Methods for testing the non-inferiority of new drugs when efficacy data follow a skewed normal distribution

Abstract

This application discloses a method for testing the non-inferiority of a new drug when efficacy data follows a skewed normal distribution. This method tests the non-inferiority of a new drug relative to a standard drug when efficacy data follows a skewed normal distribution, ensuring the reliability of the test results. S1. Acquisition of drug efficacy data in clinical trials: In a three-arm non-inferiority clinical trial, drug efficacy data are acquired from three groups of clinical trial results—the new drug group, the standard drug control group, and the placebo control group—based on drug efficacy indicators. The efficacy data follows a skewed normal distribution. S2. Calculation of the probability of committing a Type I error: In the three-arm non-inferiority hypothesis testing model, the probability of committing a Type I error is calculated using a frequentist method or a Bayesian method on the drug efficacy data from the three groups of trials. The probability of committing a Type I error is the probability of the alternative hypothesis H1 occurring when the null hypothesis H0 is true. S3. Determination of the non-inferiority of the new drug: If the probability of committing a Type I error is greater than a given reference value, the new drug is determined to be non-inferior to the standard drug. The reference value is a pre-given clinically significant threshold value.

Description

Technical Field

[0001] This application relates to the field of medical drug efficacy testing technology, and in particular to a method for testing the non-inferiority of a new drug when efficacy data follows a skewed normal distribution. Background Technology

[0002] With the continuous development of biomedical technology, in order to meet the needs of human beings in treating various diseases and promote the high-quality development of the national biomedical field, researchers must develop new drugs that are cheaper or have fewer side effects. Drug safety is paramount in new drug development. Non-inferiority evaluation of new drugs verifies that the therapeutic effect of a new drug is not inferior to that of the standard drug currently in use, given a non-inferiority margin, and can effectively evaluate whether a new drug can replace standard treatment drugs.

[0003] In clinical trials evaluating the non-inferiority of new drugs, a three-arm non-inferiority clinical trial is one that simultaneously designs a positive control group and a placebo. This method allows for comparison not only of whether the new drug is non-inferior to the standard drug, but also of whether the new drug is superior to the placebo. This approach ensures both the efficacy of the drug and provides internal and external validation of the trial results.

[0004] Current non-inferiority studies of new drugs primarily focus on efficacy data following normal, binomial, and Poisson distributions, with fewer studies addressing cases where efficacy data follows a skewed normal distribution. Given my country's historical culture, population, and medical resources, unimodal and asymmetrical distributions of efficacy data are common in clinical trials. This implies a long tail extending towards higher or lower values, suggesting a skewed normal distribution. Therefore, determining how to verify the non-inferiority of new drugs in three-arm non-inferiority clinical trials, especially when efficacy data follows a skewed normal distribution, is a pressing issue that needs to be addressed. Summary of the Invention

[0005] To overcome the aforementioned technical deficiencies, this application provides a method for testing the non-inferiority of a new drug when efficacy data follows a skewed normal distribution. This method tests the non-inferiority of the new drug when efficacy data follows a skewed normal distribution to determine its non-inferiority while ensuring the reliability of the test results. The specific technical solution is as follows:

[0006] This testing method is applicable when the null hypothesis H0 in the three-arm non-inferiority hypothesis testing model is set to: μ E -θμ R -(1-θ)μ P ≤0, alternative hypothesis H1 is set as: μ E -θμ R -(1-θ)μ P In a three-arm non-inferiority clinical trial where μ > 0 is used to test the non-inferiority of a new drug, the hypothesis is that μ < 0. E μR μ P θ represents the mean efficacy index corresponding to the treatment effect of the new drug group, the standard drug control group, and the placebo control group, respectively, and θ is the non-inferiority margin, which takes values ​​between [0.5, 1].

[0007] The testing method includes:

[0008] S1. Acquisition of drug efficacy data in clinical trials: In a three-arm non-inferiority clinical trial, drug efficacy data of the three clinical trial results of the new drug group, the standard drug control group and the placebo control group were acquired based on drug efficacy indicators. The efficacy data followed a skewed normal distribution.

