## Abstract

When comparing results over time, biologic variation must be statistically incorporated into the evaluation of laboratory results to identify a physiologic change. Traditional methods compare the difference in 2 values with the standard deviation (SD) of the biologic variation to indicate whether a “true” physiologic change has occurred.

To develop methodology to reduce the effect of biologic variation on the difference necessary to detect changes in clinical status in the presence of biologic variation.

The standard test for change compares the difference between 2 points with the 95% confidence limit, given as . We examined the effect of multiple data pairs on the confidence limit.

Increasing the number of data pairs using the formula , where *n* = number of data pairs, significantly reduces the difference between values necessary to achieve a 95% confidence limit.

Evaluating multiple paired sets of patient data rather than a single pair results in a substantial decrease in the difference between values necessary to achieve a given confidence interval, thereby improving the sensitivity of the evaluation. A practice of using multiple patient samples results in enhanced power to detect true changes in patient physiology. Such a testing protocol is warranted when small changes in the analyte precede serious clinical events or when the SD of the biologic variation is large.

Clinicians rely on laboratory tests to monitor the progression or remission of disease, or to identify pathologic alterations in physiology that may precede clinical events. Monitoring quantitative laboratory results represents a crucial component in the assessment of response to therapy. Laboratories assist the physician's clinical decision making by providing empiric values and applicable reference intervals. Although population- and laboratory-based reference intervals are useful, clinicians frequently encounter changes in analyte values that remain within or outside the reference interval. Therefore, the clinician needs to discern the statistical significance of a change in analyte values, which can be achieved by using the patient as his own control.1

Didactic and clinical training teaches physicians to develop their judgment to assess whether serial tests represent a significant clinical change (physiologic, pathologic, or therapeutic).2 An evidence-based medicine approach calculates the probability of significant change, presented as a confidence limit. The 95% confidence limit of a test is given as , where the SD refers to the biologic variation expressed as a standard deviation. The multiplier, 1.96 (*Z*), is determined by choosing an α value of .05 (other multipliers may be calculated for α values of .1 or .01 as desired). If the absolute change in analyte value exceeds a 95% confidence limit, then the difference is considered statistically significant and likely represents a true change in patient physiology. Conversely, if the absolute change in analyte does not exceed the 95% confidence limit, then there is a greater than 5% probability that the change in analyte value is not due to a change in patient physiology, but instead attributable to inherent biologic variation or laboratory analytic variation. Although this method of determining the significance of changing analyte values is satisfactory in cases where analytic and biologic variation is small compared with the measured values, it is limited to only 2 analyte values encompassing a discrete period of time (ie, one value at time A and a second at time B, the interval in question being from A to B). For many analytes and clinical situations—for example, B-natriuretic peptide (BNP) in heart failure—the biologic variation is so large that it reduces the efficiency of the test, resulting in poor discrimination and power.3 In such cases, the power of comparing 2 analyte values often remains insufficient to discern statistically significant change. Despite this obvious limitation, current literature concentrates on single-difference comparison, avoiding the mechanics and implication of using multiple differences. Determining whether calculating multiple differences would narrow the width of the confidence interval, improving the power of the test, requires examination. We investigated ways to arrange measuring multiple points to improve the power of the test, measured by demonstrating narrowing limits of detection for fixed confidence intervals. We hypothesized that the addition of one or more analyte measurements at the initial or later point of time could be statistically incorporated into the determination of the critical standard error to result in enhanced sensitivity to clinically significant physiologic change within the patient.

