Randomized clinical trials

The equipoise principle corresponds to a state of uncertainty regarding the therapeutic results of a treatment, which justifies an RCT ⁵. RCTs with a control group are prospective studies that compare the outcomes of an intervention(s) with the best available alternative. In these studies, patient safety should always be a priority, so the possible benefits, harms and treatment alternatives for the patient’s condition should be explained. Although it may have limitations, it is considered the best alternative for evaluating the efficacy or safety of an intervention ^6,7. It is characterized by: a) an intervention that is compared with a control group that can be placebo or the usual treatment, b) randomized assignment of the interventions in the population to reduce possible confusion bias by obtaining homogeneous groups and the possibility of selection bias by avoiding foreseeing the group to which the patient is assigned c) the blinding of the treatment groups can be performed both for researchers, patients or analysts, which minimizes possible information biases ^6,7.

The RCTs are divided into four phases. Phase I seeks to determine possible toxic effects, absorption, distribution and metabolism of the drug in a group of 20-80 healthy people. Phase II is conducted in a diseased population to determine the safety and efficacy of the drug, based on biological markers and evaluating adverse reactions. Phase III is performed when there is evidence on the safety and efficacy of the intervention and additional information is sought on the safety and effectiveness of the drug in a larger number of participants. The intervention is compared with the usual therapy or placebo in a long-term follow-up in order to identify possible side effects. In phase IV, after the molecule has been approved for marketing, it is compared with other existing products in the general population; pharmacovigilance is also carried out in order to look for adverse events not identified in phase III due to their low incidence or long periods of occurrence ^3,6.

In this article we will focus on phase III and IV studies, which require a sample size calculation. Additionally, we will work on parallel RCTs characterized by a simultaneous follow-up of each group to which they were assigned ³.

Inferential statistics

Inferential statistics allows estimating the behavior of the entire population from the results obtained in a sample. This behavior is summarized in measures such as means, proportions or variances, which, if obtained for the whole population, would be called parameters ⁷. There are two alternatives: confidence intervals and hypothesis tests; with consistent results, the first seeks a range of values that, with a degree of confidence, contains the parameter of interest, while the second evaluates a statement about the parameter of interest, making the decision to reject it or not.

Since this paper presents the sample size calculation in parallel RCTs to evaluate parameter statements, we will describe the process of hypothesis testing. Initially, two hypotheses are proposed, the null hypothesis (H ₀ ) which is a statement about the parameter, and the alternative hypothesis (H _a ) which is its negation; almost always the alternative hypothesis ⁸, which is related to the research question is sought to be tested; at the end a decision is made to reject or not the H ₀ . Taking into account that this decision depends on the results obtained only from a sample, there is the probability of committing two errors, type I error or significance level (α) that occurs when rejecting H ₀ when it is true, and type II error (β), occurs when not rejecting H ₀ when it is false ⁹. The opposite of type I error is the confidence level (1-α) and corresponds to the probability of not rejecting H₀ when it is true, and the opposite of type II error (1-β), which is the power, is the probability of rejecting H ₀ when it is false ⁹. When performing a hypothesis test, the probability of committing the type I and II error is low, which implies that the confidence level and power have a high probability (typically: α=0.05 and β=0.1 or 0.2). In order to guarantee these values, it is necessary to calculate the sample size.

In order to make the decision to reject H ₀ , an operation is carried with the values from the sample (test statistic) and contrasted with the behavior that should occur if H ₀ were true. If the value found by the test statistic is unlikely, this is evidence that H ₀ is false and is rejected in favor of H _a , otherwise it is considered that there is not enough evidence to reject H ₀ . The probability reflecting the evidence “for” or “against” H ₀ is called the p-value ^10,11, and is equal to the probability, assuming the null hypothesis is true, of obtaining a value of the test statistic “...as extreme or more (in the appropriate direction of H _a ) than the value actually calculated” ^11,12; finally, H ₀ is rejected under the assumption of a value of p<α. Statistical significance, commonly evaluated by means of the p-value, does not account for clinical significance; we speak of statistical significance when the premise of a value of p<α is fulfilled, while clinical significance is defined by those results that improve the physical, mental and social functionality of the patient, which can lead to an improvement in the quality or quantity of life, depending on the context ¹⁴.

