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Quantifying Uncertainty in the Ratio of Two Measured Variables: A Recap and Example
Journal of Pharmaceutical Sciences xxx (2016) 1-2
Contents lists available at ScienceDirect
Journal of Pharmaceutical Sciences journal homepage: www.jpharmsci.org
Lessons Learned
Quantifying Uncertainty in the Ratio of Two Measured Variables: A Recap and Example David M. Shackleford 1, *, Kris M. Jamsen 2 1 2
Centre for Drug Candidate Optimisation, Faculty of Pharmacy and Pharmaceutical Sciences, Monash University, Parkville, Victoria, Australia Centre for Medicine Use and Safety, Faculty of Pharmacy and Pharmaceutical Sciences, Monash University, Parkville, Victoria, Australia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 July 2016 Accepted 26 July 2016
Estimating uncertainty in the ratio of 2 measured variables can be achieved via 2 seemingly different approaches: by determining the variance of the first-order Taylor approximation to the ratio, or by the so-called “Propagation of Error” approach. This Lesson Learned shows that the 2 approaches are mathematically equivalent, and provides an example of the approach. © 2016 American Pharmacists Association®. Published by Elsevier Inc. All rights reserved.
Keywords: ADME bioavailability caco-2 cells in vitro models pharmacokinetics
Introduction In experimental biopharmaceutics there are numerous scenarios in which a function is defined by the relationship of 2 measured variables. One example is the so-called efflux ratio (ER) in cell-based permeability assays, where ER is calculated as the ratio of the apparent permeability (Papp) in the basolateral-toapical (B-A) and apical-to-basolateral (A-B) directions; that is, ER ¼ Papp(B-A)/Papp(A-B). A perusal of the literature indicates that values for ER determined within an experiment are often reported as absolute. This absolute reporting ignores uncertainty that may arise through measurement of the numerator and denominator (e.g., variability in Papp values in the B-A and A-B directions). Consequently, this can cause difficulty in applying relatively simple statistical methods to evaluate possible implications of changes in these ratios. In the ER example, such a question might be whether there is significant evidence of a compound’s efflux being reduced in the presence of a transport inhibitor.
Approach Quantifying the uncertainty in the ratio of 2 measured variables involves computing the variance of the ratio. The derivation of the formula for this variance is slightly nontrivial, requiring 2 steps: * Correspondence to: David M. Shackleford (Telephone: þ61399039065; Fax: þ61399039052). E-mail address: [email protected] (D.M. Shackleford).
1. Applying a bivariate first-order Taylor approximation to the ratio, centered at the average values of the numerator and denominator. 2. Deriving the variance of this first-order Taylor approximation. The bivariate first-order Taylor approximation of a ratio G ¼ y/x is given by
vG vG ðx mx Þ þ y my : GzG mx ; my þ dx dy
(1)
In Equation 1, mx and my represent the averages of x and y, respectively. The variance of this approximation (assuming x and y are independent) is given as follows:
vG vG ðx mx Þ þ VarðGÞzVar G mx ; my þ y my dx dy my my 1 ðx mx Þ þ ¼ Var y my mx mx m2x ¼
VarðxÞm2y m4x
þ
VarðyÞ m2x
:
Thus,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VarðxÞm2y VarðyÞ : SDðGÞ ¼ þ m2x m4x
(2)
In Equation 2, SD indicates the standard deviation. For example, if one wanted to compute the SD of Papp(B-A)/Papp(A-B), this would
http://dx.doi.org/10.1016/j.xphs.2016.07.019 0022-3549/© 2016 American Pharmacists Association®. Published by Elsevier Inc. All rights reserved.