## MATERIALS AND METHODS

For this study, we used standard mathematic and statistical tools for analysis of variance, standard deviation, coefficient of variation, and confidence intervals as employed in the laboratory setting. To assess the significance of the difference between 2 serial data points, the percent difference between 2 serial points is compared to the critical difference, or reference change value (*RCV*).4 The *RCV* is derived from an equation that incorporates the analytic variation (*CV _{A}*), biologic variation (

*CV*), and a

_{I}*z*score corresponding to the desired level of significance.4 The critical

*RCV*represents that value that divided all results into regions of significant and nonsignificant change. The critical

*RCV*is given by

where

In this equation, *Z* is the *z* score (or multiplier) corresponding to the desired α/2 value, *CV _{A}* is the analytic variation,

*CV*is the inherent or biologic variation, and

_{I}*CV*is the total combined variation of the analyte in question. This equation applies to the clinical scenario in which 2 values are separated in time, such as before and after an intervention (Figure 1). In this case, a statistically significant change between value 1 and value 2 exists if the following criteria are satisfied:

_{T}This equation describes a valid statistical approach to determining significance in the case of a difference between 2 serial data points; however, its dependence on only 1 datum of difference limits its power. Considering multiple differences between data points requires expansion of the critical reference change (*RCV _{C}*) formula. To adapt this equation to such situations, one must consider the effects of multiple data pairs on the derivation of the

*RCV*, as well as determine the effect of multiple differences on the probability that a physiologic change has occurred (Figure 1), in terms of the pretest and posttest probabilities. The pretest probability is determined by several factors, such as the prevalence of disease, the analyte values, and the proposed intervention. The null hypothesis is that no physiologic change has occurred at the end of the time interval. The alternate hypothesis is that a physiologic chance in the patient's state has occurred. The posttest probability can be derived from a

_{C}*z*test of the difference between the test values, with the combined analytic and biologic variation representing the standard deviation:

One can replace the *SD*s with the appropriate coefficient of variation (*CV*) by dividing through by the average value.

Because, for the most part, *CV _{A}*

_{,1}=

*CV*

_{A}_{,2}and

*CV*

_{BV}_{,1}=

*CV*

_{BV}_{,2}; the indices of the tests may be dropped, and the combined SD may be calculated as follows:

The range of values indicating no change in the state of the system may be expressed as a confidence limit. For the 95% confidence limit, *Z*_{α/2} = 1.96. The 95% confidence limit is defined as , where the mean value is zero. We present a formal derivation of the formulae for incorporating multiple samples of analyte to deal with biologic variation.

Graph theory may be applied to the same hypothetical situation, involving analyte measurements taken at time 1 and time 2 to illustrate the effect of multiple patient samples. A graph is an object consisting of 2 sets: one is composed of a set of points (the vertex set), and the other is composed of a set of lines connecting these points (the edge set).5 In this case, we can use a bipartite graph in which the points of the vertex set are divided into 2 columns representing those values obtained at time 1 and time 2. The edges are then defined as vectors moving from a point in the time 1 set to one in the time 2 set; all edges moved forward in time, and there are no edges within a single set of vertices on the same side (Figure 2).6

Because each difference represents a potential calculation of posttest probability, the total number of differences indicates the number to use in the central limit theorem. For any given total number of patient samples, the maximum number of differences obtained from such testing is the product of an equal or near-equal distribution of testing samples at time 1 and those at time 2. For example, for a total of 6 testing samples, the highest possible yield is 9 difference pairs; for a total of 7 possible testing samples, the highest yield is 12 difference pairs; and for a total of 10 testing samples, the highest yield is 25 difference pairs. The maximum yield of data pairs between time A and time B using multiple patient samples may be calculated as follows.

For an even total number of testing samples *n*:

For an odd total number of testing samples *n*:

where *m*_{Δ} is the maximum number of data pairs between time A and time B, and *n* is the total number of patient test samples.

For an equal number of values, equally distributed:

where is the average of the differences. See the appendix for details. Dividing the average of the differences and the standard deviation by the average forms the fractional difference and the *CV*, respectively.

To adapt the equation for *RCV _{C}* to a situation involving 2 data pairs (ie, 2 differences), one calculates the mean of

*RCV*

_{1}and

*RCV*

_{2}, which correspond to the

*RCV*calculated from one data pair and an additional data pair (for example,

_{C}*x*

_{1}–

*x*

_{2}and

*x*

_{1}–

*x*

_{3}; Figure 2). The biologic variation (intraindividual variation) is the same for

*x*

_{1}–

*x*

_{2}and

*x*

_{1}–

*x*

_{3}; thus, according to the central limit theorem, the mean of the

*RCV*is μ, the population mean, and its variance is (

_{C}*V*[

*RCV*])/

_{C}*n*.7 Accordingly,

Incorporating one additional measurement point (*x*_{3}) increases by one the data pairs available for consideration and reduces the required demonstrated change between analyte values to establish statistical significance.