[0009] S2. Calculation of the probability of committing a Type I error: In the three-arm non-inferiority hypothesis testing model, the probability of committing a Type I error is calculated by using the frequency method or Bayesian method on the drug efficacy data of the three sets of experimental results. The probability of committing a Type I error is the probability of the alternative hypothesis H1 occurring when the null hypothesis H0 is true.

[0010] In the absence of historical drug efficacy data, a frequentist approach is used to construct an approximate test statistic for the efficacy data following a skewed normal distribution using the drug efficacy data obtained in the current trial. Given the moment estimator of the unknown parameters, a Bootstrap sampling algorithm is used to sample and construct the Bootstrap test statistic, thereby calculating the probability of committing a Type I error. In the presence of historical drug efficacy data, a Bayesian approach is used to obtain the posterior distribution of the parameters by combining the historical and current drug efficacy data. The Gibbs sampling algorithm is used to sample from the posterior distribution, and according to the Bayesian decision criterion, the probability of the alternative hypothesis being true is taken as the probability of committing a Type I error.

[0011] S3. Determination of non-inferiority of new drugs: If the probability of committing a Type I error is greater than the given reference value, the new drug is determined to be non-inferior to the standard drug, where the reference value is a pre-given clinically significant threshold value.

[0012] Optionally, in one possible implementation of the test method, the location parameter, scale parameter, and skewness parameter corresponding to the drug efficacy data following a skewed normal distribution are treated as unknown parameters. The specific steps for calculating the probability of committing a Type I error using the frequency method include:

[0013] S2.1 Using the efficacy data from the three groups of trials in the current trial, calculate the first parameter moment estimators for the location parameter, scale parameter, and skewness parameter corresponding to the drug efficacy data when they follow a partially normal distribution;

[0014] S2.2 Substitute the moment estimators of the location parameter, scale parameter, and skewness parameter into the approximate test statistic constructed based on the central limit theorem to obtain the test statistic for the non-inferiority hypothesis;

[0015] S2.3. The Bootstrap sample data is obtained by sampling the first parameter moment estimate calculated in S2.1 using the Bootstrap sampling algorithm, and the moment estimates of the corresponding position parameter, scale parameter and skewness parameter in the Bootstrap sample data are calculated.

[0016] S2.4 Substitute the moment estimates of the location parameter, scale parameter, and skewness parameter calculated based on the Bootstrap sample data in S2.3 into the calculation formula of the test statistic for the non-inferiority hypothesis in S2.2 to obtain the bootstrap test statistic.

[0017] S2.5. Obtain the probability of committing a Type I error based on the test statistic of the non-inferiority hypothesis and the bootstrap test statistic.

[0018] Optionally, in one possible implementation of the test method, the location parameter, scale parameter, and skewness parameter corresponding to the drug efficacy data following a skewed normal distribution are used as the test parameters. The specific steps for calculating the probability of committing a Type I error using the Bayesian method include:

[0019] S2.6 Construct conditional posterior distributions for location parameters, scale parameters, and skewness parameters based on historical drug efficacy data and drug efficacy data obtained in the current trial.

[0020] S2.7. Use the Gibbs sampling algorithm to sample the posterior samples of the location parameters corresponding to the three sets of experimental results from the conditional posterior distribution of the parameters.

[0021] S2.8 Calculate the probability of the alternative hypothesis being true based on the posterior samples of the position parameters corresponding to the three sets of experimental results, and use it as the probability of committing a Type I error.

[0022] Optionally, in one possible implementation of the testing method, the specific steps for constructing the posterior distribution include:

[0023] S2.6.1 Determine the prior distributions of location parameters, scale parameters, and skewness parameters based on historical drug efficacy data;

[0024] S2.6.2 Establish the likelihood function of the drug efficacy data obtained in the current experiment;

[0025] S2.6.3. Combining historical drug efficacy data and drug efficacy data obtained in the current trial, Bayes' theorem is used to construct the joint posterior distribution of the parameters by combining the prior distribution and the likelihood function.

