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**Mahmoud Abdel Mohsen ^{*}**

Pharmaceutical Research Unit (PRU), Royal Scientific Society (RSS), P.O. Box 1438, Amman 11941, Jordan

- *Corresponding Author:
- Dr. Mahmoud Abdel Mohsen

Division of Biometrics

Pharmaceutical Research Unit

Royal Scientific Society

P.O. Box 1438, Amman 11941, Jordan

**Tel:**+962-6-5358290

**Fax:**+962-6-5358261

**E-mail:**[email protected]

**Received Date:** July 13, 2010; **Accepted Date:** August 24, 2010;** Published:** August 24, 2010

**Citation:** Mohsen MA (2010) A Note on the Calculation of Intrasubject Coefficient of Variation in Bioequivalence Trails. J Bioanal Biomed 2:075-078. doi:10.4172/1948-593X.1000026

**Copyright:** © 2010 Mohsen MA. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Bioanalysis & Biomedicine

Bioequivalence studies are generally performed as crossover studies and, therefore, information in the Intrasubject Coef fi cient of Variation (CV) is needed for sample size planning. Recently, a confusing point was noticed in calculating the Intrasubject Coef fi cient of Variation (CV) in crossover studies under the additive model (i.e. under normality assumption and the multiplicative mode (i.e under logarithmic distribution). The aim of this paper is to clarify this confusion. Methods used in calculating the Intrasubject Coef fi cient of Variation (CV) are reviewed in this paper from a statistical point of view.

Coefficient of variation; Bioequivalence; Crossover design; Sample size; Normal and log-normal distribution

Bioavailability (BA) and Bioequivalence (BE) studies investigate and compare pharmacological characteristics of different drug formulations in terms of the rate and the extent of absorption of active ingredients.

The most common pharmacokinetic responses of the rate and the
extent of absorption are: Area under the plasma concentration-time
curve (AUC), maximum plasma concentration (C_{max}), and time to reach
maximum plasma concentration (t_{max}).

The distributions of responses such as AUC and C_{max} are often
positively skewed and exhibit a lack of homogeneity of variances.
Therefore, normality assumption on AUC and C_{max} may not be
suitable. In this case, the assessment of average BE based on raw
data and normality assumptions may not be appropriate. Logarithmic
transformation is the commonly chosen data transformation
technique for BE studies and is also recommended by the FDA
“Guidance on Statistical Procedures for Bioequivalence Studies using
a Standard Two-Treatment Crossover Design (July 1992)”.

The most important variation in crossover designs is the intrasubject or within-subject variance, which is the variation, exhibited by a single person when given the same dose of a drug over repeated administrations.

The magnitude of the intrasubject variance depends on the pharmacokinetics of the drug itself. For instance, drugs with simple kinetics (e.g. with little metabolism or well absorption) will generally have estimated intrasubject coefficient of variations (CVs) of less than 20%, whereas drugs with complicated kinetics (e.g. highly variable absorption) can have estimated coefficient of variations (CVs) that are equal to or greater than 40%. Therefore, information on intrasubject coefficient of variation (CV) is needed for sample size planning of future studies.

We believe that (Chow and Liu, 2000) in Chapter (6) of “Transformation and Analysis of Individual Subjects Ratios” had erroneously evaluated the Intrasubject Coefficient of Variation (CV) using a certain example. In this note, we will suggest a better formula in finding the Intrasubject (CV) under the two models of normality assumptions:

1. Additive model (raw data model).

2. Multiplicative model (log-transformed model).

The Intrasubject (CV) can be estimated from the analysis of variance of the parameters: AUC_{0→8}, C_{max}, and C_{max}/ AUC_{0→8} based on the residual mean square for error.

Theorem (1): Let X_{1}, X_{2}….X_{n} be a sample from normal distribution,
that is X ~ N (µσ^{2}) then the coefficient of variation of X is defined as:

Coefficient of Variation

In BE studies,where µ_{R} is the mean of the reference
formulation. Hence, the Intrasubject Coefficient of Variation (CV)
under raw data normality assumption is

Which can be estimated by

(1)

Where:

MSE is the mean square error for within subject variability (not for subject within sequence error), is the least square (LS) mean for the reference formulation.

Multiplication of (1) by 100 gives the percent Intrasubject CV for random variable X (raw data, normality assumption).

Theorem (2): Let X_{1}, X_{2}, …..., X_{n}, be a sample from log-normal
distribution; that is Ln(X) ~ N . Then, the Coefficient of
Variation of (X) is defined as:

Proof: Now let X_{1}, X_{2}….., X_{n} be a sample from the log-normal
distribution then Y_{1}= L_{n}(X_{1}), …..Y_{n}= L_{n}(X_{n}) is a sample from a N distribution.