For bidirectional change, mean critical *RCV*, , is the absolute value of the mean of values at time 1 minus the mean of values at time 2. Therefore, for *n* data pairs, the *RCV* may be calculated as follows:

The derivation is in the appendix.

## RESULTS AND COMMENT

Advanced interpretation of monitoring tests requires an understanding of both biologic (inherent) and analytic variation. Variation is divided into analytic and biologic components. Examples of biologic variation include random subtle alterations in low-density lipoprotein (LDL) cholesterol and BNP, the fluctuation of serum cortisol with daily circadian rhythm, and the inherent periodic oscillatory behavior of serum insulin in the absence of glucose loading.8 Potential sources of analytic variation include random mechanical variation (eg, pipetting), random sensor variation, and electric white noise. One of the main responsibilities of the laboratory medical director is to ensure that the analytic variation of a test is controlled and minimized, allowing for the enhanced detection of physiologic change for a given constant biologic (inherent) variation.

In contrast to the analytic variation, the laboratory does not have the power to control, minimize, or even affect the biologic (intraindividual) variation. By minimizing the analytic variation in a given test, the laboratory is able to report values that primarily reflect physiologic change as well as the biologic variation. In this context, physiologic change refers to all changes in the patient's state, whether due to pathology, pharmacology, disease progression, or nonpharmacologic intervention. The probability that a difference in serial analyte values represents an actual change in patient physiology is a determination that most clinicians make as part of the clinical judgment process without factoring in the effects of biologic or analytic variation. For some analytes, such as serum calcium, the biologic variation is very low (*CV _{I}* = 1.0%), and small changes are more likely to reflect significant alterations in patient physiology.4 For other analytes, such as serum-conjugated bilirubin, the biologic variation is relatively high (

*CV*= 18.4%), and small changes are less likely to be indicative of true physiologic change.4 In the era of evidence-based medicine, the ability to quantify the probability of a true physiologic change provides valuable information regarding a patient's biologic response to treatment or progression of disease and plays a large role in clinical management.

_{I}An important distinction exists between the presence of a statistical significance of a change in value and the presence of a clinically significant change in patient physiology. Clinical judgment must continue to dictate the interpretation of changing lab values, but clinical judgment itself relies on the presence or absence of statistically significant changes in laboratory values. By understanding the application of statistical methods in determining the significance of change and incorporating the use of multiple serial samples in clinical care, the precision of the test is disproportionately enhanced. This means that smaller changes in analyte values can be confidently incorporated into the monitoring of patients' clinical status.

The formula for evaluating the likelihood of significance of the difference for a series of 2 laboratory values has been described elsewhere,4 although it is not commonly employed in clinical practice. As mentioned previously, the significance of a change in analyte values is determined by comparing the percentage change in analyte value to the critical difference, or *RCV*, for that particular measured analyte.

where *Z* is the *z* score (or multiplier) corresponding to the desired α value, *CV _{A}* is the analytic coefficient of variation, and

*CV*is the intraindividual inherent or biologic coefficient of variation of the analyte in question.

