Assess bioequivalence against a regulatory framework

be_assess() performs a complete bioequivalence (BE) assessment of one or more noncompartmental endpoints against a regulatory framework, including reference scaling and the pass/fail decision. It accepts the results of pk.nca() directly (a PKNCAresults object) or a tidy long data.frame, so the same workflow that produces NCA parameters flows into the regulatory decision. It is a thin wrapper over be_fit_models() that adds the be_assess class and its print/summary methods.

Usage

be_assess(object, ...)

# Default S3 method
be_assess(
  object,
  reference_col,
  reference_value,
  endpoints = c("cmax", "aucinf.obs", "aucinf.pred", "auclast"),
  regulator = "ABE",
  model_type = NULL,
  alpha = 0.1,
  subject = NULL,
  sequence = NULL,
  period = NULL,
  design = NULL,
  covariates = NULL,
  heteroscedastic = FALSE,
  widen_endpoints = NULL,
  widen_default = TRUE,
  ...
)

# S3 method for class 'PKNCAresults_sparse_bootstrap'
be_assess(
  object,
  reference_col,
  reference_value,
  endpoints = c("cmax", "aucinf.obs", "aucinf.pred", "auclast"),
  regulator = "ABE",
  model_type = NULL,
  alpha = 0.1,
  subject = NULL,
  sequence = NULL,
  period = NULL,
  design = NULL,
  covariates = NULL,
  heteroscedastic = FALSE,
  widen_endpoints = NULL,
  widen_default = TRUE,
  ...
)

Arguments

object: A PKNCAresults object or a tidy long data.frame with a PPTESTCD column of parameter names, a PPORRES/PPSTRES column of values, and subject/sequence/period/treatment columns. The results of a sparse bootstrap (from a PKNCAconc object made by sparse_bootstrap()) are compared with the percentile intervals of the bootstrap replicates (Shen and Machado 2017); see Details.
...: Arguments passed to the methods
reference_col: The column identifying the formulation/treatment.
reference_value: The value of reference_col that is the reference formulation.
endpoints: Character vector of NCA parameters (matched against PPTESTCD) to assess.
regulator: The regulatory framework (see be_regulator()); one of "ABE", "EMA", "HC", "GCC", "FDA", "NTID", "HVNTID", or "descriptive" (no acceptance limits or pass/fail).
model_type: The model for the average-BE point estimate, one of "lmer" (mixed model, for crossover/replicate designs), "anova" (fixed-effects, for parallel designs), "isc" (intra-subject contrasts, the FDA reference-scaled path), "nlme" (mixed model with treatment-specific residual variances, for crossover/replicate designs), or "gls" (generalized least squares with treatment-specific residual variances, for parallel designs). When NULL (default) it is chosen from the design, the regulator, and heteroscedastic.
alpha: The significance level; the confidence interval has level 1 - alpha (default 0.10 gives the 90% interval).
subject, sequence, period: Column names for the subject, randomization sequence, and period. When NULL they are taken from the PKNCAresults object or detected from common column names. sequence may be absent.
design: An optional be_design() object; computed from the data when NULL.
covariates: An optional character vector of column names added to every model as additive fixed effects (numeric columns as linear terms, character or factor columns as factors). They must not be missing in any analyzed row. For a PKNCAresults object they must be columns of as.data.frame(object), which means grouping columns. The within-subject variances used for reference scaling and the intra-subject contrasts do not use covariates.
heteroscedastic: Logical. When TRUE, estimate a separate residual variance for each treatment (see Details).
widen_endpoints: For the expanding-limits frameworks ("EMA", "HC", and "GCC"), endpoints whose acceptance limits widen in addition to the framework's own choice (the endpoints judged like Cmax for the EMA and GCC, like AUC for Health Canada; see be_regulator()). Use it for an endpoint that the guidance widens but PKNCA cannot recognize from its name, such as a partial AUC of a modified-release product (EMA), or an endpoint that is neither a PKNCA parameter nor a CDISC code. Naming an endpoint for another framework is an error; be_compare() passes it to the expanding-limits frameworks only, and each adds it to its own choice.
widen_default: For the expanding-limits frameworks, whether the framework's own choice of endpoints widens (the default, TRUE). With FALSE only the endpoints in widen_endpoints widen, for example none when the protocol did not specify widening in advance; Health Canada still judges Cmax by its point estimate. FALSE for another framework is an error; be_compare() applies it to the expanding-limits frameworks only.

