(12.4)
Notice that the first term for Bilirubin is squared. These results indicate that the inter- ference mechanisms are complicated. If one is not aware of the nonlinear nature of the process one might miss important details.
12.2 Statistical Testing for Significance
To test for statistical significance one can use a t-test using the following formula
(12.5)
where SE represents the standard error of the coefficient (the slope in standard linear regression). In this case the subscript ‘1’ indicates that the coefficient and the stand- ard error are for the first term in the formula, which is because it serves as the coef- ficient for the concentration of the analyte, and indicates the recovery of the signal for the analytical method. Usually the b1 coefficient will have a value very close to 1.00.
The standard error of the coefficient is calculated when one performs the multiple regression analysis . To test for bias in the recovery of the method, one would calculate assuming that the coefficient should be equal to 1.00. One would calculate t as
(12.6)
if the value for t, based on the number of standard deviations, is low enough not to reject the null hypothesis, e.g., t = 1.5 for 8 degrees of freedom, then there is no bias for the method due to some sort of calibration error. On the other hand, if t = 3.0 for 8 degrees of freedom , then, for a two-tailed test (a two-tailed test implies that the values can be higher or lower), then p <0.05 and one would reject the null hypothesis.
The null hypothesis in this case is that the coefficient is not different from 1.00. If the coefficient were less than 1.00, e.g., 0.9, one would say that there was a negative bias.
If the coefficient were greater than 1.00, e.g., 1.1, one would say that there is a positive bias.
The b2 coefficient represents the quantification of the effect that a potential inter- ferent has on the determination process as captured by the signal and that signal is transformed into a value based on the calibration of the method. If the potential interferent has no effect, which means that it does not interfere with the method, then the value of the coefficient would be 0.0, the null hypothesis in this case. If the value of the coefficient is different from 0.0, the alternate hypothesis , then that means that Result = −0.0002[Bilirubin]2+0.299[Bilirubin]+279.
t=|b1−0| SEb1
,
t=|b1−1| SEb1
,
the potential interferent does interfere with assay. To test the statistical significance of the coefficient, one uses the t-test
(12.7)
where the assumed value for no interference is zero. If the result for the t-test yields a p >0.05, then one can accept the null hypothesis and claim that there is no interfer- ence; however, if the result for the t-test yields a p <0.05, then one would reject the null hypothesis and accept the alternate hypothesis, that there is an interference from this substance. If the coefficient is positive, then that means that there is a positive interference. If the coefficient is negative, then that means that there is a negative interference. If one stopped here, then one would be analyzing the data using the standard linear regression for the model F=b1[A]+b2[I]+b0, and one assumes that the interference model is simple (Fig. 12.4).
bilirubin
analyte reagent
reagent chromophore
chromophore
Fig. 12.4: Diagram for analyte-independent interference.
The b2 coefficient would encompass any interference , whether dependent or inde- pendent, but it would be prone to errors. Evaluation of the third coefficient allows one to ascertain any contribution by analyte-dependent interference and clearly sepa- rates the contribution caused by analyte-independent interference which is indicated by the second coefficient.
The b3 coefficient represents the quantification of the contribution to the signal that a potential interferent makes when it interacts directly with the analyte or with an intermediary in the analytical reaction or even with the final product. If the poten- tial interferent has no effect, which means that it does not interact in the reaction and interfere with the method, then the value of the coefficient would be 0.0, the null hypothesis. If the value of the coefficient is different from 0.0, the alternate hypoth- esis, then that means that the potential interferent does interact with the analyte, an intermediary or final product and interfere with assay. To test the statistical signifi- cance of the coefficient, one uses the t-test
(12.8) t= |b2−0|
SEb2
,
t=|b3−0| SEb3
,
12.2 Statistical Testing for Significance 131
where the assumed value for no interaction and interference is zero. If the result for the t-test yields a p >0.05, then one can accept the null hypothesis and claim that there is no interaction leading to interference; however, if the result for the t-test yields a p <0.05, then one would reject the null hypothesis and accept the alternate hypoth- esis, that there is an interaction that leads to interference for this substance. If the coefficient is positive, then that means that there is a positive interference. If the coef- ficient is negative, then that means that there is a negative interference (Fig. 12.5).
chromophore reagent
analyte
intermediary bilirubin
intermediary
Fig. 12.5: Diagram for analyte-dependent interference.