_{I}The coefficient of variation is expressed in units of percentage, as is the reported change in patient values. To combine variation, both analytic and intraindividual (biologic) variations enter into the difference calculation. The biologic variation of an analyte can be found in medical literature and is available in laboratory medicine reference books.9 The analytic variation is lab dependent, reflecting such factors as the equipment, operating procedures, and reagents used. The *CV _{A}* is determined as part of the internal laboratory quality control protocol and is calculated based on recommended manufacturer intervals for the particular test in question. Ideally, if kept below one-half the biologic variation, the analytic variation will contribute 10% or less to the total combined

*CV*.4

As shown in Figures 3 and 4, increasing the number of data points by providing additional analyte measurements within discrete windows of time results in a substantial decrease in the critical difference used to evaluate the statistical significance of changing analyte values. This translates clinically to enhanced power of discrimination between significant and nonsignificant changes for both bidirectional and unidirectional analyses. The *RCV* to determine significant change between patient values at time 1 and time 2 is inversely proportional to the number of patient data pairs between time 1 and time 2. Interestingly, the incremental reduction in *RCV* decreases rapidly after 6 data pairs (for example, 2 measured values at time 1 and 3 values at time 2, for a total of 5 data points); however, theoretically, with an infinite number of data points at time A and time B, the *RCV* would approach zero.

### Example: LDL Cholesterol

Suppose that a patient presents to his or her primary care giver with an LDL cholesterol concentration of 180 mg/dL (to convert to millimoles per liter, multiply by 0.0259). Based on comprehensive epidemiologic studies showing a direct relationship between LDL cholesterol, coronary heart disease, and the rate of new-onset congestive heart disease,10–12 the patient's clinician decides to start treatment with a statin drug in an effort to decrease the patient's LDL level. To monitor progress, the clinician arranges for the patient to return to the office for a follow-up appointment in 3 weeks, at which time repeat labs will be drawn (Table 1). Initial LDL: 180 mg/dL. LDL at follow-up visit: 165 mg/dL.

The clinician wants to know with a 95% confidence interval whether the patient is responding to medical management; that is, whether there is a significant statistical difference (*P* = .05) in the decrease of LDL values. The biologic coefficient of variation of LDL cholesterol is 8.3%4; the analytic variation is determined based on the internal quality control runs within the laboratory, but for this example we may substitute 1%.

The *RCV* indicating the necessary percentage difference between 2 consecutive LDL values for a statistically significant change with 95% probability is calculated as follows:

For a 95% confidence interval, we substitute 1.96 for *Z*. Using our values for the analytic and intraindividual (biologic) variation, we calculate critical reference change value between the 2 analyte measurements as follows:

Therefore, a 23% change in LDL cholesterol is the minimal change required for statistical significance with a 95% confidence interval. This translates into a change of 41 mg/dL, whereas the actual measured change is only 15 mg/dL (8.3%). By this method, the patient does not demonstrate a statistically significance decrease in LDL level, and the response to therapy is indeterminate.

Instead of one sample being taken at both the initial and follow-up visits, suppose that 3 samples were taken at each of these visits. This allows for 9 pair differences (3 initial values × 3 follow-up values). Let us further suppose that the average of the LDL measurements at the initial and follow-up visits is identical to our previous set of data.

Because the average difference of all pair differences is equal to the difference between the average initial and follow-up values, the percent change over the treatment interval remains 8.3%, or an actual change of 15 mg/dL.

*RCV* is now calculated as follows:

Therefore, in order to reach statistical significance at the 95% confidence level, the change in values from the initial to follow-up visit must reach or exceed 7.7% of the initial value, or 14 mg/dL. In this scenario, the laboratory can communicate to the clinician that the patient shows a statistically significant decline in LDL values. The clinician may then confidently incorporate this change in LDL values into his or her decision process regarding further patient management.

### BNP Example

For subjects with heart failure, the *CV _{I}* for BNP during a 1-week period is 41%, whereas for amino-terminal proBNP (NT-proBNP), it is 35%.13 To calculate the 95% confidence interval for a 1-tailed test (1-tailed because one is only interested in whether the BNP or NT-proBNP concentration has decreased), one uses

*Z*= 1.645. Thus, for 1 pair of serial tests , which for BNP is 95% and for NT-proBNP, 81%. To ascertain that therapy has been successful in these patients, one needs to observe an enormous fall in the peptide values. For BNP at a value of 400 pg/mL, the concentration would need to fall to 20 pg/mL, and for NT-proBNP at a value of 600 pg/mL, it would need to fall to 114 pg/mL. These reductions are too great to be of much use. Instead, several studies have demonstrated clinical significance, as indicated by fewer patients readmitted with decompensated heart failure after discharge, when a reduction of BNP or NT-proBNP of 50% or more was observed.13,14 A reduction of 50% for BNP yields a

