrical	12	0	49	5	20	NA
HDL	Symmetrical	12	0	2.8	0.7	1.5	NA
Potassium	Symmetrical	5	1.6	6.5	3.1	4.3	NA
Magnesium	Symmetrical	8	0	1.7	0.6	0.9	NA
Sodium	Symmetrical	2	123	155	135	144	NA
Phosphate	Symmetrical	8	0	2.6	0.3	1.3	NA
Total Protein	Symmetrical	5	30	108	60	90	NA
Transferrin	Symmetrical	8	0	4.7	1	3	NA
Analyte	Distribution	RCPAQAP APS (%)	Low TL (untransformed)	High TL (untransformed)	Low TL (transformed)	High TL (transformed)	Lambda
Skewed distributions
ALP	Right skewed	12	0	250	1.4	1.52	−0.626
ALT	Right skewed	12	0	300	1.6	2	−0.380
AST	Right skewed	12	0	300	1.25	1.35	−0.670
CK	Right skewed	12	0	1000	2.5	3.5	−0.220
Creatinine	Right skewed	8	0	150	0.86	0.865	−1.150
GGT	Right skewed	12	0	250	1.9	2.5	−0.343
Glucose	Right skewed	8	0	10	0.9	1.2	−0.626
LDH	Right skewed	8	0	350	1.7	1.77	−0.545
Total Bilirubin	Right skewed	12	0	25	1.25	1.75	−0.383
Triglycerides	Right skewed	12	0	2.3	−0.6	1	0.140
Urea	Right skewed	12	0	10	1	1.7	−0.260

Note: Included are distribution properties, truncation limits applied in the comparison and Box–Cox lambda values where transformation is applied.

TL: truncation limits applied to moving averages where applicable; RCPA QAP APS: Royal College of Pathologists of Australasia Quality Assurance Program Analytical Performance specifications; ALT: alanine aminotransferase; ALP: alkaline phosphatase; AST: aspartate aminotransferase; CK: creatine kinase; GGT: gamma-glutamyl transferase; LDH: lactate dehydrogenase; HDL: high-density lipoprotein.

The laboratory services a ∼570-bed tertiary hospital in Melbourne, Australia. Samples are mainly received from hospital inpatients, including emergency and intensive care but are also received from subacute and outpatient populations.

There were no exclusions based on age or location (e.g. ward, ICU, outpatient), but QC and external quality assurance samples were excluded. The final data-set included 945,498 results and was imported into the R statistical computing environment for further analysis.¹¹ The assays were in control during this time period.

Graphical assessment of patient data distribution

The distribution of patient data was graphically visually using normal quantile plots in R. Normal Quantile Plots (Q-Q plot) for each of the analytes in Table 1 were constructed for one month of raw patient data using the qqnorm and qqline functions in R. Each Q-Q plot was inspected for symmetry (example in Figure 1(a)) and right skew (example Figure 1(b)). Only analytes with skewed distributions were subject to protocols using Box–Cox transformation as described below. A summary of the properties of each analyte distribution appears in Table 1.

Figure 1.

(a) Normal QQ Plot for sodium showing symmetrical distribution (but not strictly Gaussian – note deviation in the tails). (b) Normal QQ plot for ALT showing significantly right skewed distribution.

PBRTQC protocols

Block size

For all protocols, the block size (or filter length) used to calculate the mean was set at 50 samples, which was the optimal block size identified by Fleming and Katayev in their study.⁹ Block size can also be calculated using the method of Cembrowski et al.,¹² based on the ratio of patient population standard deviation to the analytical standard deviation. It should be noted that the study of Fleming and Katayev was performed in a large community laboratory and a block size of 50 may not be ‘optimal’ for our setting; however, a standard block size was chosen for simplicity for this comparative study. For a particular analyte and population, the block size will vary, but the validity of the observed outcomes will not be affected by choosing a constant block size.

Control limits

Control limits were set at ±2.58 standard deviations of the calculated mean of the MA/median. A standard deviation of 2.58 was chosen to allow for a 1% false rejection rate.

