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

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

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

Guidance FDA (2001) using a REML UN model. Then, this estimate is asymptotically normally distributed, unbiased with E[νˆ ] = δ +σ − (σ )− 0.04(c ) and has variance of Var[νˆ ] = 4σ δ + l + 2l − 2l + 2l 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 7.4, we obtain the value −0.24 for log(AUC) and the value −0.19 for log(Cmax). As both of these are below zero, we can declare that T and R are PBE. 7.6 ABE for a replicate design Although ABE can be assessed using a 2× 2 design, it can also be as-sessed using a replicate design. If a replicate design is used the number of subjects can be reduced to up to half that required for a 2 × 2 de-sign. In addition it permits the estimation of σ and σ . The SAS code to assess ABE for a replicate design is given in Appendix B. Using the data from Section 7.4, the 90% confidence interval for µ is (−0.1697,−0.0155) for log(AUC) and (−0.2474,−0.0505) for log(Cmax). Exponentiating the limits to obtain confidence limits for exp(µ ), gives (0.8439,0.9846) for AUC and (0.7808,0.9508) for Cmax. Only the first of these intervals is contained within the limits of 0.8 to 1.25, there-fore T cannot be considered average bioequivalent to R. To calculate the power for a replicate design with four periods and with a total of n subjects we can still use the SAS code given in Section 7.3, if we alter the formula for the variance of a difference of two obser-vations from the same subject. This will now be σ +σ instead of σ , where σ is the subject-by-formulation interaction. Note the use of σ rather than 2σ as used in the RT/TR design. This is a result of the estimator using the average of two measurements on each treatment on each subject. One advantage of using a replicate design is that the number of sub-jects needed can be much smaller than that needed for a 2×2 design. As an example, suppose that σ = 0, and we take σ = 0.355 and α = 0.05, as done in Section 7.3. Then a power of 90.5% can be achieved with only 30 subjects, which is about half the number (58) needed for the 2 × 2 design.

2003 ◽  
pp. 370-370
Keyword(s):  
Confidence Interval ◽  
Upper Bound ◽  
Variance Components ◽  
Confidence Limits ◽  
The Subject ◽  
Obser Vations ◽  
Replicate Design ◽  
Normally Distributed

Upper bounds on the energy dissipation in turbulent precession

1996 ◽  
Vol 321 ◽  
pp. 335-370 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Experimental Data ◽  
Energy Dissipation ◽  
Viscous Dissipation ◽  
Upper Bound ◽  
Dissipation Rate ◽  
Oblate Spheroid ◽  
Upper Bounds ◽  
Precession Rate ◽  
Long Cylinder

Rigorous upper bounds on the viscous dissipation rate are identified for two commonly studied precessing fluid-filled configurations: an oblate spheroid and a long cylinder. The latter represents an interesting new application of the upper-bounding techniques developed by Howard and Busse. A novel ‘background’ method recently introduced by Doering & Constantin is also used to deduce in both instances an upper bound which is independent of the fluid's viscosity and the forcing precession rate. Experimental data provide some evidence that the observed viscous dissipation rate mirrors this behaviour at sufficiently high precessional forcing. Implications are then discussed for the Earth's precessional response.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

Models of arithmetic and upper bounds for arithmetic sets

1994 ◽  
Vol 59 (3) ◽  
pp. 977-983 ◽  
Author(s):  
Alistair H. Lachlan ◽  
Robert I. Soare
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Models Of Arithmetic

AbstractWe settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

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