Types of hypotheses

There can be different research questions in an RCT that relate to four different ways of stating the H ₀ . The sample size calculation depends on the type of hypothesis to be tested; therefore, Table 1 presents their definitions along with an example.

Sample size

Generally, it is not possible to study the entire population, therefore, a specific sample size (n) is required to represent its behavior. As the sample size increases, the results approach that of the population, so that from a specific size, the results will not present large changes, making it unnecessary to continue collecting participants ¹⁵. Recruiting more subjects than necessary increases both the complexity of the logistical operation and the costs, and poses an ethical dilemma by unnecessarily assigning subjects to a treatment that has not proven its benefit. On the other hand, defining a very small sample size implies a high risk of the type II error mentioned above. The calculation of the sample size makes it possible to determine whether a study is feasible based on a priori assumptions, given the power, significance and background of previous studies addressing the same research question, taking into account the ethical considerations of subjecting people to an experiment ^13,16.

In addition, when conducting an RCT, the possibility arises of observing the results obtained as the sample is collected. This is called “interim analyses”, which should be planned from the beginning of the investigation during the preparation of the protocol. These additional analyses increase the possibility of type I and II errors, and for this reason, the sample size must be adjusted to maintain a level of confidence and overall power throughout the RCT. The above reflects the importance of the sample size calculation, therefore, this article presents how to calculate the sample size for RCTs, showing the expression from which it is obtained, and its application using the R programming language ¹⁷. Additionally, we present how to perform the adjustment for interim analysis together with an example.

Sample size calculation for a dichotomous outcome

As an example, we assume that two treatments are to be compared and the outcome of interest is the proportion of deaths. For all expressions below, we denote p _T and p _C as the proportion of deceased in the treatment and control group, respectively; ϵ is the expected difference between these two proportions (ϵ=p _T -p _C ), δ is the margin of tolerance or superiority defined by the researchers, and k is the ratio between the sample size of the treatment group and the control group (k=n _T /n _C ), i.e., n _T =kn _C . Finally, we denote α and β as the type I and II error, respectively; and z _(q) as the q percentile of the standard normal distribution function. In Table 2, we present the expressions to obtain n _C , and, in the inset, the code in the R programming language that creates a function for its implementation, along with an example where α=0.05, β =0.2 and k=1.

Table 2 Types of hypotheses with dichotomous outcome and their code in the R programming language. 

In all four hypotheses, the smaller the expected difference (ϵ) and the closer the proportions are to 0.5, the larger the sample size. When testing a non-inferiority hypothesis, if the higher the proportion of the event the greater the effectiveness, then δ<0; if the lower the proportion of the event the greater the effectiveness, then δ>0. When testing a superiority hypothesis, if the higher the proportion of the event the greater the effectiveness, then δ>0; if the lower the proportion of the outcome the greater the effectiveness, then δ<0. When testing an equivalence hypothesis, always δ>0.

Continuous outcome sample size calculation

As an example, we assume that two treatments are to be compared, and the outcome is systolic blood pressure in mmHg (SBP). For all the expressions presented below, we denote μ _T and μ _C as the mean SBP in the treatment and control group, respectively; ϵ is the expected difference between the two means ϵ=μ _T- μ _C and is the standard deviation of the two samples together. δ, k, α, β and z _(q) represent the same values as in the previous section. In Table 3, we present the expressions to obtain the code in the R programming language for implementation with an example where α=0.05, β=0.2 and k=1.

Table 3 Types of hypotheses by continuous outcome and code in R programming language. 

In all four hypotheses, higher s and lower ϵ require larger sample sizes. While testing a hypothesis of noninferiority, if the higher μ the greater the effectiveness, then δ<0; if the lower μ the greater the effectiveness, then δ>0. When testing a superiority hypothesis, if the higher μ the greater the effectiveness, then δ>0; if the lower μ the greater the effectiveness, then δ<0. When testing an equivalence hypothesis, always δ>0.

