Sample Size Calculations Based On Two One-side Tests With Quantitative Variable

Background and objectThe clinical equivalence trial is to assess the equivalence of the clinical efficacy indexes. It heads from the bioequivalence trial. They both adopt two one-sided tests. The statistical inference of the traditional hypothesis tests of assessing the effect of treatment between the two groups is often limited to infer whether the differences between the two groups was statistically significant, which cannot meet the clinical research of equivalence evaluation on the actual work requirements. Because the two one-sided tests are different from the traditional two-sided tests and the one-sided tests, the principles and methods of their sample size estimation are also different. In recent years, there are many scholars making research and discussion on sample size estimation method of clinical equivalence test, but still have some problems need to be further clear and perfect.The Bland-Altman agreement method is a method that is currently hot. The method is always consistency evaluation methods usually use confidence interval method for statistical inference. In theory, it can prove that its essential is equivalent to two one-sided tests. The Bland-Altman agreement method is proposed by Altman DG and Bland JM in 1983. It is an intuitive and simple method using a simple difference. However, the method based on the illustrated is too subjective. Altman DG and Bland JM further improved the method in 1986. In order to improve the original subjective graphic method, they established a Bland-Altman agreement method system with 95% limits of agreement (LOAs) and Bland -Altman plot. This method system has been rapid promotion and widely recognized in the medical profession, and also has a good application in other areas. Although Bland-Altman agreement method is finding wider and wider application in the field of medicine and health care year by year, the methodology research is still rare. The problem of the unreasonable of application is quite serious, which is badly in need of statistical researchers and clinical users together to solve together. Among them, about the method of sample size estimation problem has not been very good to solve.The statistical inferences of the equivalence trial and the Bland-Altman agreement assessment are both based the principle of two one-sided tests, so their inference methods in the hypothesis test and the confidence interval have certain features in common. Therefore, there must be a certain commonality in terms of sample size estimation for the equivalence trial and the Bland-Altman agreement assessment. This study aims to further clarify and clarify sample size estimation problem of the equivalence test on the basis of previous studies, and make discussion and analysis for the sample size estimation method of Bland-Altman agreement assessment which reference to the principles and methods of the equivalence trial. This study provides methodological support for promoting the correct application of two methods.In this paper, there are five chapters. In Chapter 1, we introduce the equivalence evaluation with the hypothesis test and the confidence interval estimation method in parallel group design to evaluate the equivalence for the treatment effect between two sets of quantitative data mean. And we parse the relationship between the level of type I error and confidence level, which provides the theoretical support for the next chapter of the sample size estimation method for equivalence test. In Chapter 2, according to the statistical inference principle and the statistical distribution theory, we decompose the type II error and explore the sample size estimation of equivalence clinical trials which the primary variable follow normal distribution. Based on the formula, the sample sizes were estimated under different parameter settings. Monte-Carlo simulations were performed to obtain the corresponding powers respectively. Through example analysis, we make suggestions directly to the sample size evaluation for the equivalence test and the non-inferiority test in the practical application. Chapter 3 focuses on the method introduction of Bland-Altman agreement assessment in a single measurement situation, including the statistic calculation, the graph drawing and the parameter estimation. We also analyze the different data behaviors which may arise and give the specific methods for determining the data behavior before using the Bland-Altman agreement assessment. Meanwhile, we present strategies and suggestion when the data behavior is not well. In addition to this, we present the limits of agreement and its confidence interval of the Bland-Altman method, which provides the theoretical support for the next chapter of the sample size estimation method for the Bland-Altman method. In Chapter 4, according to the Bland-Altman method, the conclusion of agreement is made based on the width of the confidence interval for LOAs (limits of agreement) in comparison to predefined clinical agreement limit. Based on the theory of statistical inference, the formulae of sample size estimation are derived, which depended on the pre-determined level of α,β,the mean and the standard deviation of differences between two measurements, and the predefined limits. With this new method, the sample sizes are calculated under different parameter settings which occur frequently in method comparison studies, and Monte-Carlo simulation is used to obtain the corresponding powers. Bland has given the sample size for a study of agreement between two methods of measurement which were available from his website. We compare the method proposed by Bland and our method by Monte-Carlo simulation to explore the correctness and the practicability of the two methods. Finally, the practicability of the sample size evaluation for Bland-Altman agreement assessment is illustrated with a clinical instance. In Chapter, we summary the main innovative points and shortcomings in this paper and propose the problems that remain to be further discussed.MethodsFirst, according to the statistical inference principle of confidence interval estimation of the mean differences for the equivalence trial, the formula of sample size estimation are derived, the specific process are as follows:Based on the statistical inference of equivalence trial, if the total type Ⅰ error is α, then the type Ⅰ error of each one-sided test is also α. The type Ⅱ error (β) can be separated into two parts. One is the first type Ⅱ error (βU) of the upper limit and the other is the second type Ⅱ error(βL) of the lower limit. We can get two equations from decomposition sketch, and derive formula of two type Ⅱ errors. According to the statistical distribution theory, it is best to calculate the type Ⅱ error (β) under the assumption of a non-central t-distribution. The sample sizes cannot be calculated by iterative operation. Monte-Carlo simulation was used to examine the validity of the proposed formulae for estimating sample size by