*Z*of 0.865, corresponding to a confidence interval of 80%, and a reduction of 50% for NT-proBNP yields a

*Z*of 1.01, corresponding to a confidence interval of 84%. Confidence intervals less than the typical 95% have an impact on the ability to use the natriuretic peptides in predicting improved outcome.15

Alternatively, if one were to use multiple samples with 4 serial pairs, then for BNP the percentage change is , and for NT-proBNP it is (600 pg/mL to 480 pg/mL). Thus, one could detect a favorable outcome with a much lower percentage change.

In addition, one could calculate the number of serial pairs necessary to achieve a change based on an arbitrary confidence interval. For a confidence interval of 90%, *Z* = 1.28 for a 1-tailed test:

where |*m*| is the fractional or percentage difference.

For a confidence interval of 90%, a percentage difference of 50% requires at least 2 pairs for both peptides, a percentage difference of 40% requires 3 serial pairs for BNP and 2 serial pairs for NT-proBNP, and a percentage difference of 30% requires 5 serial pairs for BNP and 3 for NT-proBNP.

### Effect of Reduction of Analytic Variance

Reduction of analytic variance does not have the same impact as multisampling. To illustrate, we will use CA-125 as an example, because both of its variations are large. The *CV _{A}* for CA-125 is 12.6%, and its

*CV*is 24%.16

_{I}Let *n*(*A*) represent the number of times that CA-125 is determined for the same sample, which we will call analytic measurements.

When *n*(*A*) = 1, then the combination of the analytic and intra-individual coefficients of variation at one time (*CV _{t}*) = 0.271.

Take 9 analytic measurements (ie, 1 sample analyzed 9 times at *t*_{1} and *t*_{2}, for a total of 18 determinations, then:

When *n*(*A*) = 9, then *CV _{t}* = 0.244.

When *n* = 1 and *n*(*A*) = 9, then *Z* = 2.9·|*m*|. Setting *Z* = 1.65, then |*m*| = (1.65/2.9) = 0.57 = 57%, compared with 63% when *n*(*A*) = 1, for a reduction of 9.5%.

Instead, take 3 samples at *t*_{1} and *t*_{2}, each determined once analytically; thus, *n* = 9 and *n*(*A*) = 1. The combination of analytic and intraindividual standard deviations at one time (*S _{t}*) = 0.271.

These results yield a reduction of 67%. If one were to set *n* = 9 with *n*(*A*) of 1, then the percentage change in the test to yield a significant change at a 95% confidence level would be 4.3%.

### Relevance of Biologic Variation to Testing Schedule

The timing between serial patient samples within one designated window of measurement (for example, on the first day of measurement to establish a patient's baseline, or the follow-up visit to detect the presence of change) may be a source of confusion when considering the temporal nature of a particular analyte's biologic variation. Practically speaking, the method and timing of sampling should be convenient and not necessitate any additional clinical encounters than those employed in the current paradigm of patient testing. The enhanced statistical power provided by multiple-analyte testing should not preclude common-sense approaches to patient testing. If the analytes to be monitored have a known circadian rhythm, then drawing samples within the same time window on days of initial and follow-up testing is clearly indicated. Additionally, if such factors as glucose load or physical activity alter analyte concentrations, then similar testing conditions must be provided with each patient visit.