[0026] S2.6.4 Based on the joint posterior distribution of the parameters, construct the conditional posterior distributions of the location parameter, scale parameter, and skewness parameter respectively.

[0027] Optionally, in one possible implementation of the testing method, the sampling calculation process using the Gibbs sampling algorithm is as follows:

[0028] S2.7.1. Given the initial values ​​of the position parameter, scale parameter, and skewness parameter, i.e., the initialization parameters;

[0029] S2.7.2 For each parameter, fix the other parameters and only update the current parameter. That is, the parameter update needs to be sampled from the corresponding conditional posterior distribution.

[0030] S2.7.3 Repeat the above steps multiple times until the conditional posterior distribution of the parameters is stable, i.e., take the stable posterior sample.

[0031] The technical solutions employed in this application embodiment achieve the following technical effects: For efficacy data following a skewed normal distribution, a frequentist method and a Bayesian method are provided to calculate the probability of committing a Type I error. Specifically, in the absence of historical drug efficacy data, the frequentist method is used to construct an approximate test statistic for efficacy data following a skewed normal distribution using the drug efficacy data obtained in the current trial. Given a moment estimator for the unknown parameters, a Bootstrap sampling algorithm is used to sample and construct a Bootstrap test statistic, thereby calculating the probability of committing a Type I error. In the presence of historical drug efficacy data, a Bayesian method is used to obtain the posterior distribution of the parameters by combining historical drug efficacy data and the drug efficacy data obtained in the current trial. The posterior probability calculated by sampling from the posterior distribution using the Gibbs sampling algorithm is used as the probability of committing a Type I error. Therefore, the non-inferiority of the new drug is determined based on the probability of committing a Type I error, thus solving the problem of determining whether a new drug is non-inferior to a standard drug when efficacy data follows a skewed normal distribution. Attached Figure Description

[0032] The accompanying drawings exemplify embodiments and form part of the specification, serving together with the textual description to explain exemplary implementations of the embodiments. The illustrated embodiments are for illustrative purposes only and do not limit the scope of the claims. Throughout the drawings, the same reference numerals refer to similar but not necessarily identical elements.

[0033] Figure 1 This is a flowchart illustrating a method for testing the non-inferiority of a new drug when the efficacy data in this application follow a skewed normal distribution.

[0034] Figure 2This is a flowchart illustrating the calculation of the probability of committing a Type I error using the frequency method in this application.

[0035] Figure 3 This is a flowchart illustrating the calculation of the probability of committing a Type I error using the Bayesian method in this application.

[0036] Figure 4 A frequency histogram with probability density curves;

[0037] Figure 5 This describes the characteristics of three groups of data: new drug, standard drug, and placebo. Detailed Implementation

[0038] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application. All other embodiments obtained by those skilled in the art based on the embodiments in this application without inventive effort are within the scope of protection of this application.

[0039] It should be noted that the descriptions involving "first," "second," etc., in the embodiments of this application are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined with "first" or "second" may explicitly or implicitly include at least one of that feature. Furthermore, the technical solutions of the various embodiments can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. If the combination of technical solutions is contradictory or impossible to implement, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection claimed in this application.

[0040] In the description of this application, it should be understood that the numerical labels before the steps do not indicate the order of the steps, but are only used to facilitate the description of this application and to distinguish each step, and therefore should not be construed as a limitation of this application.

[0041] First, a definition of the terminology used in this application is provided:

[0042] Bootstrap sampling algorithm: The Bootstrap method is a numerical simulation technique that estimates the distribution of sample statistics by repeatedly sampling. Its core idea is to use the original dataset to perform multiple random samplings, each time drawing a sample to simulate the sample distribution of the population, thereby obtaining an estimate of the sample statistics and the corresponding confidence interval.