2
D.M. Shackleford, K.M. Jamsen / Journal of Pharmaceutical Sciences xxx (2016) 1-2
require sample means and SDs for Papp(A-B) and Papp(B-A). Say in a hypothetical experiment the sample mean and SD for Papp(A-B) were 1.47 and 0.425, respectively, and the corresponding values for Papp(B-A) were 2.58 and 0.823, respectively. Also, Papp(A-B) was measured in one group of transwells and Papp(B-A) was measured in another group of transwells (hence Papp(A-B) and Papp(B-A) are independent). Applying Equation 2 yields an SD of 0.756. It can be shown that Equation 2 is mathematically equivalent to the “propagation of error” (POE). The formula for POE (again assuming x and y are independent) is given as follows:
POE ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u 2 my u t SDðxÞ þ SDðyÞ : mx my mx
(3)
Applying this formula to the hypothetical experimental data above yields the same SD (see You et al.1 for an application). In some experiments, the objective may be to generalize the results to the population. Specifically, the investigators may want to derive a plausible range for what the true (but unknown) ratio may be in the population (i.e., a confidence interval). This would involve computing the standard error (SE) for the ratio. This can be achieved by using Equation 2 and computing the SEs for x and y:
SDðxÞ SEðxÞ ¼ pffiffiffiffiffi nx SDðyÞ SEðyÞ ¼ pffiffiffiffiffi : ny
(4)
In Equation 4, nx and ny indicate the number of observations for x and y, respectively. To obtain the SE of G, the square of the SE terms in Equation 4 can be plugged into the respective variance terms in Equation 2. Using the hypothetical experimental data above, assuming 3 and 4 observations for Papp(A-B) and Papp(B-A), respectively, the SE for Papp(B-A)/Papp(A-B) (or equivalently the SD for mPappðBAÞ =mPappðABÞ ) is 0.405. This corresponds to a 95% confidence interval of 0.961-2.55. This short communication provides a recap on the uncertainty in the ratio of 2 measured variables. Equation 1 can be used to calculate the SD of a ratio (equivalent to the POE) or the SE of a ratio. Reference 1. You L, Qian J, Wu X, et al. Propagation of error in ocular pharmacokinetic parameters estimate of azithromycin in rabbits. J Pharm Sci. 2013;102(7):23712379.
Contents lists available at ScienceDirect
Journal of Pharmaceutical Sciences journal homepage: www.jpharmsci.org
Lessons Learned
Quantifying Uncertainty in the Ratio of Two Measured Variables: A Recap and Example David M. Shackleford 1, *, Kris M. Jamsen 2 1 2
Centre for Drug Candidate Optimisation, Faculty of Pharmacy and Pharmaceutical Sciences, Monash University, Parkville, Victoria, Australia Centre for Medicine Use and Safety, Faculty of Pharmacy and Pharmaceutical Sciences, Monash University, Parkville, Victoria, Australia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 July 2016 Accepted 26 July 2016
Estimating uncertainty in the ratio of 2 measured variables can be achieved via 2 seemingly different approaches: by determining the variance of the first-order Taylor approximation to the ratio, or by the so-called “Propagation of Error” approach. This Lesson Learned shows that the 2 approaches are mathematically equivalent, and provides an example of the approach. © 2016 American Pharmacists Association®. Published by Elsevier Inc. All rights reserved.
Keywords: ADME bioavailability caco-2 cells in vitro models pharmacokinetics
Introduction In experimental biopharmaceutics there are numerous scenarios in which a function is defined by the relationship of 2 measured variables. One example is the so-called efflux ratio (ER) in cell-based permeability assays, where ER is calculated as the ratio of the apparent permeability (Papp) in the basolateral-toapical (B-A) and apical-to-basolateral (A-B) directions; that is, ER ¼ Papp(B-A)/Papp(A-B). A perusal of the literature indicates that values for ER determined within an experiment are often reported as absolute. This absolute reporting ignores uncertainty that may arise through measurement of the numerator and denominator (e.g., variability in Papp values in the B-A and A-B directions). Consequently, this can cause difficulty in applying relatively simple statistical methods to evaluate possible implications of changes in these ratios. In the ER example, such a question might be whether there is significant evidence of a compound’s efflux being reduced in the presence of a transport inhibitor.