Individual protocols

The protocols that were assessed differed for symmetrical (e.g. calcium, sodium) and skewed distributions (e.g. alanine aminotransferase [ALT], alkaline phosphatase [ALP]) with Box–Cox transformation only being used for skewed distributions. All MAs and medians were calculated in R using the Zoo package.¹³

Symmetrical distributions

Moving average

A simple MA (block size 50) was calculated using all results without truncation or exclusion.

MA with wide truncation limits

For this approach, patient data for each analyte were censored using truncation limits corresponding to the method of Ng et al.⁶ and suggested as initial limits in recent recommendations.² Briefly, truncation limits are placed at ±4 standard deviations from the patient mean for roughly symmetrical result distributions in order to reduce the influence of outliers on the MA. Truncation limits appear in Table 1.

MA with narrow truncation limits

For this protocol, truncation limits were set by visual inspection of the normal quantile plots (Q-Q plot) for each analyte. In theory, the distribution of patient results represents a mixture of diseased and non-diseased patients, with the values from diseased patients concentrated in the upper and lower parts of the distribution with an interval containing values predominantly from non-diseased patients.^14,15 Therefore, in an attempt to minimize the influence of diseased population on the MA, limits were set based on deviation from the linear portion of the Q-Q plot (see Supplemental Figure S1).

Moving median

A simple moving median was calculated using all results without truncation or exclusion.

Skewed distributions

MA and moving median

Simple MAs and moving medians were performed for skewed distributions as described for symmetrical distributions.

MA with truncation

For this approach, patient data for each analyte with a skewed distribution were censored using truncation limits described by Ng et al.⁶ at a user-defined inflection point for which the frequency distribution curve of the patient data begins to ‘level off’.

Transformed MA and moving median

Patient data for skewed distributions were transformed by the Box–Cox transformation

y_{i} = \frac{x_{i}^{λ} - 1}{λ} if λ > 0, y_{i} = \ln x_{i} if λ = 0

The Box–Cox transformation is a type of power transformation that can be can be used to transform non-Gaussian data to approximate a normal distribution. Lambda was determined in R using the ‘boxcox’ function in the MASS package.¹⁶ Simple MAs and moving medians were then performed on the transformed data. A normal Q-Q plot for ALT before and after Box–Cox transformation appears in Supplemental Figures S2(a) and (b).

Transformed MA with narrow truncation limits

For this protocol, truncation limits on the Box–Cox transformed data were set by visual inspection of the normal Q-Q plot for each analyte as described above.

Bias simulation and detection

We assessed the performance of each PBRTQC protocol using the method of Ng et al.⁶ In this method, a bias is introduced into the patient data and the number of patient results falling between the point of bias introduction and the first PBRTQC value exceeding a control limit is calculated as the number of patient samples to error detection (NP_ed). For each analyte and PBRTQC method, this process was repeated at different points in the data with the order of obtained results being maintained. We performed 10 trials for each analyte and PBRTQC method, introducing bias spaced at 200 sample intervals in the patient data. From these trials, the average number of patient samples to error detection (ANP_ed) is calculated.

Bias was introduced as a percentage based on multiples of the Royal College of Pathologists of Australasia Quality Assurance Program’s Analytical Performance specifications (APS) for each analyte.¹⁷ These are shown in Table 1.

Results

Symmetrical distributions

For the patient data with roughly symmetrical distributions, bias detection curves for the different PBRTQC approaches appear in Figure 2. Calculated ANP_ed and standard deviations for each approach can be found in supplemental Table S1.

Figure 2.

Bias detection curves for symmetrical distributions. For all analytes see supplemental Figure 3.

From Figure 2, it can be seen that the detection of bias varies depending on the analyte, the PBRTQC protocol used and whether the bias is positive or negative. For all PBRTQC protocols, a negative bias in albumin was difficult to detect with an ANP_ed for a −1 APS shift (−6%) greater than 300 samples for each protocol. Similarly, a positive bias in bicarbonate was difficult to detect with a +1 APS shift (+10%) also requiring greater than 300 samples to detect. Error detection was better for negative shifts in bicarbonate with an ANP_ed less than 50 samples for a –1 APS shift (−10%). The error detection was most readily achieved by most methods for sodium, with ANP_ed less than 55 samples for both positive and negative single APS (±2%) shifts in sodium.

