Design and Analysis of Cross-Over Trials
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be seen, not all volunteers had sufficient data to make this possible and there are several missing values. Subject AUC AUC Cmax Cmax Test Ref Test Ref 1 79.34 58.16 2.827 2.589 3 85.59 69.68 4.407 2.480 5 . 121.84 . 5.319 8 377.15 208.33 11.808 9.634 10 14.23 17.22 1.121 1.855 11 750.79 1407.9 6.877 13.615 13 21.27 20.81 1.055 1.210 15 8.67 . 1.084 0.995 18 269.40 203.22 9.618 7.496 20 412.42 386.93 12.536 16.106 21 33.89 47.96 2.129 2.679 24 32.59 22.70 1.853 1.727 26 72.36 44.02 4.546 3.156 27 423.05 285.78 11.167 8.422 31 20.33 40.60 1.247 1.900 32 17.75 19.43 0.910 1.185 36 1160.53 1048.60 17.374 18.976 37 82.70 107.66 6.024 5.031 39 928.05 469.73 14.829 6.962 43 20.09 14.95 2.278 0.987 44 28.47 28.57 1.773 1.105 45 411.72 379.90 13.810 12.615 47 46.88 126.09 2.339 6.977 50 106.43 75.43 4.771 4.925 The subject-profile plots (see Chapter 2) for those subjects that had values in both periods for log(AUC) and for log(Cmax) are given in Fig-ure 7.2. It is clear that for Subject 4 in the TR sequence both log(AUC) and log(Cmax) are unusually low in the second period. These plots have been drawn using a modified version of the Splus-code given in Millard and Krause (2001), Chapter 7. The SAS code to calculate the confidence intervals for log(AUC) and

2003 ◽  
pp. 353-353
Keyword(s):  
Confidence Intervals ◽  
Missing Values ◽  
Sufficient Data ◽  
The Subject ◽  
Second Period

and Var[νˆ ] = 4σ + l ) (l −2(1 + c . Similarly, when σˆ ≤ 0.04, νˆ = δˆ + σˆ − 0.04(c ) (7.12) is an estimate for the constant-scaled metric in accordance with FDA Guidance (2001) using a REML UN model. This estimate is asymptoti-cally normally distributed and unbiased with E[νˆ ] = δ +σ −σ − 0.04(c ) and Var[νˆ ] = 4σ . To assess PBE we ‘plug-in’ estimates of δ and the variance components and calculate the upper bound of an asymptotic 90% confidence interval. If this upper bound is below zero we declare that PBE has been shown. Using the code in Appendix B and the data in Section 7.2, we obtain the value −1.90 for log(AUC) and the value −0.95 for log(Cmax). As both of these are below zero, we can declare that T and R are PBE. 7.5.2 PBE using a replicate design Here we fit the same REML UN model as defined in Section 7.4. Let νˆ = δˆ + σˆ + σˆ − (1 + c )(σˆ + σˆ ) (7.13) be an estimate for the reference-scaled metric in accordance with FDA Guidance (2001) when (σˆ + σˆ > 0.04 and using a REML UN model. Then, this estimate is asymptotically normally distributed, un-biased with E[νˆ ] = δ +σ − (1 + c ) and has variance of Var[νˆ ] = 4σ + l + (1 + c ) (l )+ 2l −2(1+c −2(1 + c + 2(1 + c ) (l ) When σˆ + σˆ ≤ 0.04, let νˆ = δˆ + σˆ + σˆ − (σˆ + σˆ )− 0.04(c ) (7.14) be an estimate for the constant-scaled metric in accordance with FDA

