Calculating sample size

Once the requirements of a trial have been specified then calculating sample size is fairly straightforward formulas exist for all of the commonly occurring situations. 

In all cases we need to specify the required values of the type I error and the power. Usually we set the type I error at 5 per cent and the recommended minimum value for power is 80 per cent, although for important trials 80 per cent is not enough and 90 per cent at least is recommended. 

The remaining quantities that need to be considered when calculating sample size depend upon the particular statistical test to be used  

There is usually an implicit assumption in this calculation that the standard deviations are the same in each of the treatment groups. Generally speaking this assumption is a reasonable one to make as the effect of treatment will be to change the mean with no effect on the variability. We will say a little more about dealing at the analysis stage with situations where this is not the case in a later section. The sample size calculation, however, is also easily modified, if needed, to allow unequal standard deviations. 

We commonly refer to the level of effect to be detected as the cliniMlly relevant difference (crd) what level of effect is an important effect from a clinical standpoint. Note also that crd stands for commercially relevant difference it could well be that the decision is based on commercial interests. Finally crd stands for cynically relevant difference It does happen from time to time that a statistician is asked to do a sample size calculation, oh and by the way, we want 200 patients The issue here of course is budget and the question really is what level of effect are we able to detect with a sample size of 200  

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Statistical methods are often employed to determine the study sample size and optimize power. Outlining the methods for calculating sample size and power for clinical trials is beyond the scope of this chapter. Interested readers are referred to texts by Chow and Liu (1998), Hulley and Cummings (1988), and Shuster (1990) for specific information on sample size and power estimation methods. 

Power Analysis for ANOVA Designs can be used to calculate sample size for one and two-way factorial designs with fixed effects http //evall.crc.uiuc.e du/ fp o wer. html/... 

CALCULATING SAMPLE SIZE Table 8.4 Sample sizes per group... 

Finally note that in our considerations we have worked with groups of equal size. It is straightforward to adapt the calculations for unequal randomisation schemes and the computer packages mentioned earlier can deal with these. Altman (1991), Section 15.3 provides a simple method for adapting the standard sample size calculation to unequal group sizes as follows. If N is the calculated sample size based in an equal randomisation and k represents the ratio of the number of patients in one group compared to the other group, then the required number of patients for a A to 1 randomisation is ... 

Of the four values needed to calculated sample size, the a and [> values are standardized for most scientific and industrial applications as a = 0.05, /< = 0.1-0.2 from the t table for a and [> with a given number degrees of freedom (sample size minus 1) of the data used to estimate the variance S2 and d2 are more difficult to obtain. Since methods and processes undergo change, the estimate of variance is limited to methods and products resulting from the most recent changes. Since some products are made only once or twice a year, there may not be sufficient data available to yield a reasonable estimate of overall variance. Moreover, d2 cannot be determined for future OOS results since this information is not known. This fact represents an inherent and unintended conflict in the FDA position of a predetermined sample size since to use a reliable d2 value, one must know in advance how far out the OOS result will be. This information cannot be determined until the OOS is actually observed. This is yet another argument for the use of protocols for each retest scenario, since the protocol will be written once the difference between the specification and the OOS result is known. 

When calculating sample sizes, the figure for the smallest difference to be detectable should generally match the smallest difference that would be of practical significance. 

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. 

Pezeshk H, Gittins J (2002) A fully Bayesian approach to calculating sample sizes for clinical trials with binary responses. Drug Information Journal 36 143 150. 

Jaech JL, Russell M (1991) Algorithms to calculate sample sizes for inspection sampling plans, STR-261, rev.l. IAEA, Vienna... 

Standard Practice for Calculating Sample Size to Estimate, with a Specified Tolerable Error, the Average Characteristics of a Lot or Process," in ASTM Standard E 122-00,14.02, ASTM International, 2004,... 

Sample size is 100 ml and distillation conditions are specified according to the type of sample. Temperature and volume of condensate are taken simultaneously and the test results are calculated and reported as boiling temperature as a function of the volume recovered as shown in Table 2.1. 

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. 

Example 3 Calculating Sample Weight for Screen-Size... 

Example 3 Calculating Sample Weight for Screen-Size Measurement Weight W of bulk sample for screen analysis is calculated by the Gayle model for percent retained on a specified screen with relative standard error s.e. in percent... 

In general, the larger the sample size, the more accurate will be the experimental calculations of the population parameters. A sample size A > 30 is typical, but sometimes 100 is preferred. 

Quite a few years ago, Dr. Azbel and I analyzed the operational requirements for these machines and developed some design formulae. You can find this analysis on pages 646 through 665 in Fluid Mechanics and Unit Operations, David S. Azbel and Nicholas P. Cheremisinoff, Ann Arbor Science Publishers, 1983. There are some sample calculations and sizing criteria that you can follow for some practical exercises in this publication. 

The precondition for the use of the normal distribution in estimating the random error is that adequate reliable estimates are available for the parame-rcrs ju. and cr. In case of a repeated measurement, the estimates are calculated using Eqs. (12.1) and (12,3). When the sample size iiicrease.s, the estimates m and s approach the parameters /c and cr. A rule of rhumb is that when s 30. the normal distribution can be osecl,... 

In a data set it may be desirable to ask the question Is any one value significantly different from the others in the sample A t statistic (for n — 1 degrees of freedom where the sample size is n) can be calculated that takes into account the difference of the magnitude of that one value (xj and the mean of the sample (x ) ... 

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 ... 

The percolation theory [5, 20-23] is the most adequate for the description of an abstract model of the CPCM. As the majority of polymers are typical insulators, the probability of transfer of current carriers between two conductive points isolated from each other by an interlayer of the polymer decreases exponentially with the growth of gap lg (the tunnel effect) and is other than zero only for lg < 100 A. For this reason, the transfer of current through macroscopic (compared to the sample size) distances can be effected via the contacting-particles chains. Calculation of the probability of the formation of such chains is the subject of the percolation theory. It should be noted that the concept of contact is not just for the particles in direct contact with each other but, apparently, implies convergence of the particles to distances at which the probability of transfer of current carriers between them becomes other than zero. 

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... 

The number of subjects planned to be enrolled, if more than one site the numbers of enrolled subjects projected for each trial site should be specified. Reason for choice of sample size include calculations of the statistical power of the trial, the level of significance to be used and the clinical justification. 

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