In sequential analysis, the final sample size is not known at the beginning of the study. On average, sequential analysis will lead to a smaller average sample size than that in an equivalently powered study with a fixed-sample-size design. This is a major advantage to sequential analysis and is a reason that it should be given consideration when one is planning and analyzing a small clinical trial.
For example, take the case study of sickle cell disease introduced in Chapter 1 and consider the analysis of the clinical design problem introduced in Box as an example of sequential analysis Box Data from a clinical trial accumulate gradually over a period of time that can extend to months or even years. Thus, results for patients recruited early in the study are available for interpretation while patients are still being recruited and allocated to treatment. This feature allows the emerging evidence to be used to decide when to stop the study.
In particular, it may be desirable to stop the study if a clear treatment difference is apparent, thereby avoiding the allocation of further patients to the less successful therapy. Investigators may also want to stop a study that no longer has much chance of demonstrating a treatment difference Whitehead, , For example, consider the analysis of an intervention countermeasure to prevent the loss of bone mineral density in sequentially treated groups of astronauts resulting from their exposure to microgravity during space travel Figure The performance index is the bone mineral density in grams per square centimeter of the calcaneus.
The confidence intervals for p and q are obtained after each space mission, that is, for p, P 1 , P 2 , and for q, q 1 , q 2. Unacceptable performance indices occur with less than a 75 percent success rate or more than a 5 percent failure rate. As the number of performance indices increases, level 1 performance crite-. Fifty percent of individuals living with sickle cell disease die before age The most common complications include stroke, renal failure, and chronic severe pain.
Patients who have a stroke are predisposed to having another one. Mixed donor and host stem cell chimerism e. Only 20 percent of donor RBC production and 80 percent of recipient RBC production is required to cure the abnormality. Conditioning of the recipient is required for the transplanted bone marrow stem cells to become established. The degree of HLA human leukocyte antigen mismatch as well as the sensitization state i. In patients who have an HLA-identical donor and who have not been heavily transfused, centigrays cGy of total body irradiation TBI is sufficient to establish donor engraftment establish a cure.
This dose of irradiation has been shown to be well tolerated. In heavily transfused recipients who are HLA mismatched, more conditioning will probably be required. The optimal dose of TBI for this cohort has not been established. The focus of this study is to establish the optimum dose of TBI to achieve 20 percent donor cells chimerism in patients enrolled in the protocol. How many patients must be enrolled per cohort to obtain durable bone marrow stem cell establishment engraftment?
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Patients are monitored monthly for the level of donor chimerism. When can TBI dose escalation be implemented? How many patients are required per group before an increase in dose can be made? One traditional approach to this problem is to identify an acceptable engraftment rate and to then identify the number of subjects required to ensure that the confidence interval for the true proportion is sufficiently narrow to be protective of human health.
For example, if the desired engraftment rate is 95 percent, 19 subjects will provide a 95 percent confidence interval with a width of 10 percent i. If for a particular application, this interval is too wide, a width of 5 percent can be obtained with 73 subjects 0. On the basis of these results, should 73 subjects be required for each TBI dose group?
Is a total of patients really needed for all dose groups? The answer is that a much smaller total number of patients is required by invoking a simple sequential testing strategy. For example, assume that the study begins with three patients in the lowest-dose group and it is observed that none of the patients are cured. On the basis of a binomial distribution and by use of a target engraftment proportion of 0.
Similarly, the cumulative probability of one or fewer cures is less than 15 percent. As such, after only three patients are tested, considerable information regarding whether the true cure rate is 95 percent or more is already available. Following this simple sequential strategy, one would test each dose beginning with the lowest dose with a small number of patients e. In the current example, one would clearly increase the dose if zero of three patients was cured and would most likely increase the dose to the next level even if one or two patients were cured.
If, in this example, all three patients engrafted, one would then test either 19 or 73 patients depending on the desired width of the confidence interval and determine a confidence interval for the true engraftment rate with the desired level of precision. If the upper confidence limit is less than the targeted engraftment rate, then one would proceed to the next highest TBI dose level and repeat the test.
Indeterminate I is equal to a moderate loss of 1 to 2 percent from that at the baseline.
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F is equal to the severe loss of 2 percent or more from that at the baseline Feiveson, See Box for an alternate design discussion of this case study. The use of study stopping cessation rules that are based on successive examinations of accumulating data may cause difficulties because of the need to reconcile such stopping rules with the standard approach to statistical analysis used for the analysis of data from most clinical trials.
In comparison, if the data are examined in a way that might lead to early cessation of the study or to some other change of design, then a fixed-sample analysis will not be valid.
The lack of validity is a matter of degree: if early cessation or a change of design is an extremely remote possibility, then fixed-sample methods will be approximately valid Whitehead, , For example, in a randomized clinical trial for investigation of the effect of a selenium nutritional supplement on the prevention of skin cancer, it is determined that plasma selenium levels are not rising as expected in some patients in the supplemented group, indicating a possible noncompliance problem.