With regard to the protocol used, the addition of ±4SD truncation limits (MA wide truncation) adds little benefit over a simple MA with no truncation. One notable exception is magnesium, where ANP_ed is reduced from 460 and 576 samples to 40 and 56 samples for a positive and negative single APS (8%) shift, respectively. Interestingly, it can be seen that attempting to exclude abnormal results from the protocol with narrow truncation limits paradoxically increases ANP_ed for larger shifts. This is due to an increasing number of results that fall outside the truncation limits as the introduced bias increases, thereby increasing ANP_ed.^6,18 Overall, both the MA with wide or no truncation and the median perform similarly for these analytes.

Skewed distributions

For the patient data with right skewed distributions, bias detection curves for the different PBRTQC approaches appear in Figure 3. Again, the calculated ANP_ed and standard deviations for each approach can be found in supplemental Table S1.

Figure 3.

Bias detection curves for skewed distributions. For all analytes see supplemental Figure 4.

From Figure 3, it can be seen that for these right skewed analytes, the overall error detection is poorer than that for the symmetrically distributed analytes with the few protocols being able to detect a single APS shift in less than 100 samples. Additionally, differences in error detection between each PBRTQC protocol are more marked. The protocols without transformation perform poorly with the untransformed MA failing to detect negative shifts at any multiple of APS in ALT, aspartate aminotransferase (AST), bilirubin, creatine kinase (CK), creatinine, gamma-glutamyl transferase (GGT), glucose, lactate dehydrogenase (LDH) and triglycerides. The untransformed median and the untransformed MA with truncation performed better but still failed to detect negative shifts in CK and had the poorest performance in detecting negative shifts in AST and ALT.

Box–Cox transformation of the data markedly improves the performance of a simple MA for a number analytes allowing the MA to have the best overall performance when detecting negative shifts in ALT, AST, bilirubin, LDH, creatinine and urea. Despite being a more robust measure of central tendency, the performance of the moving median is also improved by the Box–Cox transformation of the data. For example, a 1 APS negative shift in creatinine (−8%) is detected in 423 samples for the median of the transformed data versus 1653 samples for the untransformed data.

Retrospective detection of a real analytical shift

We assessed if we could replicate the improved detection of analytical shifts for skewed distributions in detecting a real instrument failure affecting creatinine and urea. Each algorithm was applied to patient results from the analyser retrospectively, for the day the analyser was noted to have a drainage failure, resulting in wash fluid back flowing into the cuvettes. This shift was originally detected by a delta check from patient results, around 1.5 h since starting the analyser, resulting in 85 samples that required re-run and amending. Urea and creatinine were most affected by this failure. Figure 4 shows the selected algorithms against time for urea and creatinine for the day of the failure. Table 2 summarizes the error detection time of the different algorithms.

Figure 4.

Retrospective detection of analytical shift due to analyser drainage failure. (a) MA of Box–Cox transformed data for creatinine, (b) MA of Box–Cox transformed data for creatinine and (c) MA of untransformed data for creatinine.

Table 2.

Retrospective analysis of patient data from the day of an analyser error.

Method	Urea (min)	Creatinine (min)
Delta check	Not detected	64
Moving average	64	Not detected
Moving average (truncated for Cr <150, urea <10)	57	21
Moving average (Box–Cox transformed)	52	21
Moving average (Box–Cox transformed and truncated)	Not detected	Not detected
Moving median	62	37
Moving median (Box–Cox transformed)	57	36
Internal QC (scheduled 8 hourly)	480	480

Note: Time (in min) until detection of analytical shift is shown for different algorithms.

From Table 2, it can be seen that the MA for the Box–Cox transformed data detected the error in the shortest time (20 min for creatinine and 52 min for urea). MA for untransformed data could only detect the shift with tight upper truncation, but did so in a similar time to the MA on Box–Cox transformed data. A simple MA did not detect the shift in creatinine and detected the urea shift in 64 min.