Straight forward computation yield:

Var (X) = exp (2µ,σ^{2}) exp ((σ^{2}) – 1),

Then by using Theorem (1), the Coefficient of Variation (CV) of X is defined as:

Coefficient of Variation

In BE studies,, where: is within subject variability.

Hence, the Intrasubject CV under log-transformed data, normality assumption is

This can be estimated by

(2)

Where: MSE is the mean square error for within subject variability (not for subject within sequence).

Multiplication of (2) by 100 gives the percent Intrasubject CV for a random variable X (log-transformed data, normality assumption).

Now we can distinguish between (formula 1) and (formula 2) in finding the Coefficient of Variation (CV) for a given random variable.

Under raw data, (formula 1) depends on two unknown parameters and *μ _{R}* . On the other hand, under log-transformed data, (formula
2) depends on one unknown parameter .

Consequently, in order to build a good BE study, we recommend using the log-transformed data transformation in crossover designs.

If one is interested in finding the (1 - α) level confidence interval for the intrasubject coefficient of variation (CV) under log-normal distribution in a 2x2 crossover design, one can use the following:

(Theorem 3): Assuming a log-normal distribution and a 2x2 crossover design, it is known that a (1 - α) level confidence interval for is [ Le, Ue ].

Where:

Where X^{2}(n_{1} + n_{2} − 2) denotes the cumulative distribution function of a central chi-squared distribution with (n_{1}+n_{2} - 2) degrees of
freedom. Thus

(3)

is the (1 - α) level confidence interval for the intrasubject coefficient of variation (CV).

**The measure of sensitivity**

The index of measure of sensitivity is defined as the ratio of the
width of the average BE interval (2Δ = θ_{U} – θ_{L}), to standard error
of the difference in least square means for the Test and the
Reference formulations, that is:

1. Under raw data, normality assumption, and according to the ± 20 rule, , therefore, ∇ can be written in terms of as follows:

(4)

(See for example in (Chow and Liu, 2000).

2. Under Log transformed data, normality assumption, and according to the equivalence limit = (Ln (0.8), Ln (1.25)), ▽ can be written in terms of as follows:

(5)

In the two cases, this index (▽) is a function that depends on the
Coefficient of Variation (CV) and the sample sizes; n_{1} and n_{2} used
in BE studies.

(Schuirmann, 1987) used this index to compare the size of the two one-sided test procedures to the power approach.

Under raw data, normality assumption, (Chow and Liu, 2000)
calculated values of ▽ for different combinations of CVs, n_{1}, and n_{2} as shown in (**Table 1**).

Nominal concentration (ng/mL) | Observed concentration (ng/mL, mean ± S.D.) |
Precision a (%) |
Accuracy b (%) |
---|---|---|---|

10 | 10.17 ± 0.21 | 2.05 | 101.67 |

20 | 19.27 ± 1.15 | 5.97 | 96.33 |

50 | 44.35 ± 1.06 | 2.39 | 88.70 |

100 | 107.67 ± 4.04 | 3.75 | 107.67 |

200 | 216.67 ± 1.53 | 0.71 | 108.33 |

500 | 489.33 ± 20.13 | 4.11 | 97.87 |

1000 | 925.33 ± 22.19 | 2.40 | 92.53 |

2000 | 1940 ± 70 | 3.61 | 97 |

a Expressed as % R.S.D. (S.D./mean) × 100.

b Calculated as (mean determined concentration/nominal concentration) × 100.

**Table 1:** Precision and accuracy data of back-calculated concentrations of
calibration samples for curcumin in rat plasma (n = 3).

Under log-transformed data, normality assumption, we
recalculated values of ▽ for different combinations of CVs, n_{1}, and
n_{2} as shown in (**Table 2**).

Total Sample Size (n_{1}+n_{2}) |
CV | ||||||
---|---|---|---|---|---|---|---|

10% | 15% | 20% | 25% | 30% | 35% | 40% | |

8 | 8.95 | 5.98 | 4.51 | 3.63 | 3.04 | 2.63 | 2.31 |

10 | 10.00 | 6.69 | 5.04 | 4.05 | 3.39 | 2.94 | 2.59 |

12 | 10.96 | 7.33 | 5.52 | 4.44 | 3.72 | 3.22 | 2.84 |

14 | 11.84 | 7.92 | 5.96 | 4.79 | 4.02 | 3.47 | 3.06 |

16 | 12.65 | 8.46 | 6.37 | 5.13 | 4.30 | 3.71 | 3.28 |

18 | 13.42 | 8.98 | 6.76 | 5.44 | 4.56 | 3.94 | 3.48 |

20 | 14.15 | 9.46 | 7.13 | 5.73 | 4.81 | 4.15 | 3.66 |

22 | 14.84 | 9.92 | 7.47 | 6.01 | 5.04 | 4.35 | 3.84 |

24 | 15.49 | 10.36 | 7.81 | 6.28 | 5.27 | 4.55 | 4.01 |

**Table 2:** Values of the Index of Sensitivity.

From the two (**Tables 1 and 2**), the results indicate that▽ increases as sample size increases and as CV decreases.