Approach Quantifying the uncertainty in the ratio of 2 measured variables involves computing the variance of the ratio. The derivation of the formula for this variance is slightly nontrivial, requiring 2 steps: * Correspondence to: David M. Shackleford (Telephone: þ61399039065; Fax: þ61399039052). E-mail address: [email protected] (D.M. Shackleford).
1. Applying a bivariate first-order Taylor approximation to the ratio, centered at the average values of the numerator and denominator. 2. Deriving the variance of this first-order Taylor approximation. The bivariate first-order Taylor approximation of a ratio G ¼ y/x is given by
vG vG ðx mx Þ þ y my : GzG mx ; my þ dx dy
(1)
In Equation 1, mx and my represent the averages of x and y, respectively. The variance of this approximation (assuming x and y are independent) is given as follows:
vG vG ðx mx Þ þ VarðGÞzVar G mx ; my þ y my dx dy my my 1 ðx mx Þ þ ¼ Var y my mx mx m2x ¼
VarðxÞm2y m4x
þ
VarðyÞ m2x
:
Thus,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VarðxÞm2y VarðyÞ : SDðGÞ ¼ þ m2x m4x
(2)
In Equation 2, SD indicates the standard deviation. For example, if one wanted to compute the SD of Papp(B-A)/Papp(A-B), this would
http://dx.doi.org/10.1016/j.xphs.2016.07.019 0022-3549/© 2016 American Pharmacists Association®. Published by Elsevier Inc. All rights reserved.
2
D.M. Shackleford, K.M. Jamsen / Journal of Pharmaceutical Sciences xxx (2016) 1-2
require sample means and SDs for Papp(A-B) and Papp(B-A). Say in a hypothetical experiment the sample mean and SD for Papp(A-B) were 1.47 and 0.425, respectively, and the corresponding values for Papp(B-A) were 2.58 and 0.823, respectively. Also, Papp(A-B) was measured in one group of transwells and Papp(B-A) was measured in another group of transwells (hence Papp(A-B) and Papp(B-A) are independent). Applying Equation 2 yields an SD of 0.756. It can be shown that Equation 2 is mathematically equivalent to the “propagation of error” (POE). The formula for POE (again assuming x and y are independent) is given as follows:
POE ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u 2 my u t SDðxÞ þ SDðyÞ : mx my mx
(3)
Applying this formula to the hypothetical experimental data above yields the same SD (see You et al.1 for an application). In some experiments, the objective may be to generalize the results to the population. Specifically, the investigators may want to derive a plausible range for what the true (but unknown) ratio may be in the population (i.e., a confidence interval). This would involve computing the standard error (SE) for the ratio. This can be achieved by using Equation 2 and computing the SEs for x and y:
SDðxÞ SEðxÞ ¼ pffiffiffiffiffi nx SDðyÞ SEðyÞ ¼ pffiffiffiffiffi : ny
(4)
In Equation 4, nx and ny indicate the number of observations for x and y, respectively. To obtain the SE of G, the square of the SE terms in Equation 4 can be plugged into the respective variance terms in Equation 2. Using the hypothetical experimental data above, assuming 3 and 4 observations for Papp(A-B) and Papp(B-A), respectively, the SE for Papp(B-A)/Papp(A-B) (or equivalently the SD for mPappðBAÞ =mPappðABÞ ) is 0.405. This corresponds to a 95% confidence interval of 0.961-2.55. This short communication provides a recap on the uncertainty in the ratio of 2 measured variables. Equation 1 can be used to calculate the SD of a ratio (equivalent to the POE) or the SE of a ratio. Reference 1. You L, Qian J, Wu X, et al. Propagation of error in ocular pharmacokinetic parameters estimate of azithromycin in rabbits. J Pharm Sci. 2013;102(7):23712379.