{"id":"e3c29a85-3816-4d30-8a0a-c6ead88ea1f2","arxiv_id":"2411.16429","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A size-adjusted multivariate equivalence test, called the multivariate α-TOST, is introduced and shown to be more powerful than the conventional multivariate TOST.","lead":"This paper proposes a corrected significance level for the standard multivariate bioequivalence test, making the test less conservative and more likely to declare equivalence when it is true. The method could make generic drug approval easier for highly variable products.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The size-α guarantee depends on computing the global supremum λ(γ,Ω) in Eq. (8), but Algorithm 1 never specifies this non-convex optimization; a local maximizer would make α* too large and the feasible test liberal.","rationale":"The reader's weakest assumption identifies the same core issue: Eq. (8) requires a global argmax over the null space, and Algorithm 1 invokes this computation without specifying a reliable method. I consider this the single most load-bearing concern because the size-α property is not merely a power enhancement; it is the defining guarantee of the proposed test. If the optimizer used in the implementation returns a suboptimal λ, the adjusted level α* solves the wrong equation and the test is anti-conservative, undermining both the theoretical claim and the practical recommendation. Other concerns raised in the reader's rationale are less decisive: the randomness of pα* does not by itself break the uniform-power claim, since for any covariance matrix the conventional multivariate TOST is conservative and hence the solution α*(Ω) lies above α whenever it exists; and the case-study outlier exclusion is an applied-data issue rather than a flaw in the method's logic. The paper's simulations are extensive and the asymptotic argument in Appendix D is plausible under the stated regularity conditions, so the concern is addressable rather than fatal: the authors could specify a global optimizer (or verify that their routine provably attains the supremum for the settings considered) and check the derivative condition in Appendix E. Since the reader already assigned CONDITIONAL, my read does not change the verdict.","tokens_in":23075,"tokens_out":15429,"duration_ms":156404,"concrete_test":"Reproduce the m = 4, ρ = 0, heteroscedastic variance scenario from Simulation 1 (Table 1, e.g., Σ3 or Σ5) using the authors' cTOST implementation. Separately compute a reference global supremum of p(γ,θ,Ω,ν,c) over the 8 faces of the null boundary (θ_j = ±c) using a dense deterministic grid (at least 10^4 points per face) or a global stochastic optimizer (e.g., differential evolution) with the same Monte Carlo seed for evaluating Eq. (7). Compare the reference supremum at γ = α and at the reported pα* against the values returned by the default λ routine. If the default λ's rejection probability is lower than the reference supremum by an amount that shifts the zero of Eq. (12) by more than 0.001 in γ, the size guarantee fails in that scenario; if the values match to within Monte Carlo error, the concern is resolved for that case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (12) defines α* as the zero of p(γ, λ(γ,Ω), Ω, ν,c) − α, where λ(γ,Ω) is the global argmax over θ outside the equivalence region of the rejection probability. This is a non-convex optimization over the boundary of an m-dimensional hypercube, and Eq. (7) shows that each evaluation of p is itself a high-dimensional integral over a Wishart-distributed covariance estimate, typically computed by Monte Carlo. Algorithm 1 invokes 'compute λ(α*(r))' in steps 2 and 10 without specifying a global optimizer or providing evidence that the routine used in the cTOST package attains the supremum. If the numerical routine returns a local maximizer λ_loc with p(γ,λ_loc) < sup_θ p(γ,θ), then the equation solved in (12) is p(γ,λ_loc,Ω,ν,c) = α, which has a larger zero than the true size-α equation. The resulting α* is too large, and the test's actual size at the true worst-case λ exceeds α, directly violating the paper's central size-control claim. The Appendix E convergence proof also relies on 0 < p'(γ) < 2 being 'usually satisfied,' but this condition is not verified for the m = 4 case study or the simulated scenarios, so the inner fixed-point iteration could fail or converge to a different point. Because the population-level guarantee is definitional only when λ is the true supremum, the missing global-optimization specification is the most load-bearing gap in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a