Discussion

In this study, we directly compare the performance of a number of PBRTQC protocols, assessing truncation limits, data transformation and type of PBRTQC algorithm (MA vs. moving median). We used the performance metric ANP_ed in our comparison, a metric that has been used extensively in investigations of PBRTQC approaches.^6,18,19 Our study highlights the importance of careful assessment of the distribution of patient results for each analyte of interest in a PBRTQC program with the optimal approaches dependent on whether the patient result distribution is symmetrical or skewed.

For analytes with symmetrical distributions, we show that ANP_ed is similar for the MA with and without four SD truncation limits and for the moving median. Narrow truncation limits selected to isolate predominantly non-pathological from pathological specimens based on the normal QQ-plot had the undesired effect of increasing ANP_ed for greater biases. This effect is well known and was shown in the excellent study of MA truncation limits by Ye et al., where ANP_ed was also used as a performance metric¹⁸ and by Ng et al.⁶ Overall, for our laboratory data, we question the importance of truncation limits for symmetrically distributed analytes. In a PBRTQC program, the use of a median or a mean truncated for extreme outliers (±4 SD) is both reasonable, and narrow limits aimed at excluding pathological samples may have the paradoxical effect of increasing the number of samples until bias is detected. For analytes with symmetrically distributed results, the overall variance of the distribution is likely low enough, even when pathological results are present, to allow the mean or median to be a robust measure in PBRTQC. This adds weight to the comment of Ng et al., that the goal of PBRTQC is to ‘monitor the process not the patients’.⁶

In contrast to analytes with symmetrically distributed result (e.g. sodium, calcium), MA performs poorly for right skewed distributions such as ALT and creatinine and functions only with a tight upper truncation limit. The requirement for tight truncation for analytes with skewed distribution was shown by Ye et al. with ALT with ‘optimal’ truncation limits having an upper limit of 42 U/L which excluded ∼20% of the patient data.¹⁸ For skewed distributions, the median outperforms the simple MA without truncation limits. As the median is a more robust indicator of central tendency, this is expected, but to our knowledge, this has not been shown directly for skewed distributions of patient results. Box–Cox transformation of the data both improves the performance of MA and allows all data points to be used. Box–Cox transformation also largely improves the performance of the moving median for skewed data. This was also confirmed for retrospective detection of a real analytical shift in creatinine and urea due to a drainage failure. To our knowledge, this is the first study to quantitatively assess the importance of transformation for skewed data in a PBRTQC program. In a study on PBRTQC, Fleming and Katayev applied transformations (natural log or square root) to visually skewed data, but did not compare the performance of PBRTQC approach with transformed data to that with untransformed data. Here, data were first assessed for skewness based on a graphical method (normal QQ-plot). For skewed distributions, we used the Box–Cox transformation, as it may be assumed that clinical chemistry analytes from normal subjects follow a Power Normal Distribution which is related to a Gaussian distribution by the Box–Cox transformation.¹⁴ Box–Cox transformation has been used previously in reference interval studies,¹⁰ but the use of the transformation in a PBRTQC is to our knowledge novel. The logarithmic transformation is more commonly available in middleware software and is a special case of the Box–Cox family of transformations with the exponent λ = 0. A direct comparison of logarithmic transformation with Box–Cox transformation is beyond the scope of this paper, but where Box–Cox transformation is unavailable, logarithmic transformation may offer similar benefits for skewed distributions. Due to the utility of the Box–Cox transformation shown here, we endorse the recommendation of Loh et al. that middleware and laboratory information system software vendors support real-time Box–Cox transformation in their software.²⁰