When defining the effect of the temporal aspect of the inherent (biologic) variation of an analyte, which is independent of the known (and controlled-for) influencing factors discussed above, one must consider the nature of the physiologic regulation of the analyte in question. Generally speaking, the extent of biologic variation for a given analyte is defined by the interaction between the drift, or random movement of a concentration of analyte from baseline and the opposing restoring forces (Φ), which return the analyte to a physiologic baseline.17 In the case of analytes that are very highly regulated, such as sodium, a random disturbance in concentration is rapidly met with an opposing restoring force that almost immediately returns the analyte to physiologic baseline. In an alternate model, a steady state exists between the production and destruction/removal of a specific analyte, and in response to a disturbance from baseline, the restoring force takes the form of increased production or increased removal of the analyte in question. In this model, the kinetics of the correction is linear and analyte-dependent, taking the form of exponential decay.

where δ_{i} is the deviation and Φ is a constant (unit of [time]^{−1}) determined by the nature of the system and the restoring force for that particular analyte.17 Examples of analytes whose regulation follows this model include prostate-specific antigen, alanine aminotransferase, and aspartate aminotransferase.17

The issue to be addressed when considering the potentially confounding effect of biologic variation in multiple-sample testing is whether serial samples taken close together will produce values that are physiologically distinct from one another while being minimally affected by the individual biologic variation of the patient being tested.

There are 2 limits to examine in this context. The lower limit defines how close together in time the samples can be taken to allow independence between samples. The upper limit defines how great a time can separate samples without the effects of biologic variation significantly altering the values.

The lower limit may be found by considering the movement of blood by the collection site. The total volume of blood in the intravascular space in an average adult male is 5.5 L, and the resting cardiac output is, on average, 5 L/min.18 That means that 90% of the body's blood volume is recirculated each minute at a rate of more than than 80 mL/s. If the arm from which the sample is being taken is not subject to a tourniquet, only a few seconds are needed between sample draws to collect samples that are geographically and physiologically distinct from each other. Therefore, the temporal lower limit between sampling is on the order of seconds, allowing samples to be drawn via the same venipuncture and with only a brief loosening of the tourniquet between draws. Such an effect has been demonstrated with rapidly changing insulin concentrations, with samples obtained every minute.19

The upper temporal limit between sampling may be calculated by determining the acceptable percent of the analyte's biologic variation expressed within the time window during which samples are drawn. The degree to which an analyte expresses its biologic variability within any given time period may be calculated and is dependent on the related variables of the analyte's biologic variation and restoring force (Φ, expressed as unit of [time]^{−1}).17 As the restoring force of an analyte increases, the time to reach the maximum biologic variation decreases. The time needed to reach a maximum biologic variation is not known for all analytes, but values can be mathematically determined.17 The linear relationship between an analyte's inherent biologic variation and the proportion of that biologic variability that is expressed within a 5-minute window may be expressed as *Y* = 0.16*X* + 0.0003 (*R*^{2} = 0.999), where *Y* equals percent biologic variation expressed within 5 minutes, and *X* is the restoring force (Ф; units of [hours]^{−1}). The relationship between the time an analyte takes to achieve 99% of its biologic variation to the amount of biologic variation expressed in 5 minutes is an inverse function, *Y* = 0.3939*X* − 0.0004 (*R*^{2} = 0.999), in which *Y* equals the percent of the analyte's biologic variation expressed within 5 minutes, and *X* is the inverse of the hours needed for the analyte to express 99% of its biologic variability. Even at a high restoring force, where it may take 5 hours for an analyte to reach its maximum biologic variability, at 5 minutes, only 0.08% of that analyte's biologic variability has been achieved. Therefore, for many analytes, samples drawn within a time period between several seconds and 5 minutes would produce satisfactory values representing physiologically and geographically distinct samples with the minimal confounding influence of the analyte's inherent biologic variability (Figures 5 and 6; Table 2). Use of multisampling can have a profound effect on samples with large biologic variations, reducing the *RCV* (Figure 7).

The laboratory may incorporate multisampling into practice by establishing an order set specifying the collection of multisamples, first at t_{1} and then at time t_{2}, based on the known interval desired for the biologic variation and the desired difference between the 2 times (eg, 1 day apart for BNP). Upon collection of the second set of data, the laboratory information system can generate a report showing the raw data, the calculated difference, the *CV _{I}* and

*CV*, and the confidence interval for change. Because many factors may complicate the results, a laboratory physician or scientist should provide interpretation of the results.

