Sample size example calculation

Instead of calculations, practical work can be done with scale models (33). In any case, calculations should be checked wherever possible by experimental methods. Using a Monte Carlo method, for example, on a shape that was not measured experimentaUy, the sample size in the computation was aUowed to degrade in such a way that the results of the computation were inaccurate (see Fig. 8) (30,31). Reversing the computation or augmenting the sample size as the calculation proceeds can reveal or eliminate this source of error. 

In this example Q calculated is 0.727 and Q critical, for a sample size of four, is 0.831. Hence the result 3.2 jug g 1 should be retained. If, however, in the above example, three additional measurements were made, with the results ... 

For a more realistic sample size than that in Example 7.7, one that contains 1.00 mol CO, corresponding to 6.02 x 1023 CO molecules, each of which could be oriented in either of two ways, there are 2602x10 (an astronomically large number) different microstates, and a chance of only 1 in 2< 02x l0" of drawing a given microstate in a blind selection. We can expect the entropy of the solid to be high and calculate that... 

This means that the necessary sample size will be very large if the systems 0 and 1 differ too much. For example, for AS/kB = —15, N > 3 x 108 is needed to reach a 95% accuracy level. In this case, it is better to introduce intermediates and perform multistage calculations instead. 

This example shows that the standard deviation of the sampling distribution is less than that of the population. In fact, this reduction in the variability is related to the sample size used to calculate the sample means. For example, if we repeat the sampling experiment, but this time based on 15 rather than 10 random samples, the resulting standard deviation of the sampling is 0.159, and on 25 random samples it is 0.081. The precise relationship between the population standard deviation a and the standard error of the mean is ... 

Machin et al. (1997) provide extensive tables in relation to sample size calculations and include in their book the formulas and many examples. In addition there are several software packages specifically designed to perform power and sample size calculations, namely nQuery (www.statsol.ie) and PASS (www.ncss.com). The general statistics package SPLUS (www.insightful.com) also contains some of the simpler calculations. 

Example 13.4 Sample size calculation for survival data... 

Fig. 1. An example of power calculation to detect the indicated odds ratio for a range of risk factor prevalence and event rate with a sample size of 300 patients.

Perhaps the ultimate failing of the 0SHA/NI0SH scheme is that it bases important decisions on relatively small amounts of data. Intuitively, such a scheme would lead to incorrect conclusions in many cases. Table III gives the number of samples expected to be required for making decisions in various environments (calculated from the relationship derived in the appendix). As in the previous example the PEL is 10 and the AL is 5. In virtually all cases the number of samples is two or less. With such small sample sizes accurate prediction of the long-term rates of exposure is impossible without additional information or assumptions. Stated in slightly different terms, the interday variability of 8-hr TWA values cannot be measured or controlled for with information based strictly on such small sample sizes. 

Note that f-statistics should be followed when the sample size is small, i.e., <30. In the MDL measurements, the number of replicate analyses are well below 30, generally 7. For example, if the number of replicate analyses are 7, then the degrees of freedom, i.e., the ( -1) is 6, and, therefore, the t value for 6 should be used in the above calculation. MDL must be determined at the 99% confidence level. When analyses are performed by GC or GC/MS methods, the concentrations of the analytes to be spiked into the seven aliquots of the reagent grade water for the MDL determination should be either at the levels of their IDL (instrument detection limit) or five times the background noise levels (the noise backgrounds) at or near their respective retention times. 

Unlike solids and liquids, the density of a gas depends very strongly on the temperature and pressure. Also, unlike solids and liquids, we can easily calculate the density of a gas if we know the temperature and pressure. For example, what is the density of air under normal conditions (25°C and 750 torr) Recall that air is approximately 79% nitrogen and 21% oxygen (by volume). If we want to calculate the density of a sample, we need to know its mass and volume. Since density is an intrinsic physical property, we can take any sample size we want, so let s take a sample volume of 1.0 L. 

Consider two examples from previous chapters that illustrate this. In Chapter 9, discussions of sample-size estimation emphasized that the process is indeed one of estimation rather than pure calculation. A calculation is certainly executed, but the values that are placed into the appropriate formula are chosen by the sponsor. On each occasion, the sponsor must consider the influences of the choices that are made and make the most appropriate decision in the specific context of that trial. In Chapter 11, equivalence and noninferiority designs were discussed. In addition to the calculations that are involved using the data collected in a trial, equivalence or noninferiority margins must be established before the trial commences. Their choice is a clinical choice, not a statistical choice, and subjectivity is necessarily involved in this choice. Thus, the discipline of Statistics certainly involves using informed judgments. Statistics really is an art as well as a science, a sentiment well expressed by Katz (2001) as cited in Chapter 13. 

In the example, all of the results are for the given sample size of 1 liter and the quantities estimated have units reflecting that basis. This basis volume is arbitrary, but use of the calculated quantities requires care in defining this basis consistently in corresponding mass and population balances. The volume of clear liquor in the sample is an alternative, and sometimes more convenient, basis. 

It is worth noting that the critical t value for an infinite number of test objects at the 95% confidence limit is 1.96 and here, with a sample size of n = 3, the value is 4.3, so clearly the larger the number of test objects, the smaller the t critical value becomes. For example, for a sample size of n = 6 (and therefore 5 degrees of freedom), the t critical value is 2.57. This is useful, as n = 6 is a very common number of test objects to run in an analytical test, and so remembering the critical value saves one from hunting statistical tables. If the calculated value for a data set is less than 2.57, the null hypothesis is retained, and if it is greater than 2.57 the null hypothesis is rejected. 

A common approach in statistics is to ask What would happen if we were to repeat a sampling procedure many times In this case, the question we ask is What would happen if we were to take repeated samples and calculate the mean of each sample Fortunately, we do not actually have to take real repeated samples. We can calculate what would happen if we did, based on the fact that we know sampling error is dependent upon sample size and SD. An example of hypothetical repeated resampling is shown in Figure 4.2. Note that the horizontal axis represents the mean values of samples, not the individual values that go into the samples. The sample means mainly cluster around the true population mean, but there are a few outlying results badly above and below. These sample means themselves form a normal... 

Based on published variability in pharmacokinetic studies of ethinylestradiol in lean subjects, taking confidence intervals of 80-125%, residual variance ranged between 10 and 33%. Based on these residual variance values, calculated samples sizes ranged between 6 and 30 (subjects). For example, based on a residual variance value of 17.5%, a sample size of 14 was calculated. 

---