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Statistical Methods for Obtaining Confidence Intervals for Individual and Population Bioequivalence Criteria

Individual Bioequivalence Method 1 -- Constrained REML

Statistical Model:

  • Mixed effects ANOVA model in natural log scale
  • Subjects within sequence as random effects
  • Within- and between-subject variances allowed to differ by formulation
  • Fixed effects are formulation, period, sequence, and period*sequence interaction (nested within formulation)

Parameter Estimation:

  • Restricted maximum likelihood (REML) estimates of random effects
  • Choose estimation procedure so that the between-subject covariance is non-negative definite; i.e., so that the correlation does not exceed 1.0 (This is the constraint in "constrained REML". REML without the constraint, similar to Method 2, is possible but has not been evaluated.)
  • Generalized least squares estimates of fixed effects. (Type III coefficients)

Confidence Intervals:

  • 95% upper confidence bounds using nonparametric percentile bootstrap confidence interval procedure (upper bound of the 90% two-sided confidence interval)
  • Use minimum of 1500 (2000 recommended) bootstrap samples that preserve the number of subjects per sequence

Example of an Implementation -- SAS:

  • SAS PROC MIXED version 6.10 for Windows,™ 6.09 maintenance release for UNIX, or equivalent or later release (needed for csh covariance structure)
  • The following is some SAS code for the above model and four-period designs:
    proc mixed method=reml maxiter=200 ;
    class formulat subj_id period sequence ;
    model lnmetric = formulat period sequence period*sequence(formulat) ;
    random formulat / subject=subj_id type=csh ;
    repeated / group=formulat ;
    estimate 'T - R' formulat -1 1 ;
  • For three-period designs, the simple estimate statement is usually not enough. The coefficients for the estimable function need to be specified.
  • Note that the type=un covariance structure is nominally the same as csh for this model (with a 2x2 covariance matrix). However there are some differences. Csh forces the covariance matrix to be non-negative definite and is the approach that has been evaluated. Un may yield estimates of correlation greater than 1.0 and, hence, estimates of the subject-by formulation interaction (s D) that are negative. However, the un structure sometimes finds better estimates than csh (in terms of likelihood) when it does find a positive-definite covariance matrix.
  • Bootstrapping uses SAS macros. (Do not use a BY statement to bootstrap PROC MIXED.)


Individual Bioequivalence Method 2 -- Method of Moments

Note: This method is not yet fully evaluated. It is included here due to its relative simplicity and good results to date.

References: Chinchilli, 1996 J Biopharmaceutical Statistics; Chinchilli and Esinhart, 1996 Statistics in Medicine; the method is also related to a suggestion of Schall and Luus, but is less ad hoc for handling period effects than Schall and Luus’ approach.

Parameter Estimation (complete data):

  • Use method of moments to obtain unbiased estimates of the components of the criterion, difference of means, sum of variance terms in numerator, and the within-subject variance of the reference, and then bootstrap to obtain confidence intervals. The implied estimate of the subject-by-formulation term may then be negative ("unconstrained"). The between-subject components of variance are not estimated.
  • Chinchilli provides estimates of the means.
  • Variance estimates are standard unbiased estimates pooled across sequences.
  • Formulas depend on design and so are not given here.
  • Much simpler to implement than Method 1. For example, implementation in SAS requires only PROCs MEANS, SUMMARY, and TRANSPOSE.
  • Compared to Method 1, Method 2 tends to yield larger estimates of the within-subject variances and smaller estimates of the subject-by-formulation interaction.
  • This approach is possible with three-period designs that replicate only the reference. However, evaluation has considered only four-period designs.

Parameter Estimation (incomplete data): 

  • One approach to missing data with Method 2 is to use the methods described above for complete data with the following additions:
  • Any subject missing a formulation metric will not be used in the calculation of intra-subject variance component for that formulation.
  • Any subject with any missing T or R will not be used in the estimate of subject by formulation interaction.
  • In all cases, the denominator of the variance estimate will be corrected to reflect the number of subjects actually used in calculations for the particular variance component.
  • There may be alternate approaches for handling missing data with Method 2. If there is more than minimal missing data, Method 1 is likely preferable to the approach described here for Method 2.


Population Bioequivalence

Parameter Estimation: 

  • As for individual bioequivalence method 2, obtain unbiased estimates of the components of the criterion, difference of means, sum of variances in numerator, and the total variance of the reference, and then bootstrap to obtain confidence intervals.
  • Use any available method, such as SAS Proc GLM, to obtain unbiased estimates of the difference of means.
  • For standard, two-period, two-sequence crossover designs, the variance estimates are the standard unbiased estimates pooled across sequences.

Some Examples for Individual BE, 95% Upper Confidence Bounds for Q S

Allowable Upper Limit 2.495
(e = .05)
ConstrainedREMLMethod ofMoments

Scaled to:




















Shaded area shows criterion selected by mixed scaling. Figures in bold satisfy the individual bioequivalence criterion at the 5% level.


Some Examples for Individual BE, Parameter Estimates and 90% CI using Constrained REML

   Intra-SubjectSD (s W)Inter-SubjectSD (s B) 





Ratio of Means

(original scale)








Interaction (s D)

Furosemide,1 all



(81.2 - 116.9)


(.108 - .263)


(.543 - 1.813)


(.000 - .157)


(.665 - 6.725)


(.033 - .373)

Subj. 8, per. 1 removed



(91.6 - 120.5)


(.111 - .263)


(.432 - 1.581)


(.000 - .167)


(.000, 3.563)


(.012 - .238)




(91.1 - 106.0)


(.172 - .278)


(.924 - 1.700)


(.348 - .594)


(.705 - 1.070)


(.005 - .126)




(106.0 - 127.3)


(.137 - .392)


(.493 - 1.463)


(.000 - .634)


(.635 - 1.058)


(.008 - .147)

Mean ratios in bold satisfy the 80/125 average bioequivalence criterion at the 5% level.


  1. Ekbohm G., Melander H. (1989). The Subject-by-Formulation Interaction as a Criterion of Interchangeability of Drugs, Biometrics, 45, 1249-1254. (Furosemide -- Lasix and Furix)
  2. Esinhart J., Chinchilli V. (1994). Extension to the Use of Tolerance Intervals for the Assessment of Individual Bioequivalence, Journal of Biopharmaceutical Statistics, 4, 39-52. (Verapamil)
  3. Ryde, M., Huitfeldt, B., and Pettersson, R. (1991). Relative bioavailability of Olsalazine from tablets and capsules: a drug targeted for local effect in the colon, Biopharm. & Drug Disposition, 12, 233-246. Also, in Chow and Liu, p 280. (N-acetyl-5-aminosalicyclic acid)


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