multivariate extension of the univariate α-TOST procedure for average equivalence testing. For a given covariance matrix Ω, the adjusted level α*(Ω) is defined in Eq. (12) as the zero of the size function p(γ, λ(γ, Ω), Ω, ν, c) − α, where λ(γ, Ω) is the supremum point of the rejection probability over the null parameter space. The authors claim that, when Ω is the true covariance, the resulting procedure is exactly size α, that the estimated version satisfies pα* = α* + op(ν^{−1}), and that, because α*(Ω) ≥ α, the multivariate α-TOST is uniformly more powerful than the conventional multivariate TOST. The paper presents an iterative algorithm, a simulation study with m ∈ {2,4}, and a re-analysis of the ticlopidine hydrochloride bioequivalence case study with m = 4.","tokens_in":23324,"tokens_out":5309,"duration_ms":55650,"significance":"If the computational and regularity premises hold, the paper gives practitioners a finite-sample size-corrected multivariate equivalence test with power superior to the conventional TOST, which is known to become very conservative as m grows. The extension is nontrivial because the argmax point λ in Eq. (8) depends on the candidate level γ, requiring a nested optimization. The paper provides a formal asymptotic argument in Appendix D, a convergence argument in Appendix E, and an extensive simulation study, and it makes an R implementation available in the cTOST package. However, the central size guarantee is conditional on the ability to compute the global supremum in Eq. (8), and this step is not specified or verified. The claimed uniform power advantage is rigorously justified only for the population-level adjustment, not for the feasible estimated version.","major_comments":[{"comment":"The size-α property is only as strong as the computation of λ(γ, Ω) in Eq. (8), which is a non-convex supremum over the null boundary of the m-dimensional hypercube. Algorithm 1 invokes 'compute λ(α*(r))' in steps 1 and 10 (and Algorithm 2 does the same in steps 2 and 14) without specifying a global optimizer or providing evidence that the routine used attains the supremum. Moreover, each evaluation of p in Eq. (7) is a high-dimensional integral over a Wishart distribution, typically approximated by Monte Carlo, so the fixed-point iteration is solving a random equation unless the Monte Carlo error is controlled. If the optimization returns a local maximizer λ_loc with p(γ, λ_loc) < sup_{θ ∉ Θ₁} p(γ, θ), then the zero of Eq. (12) is larger than the true size-α level, and the resulting test is liberal at the actual worst-case null parameter. The paper should either specify a certified global optimization method, provide a deterministic or error-bounded procedure for Eq. (7), or include a numerical verification for the m = 4 case study and the simulation scenarios.","section":"§4.2, Algorithm 1; Eqs. (8)–(12)"},{"comment":"The convergence and uniqueness proof for the inner fixed-point iteration relies on the assumption that 0 < p'(γ) < 2 on the solution space A, which is stated as 'usually satisfied' without verification. This condition is not checked for the m = 4 case study or for any of the simulation settings in Table 1. The existence of α* is also addressed only through a sufficient condition derived under independence, Eq. (13), while the correlated and heteroscedastic settings are justified heuristically. Without a verifiable condition or a fallback numerical procedure, the algorithm could fail to converge or converge to a point that is not the unique solution of Eq. (12), which would break the claimed size control.","section":"Appendix E, condition 0 < p'(γ) < 2"},{"comment":"The statement that 'its rejection region cannot be smaller than that of the multivariate TOST, which makes the former uniformly more powerful' is proved for the population-level adjusted level α*(Σ), where α*(Σ) ≥ α follows by construction. For the feasible procedure based on pα*, the paper does not establish that pα* ≥ α almost surely, nor does it provide a finite-sample bound on the difference. Section 5 only reports empirical size below α in selected scenarios and says that this conservative behavior is conjectured to extend to the multivariate framework. The uniform-power claim should be explicitly restricted to the population-level version, or a proof or rigorous bound for the feasible version should be supplied.","section":"§4.1, 'uniformly more powerful' claim"},{"comment":"The asymptotic size proof in Appendix A assumes that, when max_{i≠j} |ρ_{ij}| < 1, the vector λ contains m−1 components in (−c, c) and exactly one component equal to c. This