In some cases, the interference caused by an interferent, when plotted on a graph, does not show a straight line. Interferences caused by analyte-interferent interactions are more likely than simple analyte-independent interferences to demonstrate devia- tion from a straight line. Deviation from a straight line represents a nonlinear effect.
Interactions between two reacting species are prone to nonlinearities. A require- ment for methods to be linear is that the concentrations of the reagents used in the determination of the analyte have to be much greater than the concentration of the analyte. Keeping the concentration of the reagents greater than the concentration of the analyte means that as the reaction proceeds, the apparent concentration of the reagents does not appear to decrease. Usually, the ratio of the concentration of the reagents compared with the concentration of the analyte is at least 20–1. In enzymatic reactions, the substrate’s concentration must be kept high, otherwise in specimens with extremely high concentrations of the analyte substrate depletion can occur.
When the interferent reacts with an intermediary in the reaction, the concentration of the intermediary is very close to that of the analyte. The reaction can be described using an equilibrium constant, K:
(12.9)
but as the reaction proceeds, the amount of interferent and intermediary decrease as the product is consumed. Using P as the product, and ITM as the intermediary, with subscripts of 0 to indicate the initial concentration,
K =[interferent−intermediary−product] [interferent][intermediary]
(12.10)
and the concentration of product appears in both the numerator and the denomina- tor in a nonlinear fashion. With increasing concentrations of interferent the amount of product decreases. Often this pattern follows a pattern that looks like substrate exhaustion. For the simpler formula for K where the interferent is not being greatly consumed,
(12.11)
and the equation for the production of the product becomes
(12.12)
which is a hyperbolic curve and clearly nonlinear (see the lower graph in Fig. 12.3).
Nonlinear relationships can be successfully captured in the multiple regression anal- ysis using polynomial regressions [4].
For example, in peroxidase based reactions, hydrogen peroxide is produced and aminoantipyrine reacts with a phenol derivative to produce a chromophore . If bili- rubin reacts directly with hydrogen peroxide or with an intermediary, the apparent analytes (creatinine, uric acid, triglycerides, cholesterol, etc.) would appear to be decreased. The concentration of bilirubin needs to be only one-tenth of the concentra- tion of analyte to produce this nonlinearity. As examples, the normal concentration of bilirubin has a limit at about 20 μmol/L. At ten times this concentration, the concen- tration of bilirubin would be 200 μmol/L. A mildly elevated concentration of triglyc- erides is 2.26 mmol/L, which has a ratio of about 11 and could affect the total amount of the interference. A mildly elevated uric acid has a concentration of 500 μmol/L and with a bilirubin of 200 μmol/L, that ratio is 2.5 to 1. And for creatinine, a mildly elevated concentration would be 150 μmol/L and with a bilirubin concentration of 200 μmol/L, the ratio of creatinine to bilirubin is 0.75 to 1, enough to readily reduce the amount of apparent creatinine, but also to produce a major nonlinearity.
The complexity of the nonlinear interactions between conjugated bilirubin and creatinine has been documented in detail. Using an enzymatic method that employs creatininase to form creatinine, creatinase to form sarcosine, sarcosine oxidase to form glycine and hydrogen peroxide and finally hydrogen peroxide with a phenol derivative, 4-aminophenazone and peroxidase (PAP) to form a red benzoquinon- eimine dye , Eng et al. showed that one needed to use polynomial regressions to find the best fit of the data for the interference [5]. Stepwise regression analysis showed that the best fitting of the data occurred when using models that included terms such
K = [P]
([I]0−[P])([ITM]0−[P])
K = [P]
([I])([ITM]0−[P])
[P]= K[I] [ITM]0
1+K[I] ,
12.4 Advantages of Using Multiple Regression Analysis 133
as [creatinine]2[bilirubin] or even higher terms, and that it did not matter whether one used an end-point or a kinetic method with the PAP reagents. The nonlinearities in the data set were so large that fitting the data to a linear model produced large errors.
Also, it suggests that there may be multiple interactions occurring between bilirubin and the intermediaries [5].