Interim Analysis

In an RCT, the study hypothesis can be tested sequentially as the sample is collected, giving the possibility of interrupting the collection if a clear benefit of the intervention is identified early. Depending on the number of evaluations (R) that are programmed, it is necessary to adjust the initial sample size to maintain the overall significance level of the study, and to establish the critical values on the distribution of the test statistic to reject or accept the null hypothesis in each evaluation. R evaluations are performed as n/R subjects accumulate, and the test statistic z _r (r=1,2,...,R) for a dichotomous outcome is equal to:

where n _r , p̂ _T,r and p̂ _C,r are the sample size per intervention group and the estimated proportions of the outcome at the r time of assessment of the treatment group and the control group, respectively.

For a continuous outcome, the test statistic is equal to:

where n_r, and are the sample size in each intervention group, and the estimated variances at the time of the r_th evaluation of the treatment group and the control group, respectively. x _Tj and x _Cj are the observed values of the outcome in each subject collected until time r.

We present four methods that allow the adjustment of the sample size depending on the number of programmed evaluations, the significance level and the power established in a hypothesis of equality. First, the Pocock method, in which the sample size is adjusted by multiplying the sample size initially obtained from the expressions presented in the previous section, by the coefficients included in Annex 1, depending on the number of evaluations and the significance and power levels established. Now, if |z _r |>CP _(r,α), the H ₀ is rejected and data collection is suspended, otherwise, the collection continues. The critical values CP _(r,α) are presented in Annex 1 for defined R and α. The second method is that of O’Brien and Fleming and the coefficients to perform the initial sample size adjustment are presented in Annex 2. In this approach H ₀ is rejected at each evaluation if |z _r |>COF _(r,α) √(R/r), otherwise it continues. The critical values COF _(r,α) are presented in Annex 2 according to the number of evaluations and significance level. The third method is that of Wang and Tsiatis, which includes a new parameter Δ ; coefficients for sample size adjustment for α=0.05 are included in Appendix 3. In this method H ₀ is rejected if otherwise it continues. The critical CWT _(r,α,Δ) values are presented in Annex 3 for α=0.05. The methods of Pocock and O’Brien and Fleming are particular cases of the method of Wang and Tsiatis when Δ=0.5 and Δ=0, respectively, therefore, the critical values for these values of Δ are obtained from Annexes 1 and 2.

Finally, we present the Inner Wedge method; in this method unlike the previous three, two critical values are proposed: if |zr| ≥b _r rejects H ₀ and collection is suspended, the conclusion is that a significant treatment effect was found, on the other side, if |zr| <a _r does not reject H ₀ and collection is suspended the conclusion is that no differences between treatment and control are going to be found, otherwise, collection continues.

The critical values a _r and b _r are equal to:

Annex 4 presents the values of Cw1 and Cw2 for an α=0.05 and a power of 0.8 and 0.9, and the columns coef.fit include the coefficients, by which the original sample must be multiplied to perform the R evaluations.

As an example, consider that you want to compare drug A vs. placebo and the outcome is the proportion of deaths at the end of follow-up. In a hypothesis of equality, assuming α=0.05, β=0.1, p _T =0.1 and p _C =0.2 (i.e. ϵ=0.1), for two groups of the same size, k=1, the required sample size in each group is 263 subjects. If we plan to perform R=5 evaluations, the adjusted sample size by Pocock’s method is 263 x 1.207=318 for each group and the critical value in each evaluation is CP _0.05,r=2.413. The adjusted sample size by the method of O'Brien and Fleming is 263 x 1,026=270 for each group and the critical values for each evaluation are COF _(r,0,05) =4.562; 3.226; 2.634; 2.281 and 2.040. The adjusted sample size with the method of Wang and Tsiatis for each group, with Δ=0.25, would be 263 x 1.066=281, and the critical values at each evaluation would be CWT _{(r;0.05;0.25)} =3.194; 2.686; 2.427; 2.259 and 2.136. Finally, the adjusted sample size with the Inner Wedge method for each group would be, with Δ=0.25, 263 x 1.199=316, and the critical values for each evaluation are a _r =0;0.388; 1.072; 1.613 and 2.073 and b _r =3.1; 2.607; 2.355; 2.192 and 2.073.