structural fact about the argmax in Eq. (8) is asserted without proof for general covariance matrices. It underlies both the asymptotic size calculation and the κλ trajectories used in the simulations. The authors should either prove this characterization or state it as a lemma with the needed regularity conditions on Σ.","section":"Appendix A, structure of λ"}],"minor_comments":[{"comment":"The 'Results in' row of Table 1 refers to 'Figure 7' for Simulation 1, but the simulation results are displayed in Figure 6, while Figure 7 contains the case-study scatterplots; this cross-reference should be corrected.","section":"Table 1"},{"comment":"The horizontal axis in the simulation plots is labeled 'Index NA' in the displayed version; it should be labeled 'κ' uniformly across panels.","section":"Figures 6, A.1–A.3"},{"comment":"The discussion after Eq. (13) uses the independence-based condition to argue existence of α* in the correlated case study; the logical status of this inference should be clarified, since the condition is sufficient but not necessary in that setting.","section":"§4.1, existence discussion"},{"comment":"The statement that computing the multivariate α-TOST requires 'between 1 to 10 seconds' would be more useful with details on the machine, the Monte Carlo sample size, and the optimization tolerance used.","section":"§4.2, computation time"},{"comment":"The data and code are available in a GitHub repository, but a versioned release or a persistent archive would improve reproducibility; the authors should also report the versions of R and the cTOST package used for the results.","section":"Data availability statement"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and addresses a real practical problem, but the main size-control guarantee is conditional on a global optimization step that the manuscript does not specify or verify. I would be willing to accept after the authors either provide a certified optimization method, a verification study, or substantially weaken the claims to what is rigorously established. The reliance on the authors' own univariate α-TOST is natural and appropriately referenced; I do not see a novelty disclosure issue. The missing global-optimization specification is the key technical gap to resolve."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid extension of the univariate α-TOST to multivariate equivalence testing, with a sensible definition of the adjusted level α* and a thorough simulation study. The main thing you should know is that the size guarantee is only as good as the computation of λ(γ,Ω), the worst-case null vector in Eq. (8), and Algorithm 1 does not specify how that global supremum is computed. If the routine in cTOST returns a local maximizer, the fixed-point equation (12) solves the wrong size equation and the feasible test is liberal. That is the load-bearing gap.\n\nWhat is new and good: the recursive definition of α* in the multivariate setting, the existence conditions (Appendix C), and the op(ν^{-1}) asymptotic result for the estimated version (Appendix D). The analytic size under independence, Eq. (11), is a clean derivation. The simulations are extensive — four settings, varying ν, m, variance structure, and correlation — and they convincingly show the α-TOST recovers power where the TOST is near zero. The ticlopidine case study is a nice illustration, though the post hoc exclusion of four t1/2 outliers (Appendix G) deserves a sensitivity analysis. The paper is honest: it explicitly states that feasible size control is not guaranteed, only conjectured from simulations, and this conjecture is stated as such.\n\nSoft spots in order of importance. First, the global-optimization issue above. 'Compute λ(α*(r))' is not an algorithm; there is no optimizer specified, no grid check, no diagnostic that the true supremum was attained. Appendix E's condition 0 < p'(γ) < 2 is 'usually satisfied' but not checked for m ≥ 4. Both are fixable, but they are exactly what a referee should demand. Second, the 'uniformly more powerful' wording is too strong for the estimated version; it is proven for known covariance only, and the paper's own simulations show the feasible version is slightly conservative. That should be qualified in the abstract. Third, the Monte Carlo integration in Eq. (7) enters without discussion of its own numerical error; with B = 5e4 per scenario it is presumably fine, but a sentence on tolerance would help.