[0043] Gibbs sampling algorithm: a special Markov chain Monte Carlo (MCMC) method used for sampling from a multivariate joint distribution, especially suitable for sampling from a posterior distribution. When direct sampling is difficult, Gibbs sampling updates parameters iteratively, utilizing a conditional distribution for sampling.

[0044] Test statistic: A test statistic is a statistical measure used in hypothesis testing. Under the null hypothesis, this statistic follows a given probability distribution, while under another hypothesis, it does not. Therefore, if the value of the test statistic falls outside the critical value of the aforementioned distribution, it can be considered that the null hypothesis may not be true. The main function of the test statistic is to play a crucial role in hypothesis testing. By calculating the test statistic, researchers can determine whether the sample data supports or rejects the null hypothesis. Specifically, the test statistic helps determine the relationship between the sample data and population parameters, thereby making statistical inferences.

[0045] The probability of committing a Type I error is the probability of incorrectly rejecting the null hypothesis in a hypothesis test, that is, the probability that the test result incorrectly supports the alternative hypothesis when the null hypothesis is true.

[0046] As described in the background section, in view of the actual situation in my country, this application has designed a technical solution for testing the non-inferiority of new drugs in three-arm non-inferiority clinical trials when the efficacy data follows a skewed normal distribution. It should be noted that the non-inferiority of new drugs mentioned in this article is an evaluation of the biological non-inferiority of new drugs, which can also be called biological non-inferiority, or non-inferiority, etc., all of which have the same meaning. In the following text, it will be referred to as non-inferiority of new drugs.

[0047] Secondly, to facilitate understanding of the technical solutions provided in the embodiments of this application by those skilled in the art, the relevant technologies are described below:

[0048] Example 1

[0049] The technical solution of this application is designed to test the non-inferiority of a new drug in a three-arm non-inferiority clinical trial when the efficacy data follows a skewed normal distribution. The clinical trial is designed with a new drug group, a standard drug control group, and a placebo control group, and the efficacy data are tested based on the results of the above three groups of clinical trials.

[0050] See Figure 1 The method for testing the non-inferiority of a new drug when the efficacy data follows a skewed normal distribution, as described in this application embodiment, includes:

[0051] S1. Obtaining drug efficacy data in clinical trials:

[0052] In a three-arm non-inferiority clinical trial, drug efficacy data were obtained from three groups of clinical trial results, namely the new drug group, the standard drug control group, and the placebo control group, based on drug efficacy indicators. The efficacy data followed a skewed normal distribution.

[0053] The efficacy data are determined based on the corresponding drug efficacy indicators of the new drug, and different new drugs may have different drug efficacy indicators.

[0054] A three-arm non-inferiority clinical trial is a clinical trial designed to test the non-inferiority of a new drug. It includes two control groups: a standard drug control group (i.e., a positive drug control group) and a placebo control group. The standard drug or positive drug is the target drug that the new drug is intended to replace.

[0055] In the three-arm non-inferiority hypothesis testing model for three-arm non-inferiority clinical trial designs, the null hypothesis H0 is set as: μ E -θμ R -(1-θ)μ P ≤0, alternative hypothesis H1 is set as: μ E -θμ R -(1-θ)μ P >0, μ E μ R μ P θ represents the mean efficacy index of the treatment effect in the experimental drug group, the standard drug control group, and the placebo control group, respectively. θ is a given constant between [0.5, 1], also known as the non-inferiority margin.

[0056] S2. Calculation of the probability of committing a Type I error:

[0057] In the three-arm non-inferiority hypothesis testing model, the probability of committing a Type I error is calculated by using the frequency method or Bayesian method on the drug efficacy data of the three sets of experimental results. The probability of committing a Type I error is the probability of the alternative hypothesis H1 occurring when the null hypothesis H0 is true.

[0058] Generally, before non-inferiority testing of a new drug, technicians already have prior knowledge of historical data related to the drug's efficacy, meaning a large data sample is available. However, there are also situations where no prior testing has been conducted, and historical data related to the drug's efficacy is unknown, resulting in a smaller data sample size. Therefore, for both scenarios, either a frequency-based method or a Bayesian method is chosen for probability calculation.