This study has some important limitations that may influence the wider applicability of the findings. Control limits of ±2.58 SD of the calculated mean for each algorithm were used to achieve a ≤ 1% false rejection rate. This was deemed appropriate for this comparative study, and a false rejection rate of ≤ 1% in a PBRTQC program has also been targeted by other groups.⁹ However, given the large number of results being produced, these control limits may still result in excessive false alarms. In a clinical setting, initial control limits are often adjusted empirically to achieve the tolerable false rejection rate and wider control limits may be more appropriate. Although widening control limits will increase ANP_ed and the relative performance of each algorithm, our conclusions are unlikely to be affected. Control limits based on SD were used in this study. These are commonly used in PBRTQC programs allowing plotting on interpretation of algorithms on a Levey-Jennings like plot. However, using symmetrical SD-based control limits for the untransformed or untruncated, MA and median for non-symmetrical distributions will have contributed to the poor performance of these algorithms, particularly on the low side. For untransformed MAs or medians, ideally, asymmetric control limits should be used, but these are difficult to specify and also plot on Levey-Jennings like plot. One of the benefits of data transformation is the ability to use symmetrical SD-based control limits. Alternatively, percentile-based control limits could be considered for skewed distributions. Another limitation is that the various protocols we are assessed on their ability to detect simulated bias in real patient data, introduced as percentage shift at a particular sample. This method has been used in simulation studies previously and is simple to implement computationally.^6,18 We also confirmed our findings for skewed distributions on a true analytical shift due to a drainage failure. The findings may, however, not extend to the detection of gradually introduced bias or increased imprecision in an assay. The patient data in our study represent mainly hospital inpatients in a major tertiary referral centre in Melbourne, Australia and the conclusion here may not apply to laboratories servicing a different population. However, given our patient population will contain a high proportion of samples with disease present and be influenced by outpatient clinics and phlebotomy schedules, our findings are likely to be robust for settings with less variation.

There is growing interest in PBRTQC with a number of approaches that can be applied.² There are currently a number of barriers to widespread implementation of PBRTQC with time, statistical skills and software support required to optimize and implement a PBRTQC program.² We show that managing the ‘population problem’ is not so much about excluding portions of the population but identifying where a population distribution is skewed and variance can be reduced by appropriate transformation. We hope our findings will assist those both selecting PBRTQC approaches for implementation and those designing software for PBRTQC, allowing the use of the most optimal methods and wider adoption.

Supplemental Material

ACB902174 Supplemental Figure1 - Supplemental material for A direct comparison of patient-based real-time quality control techniques: The importance of the analyte distribution

Supplemental material, ACB902174 Supplemental Figure1 for A direct comparison of patient-based real-time quality control techniques: The importance of the analyte distribution by Joel D. Smith, Tony Badrick and Francis Bowling in Annals of Clinical Biochemistry

Supplemental Material

ACB902174 Supplemental Figure2 - Supplemental material for A direct comparison of patient-based real-time quality control techniques: The importance of the analyte distribution

Supplemental material, ACB902174 Supplemental Figure2 for A direct comparison of patient-based real-time quality control techniques: The importance of the analyte distribution by Joel D. Smith, Tony Badrick and Francis Bowling in Annals of Clinical Biochemistry

Supplemental Material

ACB902174 Supplemental Figure3 - Supplemental material for A direct comparison of patient-based real-time quality control techniques: The importance of the analyte distribution

Supplemental material, ACB902174 Supplemental Figure3 for A direct comparison of patient-based real-time quality control techniques: The importance of the analyte distribution by Joel D. Smith, Tony Badrick and Francis Bowling in Annals of Clinical Biochemistry

Supplemental Material

ACB902174 Supplemental Figure4 - Supplemental material for A direct comparison of patient-based real-time quality control techniques: The importance of the analyte distribution

Supplemental material, ACB902174 Supplemental Figure4 for A direct comparison of patient-based real-time quality control techniques: The importance of the analyte distribution by Joel D. Smith, Tony Badrick and Francis Bowling in Annals of Clinical Biochemistry

Supplemental Material

ACB902174 Supplemental Tables - Supplemental material for A direct comparison of patient-based real-time quality control techniques: The importance of the analyte distribution

Supplemental material, ACB902174 Supplemental Tables for A direct comparison of patient-based real-time quality control techniques: The importance of the analyte distribution by Joel D. Smith, Tony Badrick and Francis Bowling in Annals of Clinical Biochemistry

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Ethical approval

This retrospective quality assurance (QA) project was approved by the Melbourne Health Office for Research (Project Number QA2019134).

Guarantor

JDS.

Contributorship

JDS, FB and TB conceived the study. JDS performed the simulation and data analysis. JDS wrote the first draft of the article. All authors reviewed and edited the article and approved the final version of the article.

ORCID iD

Joel D Smith

Supplemental material

Supplemental material is available for this article online.

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Supplementary Material

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