Sample size conventions used

As an example to help understand the effect of equivalence on sample size, consider a case where we wish to show that the difference in FEVi between two treatments is not greater than 200 ml and where the standard deviation is 350 ml for conventional type I and type II errors rates of 0.05 and 0.2. If we assume that the drugs are in fact exactly identical, the sample size needed (using a Normal approximation) is 53. If we allow for a true difference of 50 ml this rises to 69. On the other hand, if we wished to demonstrate superiority of one treatment over another for a clinically relevant difference of 200 ml with the other values as before, a sample size of 48 would suffice. Thus, in the best possible case a rise of about 10% in the sample size is needed (from 48 to 53). This difference arises because we have two one-sided tests each of which must be significant in order to prove efficacy. To have 80% power each must (in the case of exact equality) have approximately 90% power(because 0.9 x 0.9 = 0.8). The relevant sum of z-values for the power calculation is thus 1.2816-1-1.6449 = 2.93 as opposed to for a conventional trial 0.8416 -I-1.9600 = 2.8. The ratio of the square of 2.93 to 2.8 is about 1.1 explaining the 10% Increase in sample size. 

The principal advantage of conventional GC-IRMS in this context is its ease of use and lower sample size requirements, while the IRMS approach provides superior accuracy and precision, particularly at lower enrichments. IRMS data are usually expressed using the 6yX notation, given as... 

Another possible solution to the problem of analyzing multiple-layered membrane composites is a newly developed method using NMR spin-lattice relaxation measurements (Glaves 1989). In this method, which allows a wide range of pore sizes to be studied (from less than 1 nm to greater than 10 microns), the moisture content of the composite membrane is controlled so that the fine pores in the membrane film of a two-layered composite are saturated with water, but only a small quantity of adsorbed water is present in the large pores of the support. It has been found that the spin-lattice relaxation decay time of a fluid (such as water) in a pore is shorter than that for the same fluid in the bulk. From the relaxation data the pore volume distribution can be calculated. Thus, the NMR spin-lattice relaxation data of a properly prepared membrane composite sample can be used to derive the pore size distribution that conventional pore structure analysis techniques... 

Although conventional p-values have no role to play in equivalence or noninferiority trials there is a p-value counterpart to the confidence intervals approach. The confidence interval methodology was developed by Westlake (1981) in the context of bioequivalence and Schuirmann (1987) developed a p-value approach that was mathematically connected to these confidence intervals, although much more difficult to understand It nonetheless provides a useful way of thinking, particularly when we come later to consider type I and type II errors in this context and also the sample size calculation. We will start by looking at equivalence and use A to denote the equivalence margins. 

Most of the analyses of Antarctic coal were based on blocks of transportable size taken from surface outcrops. In earlier work, sections of purer coal were cut from the blocks and submitted for analysis. In later work, selected blocks were crushed, and a float fraction of purer coal was submitted to provide analytic data from each deposit. These samples, while not useful for determining coal grade, may be adequate for indicating rank. Practically all of the Antarctic coal, with the exception of that at Amery (which may represent high volatile bituminous coal), corresponds to medium volatile bituminous or higher rank. The high apparent rank Antarctic coal may be classified according to conventional ASTM standards of rank based on proximate analysis, but it seems clear that these results serve only as a first approximation. 

Near-infrared chemical imaging using multichannel detectors inherits many of the attributes of conventional NIR spectroscopy using a single-channel detector. In addition to its well-documented capabilities as a spectroscopic technique, in comparison to other vibrational imaging approaches, it has unparalleled flexibility in terms of managing widely varying sample size, placement, shape and color. As has been shown with the three examples presented in this chapter, the technique easily... 

---