_{A}### Conclusion

Laboratory medicine physicians or scientists are in a unique position to influence the paradigm of patient care. Because the clinician does not have access to the analytic *CV* and has limited access to the intraindividual variation, the laboratory medical director is able to provide a valuable expertise; that is, the determination of significance of changing quantitative values.

Currently, the intuitive clinical decision making of clinicians often relies on past experience, with incomplete usage of appropriate statistical tools. In such cases, the clinical thought process skips the statistical verification of significance and relies on an intuitive approach to test interpretation. One may enhance one's ability to detect significant change by using multisample tests. As shown in the example of LDL cholesterol levels discussed previously, in both testing paradigms, the preintervention and postintervention values were the same. Therefore, an interpreting physician may be tempted to interpret the difference as significant or not in his or her personal judgment, regardless of the method of testing (ie, using multiple samples). It will only be through a regular and active educational process that a new relationship between the laboratory and the clinicians will develop to take maximal advantage of such testing paradigms. Indeed, this approach has the potential to contribute to the personalization of health care delivery by redefining common thresholds in clinical patient management. The decision to treat a patient with a more aggressive therapeutic regimen may be reconsidered when a change in testing protocol can show that a small but highly statistically significant change has taken place. Such a result may lead to the postponement of switching to a new therapeutic regimen until the next testing time, where the initial small but significant change in test values has now grown to be both statistically and clinically relevant. In such an example, the switch to a new, potentially deleterious regime has been avoided by the knowledge that although the initial response to therapy was low, it represented a real physiologic change toward a therapeutic level, and more time may be needed to see optimal response.

Evaluating the clinical significance of multiple paired sets of patient data, rather than a single pair, results in a substantial decrease in the reference change value (critical difference), thereby improving the sensitivity of the evaluation. A practice of using multiple patient samples results in enhanced power to detect true changes in patient physiology. This testing protocol is warranted when small changes in the analyte precede serious clinical events or when the *CV* of the biologic variation is large. We have shown that such a change in testing practice could meaningfully alter the clinical management of patients in a variety of situations. This testing approach may also alter patient care when used to expedite processing and development of new drugs. Testing multiple analyte samples as outlined in this paper has the potential to fundamentally alter the way in which laboratory medicine is practiced; further research into the feasibility and cost of multiple analyte testing paradigms in the clinical setting is warranted.

## References

### Appendix: Derivation of Multisample Formula for Biologic Variation

Calculate *Z* from the following formula:

Number20 of measurements in the sample = *n*. Mean fraction of the sample measurements = *m*. Mean fraction of the large parent group = *M*. Standard deviation of the large parent group = *S*. Coefficient of variation for the large parent group = *CV*. Let *CV _{t}* = the standard deviation of any given measurement at a particular time; it is given as

*CV* refers to the coefficient of variation for the measurement, *A* to the analytic component, and *I* the individual component (biologic variation). *CV* is a combination (the square root of the sum of the squares of the standard deviations) of *CV _{t}* for a

*t*

_{1}and a

*t*

_{2}. It is given as

Because the analytic and biologic variations at the 2 different times frequently are equivalent, its value is often reduced to

when the biologic variation is much greater than the analytic variation.

*M* is the fractional difference between a value measured at time 1 and that measured at time 2. One must sum up all of the difference pairs.

Number of measurements in the parent population = *N*.

Number of measurements in the parent population = *N*. *M* = 0 if there is no difference between the state of the system at the 2 different times, because the sum of the differences adds up to zero.

Thus,

The number of differences is given as *n* and is given by the formula *n* = *r*·*p*. The calculation of *m* can be simplified to

where the bar indicates the average for the values.

One can rearrange this formula to find the fraction (percentage) change,

when number of samples at time 1 (*r*) and the number of samples at time 2 (*p*) are not the same, and

when they are. When *n* = 1, then the formula becomes .

## Author notes

From the Division of Pathology and Laboratory Medicine, Boston Medical Center, Boston University School of Medicine, Boston, Massachusetts.

The author has no relevant financial interest in the products or companies described in this article.