2003 ◽  
pp. 369-369
Keyword(s):  
Confidence Interval ◽  
Upper Bound ◽  
Variance Components ◽  
Fda Guidance ◽  
Replicate Design ◽  
Normally Distributed

ances and covariances obtained from REML are normally distributed with expectation vector and variance-covariance matrix equal to the fol-low  ing, r  espectiv    ely,   When σˆ > 0.04, let νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − (1 + c (7.6) be an estimate for the (7.3) reference-scaled metric in accordance with FDA Guidance (2001) and using a REML UN model. Then (Patter-son, 2003; Patterson and Jones, 2002b), this estimate is asymptotically normally distributed and unbiased with E[νˆ ] = δ +σ − (1 + c and Var[νˆ ] = 4σ + l + 4l + (1 + c ) (l )+ 2l −2(1+c − 2(1+c +4(1+c −2(1+c . Similarly, for the constant-scaled metric, when σˆ ≤ 0.04, νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − σˆ − 0.04(c ) (7.7) E[νˆ ] = δ +σ − 0.04(c ) Var[νˆ ] = 4σ + l + 4l + 2l − 2l − 4l + 4l − 2l . The required asymptotic upper bound √ of the 90% confidence interval can √ then be calculated as νˆ + 1.645× V̂ ar[νˆ ] or νˆ + 1.645× V̂ ar[νˆ ], where the variances are obtained by ‘plugging in’ the estimated values of the variances and covariances obtained from SAS proc mixed into the formulae for Var[νˆ ] or Var[νˆ ]. The necessary SAS code to do this is given in Appendix B. The output reveals that σˆ = 0.0714 and the upper bound is−0.060 for log(AUC). For log(Cmax), σˆ = 0.1060 and the upper bound is −0.055. As both of these upper bounds are below zero, IBE can be claimed.

2003 ◽  
pp. 367-367
Keyword(s):  
Confidence Interval ◽  
Covariance Matrix ◽  
Upper Bound ◽  
Upper Bounds ◽  
Fda Guidance ◽  
Asymptotic Upper Bound ◽  
Variance Covariance Matrix ◽  
Normally Distributed

data a; * significance level; a=0.05; * variance of difference of two observations on the log scale; * sigmaW = within-subjects standard deviation; sigmaW=0.355; s=sqrt(2)*sigmaW; * total number of subjects (needs to be a multiple of 2); n=58; * error degrees of freedom for AB/BA cross-over with n subjects in total; n2=n-2; * ratio = mu_T/mu_R; ratio=1.00; run; data b; set a; * calculate power; t1=tinv(1-a,n-2); t2=-t1; nc1=(sqrt(n))*((log(ratio)-log(0.8))/s); nc2=(sqrt(n))*((log(ratio)-log(1.25))/s); df=(sqrt(n-2))*((nc1-nc2)/(2*t1)); prob1=probt(t1,df,nc1); prob2=probt(t2,df,nc2); answer=prob2-prob1; power=answer*100; run; proc print data=b; run; As an example of using this SAS code, suppose µ = 1, σ = 0.355, α = 0.05 and n = 58. The power (as a percentage) is calculated as 90.4. The required number of subjects to achieve a given power can easily be obtained by trial and error using a selection of values of n. An alternative approach is to use trial and error directly on the sample size n for a given power. For more on this see Phillips (1990) and Diletti et al. (1991), for example. 7.4 Individual bioequivalence As noted in Section 7.1, individual bioequivalence (IBE) is a criterion for deciding if a patient who is currently being treated with R can be

2003 ◽  
pp. 362-362
Keyword(s):  
Standard Deviation ◽  
Sample Size ◽  
Degrees Of Freedom ◽  
Trial And Error ◽  
Significance Level ◽  
Individual Bioequivalence ◽  
Alternative Approach ◽  
Within Subjects ◽  
Log Ratio ◽  
Selection Of

3 Power and sample size for ABE in the 2× 2 design Here we give formulae for the calculation of the power of the TOST procedure assuming that there are in total n subjects in the 2× 2 trial. Suppose that the null hypotheses given below are to be tested using a 100(1 − α)% two-sided confidence interval and a power of (1 − β) is required to reject these hypotheses when they are false. H :µ ≤− ln 1.25 H :µ ≥ ln 1.25. Let us define x to be a random variable that has a noncentral t-distribution with df degrees of freedom and noncentrality parameter nc, i.e., x ∼ t(df,nc). The cumulative distribution function of x is defined as CDF(t,df,nc) = Pr(x≤ t). Assume that the power is to be calculated using log(AUC). If σ is the common within-subject variance for T and R for log(AUC), and n/2 subjects are allocated to each of the sequences RT and TR, then 1−β = CDF(t , df,nc )−CDF(t , df,nc ) (7.1) where √ n(log(µ )− log(0.8)) nc = 2σ √ n(log(µ )− log(1.25)) nc = 2σ √ (n− 2)(nc −nc = ) df 2t and t is the 100(1 − α)% point of the central t-distribution on n− 2 degrees of freedom. Some SAS code to calculate the power for an ABE 2 × 2 trial is given below, where the required input variables are α, σ value of the ratio µ and n, the total number of subjects in the trial.