Value

An object of class be_assess (a data.frame), ordered by endpoint (in the order requested) then by test formulation (reference-first), with one row per endpoint and test formulation and the columns endpoint, test, n (the number of subjects behind the ratio shown; see Details), design, units (the endpoint's measurement units; the column is omitted with a warning when units are not provided), the reference and test geometric means with their 90% confidence intervals on the measurement scale (gm_reference, gm_reference_lower, gm_reference_upper, gm_test, gm_test_lower, gm_test_upper), gmr_percent, ci_lower, ci_upper, cvwr_percent, cvwt_percent, swr, rule, limit_lower, limit_upper, criterion, regulator, model_type, and pass. rule names the criterion the row's decision used: "ABE" (the confidence interval within 80.00-125.00%), "ABEL" (the confidence interval within widened limits), "PE" (the point estimate alone within 80.00-125.00%), "RSABE" (the reference-scaled bound), "NTID", or "HVNTID". limit_* are NA for the RSABE criterion and criterion is NA for the limit-based frameworks. For regulator = "descriptive" the rule, limit_lower, limit_upper, criterion, and pass columns are omitted. A caption attribute documents the model, the decision rule for each endpoint, and its source guidance, or states that no regulatory decision was applied.

Details

The model type is selected automatically from the design and the regulator. By design, a parallel study (one measurement per subject) uses a fixed-effects ANOVA ("anova"), while a crossover or replicate study (repeated measures per subject) uses the mixed model ("lmer") for the geometric mean ratio and its confidence interval. The FDA reference-scaled frameworks (FDA RSABE, NTID, HVNTID) always use intra-subject contrasts ("isc"), as the guidances specify, regardless of design. Each subject's contrast is its mean log test minus its mean log reference, from the subjects with every test and reference administration their sequence calls for; the ratio is the unweighted mean of the sequence means of the contrasts, with n - m degrees of freedom for n subjects in m sequences (FDA draft guidances on progesterone and warfarin sodium). Without a sequence column, a subject's sequence is its treatment-by-period pattern. The unscaled 80.00-125.00% criterion of these frameworks (FDA RSABE when swR < 0.294, NTID, and HVNTID) uses the model of model_type; the default "isc" is the guidance mixed-effects model for replicate designs: fixed sequence, period, and treatment effects, correlated treatment random effects for each subject, a residual variance per treatment, REML, and Satterthwaite degrees of freedom (SAS RANDOM TRT/TYPE=FA0(2), REPEATED/GRP=TRT). A formulation that no subject receives twice has its residual variance folded into its between-subject variance, where the guidance model is over-specified, and a subject missing a period forms its own sequence when the sequence is derived from the treatment pattern. When the mixed model cannot be fit (for example, a formulation whose replicates agree exactly), only a decision that needs the unscaled criterion stops with an error. Each row shows the ratio and interval its decision used: the intra-subject contrasts for the reference-scaled criterion and the model for the unscaled one, and n counts the subjects behind that ratio: for the intra-subject contrasts, the subjects with every test and reference administration their sequence calls for; for the model, every subject with the test or the reference formulation. Pass model_type to override ("lmer", "anova", "isc", "nlme", or "gls"); with "isc" the FDA mixed model also gives the ratio for the other frameworks, and a design without a replicated formulation keeps the fixed-effects ANOVA ratio. Within-subject variability for the FDA frameworks uses the FDA estimator from within-sequence differences between replicate administrations (model_type = "isc" in be_within_var()); for the other frameworks it uses the EMA ANOVA estimator for the "lmer"/"anova"/"isc" paths and the treatment-specific mixed-model estimator for "nlme".

With heteroscedastic = TRUE the residual variance is estimated separately for each treatment: a crossover or replicate study uses nlme::lme() with varIdent(~ 1 | treatment) ("nlme"), and a parallel study uses nlme::gls() with the same variance structure ("gls"). lme4::lmer() cannot estimate treatment-specific residual variances (its weights are fixed prior weights), so heteroscedastic = TRUE with model_type = "lmer" is an error. Without reference scaling, "nlme" accepts non-replicated designs and several test formulations; with reference scaling it needs a fully replicated design and one test formulation, because the per-formulation variances are then used as the within-subject variances.

regulator = "descriptive" estimates the geometric means, their ratio, and its confidence interval exactly as for "ABE" but applies no acceptance limits and reports no pass/fail, for comparisons such as food effect and drug-drug interaction studies.