[0059] Specifically, in the absence of historical drug efficacy data, a frequentist approach is used to construct an approximate test statistic based on the drug efficacy data obtained in the current trial, assuming the efficacy data follows a skewed normal distribution. Given the moment estimator of the unknown parameters, a Bootstrap sampling algorithm is used to sample and construct the Bootstrap test statistic, thereby calculating the probability of committing a Type I error. When historical drug efficacy data exists, a Bayesian approach is used to obtain the posterior distribution of the parameters by combining the historical and current drug efficacy data. The posterior probability calculated by sampling from the posterior distribution using the Gibbs sampling algorithm is then used as the probability of committing a Type I error.

[0060] S3. Non-inferiority assessment of new drugs:

[0061] If the probability of committing a Type I error is greater than a given reference value, then the new drug is determined to be non-inferior to the standard drug, where the reference value is a pre-given clinically significant threshold.

[0062] In clinical trials, the choice of cutoff value is highly causal. A reasonable default value might be 0.5, which essentially means that the hypothesis with the higher posterior probability is retained between the null and alternative hypotheses.

[0063] In this embodiment, for efficacy data following a skewed normal distribution, a frequentist method and a Bayesian method are provided to calculate the probability of committing a Type I error. Specifically, in the absence of historical drug efficacy data, the frequentist method is used to construct an approximate test statistic for efficacy data following a skewed normal distribution using the drug efficacy data obtained in the current trial. Given a moment estimator for the unknown parameters, a Bootstrap sampling algorithm is used to sample and construct a Bootstrap test statistic, thereby calculating the probability of committing a Type I error. In the presence of historical drug efficacy data, the Bayesian method is used to obtain the posterior distribution of the parameters by combining the historical and current drug efficacy data. A Gibbs sampling algorithm is used to sample from the posterior distribution, and according to the Bayesian decision criterion, the probability of the alternative hypothesis being true is taken as the probability of committing a Type I error. Thus, the non-inferiority of the new drug is determined based on the probability of committing a Type I error, thereby solving the problem of determining whether a new drug is non-inferior to a standard drug when efficacy data follows a skewed normal distribution.

[0064] To facilitate a more detailed explanation of the technical solutions of the frequency method and the Bayesian method described below, we will first provide a description of the drug efficacy data in the three-arm non-inferiority hypothesis testing model of this application.

[0065] The three-arm non-inferiority test includes three drug groups: the investigational drug (i.e., the new drug) group is denoted as E (experimental), the standard drug control group is denoted as R (reference), and the placebo group is denoted as P (placebo). Let:

[0066] Where X kj SN(μ, σ) represents the therapeutic effect value of the j-th individual in the k-th group. 2 λ) represents a mean of μ and a variance of σ. 2 A skewed normal distribution with skewness λ, μ k n represents the efficacy value of the k-th group. k This represents the sample size of the k-th group. Let λ represent the variance of the k-th group. k This represents the skewness of the k-th group, where k = E, R, P; j = 1, 2, ..., n k Without loss of generality, we assume μ k The larger the value, the better the treatment effect of the k-th group.

[0067] Example 2

[0068] Next, we will explain the technical solution for calculating the probability of committing a Type I error using the frequency method when there is no historical data on the efficacy of a new drug. The details are as follows:

[0069] like Figure 2 As shown, when drug efficacy data follows a skewed normal distribution, the location parameter, scale parameter, and skewness parameter are used as parameters for the test. The specific steps for calculating the probability of committing a Type I error using the frequency method include:

[0070] S2.1 Calculate the parametric moment estimates of the location parameter, scale parameter, and skewness parameter corresponding to the drug efficacy data when the drug efficacy data follows a skewed normal distribution, using the efficacy data of the three groups of experimental results in the current experiment.

[0071] Specifically, let: And δ k ∈(-1, 1),

[0072] like Then the parameter The moment estimator can be expressed as:

[0073] in

[0074] k = E, R, P, b = (2 / π) 1 / 2 c = [2 / (4-π)] 1 / 3 , S k2 S k3 They represent X respectivelyk The sample mean, the second central distance, and the third central distance.