calculating empirical powers. Considering the possibilities of practical application, simulation data were generated on the basis of normal distribution by considering typical situations under different parameter settings:Define type Ⅱ errors β, △/σ and δ/σ (△ is the limit of equivalence, σ is the mean differences of two populations), and α=0.05. By the formula, the sample sizes were calculated under different settings, and Monte-Carlo simulation was used to obtain the corresponding powers. For a specific sample size, the simulation steps are as follows:First, we define the means of two populations as μA、μB, the standard deviations are both a and the pre-defined clinical agreement limit (△). And then we produce two sets of normal distribution random numbers with the same sample size and estimate the 90% confidence interval of the mean difference of two samples. If they lie within the pre-defined clinical equivalence limits[-△,△], then we draw a conclusion that the two methods are equivalent. We repeat the procedure above 10,000 times and compute the times (t) that draw equivalence conclusion. The value of t/10000×100% is the achieved power.According to the Bland-Altman method, the conclusion of agreement is made based on the width of the confidence interval for LOAs (limits of agreement) in comparison to predefined clinical agreement limit. Based on the theory of statistical inference, the formulae of sample size estimation are derived, which depended on the pre-determined level of α, β,the mean and the standard deviation of differences between two measurements, and the predefined limits. With this new method, the sample sizes are calculated under different parameter settings which occur frequently in method comparison studies, and Monte-Carlo simulation is used to obtain the corresponding powers. Because the Bland-Altman agreement assessment and the equivalence trial are both based on the two one-sided tests, their methods of sample size estimation are similar. For a specific sample size, the simulation steps are as follows:First, we define the different standard limits as δd/σd (σd is the mean of the differences between two populations and σd is the standard deviations of the difference), which is from 0 to 0.9 step by 0.1, the standard agreement limits as △d/σd(△d is the clinical agreement limits) from 2.0 to 3.0 step by 0.1. And then we estimate the sample size with the formula, and calculate the 95% confidence interval for the 95% LOAs. Second, we produce two sets of normal distribution random numbers with the same sample size and estimate the 95%c onfidence interval of 95% LOAs. If they lie within the pre-defined clinical agreement limits [-△d,△d], then we draw a conclusion that the two methods are agree, otherwise they disagree. We repeat the procedure above 10,000 times and compute the times (t) that draw agreement conclusion. The value of t/10000×100% is the achieved power.Bland has given the sample size for a study of agreement between two methods of measurement which were available from his website. In the 1986 Lancet paper they gave a formula for the confidence interval for the 95% limits of agreement. In order to compare the sample size estimation proposed by Bland and our method in this article, we use Monte-Carlo to simulate sample sizes and powers under different parameters, the specific steps as follow:we set α=0.05,β=0.20,δd=-0.4～0.4, σd=1,△d=2.7, and pre-specified power=80%. And then we produce a set of normal distribution random numbers and estimate the 95% confidence interval of 95% LOAs. If they lie within the pre-defined clinical agreement limits [-△d,△d], then we draw a conclusion that the two methods agree. We repeat the procedure above 10,000 times and compute the times (t) that draw agreement conclusion. The value of t/10000×100% is the achieved powerResultsThrough the analysis of I type error and confidence level of two one-sided test, if the total type I error is a. then the type I error of each one-sided test is also a and the confidence interval estimation based on 100(1-2α)% confidence level.The results of Monte-Carlo simulation of the equivalence trial and the Bland-Altman agreement assessment showed that the achieved powers could coincide with the pre-determined level of powers, thus validating the method.The results of the sample sizes and powers of B-A method and our new method under different parameter settings shows that the sample sizes calculated by Bland’s method is still fewer than the sample sizes by our new method. As the difference mean is larger, so the difference of the sample sizes between two methods is larger. With the method proposed by Bland, the sample size is calculated without considering the power of the statistical procedure, and so the probability of obtaining the required width is only 0.50. With the new method, the achieved power is generally close to the pre-specified power. The sample size of the Bland-Altman method proposed by Bland is determined on the basis of the expected width of a confidence interval. It fails to explicitly consider the probability of achieving the desired interval width and may thus provide sample sizes that are too small to have enough power. However, the new method is more appropriate, because it can ensure an adequate probability of achieving the desired precision.ConclusionsBased on the principle of the two one-sided of the equivalence test, we analyze the relationship between the level of type I error and confidence level.-Though the decomposition of the total II error, we further clarify the sample size estimation of equivalence trial. Using the principle of two one-sided test of the equivalence test, we can derive the sample size estimation formula for Bland-Altman agreement assessment. Because the methods are both involved the non-central t-distribution, the sample sizes should be calculated by iterative operation. It is relatively cumbersome, but it is no problem with the aid of computer operation. For the convenience of application, we present the reference tables for clinical researchers to estimate the sample size in the equivalence trial and the Bland-Altman method under different parameters. The sample size estimation method for Bland-Altman method fills the gaps in methodology on the field.About the sample size estimation of the equivalence trial, we just consider the simplest case of quantitative data. The other data and design types are not involved in this paper. About the sample size estimation of the Bland-Altman method, it is just appropriate for the data which are well-behaved. If the data behavior is not very well, such as non-normality or non-constant variance of the differences (heteroscedasticity) and proportional bias, the formulae are not suitable to solve the problem of estimating sample size. In addition, we just consider the circumstances of single measurement design without the circumstance of repeated measurement design. The sample size estimation when the data behavior is not very well and the circumstance of repeated measurement design need to further discuss.