The expanding-limits frameworks widen only some endpoints, as their guidances specify (see be_regulator() for the sources and for which parameters, by PKNCA name or CDISC code, are judged like Cmax and like AUC): the EMA and the GCC widen Cmax and the parameters judged like it (such as Ctau,ss and Rmax), and Health Canada widens AUC and the parameters judged like it and judges the Cmax group by its point estimate alone. Every other endpoint needs its confidence interval within 80.00-125.00%. The rule column names the criterion each row used. widen_endpoints adds endpoints to those that widen, for example "aucint.last" for a partial AUC of a modified-release product under the EMA modified-release guideline. widen_default = FALSE drops the framework's own choice, for example when the protocol did not specify widening in advance, which the EMA and the GCC require; the endpoints in widen_endpoints still widen. Health Canada judges the Cmax group by its point estimate unless an endpoint is named in widen_endpoints.

For the results of a sparse bootstrap (Shen and Machado 2017; see sparse_bootstrap()), there is one estimate per treatment, from the sparse data, rather than one per subject. The ratio of the test and reference estimates is the point estimate, and its confidence interval is the percentile interval of the ratios of the bootstrap replicates (model_type = "bootstrap"). The gm_* columns hold the estimates of each treatment with their percentile intervals. A parallel design resamples the treatments independently; for a crossover, give the treatment column as paired to sparse_bootstrap(). There is no within-subject variance, so only the "ABE" and "descriptive" frameworks apply, and the results may have no groups other than reference_col (filter them first).

Methods (by class)

be_assess(PKNCAresults_sparse_bootstrap): Bioequivalence from the percentile intervals of a sparse bootstrap (Shen and Machado 2017)

References

Shen M, Machado SG. Bioequivalence evaluation of sparse sampling pharmacokinetics data using bootstrap resampling method. Journal of Biopharmaceutical Statistics. 2017;27(2):257-264. doi:10.1080/10543406.2016.1265543

Examples

# A replicate-design crossover in long (PPTESTCD/PPORRES) format
set.seed(1)
nsub <- 24
seqs <- rep(c("TRTR", "RTRT"), length.out = nsub)
b <- stats::rnorm(nsub, sd = 0.4)
d <- do.call(rbind, lapply(seq_len(nsub), function(i) {
  trt <- strsplit(seqs[i], "")[[1]]
  data.frame(
    subject = i, sequence = seqs[i], period = seq_along(trt), treatment = trt,
    PPTESTCD = "cmax",
    PPORRES = exp(log(100) + ifelse(trt == "T", 0.04, 0) + b[i] +
                    stats::rnorm(length(trt), sd = 0.45))
  )
}))
# The EMA widens the limits of Cmax
be_assess(d, reference_col = "treatment", reference_value = "R",
          endpoints = "cmax", regulator = "EMA")
#> Warning: Units were not found in the data; omitting the `units` column. Supply units via a PKNCAresults object or `PPSTRESU`/`PPORRESU` columns.
#> Bioequivalence assessment: EMA (model_type lmer, 90% CI)
#> Design: full_replicate
#> 
#>  endpoint test  n         design gm_reference gm_reference_lower
#>      cmax    T 24 full_replicate       114.44              95.46
#>  gm_reference_upper gm_test gm_test_lower gm_test_upper gmr_percent ci_lower
#>               137.2  112.15         93.55        134.46          98    85.72
#>  ci_upper cvwr_percent cvwt_percent  swr rule limit_lower limit_upper criterion
#>    112.04        34.11        47.67 0.33 ABEL       77.71      128.68        NA
#>  regulator model_type pass
#>        EMA       lmer TRUE
#> 
#> Caption: EMA bioequivalence assessment (90% CI). Geometric means and the geometric mean ratio are least-squares means from a mixed-effects model (lmerTest::lmer, Satterthwaite degrees of freedom); the 90% CI comes from exponentiated least-squares-mean differences (emmeans). The confidence-interval limits of cmax widen with the within-reference variability (ABEL) when CVwR exceeds 30%, capped per the regulator. Source: EMA Guideline on the Investigation of Bioequivalence (CPMP/EWP/QWP/1401/98 Rev. 1), sections 4.1.8 and 4.1.10. Within-formulation variances (swR, swT) are estimated by ANOVA on each formulation's replicates.
#>