[0075] S2.2 Substitute the moment estimators of the location parameter, scale parameter, and skewness parameter into the approximate test statistic constructed based on the central limit theorem to obtain the test statistic for the non-inferiority hypothesis.

[0076] Specifically, the formula for calculating the test statistic, constructed using the central limit theorem, is as follows:

[0077]

[0078] S2.3. The Bootstrap sample data is obtained by sampling the parameter distance estimate calculated in S2.1 using the Bootstrap sampling algorithm, and the moment estimates of the corresponding position parameter, scale parameter and skewness parameter in the Bootstrap sample data are calculated.

[0079] Specifically, the Bootstrap sample data obtained by sampling Bootstrap is as follows:

[0080] in for The moment estimates of the parameters are obtained, and then the moment estimates of the position parameter, scale parameter, and skewness parameter are calculated based on the Bootstrap sample data as follows:

[0081] and In the formula,

[0082] S2.4 Substitute the moment estimates of the location parameter, scale parameter, and skewness parameter calculated based on the Bootstrap sample data in S2.3 into the calculation formula of the test statistic for the non-inferiority hypothesis in S2.2 to obtain the bootstrap test statistic.

[0083] Specifically, the formula for calculating the bootstrap test statistic is:

[0084]

[0085] S2.5. Obtain the probability of committing a Type I error based on the test statistic of the non-inferiority hypothesis and the bootstrap test statistic.

[0086] Specifically, parameter estimates can be obtained from efficacy data and Bootstrap sampling, and these estimates can be substituted into the test statistics T and T0. B The corresponding test statistic can be obtained from this, and T can be calculated. B The probability ≥ T can be used to obtain the P-value for the test.

[0087] The probability value obtained in this application is expressed as: p = P(T) B ≥T), which is the probability that the bootstrap test statistic is greater than or equal to the test statistic of the non-inferiority hypothesis, i.e., sample T. B The frequency of occurrence of ≥T is used to determine the probability of committing a Type I error. If p < α, the null hypothesis H0 is rejected at the significance level α, meaning the experimental treatment is considered non-inferior to the positive control treatment.

[0088] In the technical solution of this application embodiment, when the efficacy data follows a skewed normal distribution, the test statistic constructed based on the central limit theorem does not follow a specific distribution function form when the sample size is small. Therefore, the probability of the alternative hypothesis occurring cannot be directly calculated from the probability distribution function form of the test statistic. Using the Bootstrap sampling method, the frequency of sample occurrence can be used to replace the probability of the alternative hypothesis occurring, thereby determining the biological non-inferiority of the new drug. Therefore, the approximate test statistic based on the Bootstrap method can more accurately determine the non-inferiority of the new drug relative to the standard drug.

[0089] Example 3

[0090] Bayesian methods are a type of statistical approach that updates our understanding of an unknown parameter by combining prior knowledge with observed data. At the heart of this method is Bayes' theorem, which can be used to calculate the posterior distribution, that is, the probability distribution of the parameter given the data.

[0091] Next, for cases where new drugs have historical efficacy data, a technical solution using Bayesian methods to calculate the probability of committing a Type I error will be explained, as follows:

[0092] like Figure 3 As shown, when drug efficacy data follows a skewed normal distribution, the location parameter, scale parameter, and skewness parameter are treated as unknown parameters. The specific steps for calculating the probability of committing a Type I error using the Bayesian method include:

[0093] S2.6 Construct conditional posterior distributions for location parameters, scale parameters, and skewness parameters based on historical drug efficacy data and drug efficacy data obtained in the current trial.

[0094] Specifically, the steps for constructing the posterior distribution include:

[0095] S2.6.1 Determine the prior distributions of location parameters, scale parameters, and skewness parameters based on historical drug efficacy data.

[0096] The prior distribution is the assumption we make about the parameters before observing the data.

