Sample size calculations

Similarly, the number and types of tests completed will influence the cost in ways that can be very complex. Recently, the author was involved in designing a proof-of-principle study intended to assess the ability of a dietary supplement to enhance weight loss among subjects instructed to follow a reduced energy diet. Sample size calculations were run for two scenarios, the first using change in body weight as the primary outcome variable, the... 

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. 

It is generally the case that when more complex statistical analysis strategies and designs are under consideration, standard sample size calculations are inadequate to cover them. In such circumstances simulation is often used to determine the t)rpe-I and type-II errors of the proposed studies for a given sample size. 

Also, the analysis plan should identify the statistical methods that will be used and how hypotheses will be tested (e.g., a p value cutoff or a confidence interval for the difference that excludes 0). And the plan should prespecify whether interim analyses are planned, indicate how issues of multiple testing will be addressed, and predefine any subgroup analyses that will be conducted. Finally, the analysis plan should include the results of power and sample size calculations. 

Shao Y, Tseng CH (2007) Sample size calculation with dependence adjustment for FDR-control in microarray studies. Stat Med 26 4219-4237. doi 10.1002/sim.2862... 

This null hypothesis is saying that the treatment difference/effect is consistent. If the p-value from this test is significant then we talk in terms of having a significant treatment-hy-centre (or a treatment x centre) interaction. Power and sample size calculations (see later chapter on this topic) will have focused... 

Does basing the sample size calculation on the per-protocol set and then increasing the sample size to allow for dropouts ensure that the per-protocol set will not be subject to bias ... 

Before moving on to discuss sample size calculations in more detail, it is worth noticing that the power curve does not come down to zero at a difference of 0.0,... 

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 ... 

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. 

It is generally true that sample size calculations are undertaken based on simple test procedures, such as the unpaired t-test or the test. In dealing with both continuous and binary data it is likely that the primary analysis will ultimately be based on adjusting for important baseline prognostic factors. Usually such analyses will give higher power than the simple alternatives. These more... 

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 ... 

It must be noted, however, that even if the sample size calculation gives enough power for the per-protocol analysis the potential for bias in that analysis still remains. 

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... ... 

A detailed statement of the basis of the sample size calculation should be included in the protocol and in the final report. This statement should contain the following ... 

The CONSORT statement (Moher et al. (2001)) sets down standards for the reporting of clinical trials and their recommendations in relation to the sample size calculation are in line with these points. 

Which statistical test was to be used for the comparison of the treatment groups in terms of the primary endpoint do you think This is a comparison between two independent groups in a parallel group trial and the primary endpoint is continuous so the sample size calculation will undoubtedly have been based on the two-sample t-test (although this is not specified). 

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. 

Although conventional p-values have no role to play in equivalence or noninferiority trials there is a p-value counterpart to the confidence intervals approach. The confidence interval methodology was developed by Westlake (1981) in the context of bioequivalence and Schuirmann (1987) developed a p-value approach that was mathematically connected to these confidence intervals, although much more difficult to understand It nonetheless provides a useful way of thinking, particularly when we come later to consider type I and type II errors in this context and also the sample size calculation. We will start by looking at equivalence and use A to denote the equivalence margins. 

We will focus our attention to the situation of non-inferiority. Within the testing framework the type I error in this case is as before, the false positive (rejecting the null hypothesis when it is true), which now translates into concluding noninferiority when the new treatment is in fact inferior. The type II error is the false negative (failing to reject the null hypothesis when it is false) and this translates into failing to conclude non-inferiority when the new treatment truly is non-inferior. The sample size calculations below relate to the evaluation of noninferiority when using either the confidence interval method or the alternative p-value approach recall these are mathematically the same. 

The sample size calculation requires pre-specification of the following quantities ... 

The power of a study where the primary endpoint is time-to-event depends not so much on the total patient numbers, but on the number of events. So a trial with 1000 patients with 100 deaths has the same power as a trial with only 200 patients, but with also 100 deaths. The sample size calculation for survival data is therefore done in two stages. Firstly, the required number of patients suffering events is... 

Example 13.4 Sample size calculation for survival data... 

In Chapter 8 we spoke about the calculation of sample size and in Section 8.5.3 revisiting this sample size calculation as the trial data accumulates. 

Do we need to revisit the sample size calculation at some interim stage in the trial ... 

The precise content is too detailed to give complete coverage here, but just to give an impression we will consider two areas the sample size calculation (item 7) and participant flow (item 13). 

The sample size calculation should be detailed in the trial publication, indicating the estimated outcomes in each of the treatment groups (and this will define, in particular, the clinically relevant difference to be detected), the type I error, the type II error or power and, for a continuous primary outcome variable in a parallel group trial, the within-group standard deviation of that measure. For time-to-event data details on clinically relevant difference would usually be specified in terms of the either the median event times or the proportions event-free at a certain time point. 

The clinical endpoint is a clinically meaningful measure of how patients feel, function or survive. Investigator-rated or self-assessed rating instruments are the most frequently used clinical endpoints. A primary endpoint is the main outcome that a study protocol is designed to evaluate. The statistical power and the sample size calculation of a particular trial are determined by the primary endpoint. Depending on the purpose of a study the primary endpoint can be... 

Chow, S-C., Shao, J., and Wang, H., 2003, Sample size calculations in clinical research, CRC/Taylor Francis. 

---