2003 ◽  
pp. 361-361
Keyword(s):  
Distribution Function ◽  
Confidence Interval ◽  
Degrees Of Freedom ◽  
Random Variable ◽  
Noncentrality Parameter ◽  
Cumulative Distribution ◽  
Total N ◽  
The Common ◽  
Input Variables ◽  
T Distribution

population of potential patients, but be such that they produce different effects when a patient is switched from formulation T to formulation R or vice-versa. In other words there is a significant subject-by-formulation interaction. To show that this is not the case T and R have to be shown to be IBE, i.e., individually bioequivalent. The measure of IBE that has been suggested by the regulators is an aggregate measure involving the means and variances of T and R and the subject-by-formulation inter-action. We will describe this measure in Section 7.4. In simple terms PBE can be considered as a measure that permits patients who have not yet been treated with T or R to be safely prescribed either. IBE, on the other hand, is a measure which permits a patient who is cur-rently being treated with R to be safely switched to T (FDA Guid-ance, 1997, 1999a,b, 2000, 2001). It is worth noting that if T is IBE to R it does not imply that R is IBE to T. The same can be said for PBE. An important practical implication of testing for IBE is that the 2×2 cross-over trial is no longer adequate. As will be seen, the volunteers in the study will have to receive at least one repeat dose of R or T. In other words, three-or four-period designs with sequences such as [RTR,TRT] and [RTRT,TRTR], respectively, must be used. The measures of ABE, PBE and IBE that will be described in Sec-tions 7.2, 7.5 and 7.4 are those suggested by the regulators. Dragalin and Fedorov (1999) and Dragalin et al. (2002) have pointed out some drawbacks of these measures and suggested alternatives which have more attractive properties. We will consider these alternatives in Section 7.7. All the analyzes considered in Sections 7.2 to 7.4 are based on sum-mary measures (AUC and Cmax) obtained from the concentration-time profiles. If testing for bioequivalence is all that is of interest, then these measures are adequate and have been extensively used in practice. How-ever, there is often a need to obtain an understanding of the absorb-tion and elimination processes to which the drug is exposed once it has entered the body, e.g., when bioequivalence is not demonstrated. This can be done by fitting compartmental models to the drug con-centrations obtained from each volunteer. These models not only pro-vide insight into the mechanisms of action of the drugs, but can also be used to calculate the AUC and Cmax values. In Section 7.8 we de-scribe how such models can be fitted using the methods proposed by Lindsey et al. (2000a). The history of bioequivalence testing dates back to the late 1960s and early 1970s. Two excellent review articles written by Patterson (2001a, 2001b) give a more detailed description of the history, as well as a more extensive discussion of the points raised in this section. The regulatory

2003 ◽  
pp. 351-351
Keyword(s):  
Practical Implication ◽  
The Body ◽  
Repeat Dose ◽  
Extensive Discussion ◽  
Time Profiles ◽  
History Of ◽  
The Subject ◽  
Bioequivalence Testing ◽  
To Receive ◽  
Insight Into

