Exact and Approximate Power and Sample Size Calculations for Analysis of Covariance in Randomized Clinical Trials With or Without Stratification

<p>Analysis of covariance (ANCOVA) is commonly used in the analysis of randomized clinical trials to adjust for baseline covariates and improve the precision of the treatment effect estimate. We derive the exact power formulas for testing a homogeneous treatment effect in superiority, noninferiority, and equivalence trials under both unstratified and stratified randomizations, and for testing the overall treatment effect and treatment × stratum interaction in the presence of heterogeneous treatment effects when the covariates excluding the intercept, treatment, and prestratification factors are normally distributed. These formulas also work very well for nonnormal covariates. The sample size methods based on the normal approximation or the asymptotic variance generally underestimate the required size. We adapt the recently developed noniterative and two-step sample size procedures to the above tests. Both methods take into account the nonnormality of the <i>t</i> statistic, and the lower order variance term commonly ignored in the sample size estimation. Numerical examples demonstrate the excellent performance of the proposed methods particularly in small samples. We revisit the topic on the prestratification versus post-stratification by comparing their relative efficiency and power. Supplementary materials for this article are available online.</p

Supplemental material for A monotone data augmentation algorithm for longitudinal data analysis via multivariate skew-t, skew-normal or <i>t</i> distributions

Supplemental Material for A monotone data augmentation algorithm for longitudinal data analysis via multivariate skew-t, skew-normal or t distributions by Yongqiang Tang in Statistical Methods in Medical Research</p

Online-Appendix - Supplemental material for A monotone data augmentation algorithm for longitudinal data analysis via multivariate skew-t, skew-normal or <i>t</i> distributions

Supplemental material, Online-Appendix for A monotone data augmentation algorithm for longitudinal data analysis via multivariate skew-t, skew-normal or t distributions by Yongqiang Tang in Statistical Methods in Medical Research</p

Notes on Exact Power Calculations for <i>t</i> Tests and Analysis of Covariance

Tang derived the exact power formulas for t tests and analysis of covariance (ANCOVA) in superiority, noninferiority (NI), and equivalence trials. The power calculation in equivalence trials can be simplified by using Owen’s Q function, which is available in standard statistical software. We extend the exact power determination method for ANCOVA to unstratified and stratified multi-arm randomized trials. The method is applied to the design of multi-arm trials and gold standard NI trials. Supplementary materials for this article are available online.</p

Effects of Cationic Ammonium Gemini Surfactant on Micellization of PEO–PPO–PEO Triblock Copolymers in Aqueous Solution

Effects of cationic ammonium gemini surfactant hexamethylene-1,6-bis­(dodecyldimethylammonium bromide) (12–6–12) on the micellization of two triblock copolymers of poly­(ethylene oxide)–poly­(propylene oxide)–poly­(ethylene oxide), F127 (EO₉₇PO₆₉EO₉₇) and P123 (EO₂₀PO₇₀EO₂₀), have been studied in aqueous solution by differential scanning calorimetry (DSC), dynamic light scattering (DLS), isothermal titration calorimetry (ITC), and NMR techniques. Compared with traditional single-chain ionic surfactants, 12–6–12 has a stronger ability of lowering the CMT of the copolymers, which should be attributed to the stronger aggregation ability and lower critical micelle concentration of 12–6–12. The critical micelle temperature (CMT) of the two copolymers decreases as the 12–6–12 concentration increases and the ability of 12–6–12 in lowering the CMT of F127 is slightly stronger than that of P123. Moreover, a combination of ITC and DLS has shown that 12–6–12 binds to the copolymers at the temperatures from 16 to 40 °C. At the temperatures below the CMT of the copolymers, 12–6–12 micelles bind on single copolymer chains and induce the copolymers to initiate aggregation at very low 12–6–12 concentration. At the temperatures above the CMT of the copolymers, the interaction of 12–6–12 with both monomeric and micellar copolymers leads to the formation of the mixed copolymer/12–6–12 micelles, then the mixed micelles break into smaller mixed micelles, and finally free 12–6–12 micelles form with the increase of the 12–6–12 concentration

Aggregation Behavior of Sodium Lauryl Ether Sulfate with a Positively Bicharged Organic Salt and Effects of the Mixture on Fluorescent Properties of Conjugated Polyelectrolytes