\n\nOverall, the central idea holds up and the contribution is meaningful for bioequivalence of highly variable drugs. The missing numerical details are addressable. I would send this to a serious referee rather than desk reject, with a request to specify or benchmark the global optimizer and to soften the uniformity claim.","headline":"Multivariate α-TOST is a real improvement over multivariate TOST for bioequivalence, but the size guarantee rests on an unspecified global optimization of the worst-case null vector.","tokens_in":23931,"tokens_out":4784,"would_cite":false,"duration_ms":44276,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F03","62H15","62P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the conservative multivariate TOST for average equivalence can be made exactly size $\\alpha$ in finite samples by raising the test level to a corrected $\\alpha^*$ that solves a fixed-point equation, yielding a…","keywords":["multivariate equivalence testing","two one-sided tests (TOST)","finite-sample adjustment","significance level correction","bioequivalence","test size","type I error control"],"falsifier":"Run Algorithm 1 on a heteroscedastic $m = 4$ configuration similar to the case study's covariance, but with the $\\lambda$-search performed from many random starting points on the null boundary. If different restarts return measurably different $\\hat\\alpha^*$ values whose associated true suprema of $p(\\hat\\alpha^*, \\theta, \\Sigma, \\nu, c)$ over $\\theta \\notin \\Theta_1$ (evaluated by dense grid or large Monte Carlo) deviate from $\\alpha$ by more than Monte Carlo error, the global-optimization premise fails and the exact-size claim is not delivered by the algorithm as written. A second check: numerically evaluate $0 < p'(\\gamma) < 2$ along the iteration path in that setting — a violation means the exponential-convergence proof does not apply and the output level's exactness is unsupported.","tokens_in":58,"feed_emoji":"⚖️","tokens_out":10978,"duration_ms":141919,"temperature":0.7,"pith_summary":"The paper tries to establish that the standard multivariate TOST for average equivalence — checking, outcome by outcome, whether each $100(1-2\\alpha)\\%$ confidence interval for a mean difference falls inside pre-specified equivalence bounds — is needlessly conservative, and that a corrected significance level $\\alpha^*$ restores the test to exactly its nominal size (its intended maximum type-I error rate) at finite samples. The correction is defined as the level at which the worst-case rejection probability over the null space equals $\\alpha$, which makes the adjustment depend on the arbitrary correlation between outcomes and on the largest outcome variance. With the true covariance the adjusted test is exactly size $\\alpha$; with an estimated covariance the size differs from $\\alpha$ by a term of order $o_p(\\nu^{-1})$, smaller than the estimation error of the means and variances themselves. Because the correction only ever raises the level, the $\\alpha$-TOST's rejection region contains the conventional TOST's, making it uniformly more powerful; simulations across small samples, heterogeneous variances, and correlation structures confirm the gains, and a re-analysis of a ticlopidine hydrochloride bioequivalence study flips a non-equivalence verdict to equivalence. If the claims hold, practitioners get a drop-in finite-sample adjustment that recovers power in multivariate bioequivalence assessment without abandoning the per-outcome interval-inclusion logic.","feed_headline":"Adjusted level restores power in multivariate equivalence tests","feed_subtitle":"Correcting the test level to α* makes the multivariate TOST hit nominal size — and beat the conventional version in power.","key_machinery":"The load-bearing object is the size map of the multivariate TOST, $p(\\alpha, \\theta, \\Sigma, \\nu, c) = \\Pr\\left(\\bigcap_{j=1}^m \\{|\\hat\\theta_j| \\leq c - t_{\\alpha,\\nu} \\hat\\sigma_j\\}\\right)$, computed by integrating a multivariate normal density over outcome-dependent limits against a Wishart density for the covariance estimate (equation 7). Its supremum over the null space $\\theta \\notin \\Theta_1$ defines the test size, and the argument achieving that supremum, $\\lambda(\\alpha, \\Sigma)$ in equation (8), is the quantity that makes the multivariate problem hard: unlike in one dimension, $\\lambda$ moves with the level $\\alpha$ and with the covariance structure, and it generally has no closed form. The adjustment $\\alpha^*(\\Omega)$ is the fixed point of the level-to-size map defined by equation (12), and the paper computes it with a two-loop iteration: an inner loop applies the first-order update $\\alpha_k = \\alpha_{k-1} + \\alpha - p(\\alpha_{k-1}, \\lambda, \\Omega, \\nu, c)$, which converges exponentially