(Chow and Liu, 2000) in (Chapter 6) of “Transformation and Analysis of Individual Subjects Ratios”, considered the following example from (Clayton and Leslie, 1981).

Two Erythromycin formulations were compared using the towsquare, two-period, two-treatment crossover design involving 18 healthy subjects. The two formulations were:

T: Erythromycin stearate [Erythrocin®, 500 mg Ovaloid Tablets, 6316, Abbott Australasia pty. Ltd.], and

R: Erythromycin base [Eryc®, 2X250 mg Capsules, containing enteric-coated pellets, F.H Faulding & Co. Ltd.].

The study technique of the analysis of variance was used to test for treatment differences. However, (Clayton and Leslie, 1981) did not provide the information for the sequence assignment of subjects. For the purpose of illustration, we will adapt the order of periods and sequences from (Chow and Liu, 2000).

Nine subjects were randomized to sequence (TR) (1-9), while the
remaining nine subjects were randomized to sequence (RT) (10-18). A
one-week washout period separated the two periods. Blood samples
were taken just before drug administration and then at: 0.50, 1.00,
1.50, 2.00, 3.00, 4.00, 6.00, and 8.00 hours post-dosing. The primary
summary variables of interest, were AUC_{(0-8)} and C_{max}. The data are
given in the tables, Bioequivalence data set from (Clayton and Leslie,
1981).

The results of the Analysis of Variance (ANOVA) on log-transformed
and untransformed data are given in (**Tables 4, 5, 6, and 7**) for the two
parameters; AUC_{(0-8)} and C_{max}.

