Sample size determination

In both of these examples, the task of the statistician has widened considerably beyond that of simply determining the sample size. Of course, some of the points the statistician made to the physician could have been appreciated without his help. No one profession has a monopoly of logic. The point is that these sorts of matters are more likely to be discovered and explored if statistician and physician work together on all aspects of trial design and the statistician s responsibility is not narrowly defined. 

Notwithstanding the above, sample size determination is an important part of trial design. One of the simplest techniques is frequently overlooked. If many successful trials of a given type have been run in a given indication, then the typical size of such a trial gives a good indication of what is likely to be successful. More formal methods are available and a common approach is sketched below. 

the statistician and physician together need to identify a particular outcome variable Including the particular time-point (or combination) which will be used. Next the statistician must identify an appropriate statistical test. A size of this test, or nominal significance level, (see Chapter 4) must also be identified. It will also be important to have obtained from previous trials, or some appropriate source, an estimate (sometimes no more than a guesstimate) of the likely variability of the particular measure chosen. The statistician also needs to establish a working alternative hypothesis. This is not a 

Bonus A means of rewarding employees for that serendipity that outweighs incompetence. 

---

Estimate efficacy information necessary to make sample size determinations for Phase II studies and establish adequate duration of treatment. 

Hahn, G.J. (1979b), Sample Size Determines Precision, CHEMTECH, 9, 294-295. 

Many sample size determinations take the form... 

Detection level in the range of 1 pg/L for a 25-mL sample size determined by purge and trap-GC-FID method. 

Tan, M., Fang, H. B., Tian, G. L., and Houghton, P. J. (2003). Experimenal design and sample size determination for testing synergism in drug combination studies based on uniform measures. Statistics in Medicine, 22, 2091-2100. 

In Sections 2 to 4, we review the technology of synthetic oligonucleotide microarrays and describe some of the popular statistical methods that are used to discover genes with differential expression in simple comparative experiments. A novel Bayesian procedure is introduced in Section 5 to analyze differential expression that addresses some of the limitations of current procedures. We proceed, in Section 6, by discussing the issue of sample size and describe two approaches to sample size determination in screening experiments with microarrays. The first approach is based on the concept of reproducibility, and the second approach uses a Bayesian decision-theoretic criterion to trade off information gain and experimental costs. We conclude, in Section 7, with a discussion of some of the open problems in the design and analysis of microarray experiments that need further research. 

In observational studies, the main design issue is the choice of the sample size, whereas sample size determination and treatment choice are the primary design issues in factorial experiments. Sample size determination depends on the analytical method used to identify the genes with different expression and the optimality requirements selected for the study. These topics are examined in the next two sections. 

Sample size determination based on reproducibility does not take the experimental costs into account. In this section, we introduce a formal decision-theoretic... 

In the software badge, we account for model uncertainty by averaging the results of the posterior inference conditional on the Gamma and lognormal distributions for the gene expression data. As a parallel with the sample size determination when the inference process is based on model averaging, we therefore introduce the Average Entropy, denoted by Enta(-), and defined as... 

The importance of this result is that it leads to an overall objective criterion for sample size determination that averages criteria based on specific model assumptions. Thus it provides a solution that is robust to model uncertainty. Closed-form calculations of (8) are intractable, so we have developed numerical approximations to the conditional entropies Ent(6k n, yk, MLk) and Ent(9k n, yk, MGk). The computations of the expected Bayes risk are performed via stochastic simulations and the exact objective function is estimated by curve fitting as suggested by Miiller and Parmigiani (1995). These details are available on request from the authors. 

One of the most important decisions that is left to the analyst when operating a liquid chromatograph is the choice of detector sensitivity. In some instruments the output from the sensor is monitored continuously over its entire dynamic range and so sensitivity is not an optional experimental parameter. Nevertheless, in this case, the sample size determines the concentration range over which the eluted solutes are monitored and thus an optimum sample size must be chosen. The detector should never be operated at its maximum sensitivity unless such conditions are enjoined by limited sample size or column geometry. Provided that there is adequate sample available, and the sample concentration when eluted is within the linear dynamic range of the detector, the maximum sample size that the column can tolerate should be used. This ensures that the detector noise is always minimal... 

We have already seen through a number of examples the interplay between sample size, variability and the performance of the statistical procedures employed to analyze the data. The sample size determines the amount of information that will be available at the end of the trial. Therefore, the determination of an adequate sample size is one of the most important aspects of the trial design. A trial accumulating inadequate amount of information is hopelessly flawed, as it will not enable the researcher to answer the questions the trial is intended to answer. 

Lachin JM. Introduction to sample size determination and power analysis for clinical trials. Controlled Clinical Trials 1981 2 93-113. 

If the lot size is >200 units, then n = 20. However, there are many variations in sample size determination, and it is important to consult the USP, European Pharmacopeia (EP), or Japanese Pharmocopeia (JP) for detailed directions. 

As a final result the producer issues a material certified for a number of parameters, based on the minimum sample size determined in the homogeneity study. This information must be given to the end user in the form of a certificate. The certificate or the accompanying report must also state the uncertainty and the way it is calculated. 

Hauschke, D., M. Kieser, E. Diletti, and M. Burke. 1999. Sample size determination for proving equivalence based on the ratio of two means for normally distributed data. Statistics Med. 15 93-105. 

K. Dobbin, R. Simon, Sample size determination in microarray experiments for class comparison and prognostic classification, Biostatistics 2005, 6, 27-38. 

Hale, W.E., 1972. Sample size determination for the log-normal distribution. Atmos. Environ. 6, 419 22. 

In what follows we shall assume that the sample size is going to be determined as a function of the other factors. We shall take the example of a two-arm parallel-group trial comparing an active treatment with a placebo for which the outcome measure of interest is continuous and will be assumed to be Normally distributed. It is assumed that analysis will take place using a frequentist approach and via the two independent-samples t-test. A formula for sample size determination will be presented. No attempt will be made to derive it. Instead we shall show that it behaves in an intuitively reasonable manner. 

We shall present an approximate formula for sample size determination. An exact formula introduces complications which need not concern us. In discussing the sample size requirements we shall use the following conventions ... 

In practice there are, of course, many different formulae for sample size determination. If the trial is not a simple parallel-group trial, if there are more than two treatments, if the outcomes are not continuous (for example, binary outcomes, or length of survival... 

A helpful tutorial on sample size issues is the paper by Steven Julious in Statistics in Medicine (Julious, 2004) a classic text is that of Desu and Raghavarao (1990). Nowadays, the use of specialist software for sample size determination such as NQuery, PASS or Power and Precision is common. 

A very unsatisfactory feature of conventional approaches to sample size calculation is that there is no mention of cost. This means that for any two quite different indications with the same effect size, that is to say the same ratio of clinically relevant difference to standard deviation, the sample size would be the same whatever the cost or difficulty of recruiting and treating patients. This is clearly illogical and trialists probably manage this issue informally by manipulating the clinically relevant difference in the way discussed in Section 13.2.3. Clearly, it would be better to include the cost explicitly, and this suggests decision-analytic approaches to sample size determination. There are various Bayesian suggestions and these will be discussed in the next section. 

An appropriate approach to sample size determination is to calculate assurance... 

Kieser M, Rohmel J, Friede T (2004) Power and sample size determination when assessing the clinical relevance of trial results by responder analyses . Statistics in Medicine 23 3287-3305. Lindley DV (1957) A statistical paradox. Biometrika 44 187 192. 

---