the ‘Area Under the Curve’ or AUC. The AUC is taken as a measure of exposure of the drug to the subject. The peak or maximum concen-tration is referred to as Cmax and is an important safety measure. For regulatory approval of bioequivalence it is necessary to show from the trial results that the mean values of AUC and Cmax for T and R are not significantly different. The AUC is calculated by adding up the ar-eas of the regions identified by the vertical lines under the plot in Figure 7.1 using an arithmetic technique such as the trapezoidal rule (see, for example, Welling, 1986, 145–149, Rowland and Tozer, 1995, 469–471). Experience (e.g., FDA Guidance, 1992, 1997, 1999b, 2001) has dictated that AUC and Cmax need to be transformed to the natural logarithmic scale prior to analysis if the usual assumptions of normally distributed errors are to be made. Each of AUC and Cmax is analyzed separately and there is no adjustment to significance levels to allow for multiple testing (Hauck et al., 1995). We will refer to the derived variates as log(AUC) and log(Cmax), respectively. In bioequivalence trials there should be a wash-out period of at least five half-lives of the drugs between the active treatment periods. If this is the case, and there are no detectable pre-dose drug concentrations, there is no need to assume that carry-over effects are present and so it is not necessary to test for a differential carry-over effect (FDA Guidance, 2001). The model that is fitted to the data will be the one used in Section 5.3 of Chapter 5, which contains terms for subjects, periods and treatments. Following common practice we will also fit a sequence or group effect and consider subjects as a random effect nested within sequence. An example of fitting this model will be given in the next section. In the following sections we will consider three forms of bioequivalence: average (ABE), population (PBE) and individual (IBE). To simplify the following discussion we will refer only to log(AUC); the discussion for log(Cmax) is identical. To show that T and R are average bioequivalent it is only necessary to show that the mean log(AUC) for T is not significantly different from the mean log(AUC) for R. In other words we need to show that, ‘on average’, in the population of intended patients, the two drugs are bioequivalent. This measure does not take into account the variability of T and R. It is possible for one drug to be much more variable than the other, yet be similar in terms of mean log(AUC). It was for this reason that PBE was introduced. As we will see in Section 7.5, the measure of PBE that has been recommended by the regulators is a mixture of the mean and variance of the log(AUC) values (FDA Guidance, 1997, 1999a,b, 2000, 2001). Of course, two drugs could be similar in mean and variance over the

2003 ◽  
pp. 350-350
Keyword(s):  
Multiple Testing ◽  
Random Effect ◽  
Area Under The Curve ◽  
Mean Values ◽  
Fda Guidance ◽  
Significance Levels ◽  
Mean And Variance ◽  
The Mean ◽  
Carry Over ◽  
The One

5 Population bioequivalence As noted in Section 7.1, population bioequivalence (PBE) is concerned with assessing whether a patient who has not yet been treated with R or T can be prescribed either formulation. It can be assessed using the following aggregate metric (FDA Guidance, 1997). (µ (7.8) max(0.04,σ ) where σ and σ . As long as an appropriate mixed model is fitted to the data, this metric can be calculated using data from a 2×2 design or from a replicate design. Using data from Sections 7.2 and 7.4, we will illustrate the calculation of the metric in each of the two designs. 7.5.1 PBE using a 2× 2 design As in the previous section we will test for equivalence using a linearized version of the metric and test the null hypotheses: H : ν = δ +σ − (1 + c when σ > 0.04 or H : ν = δ +σ −σ (0.04) ≥ 0, (7.10) when σ > 0.04, where σ and σ are the between-subject variances of T and R, re-spectively. Let ω denote the between-subject covariance of T and R and σ denote the variance of δˆ = µˆ . The REML estimates of σ , o  btained from using the SAS code in Appendix B, are asymptoti-cally normally distributed with expecta  tion vector   σ   l lT×ω σ and variance-covariance matrix l lT×ω l lω Then νˆ = δˆ + σˆ − (1 + c )σˆ (7.11) is an estimate for the reference-scaled PBE metric in accordance with FDA Guidance (2001) when σˆ > 0.04 and using a REML UN model. This estimate is asymptotically normally distributed and unbiased (Pat-terson, 2003; Patterson and Jones, 2002b) with E[νˆ ] = δ +σ

2003 ◽  
pp. 368-368
Keyword(s):  
Covariance Matrix ◽  
Mixed Model ◽  
Fda Guidance ◽  
Population Bioequivalence ◽  
Using Data ◽  
Replicate Design ◽  
Variance Covariance Matrix ◽  
Normally Distributed

APBC 355 310 295

2003 ◽  
pp. 355-355
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