Sample size assumptions

O. In actual practice, this is a robust test, in the sense that in most types of problems it is not sensitive to the normality assumption when the sample size is 10 or greater. 

The population of differences is normally distributed with a mean [L ansample size is 10 or greater in most situations. 

Clinical trials are costly to conduct, and results are often critical to the commercial viability of a phytochemical product. Seemingly minor decisions, such as which measurement tool to use or a single entry criterion, can produce thousands of dollars in additional costs. Likewise, a great deal of time, effort and money can be saved by having experts review the study protocol to provide feedback regarding ways to improve efficiency, reduce subject burden and insure that the objectives are being met in the most scientifically sound and cost-effective manner possible. In particular, I recommend that an expert statistician is consulted regarding sample size and power and that the assumptions used in these calculations are reviewed carefully with one or more clinicians. It is not uncommon to see two studies with very similar objectives, which vary by two-fold in the number of subjects under study. Often this can be explained by differences in the assumptions employed in the sample size calculations. 

H0 = assumption or null hypothesis regarding the population proportion H1 = alternative hypothesis a = significance level, usually set at. 10,. 05, or. 01 z = tabled Z value corresponding to the significance level a. The sample sizes required for the z approximation according to the magnitude of p0 are given in Table 3-6. 

This problem has been discussed by Snyder (8,10) and Scott 9, II). A general solution is difficult to give since it would depend oh the composition of the sample mixture, e.g., the concentration of the la eluting components, and the detection limit, which varies in liquid chromatography with the chemical nature of the sample component. Therefore some arbitrary assumptions have to be made. From Eq. (26), using the maximum permissible sample size given by Eq. (73), we can write foi the retention volume, Vr, the following expression ... 

Certain methods associated with normality, such as the t interval for the mean, are equally valid at aU sample sizes, so long as normality and other assumptions are accurate, and in most situations will improve in performance with increasing n. With small n one has low power for testing distributional assumptions. 

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. 

In tong term trials there will usually be an opportunity to check the assumptions which underlay the original design and sample size calculations. This may be particularly important if the trial specifications have been made on preliminary and/or uncertain information. An interim check conducted on the blinded data may reveal that overall response variances, event rates or survival experience are not as anticipated. A revised sample size may then be calculated using suitably modified assumptions... ... 

Most of the elements are contained within the sample size section according to the requirements set down in the CONSORT statement the only omissions seem to be specification of the statistical test on which the sample size calculation was based, the assumed standard deviation of the primary endpoint and the basis of that assumption. 

Clearly the main advantage of a non-parametric method is that it makes essentially no assumptions about the underlying distribution of the data. In contrast, the corresponding parametric method makes specific assumptions, for example, that the data are normally distributed. Does this matter Well, as mentioned earlier, the t-tests, even though in a strict sense they assume normality, are quite robust against departures from normality. In other words you have to be some way off normality for the p-values and associated confidence intervals to be become invalid, especially with the kinds of moderate to large sample sizes that we see in our trials. Most of the time in clinical studies, we are within those boundaries, particularly when we are also able to transform data to conform more closely to normality. 

In an anti-infective non-inferiority study it is expected that the true cure rates for both the test treatment and the active control will be 75 per cent. A has been chosen to be equal to 15 per cent. Using the usual approach with a one-sided 97.5 per cent confidence interval for the difference in cure rates a total of 176 patients per group will give 90 per cent power to demonstrate non-inferiority. Table 12.1 gives values for the sample size per group for 90 per cent power and for various departures from the assumptions. 

The sample size methodology above depends upon the assumption of proportional hazards and is based around the logrank test as the method of analysis. If this assumption is not appropriate and we do not expect proportional hazards then the accelerated failure time model may provide an alternative framework for the... 

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. 

A basic assumption in DSC kinetics is that heat flow relative to the instrumental baseline is proportional to the reaction rate. In the case of temperature scanning experiments the heat capacity of the sample contributes to the heat flow (endothermic), and this is compensated by the use of an appropriate baseline under the exo- or endothermic peak produced by the reaction. It is also assumed that the temperature gradient through the sample and the sample-reference temperature difference are small. Careful control of the sample size and shape, and the operating conditions are necessary in order to justify these assumptions. 

The analysis of variance technique for testing equality of means is a rather robust procedure. That is, when the assumption of normality and homogeneity of variances is slightly violated the F-test remains a good procedure to use. In the one-way model, for example, with an equal number of observations per column it has been exhibited that the F-test is not significantly effected. However, if the sample size varies across columns, then the validity of the F-test can be greatly affected. There are various techniques for testing the equality of k variances Oi, 02,..., crj,. We discuss... 

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