once the size curve satisfies $0 < p'(\\gamma) < 2$, and an outer loop re-computes $\\lambda$ at each new level until the size equals $\\alpha$. Existence of the solution is governed by a threshold on the largest variance, which in the independence case reads $\\sigma_{\\max} < 2c/\\Phi^{-1}(\\alpha^{1/m} + 1/2)$ and is argued to be easily satisfied in practice.","core_discovery":"On the paper's own terms, the discovery is that the multivariate TOST size — the supremum of the probability of declaring equivalence over the null hypothesis space — is strictly below $\\alpha$, decays sharply as the number of outcomes grows, and is driven mainly by the largest outcome variance and the weakest dependence, so that power often collapses to zero. The proposed fix is the multivariate $\\alpha$-TOST: replace $\\alpha$ by $\\alpha^*(\\Omega)$, defined in equation (12) as the value in $[\\alpha, 0.5)$ at which $p(\\gamma, \\lambda(\\gamma,\\Omega), \\Omega, \\nu, c) = \\alpha$, where $\\lambda(\\gamma,\\Omega)$ is the set of null-space parameter values maximizing the rejection probability. For $\\Omega = \\Sigma$ the paper proves the procedure is exactly size $\\alpha$ (existence conditions supplied, most explicitly for the independence case as $\\sigma_{\\max} < 2c/\\Phi^{-1}(\\alpha^{1/m} + 1/2)$); for $\\Omega = \\hat\\Sigma$ it proves $\\hat\\alpha^* = \\alpha^* + o_p(\\nu^{-1})$, i.e., the estimated adjustment converges faster than the other random quantities in the problem, and simulations show the feasible procedure's empirical size stays at or below $\\alpha$. The monotonicity $\\alpha^* \\geq \\alpha$ then implies the rejection region of the $\\alpha$-TOST contains that of the multivariate TOST, so the test is uniformly more powerful, a property the paper exhibits both in an extensive simulation study and in a re-analysis of the ticlopidine hydrochloride bioequivalence data, where the conventional test fails to declare equivalence and the corrected test succeeds.","pith_inferences":["The same fixed-point scheme transfers to asymmetric equivalence margins or to other intersection-union tests, as long as the null-space argmax $\\lambda$ can still be evaluated; the paper only develops the symmetric-margin case.","The practical guarantee depends on the $\\lambda$-search being global; a cheap robustness check for applied users is to restart the optimization from several boundary points and confirm $\\hat\\alpha^*$ is stable, a diagnostic the paper's implementation does not report.","The paper's conjecture that the feasible procedure errs on the conservative side (empirical size below $\\alpha$) is itself a regulatory asset: for approval decisions, a size-controlled test that is slightly conservative is safer than one that is exact but numerically fragile.","Replacing $\\hat\\Sigma$ by a robust covariance estimate in the $\\alpha^*$ map is a natural follow-up, since the paper flags outliers as the main open threat to the size guarantees."],"forward_implications":["A multivariate bioequivalence assessment (e.g., $C_{\\max}$ and AUC jointly) can use the $\\alpha$-TOST at level $\\alpha^*$ instead of $\\alpha$, recovering power that the per-outcome interval check loses, without changing the interpretation of the intervals.","The method is finite-sample: at the population level it is exact for any $\\nu$, and the estimated version's size error is negligible relative to the estimation error of means and variances.","Settings with large variances on any outcome — the highly-variable-drug scenario — gain the most, since the size gap that $\\alpha^*$ corrects is driven by the largest $\\sigma_j$.","In the case study the correction is mild ($\\hat\\alpha^* \\approx 0.058$) yet sufficient: the $\\alpha$-TOST declares bioequivalence of the two ticlopidine formulations where the conventional TOST cannot.","Because $\\alpha^* \\geq \\alpha$ always, the $\\alpha$-TOST is a uniform power improvement, so no scenario exists in which the corrected test is harder to pass than the conventional one (at the population level)."],"supporting_citations":[{"why":"The univariate α-TOST this work extends; supplies the adjustment logic, the convergence-rate results in equations (A.11)-(A.12), and the precedent for the conservative empirical size conjecture.","marker":"[38]"},{"why":"Introduces the TOST procedure whose multivariate extension is the baseline being corrected.","marker":"[32]"},{"why":"Documents the sharp size and power loss of the multivariate TOST as m grows and concludes simultaneous confidence sets cannot beat it; the power comparisons and case study build on it.","marker":"[33]"},{"why":"Gives