Sample size estimation

An important part of study design is the determination of the required sample size. Before starting, we should note that we prefer the term 

It is also appropriate to note that not all clinical trials utilize formal sample size estimation methods. In many instances (for example, FTIH studies) the sample size is determined on the basis of logistical constraints and the size of the study thought to be necessary to gather sufficient evidence (for example, pharmacokinetic profiles) to rule out unwanted effects. However, when the objective of the clinical trial (for example, a superiority trial) is to claim that a true treatment effect exists while at the same time limiting the probability of committing type I or II errors (a and P), there are computational methods used to estimate the required sample size. The use of formal sample size estimation is required in therapeutic confirmatory trials, this book s major focus, and strongly suggested in therapeutic exploratory trials. 

1 Sample size for continuous outcomes in superiority trials 

Consider the simple case of a superiority trial of an investigational drug (the test treatment) being compared with placebo with respect to a continuous outcome (for example, change from baseline SBP). The null hypothesis typically tested in such a trial and its complementary alternate hypothesis are  

Chapter 1 2 Additional statistical considerations in clinical trials 

---

Sample quantity to estimate moisture for specific material is influenced to various levels of significance by properties such as particle-size range as well as relative amounts or moisture distributed among denoted forms of retention. Practical sample size estimates require background knowledge of parameters derived from experience for specific materials. More detailed examination of moisture-sampling aspects is provided in reference texts (Pitard). 

One algorithm for blindly approximating physical states has already been proposed [36], although the method requires the number of states to be input. In work to be reported soon, Zhang and Zuckerman developed a simple procedure for approximating physical states that does not require input of the number of states. In several systems, moreover, it was found that sample-size estimation is relatively insensitive to the precise state definitions (providing they are reasonably physical, in terms of the timescale discussion above). The authors are therefore optimistic that a "benchmark" blind, automated method for sample-size characterization will be available before long. 

The size required of the sample to identify a meaningful economic difference is frequently problematic. Often those setting up clinical trials focus on the primary clinical question when developing sample-size estimates. They fail to consider the fact that the sample required to address the economic questions posed in the trial may differ from that needed for the primary clinical question. In some cases the sample size required for the economic analysis is smaller than that required to address the clinical question. More often, however, the opposite is true, in that the variances in cost and patient preference data are larger than those for clinical data. Then one needs to confront the question of whether it is either ethical... 

Have sample-size estimates been conducted appropriately and is the study powered as needed ... 

Sample-size estimation therefore has an important ethical component. There are ethical issues involved in recruiting both too few and too many subjects (Matthews, 2006). Recruiting too few subjects means that the study may be underpowered and unable to detect a treatment effect of interest that actually exists. Such a design is scientifically inadequate to answer the research question of interest (i.e., to address the primary objective of the trial). It is also unethical. Subjects may have taken part in a study that did not have a chance of detecting a treatment effect that may have existed, and thus their expectation that participation may add to the knowledge base about the investigational drug was violated. 

Sample-size estimation therefore takes on a special significance in clinical trials. As noted in Section 9.1, this process of estimation does not produce the right answer, so it is not possible to specify precisely what constitutes too few or too many subjects. However, it is imperative to estimate a reasonable sample size based on the best evidence that is available at the time and with full knowledge of the implications of this estimate. 

Several variables need to be considered in the process of sample-size estimation. The values of these variables in any given case can be chosen by the sponsor based on several considerations. Some terms that will be useful for present discussions are ... 

Choosing the Variables Needed for Sample-Size Estimation... 

As noted in Section 9.1, several variables are needed for sample-size estimation, and the researcher can choose the values to be used in the formula that will yield the sample size, N. These are a, p, the estimated treatment effect, and its variance. 

Sample-size estimation can be performed for any study design. In each case, the respective formula will be used to estimate the sample size required (see Chow et al., 2003). For the formula used in the type of study design that we are using as our ongoing example, each of the variables we have discussed will have certain influences on the sample size, N, that will be given by the formula. These influences, i.e., their relationships with N given that all of the others remain the same, can be summarized as follows ... 

As we have discussed, when conducting a sample-size estimation, the researcher has to choose values for a and p and has to come up with a standardized treatment effect, which is in turn the result of finding the best possible estimates of a clinically significant difference and its variation. What are the influences that lead the sponsor to choose certain values for a, p, and the standardized treatment effect ... 

While either action, i.e., reducing a or p, will increase the value of N given by the sample-size estimation and therefore result in additional cost to the sponsor, the sponsor may well decide that, in the overall balancing act of estimating sample size, there are good reasons to do this in cases such as these examples. 

A sample-size estimation must be based on a specific objective in a clinical trial s study protocol. By the time sample-size estimation becomes particularly meaningful, i.e., in later-stage clinical trials designed to demonstrate efficacy, it is a very good idea to have a single objective (the primary objective) and a single... 

This approach, however, raises issues of multiplicity (see Section 7.10). Accordingly, lower p-values may be required to be able to declare a result as statistically significant. This means that an adjustment to the sample-size estimation formula is appropriate, with the precise nature of the adjustment being related to the number of outcomes to be tested. This adjustment raises the magnitude of the estimated sample size (Machin and Campbell, 2005). 

It is useful to keep several other issues in mind when conducting sample-size estimations, including the following ... 

Safety analyses are not typically prespecified in the study protocol and/or the study analysis plan. Studies are typically powered on efficacy outcomes (the primary objective in therapeutic confirmatory trials see Chapter 9), and the sample size that results from this sample-size estimation may be considerably smaller than would be needed for a thorough investigation of safety data. 

By the time a therapeutic confirmatory trial is appropriate, it should be possible to state a single primary objective (or perhaps two if the sponsor really feels that this is appropriate) that is clinically relevant and biologically plausible. One primary objective also means that sample-size estimation can be based on that objective and the associated estimated treatment effect of interest (recall the discussions in Chapter 9). 

Sample-size estimation. A study design requires sufficient subjects but not an unnecessarily large subject sample. 

Consider two examples from previous chapters that illustrate this. In Chapter 9, discussions of sample-size estimation emphasized that the process is indeed one of estimation rather than pure calculation. A calculation is certainly executed, but the values that are placed into the appropriate formula are chosen by the sponsor. On each occasion, the sponsor must consider the influences of the choices that are made and make the most appropriate decision in the specific context of that trial. In Chapter 11, equivalence and noninferiority designs were discussed. In addition to the calculations that are involved using the data collected in a trial, equivalence or noninferiority margins must be established before the trial commences. Their choice is a clinical choice, not a statistical choice, and subjectivity is necessarily involved in this choice. Thus, the discipline of Statistics certainly involves using informed judgments. Statistics really is an art as well as a science, a sentiment well expressed by Katz (2001) as cited in Chapter 13. 

General statistical methods such as sample size estimation, determination of practical significance and one-sided testing can be applied to the paired f-test in the same manner that we have already seen for the two-sample f-test. 

The book is aimed at those who have to use statistics, but have no ambition to become statisticians per se. It avoids getting bogged down in calculation methods and focuses instead on crucial issues that surround data generation and analysis (sample size estimation, interpretation of statistical results, the hazards of multiple testing, potential abuses, etc.). In this day of statistical packages, it is the latter that cause the real problems, not the number-crunching. 

---