The aggregation behavior of anionic single-chain surfactant sodium lauryl ether sulfate containing three ether groups (SLE3S) with positively bicharged organic salt 1,2-bis­(2-benzylammoniumethoxy)­ethane dichloride (BEO) has been investigated in aqueous solution, and the effects of the BEO/SLE3S aggregate transitions on the fluorescent properties of anionic conjugated polyelectrolyte MPS-PPV with a larger molecular weight and cationic conjugated oligoelectrolyte DAB have been evaluated. Without BEO, SLE3S does not affect the fluorescent properties of MPS-PPV and only affects the fluorescent properties of DAB at a higher SLE3S concentration. With the addition of BEO, SLE3S and BEO form gemini-like surfactant (SLE3S)₂-BEO. When the BEO/SLE3S molar ratio is fixed at 0.25, with increasing the BEO/SLE3S concentration, the BEO/SLE3S mixture forms large, loosely arranged aggregates and then transforms to closely packed spherical aggregates and finally to long thread-like micelles. The photoluminescence (PL) intensity of MPS-PPV varies with the morphologies of the BEO/SLE3S aggregates, while the PL intensity of DAB is almost independent of the aggregate morphologies. The results demonstrate that gemini-like surfactants formed through intermolecular interactions can effectively adjust the fluorescent properties of conjugated polyelectrolytes

Coassembly of Poly(ethylene glycol)-<i>block</i>-Poly(glutamate sodium) and Gemini Surfactants with Different Spacer Lengths

The coassembly of poly­(ethylene glycol)-<i>b</i>-poly­(glutamate sodium) copolymer (PEG₁₁₃-PGlu₁₀₀) with cationic gemini surfactants alkanediyl-α,ω-bis-(dodecyldimethylammonium bromide) [C₁₂H₂₅(CH₃)₂N­(CH₂)_<i>S</i>N­(CH₃)₂C₁₂H₂₅]­Br₂ (designated as C₁₂C_<i>S</i>C₁₂Br₂, <i>S</i> = 3, 6, and 12) have been studied by isothermal titration microcalorimetry, cryogenic transmission electron microscopy, circular dichroism, small-angle X-ray scattering, zeta potential, and size measurement. It has been shown that the electrostatic interaction of C₁₂C_<i>S</i>C₁₂Br₂ with the anionic carboxylate groups of PEG₁₁₃-PGlu₁₀₀ leads to complexation, and the C₁₂C_<i>S</i>C₁₂Br₂/PEG₁₁₃-PGlu₁₀₀ complexes are soluble even at the electroneutral point. The complexes display the feature of superamphiphiles and assemble into ordered nanosheets with a sandwich-like packing. The gemini molecules which were already bound with PGlu chains associate through hydrophobic interaction and constitute the middle part of the nanosheets, whereas the top and bottom of the nanosheets are hydrophilic PEG chains. The size and morphology of the nanosheets are affected by the spacer length of the gemini surfactants. The average sizes of the aggregates at the electroneutral point are 81, 68, and 90 nm for C₁₂C₃C₁₂Br₂/PEG₁₁₃-PGlu₁₀₀, C₁₂C₆C₁₂Br₂/PEG₁₁₃-PGlu₁₀₀, and C₁₂C₁₂C₁₂Br₂/PEG₁₁₃-PGlu₁₀₀, respectively. Both C₁₂C₃C₁₂Br₂/PEG₁₁₃-PGlu₁₀₀ and C₁₂C₁₂C₁₂Br₂/PEG₁₁₃-PGlu₁₀₀ mainly generate hexagonal nanosheets, while the C₁₂C₆C₁₂Br₂/PEG₁₁₃-PGlu₁₀₀ system only induces round nanosheets

Baseline demographics and clinical characteristics.

<p>BMI, body mass index; i.v., intravenous; N/A, not applicable; s.c., subcutaneous; SD, standard deviation; SF-MPQ, Short-Form McGill Pain Questionnaire; VAS, visual analog scale.</p><p>Baseline demographics and clinical characteristics.</p

BG00010 serum concentrations over time.

<p>Mean (standard deviation) BG00010 serum concentrations over (a) 120 h and (b) 15 h (expanded time axis) following i.v. administration of BG00010. Note that data were only available for two subjects treated with BG00010 25 μg/kg at 9, 12, 18 and 48 h. Where data points are not shown, the mean BG00010 serum concentration was equal to 0.00 ng/ml. h, hours; i.v., intravenous.</p

Subject disposition. i.v., intravenous; s.c., subcutaneous.

<p>Subject disposition. i.v., intravenous; s.c., subcutaneous.</p