the power function for multiple TOST instances that underlies the general rejection-probability expression in equation (7).","marker":"[44]"},{"why":"Berger's intersection-union result justifies assessing multivariate equivalence through the per-outcome univariate tests.","marker":"[41]"},{"why":"Provides the canonical multivariate normal-Wishart equivalence framework in equation (1).","marker":"[39]"},{"why":"The ticlopidine hydrochloride bioequivalence data re-analyzed in the case study.","marker":"[27]"}],"fun_headline_variants":["Alpha-star fix makes multivariate TOST more powerful","Adjusted significance level restores power in multivariate equivalence","Uniformly more powerful: multivariate alpha-TOST beats TOST","Corrected alpha rescues multivariate equivalence tests from power loss"],"cache_read_input_tokens":25984,"weakest_assumption_plain":"Everything rests on reliably finding the global worst-case parameter $\\lambda$ that maximizes the rejection probability over the null boundary — the algorithm invokes this search without a guarantee of global optimality — together with the 'usually satisfied' condition $0 < p'(\\gamma) < 2$ that makes the fixed-point iteration converge to the unique $\\alpha^*$; neither is verified for the $m = 4$ settings of the case study.","fun_headline_variants_meta":{"raw":{"variants":["Alpha-star fix makes multivariate TOST more powerful","Adjusted significance level restores power in multivariate equivalence","Uniformly more powerful: multivariate alpha-TOST beats TOST","Corrected alpha rescues multivariate equivalence tests from power loss"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000577,"raw_usage":{"total_tokens":2853,"prompt_tokens":1211,"completion_tokens":1642,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":827,"completion_tokens_details":{"reasoning_tokens":1576}},"tokens_in":827,"tokens_out":1642,"duration_ms":11091,"temperature":1.0,"reasoning_tokens":1576,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:08:08.803432+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm 1 on a heteroscedastic $m = 4$ configuration similar to the case study's covariance, but with the $\\lambda$-search performed from many random starting points on the null boundary. If different restarts return measurably different $\\hat\\alpha^*$ values whose associated true suprema of $p(\\hat\\alpha^*, \\theta, \\Sigma, \\nu, c)$ over $\\theta \\notin \\Theta_1$ (evaluated by dense grid or large Monte Carlo) deviate from $\\alpha$ by more than Monte Carlo error, the global-optimization premise fails and the exact-size claim is not delivered by the algorithm as written. A second check: numerically evaluate $0 < p'(\\gamma) < 2$ along the iteration path in that setting — a violation means the exponential-convergence proof does not apply and the output level's exactness is unsupported.","supporting_citations":[{"cited_title":"Finite sample corrections for average equivalence testing","cited_arxiv_id":null,"evidence_quote":"The univariate α-TOST this work extends; supplies the adjustment logic, the convergence-rate results in equations (A.11)-(A.12), and the precedent for the conservative empirical size conjecture."},{"cited_title":"A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability","cited_arxiv_id":null,"evidence_quote":"Introduces the TOST procedure whose multivariate extension is the baseline being corrected."},{"cited_title":"and Jaki, T","cited_arxiv_id":null,"evidence_quote":"Documents the sharp size and power loss of the multivariate TOST as m grows and concludes simultaneous confidence sets cannot beat it; the power comparisons and case study build on it."},{"cited_title":"Power for testing multiple instances of the two one-sided tests procedure","cited_arxiv_id":null,"evidence_quote":"Gives the power function for multiple TOST instances that underlies the general rejection-probability expression in equation (7)."},{"cited_title":"Multiparameter hypothesis testing and acceptance sampling","cited_arxiv_id":null,"evidence_quote":"Berger's intersection-union result justifies assessing multivariate equivalence through the per-outcome univariate tests."},{"cited_title":"Statistical tests for multivariate bioequivalence","cited_arxiv_id":null,"evidence_quote":"Provides the canonical multivariate normal-Wishart equivalence framework in equation (1)."},{"cited_title":"Bioequivalence of ticlopidine hydrochloride administered in single dose to healthy volunteers","cited_arxiv_id":null,"evidence_quote":"The ticlopidine hydrochloride bioequivalence data re-analyzed in the case study."}],"review_version":1}