In this case, the failure of some subjects to receive the prescribed amount of selenium supplement would have led to a loss of power to detect a significant benefit, if one was present. One could then initiate a prestudy. FIGURE Parameters for a clinical trial with a sequential design for prevention of loss of bone mineral density in astronauts. Group sample sizes available for clinical study. Establishment of repeated confidence intervals for a clinical intervention for prevention of loss of bone mineral density for determination of the success S or failure F of the intervention.
As an illustration of sequential testing in small clinical studies, consider the innovative approach to forensic drug testing proposed by Hedayat, Izenman, and Zhang Suppose that N units such as pills or tablets or squares of lysergic acid diethylamide LSD are obtained during an arrest and one would like to determine the minimal number that would have to be screened to state with 95 percent confidence that at least N 1 of the total N samples will be positive.
To solve the problem, define m as the expected number of negative units in the initial random sample of n units and X as the observed number of negative units in a sample of size n. Typically, the forensic scientist assumes that m is equal to 0, n samples are collected, and the actual number of negative samples X is determined.
Next, define k as the minimum number of positive drug samples that are needed to achieve a conviction in the case. The question is: what is the smallest sample size n needed?
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Hedayat and co-workers showed that the problem can be described in terms of the inequality. This is a cumulative probability of the hypergeometric distribution, that can be expressed as. Another reason for early examination of study results is to check the assumptions made when designing the trial. For example, in an experiment where the primary response variable is quantitative, the sample size is often set assuming this variable to be normally distributed with a certain variance.
For binary response data, sample size calculations rely on an assumed value for the background incidence rate; for time-to-event data when individuals enter the trial at staggered intervals, an estimate of the subject accrual rate is important in determining the appropriate accrual period. An early interim. For example, assume that the total number of units under investigation is and suppose that one wants to claim with 95 percent confidence that the number of positive units, N 1 , is at least If one assumes that there will be no negative units in the initial sample i.
The investigators draw a random sample of 25, and if no negative units are found, the investigators can conclude with 95 percent confidence that the total number of positive units N 1 is greater than Note that if one actually observes X to be equal to 1 negative unit, one then can determine what value of N 1 is feasible or recompute n. For example, say that one observes X is equal to 2 negative units among 25 initial samples. With 95 percent confidence one can claim that k equal to positive units will be found.
A useful example in clinical trials is the comparison of a new drug with a standard drug for the treatment of a rare disease. For example, it may be known that the rate of response to an existing drug is 80 percent; however, the drug has serious side effects. A new drug without the side effect profile of the old drug has been developed, but it is not known whether it is equally efficacious. Power computations revealed that subjects are required to document that the response rate is at least 90 percent with 95 percent confidence i. Unfortunately, subjects are not available.
Using the strategy developed by Hedayat and colleagues , one can examine 25 patients, and if they all respond, then one can conclude with 95 percent confidence that the total number of responders is at least among the patients that the investigators would have liked to test. There are numerous applications of this type of sequential testing strategy in small clinical trials. Sequential methods typically lead to savings in sample size, time, and cost compared with those for standard fixed-sample procedures Box However, continuous monitoring is not always practical.
Hierarchical models can be quite useful in the context of small clinical trials in two regards. First, hierarchical models provide a natural framework for combining information from a series of small clinical trials conducted. In the case where the data are complete, in which the same response measure is available for each individual, hierarchical models provide a more rigorous solution than meta-analysis, in that there is no reason to use effect magnitudes as the unit of observation.
Note, however, that a price must be paid i.
Second, hierarchical models also provide a foundation for analysis of longitudinal studies, which are necessary for increasing the power of research involving small clinical trials. By repeatedly obtaining data for the same subject over time as part of a study of a single treatment or a crossover study, the total number of subjects required in the trial is reduced. The reduction in the sample size number is proportional to the degree of independence of the repeated measurements.
A common theme in medical research is two-stage sampling, that is, sampling of responses within experimental units e. For example, in prospective longitudinal studies patients are repeatedly sampled and assessed in terms of a variety of endpoints such as mental and physical levels of functioning or in terms of the response of one or more biological systems to one or more forms of treatment.
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These patients are in turn sampled from a population, often stratified on the basis of treatment delivery, for example, in a clinic, in a hospital, or during space missions. Like all biological and behavioral characteristics, the outcome measures exhibit individual differences.
ragarhacossi.cf Investigators should be interested in not only the mean response pattern but also the distribution of these response patterns e. One can then address the number or proportion of patients who are functioning more or less positively at a specific rate. One can then describe the treatment-outcome relationship not as a fixed law but as a family of laws, the parameters of which describe the individual biobehavioral tendencies of the subjects in the population Bock, This view of biological and behavioral research may lead to Bayesian methods of data analysis.
The relevant distributions exist objectively and can be investigated empirically. In medical research, a typical example of two-stage sampling is the longitudinal clinical trial, in which patients are randomly assigned to different treatments and are repeatedly evaluated over the course of the study. Despite recent advances in statistical methods for longitudinal research, the cost of medical research is not always commensurate with the quality of the analyses.
Reports of such studies often consist of little more than an end-.