Sequence | Subject | Period I | Period II | ||
---|---|---|---|---|---|

AUC | C_{max} |
AUC | C_{max} |
||

TR | 1 | 2.52 | 1.04 | 5.47 | 1.58 |

2 | 8.87 | 3.80 | 4.48 | 1.35 | |

3 | 0.79 | 0.12 | 2.25 | 0.77 | |

4 | 1.68 | 0.62 | 1.82 | 1.23 | |

5 | 6.95 | 2.92 | 7.87 | 2.25 | |

6 | 1.05 | 0.16 | 3.25 | 0.85 | |

7 | 0.99 | 0.28 | 12.39 | 3.00 | |

8 | 5.60 | 2.62 | 4.77 | 2.05 | |

9 | 3.16 | 1.51 | 1.88 | 0.60 | |

RT | 10 | 4.98 | 1.61 | 3.19 | 1.37 |

11 | 7.14 | 1.98 | 9.83 | 3.74 | |

12 | 1.81 | 0.56 | 2.91 | 1.16 | |

13 | 7.34 | 2.40 | 4.58 | 1.79 | |

14 | 4.25 | 1.04 | 7.05 | 2.51 | |

15 | 6.66 | 2.67 | 3.41 | 1.55 | |

16 | 4.76 | 1.47 | 2.49 | 0.82 | |

17 | 7.16 | 1.90 | 6.18 | 2.57 | |

18 | 5.52 | 2.47 | 2.85 | 1.07 |

**Table 3**

Source | D.F | SS | MS | F-value | p-value |
---|---|---|---|---|---|

Subjects |
17 | 9.64872 | 0.56757 | 1.8368 | 0.11543 |

Sequence |
1 | 1.30529 | 1.30529 | 2.5031 | 0.13319 |

Subject (sequence) |
16 | 8.34343 | 0.52146 | 1.6876 | 0.15275 |

Periods |
1 | 0.19586 | 0.19586 | 0.6338 | 0.43762 |

Treatments |
1 | 1.04699 | 1.04699 | 3.3883 | 0.08428 |

Residual |
16 | 4.94398 | 0.30899 | - | - |

Total |
35 | 15.83555 |

**Table 4:** Analysis of Variance for in (AUC_{(0-8)}).

Source | D.F | SS | MS | F-value | p-value |
---|---|---|---|---|---|

Subjects |
17 | 149.64430 | 8.80260 | 1.4690 | 0.22358 |

Sequence |
1 | 7.07560 | 7.07560 | 0.7941 | 0.38607 |

Subject (sequence) |
16 | 142.56870 | 8.91054 | 1.4870 | 0.21808 |

Periods |
1 | 0.93444 | 0.93444 | 0.1559 | 0.69817 |

Treatments |
1 | 11.17788 | 11.17788 | 1.8653 | 0.19090 |

Residual |
16 | 95.87808 | 5.99238 | - | - |

Total |
35 | 257.63470 |

**Table 5:** Analysis of Variance for AUC_{(0-8)}.

Source | D.F | SS | MS | F-value | p-value |
---|---|---|---|---|---|

Subjects |
17 | 13.59333 | 0.79961 | 1.6484 | 0.16186 |

Sequence |
1 | 1.85384 | 1.85384 | 2.5266 | 0.13150 |

Subject (sequence) |
16 | 11.73949 | 0.73372 | 1.5125 | 0.20841 |

Periods |
1 | 0.60542 | 0.60542 | 1.2481 | 0.28042 |

Treatments |
1 | 0.54057 | 0.60542 | 1.1144 | 0.30681 |

Residual |
16 | 7.76141 | 0.48509 | - | - |

Total |
35 | 22.50073 |

**Table 6:** Analysis of Variance for in C_{max}.

Source | D.F | SS | MS | F-value | p-value |
---|---|---|---|---|---|

Subjects |
17 | 18.21633 | 1.07155 | 1.2875 | 0.30888 |

Sequence |
1 | 0.97680 | 0.97680 | 0.9066 | 0.35518 |

Subject (sequence) |
16 | 17.23952 | 1.07747 | 1.2946 | 0.30581 |

Periods |
1 | 0.03003 | 0.03003 | 0.0397 | 0.84458 |

Treatments |
1 | 0.00047 | 0.00047 | 0.0006 | 0.98076 |

Residual |
16 | 13.31608 | 0.83225 | - | - |

Total |
35 | 31.56588 |

**Table 7:** Analysis of Variance for C_{max}.

Parameter | % CV | Ñ | ||
---|---|---|---|---|

C_{max} |
1.65444 | 0.83225 | 55.14 | 217.63 |

ln (C) _{max} |
0.39249 | 0.48509 | 79.01 | 1.92 |

AUC_{(0-8)} |
5.23111 | 5.99238 | 46.80 | 256.41 |

ln (AUC_{(0-8)}) |
1.52090 | 0.30899 | 60.17 | 2.41 |

**Table 8**

**Comments on the results obtained by (Chow and Liu, 2000)
concerning the analysis of variance of the log-transformed
data for AUC _{(0-8)}**

(Chow and Liu, 2000) concluded the following statement: “The estimated CV is about 36.55%, which is high, but not unusual in bioavailability and bioequivalence studies. The index of sensitivity defined in (Equation 5.3.10) is estimated to be about 3.28. Therefore, as discussed in (section 5.3.2), this study has little power to conclude bioequivalence if (Schuirmann, 1987) two one sided test procedures and the ± 20% rule are applied.”

(Chow and Liu, 2000) assumed that the results obtained from the analysis of variance for log-transformed data are as the results obtained from the analysis of variance of raw data. Accordingly, they computed the intrasubject coefficient of variation (CV) and the measure of sensitivity using (Equations 1 and 4):

On the other hand, different results for CV and ∇ are obtained using (Equations 2 and 5):

Also in (page 195), (Chow and Liu, 2000) gave summary of points and
interval estimation of intrasubject variability for data sets from the
Analysis of Variance of ln (AUC_{(0-8)}). That is; for the parameter ,
the point estimate = 0.3090 and 90% confidence interval for is
[Le, Ue] = [0.17, 0.72] if one interested to find the 95% confidence
interval for the parameter CV, the point estimation of CV = 60.17%,
then according to (Equation 3), the 95% confidence interval for CV is:

In this note, we try to clarify the confusion in finding the intrasubject coefficient of variation (CV) based on the additive model (raw data model) and the multiplicative model (log-transformed model). Since the information on intrasubject coefficient of variation is still needed for sample size planning in future studies, one must specify the model (raw data or log-transformed data) in the crossover design and then choose the suitable equation for the calculation of coefficient of variation (CV).

- Chow SC, Liu J (2000) Design and Analysis of Bioavailability and Bioequivalence
Studies. (2
^{nd}edn), New York, Marcel Dekker. - Clayton D, Leslie A (1981) The bioavailability of erythromycin stearate versus enteric-coated erythromycin base when taken immediately before and after food. J Int Med Res 9: 470-477.
- Schuirmann DJ (1987) A Comparison of the two one-sided tests procedures and the power approach for assessing the equivalence of average bioavailability. J Pharmacokinet Biopharm 15: 657-680.

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