Here I will illustrate the use of the formulae for the required sample size (Box 3.8). In the spider web example of Elgar et al. (1996, see Box 3.7), the standard deviation of the posterior for the difference in web size under high and low light was approximately 23 cm for both the horizontal and vertical dimensions. The standard deviation of the data (s) is estimated to be 94.8 cm (= 23 x VT7). Therefore, if we wished to reduce the standard deviation of the posterior to 10 cm and used the data from the study of Elgar et al. (1996) as the prior, the number of new spiders required would equal:
94.82 x(1/102 — 1/232) = 94.82 x(0.01 — 0.00189) = 72.9.
Therefore, approximately 73 additional spiders would be needed to increase the precision of the estimated effect of light on the size of spider webs to the required level.
the probability distribution for the variance can be calculated. The probability distribution of the variance of the data will lead to a probability distribution of the required sample size. To ensure (with reasonable certainty) that the required precision is achieved, a greater sample size is needed to account for the possibility that the variance of the data will be greater than the value that was assumed. This feature is used in Box 3.10 to calculate the probability distribution of the required sample size for the example in Box 3.9.
The important result of the analysis in Box 3.10 is that the required sample size is underestimated if uncertainty in the standard deviation is ignored. As the sample size used to estimate the standard deviation of the data increases, the precision of the required sample size also increases and the bias decreases. However, for sample sizes that are likely to be used in pilot studies (say, n = 10 to 20), the bias may be important.
Adcock (1997) reviews a number of other approaches to determine sample sizes, including the use of a utility function (Lindley, 1997). Lindley's (1997) approach recognizes that extra sampling entails costs.
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