[0097] S2.6.2 Establish the likelihood function of the drug efficacy data obtained in the current experiment.

[0098] The likelihood function is the relationship between observed data and parameter values, representing the probability of observing data given the parameters.

[0099] S2.6.3. Based on historical data on the efficacy of the compound and drug efficacy data obtained in the current trial, Bayes' theorem is used to construct a joint posterior distribution of the parameters by combining prior distribution and likelihood function.

[0100] Since the parameters include location, scale, and skewness parameters, the constructed joint posterior distribution includes a joint posterior distribution of location, scale, and skewness parameters.

[0101] S2.6.4 Based on the joint posterior distribution of the parameters, construct the conditional posterior distributions of the location parameter, scale parameter, and skewness parameter respectively.

[0102] S2.7. Use the Gibbs sampling algorithm to sample the posterior samples of the location parameters corresponding to the three sets of experimental results from the conditional posterior distribution of the parameters.

[0103] Specifically, the sampling calculation process using the Gibbs sampling algorithm is as follows:

[0104] S2.7.1 Provide the initial values ​​for the position parameter, scale parameter, and skewness parameter, i.e., the initialization parameters.

[0105] S2.7.2 For each parameter, fix the other parameters and only update the current parameter. That is, the parameter update needs to be sampled from the corresponding conditional posterior distribution.

[0106] S2.7.3 Repeat the above steps multiple times until the conditional posterior distribution of the parameters is stable, i.e., take the stable posterior sample.

[0107] S2.8 Calculate the probability of the alternative hypothesis being true based on the posterior samples of the position parameters corresponding to the three sets of experimental results, and use it as the probability of committing a Type I error.

[0108] Within the Bayesian framework, the Bayesian decision criterion for a three-arm non-inferiority test model of a new drug relative to a standard drug is: P(H1:μ) E -θμ R -(1-θ)μ P >0|μ R -μ P >0, Data)>R NI ;

[0109] If the posterior probability value is greater than or equal to the given critical value R NI If the new drug is non-inferior to the standard drug, then it is said that the new drug is not inferior to the standard drug.

[0110] In the technical solutions of this application, the Bayesian method allows the integration of historical data information into probability calculations. This means that if there is already some information or assumptions about the problem, this knowledge can be fused into the analysis in the form of prior probability distributions. This is particularly useful when sample data is scarce or incomplete, allowing for the updating of sample information and thus providing a more comprehensive analysis.

[0111] For example, for the three-arm non-inferiority hypothesis testing model:

[0112] H0: μ E -θμ R -(1-θ)μ P ≤0 vs. H1: μ E -θμ R -(1-θ)μ P >0, θ∈[0.5,1];

[0113] Where μ E μ R μ P Let represent the mean efficacy of the experimental group, the positive control group, and the placebo group, respectively, and θ is a given constant between [0.5, 1].

[0114] This application provides the following procedure for testing this hypothesis:

[0115] Based on the stochastic representation of the skewed normal distribution introduced by Azzalini (1986), the above three-arm non-dominance hypothesis testing model is rewritten as a stratified model, as follows:

[0116]

[0117] in The mean is μ k +σ k δ k U kj The variance is The normal distribution is TN(0,1)I{u kj >0} is a normal distribution with a mean of 0 and a variance of 1, but its values ​​are truncated to the part greater than 0, that is, only positive values ​​are allowed. Where U kj It is a latent variable and X kj μ k , σ k , λ k The meaning is the same as before.

[0118] To simplify the calculation, let The above hierarchical model can then be rewritten as follows:

[0119]

[0120] In Bayesian estimation, the choice of prior distribution is crucial for determining the posterior distribution. In this case, we can consider that the location and scale parameters follow a Jeffreys prior, a non-informative prior used to express the parameter uncertainty when prior information is lacking. The prior distribution of the skewness parameter, however, follows a t-distribution due to its thicker tail. Such a prior setting helps to more comprehensively reflect the prior knowledge of the parameters and derive a reasonable posterior distribution. Considering that the location and scale parameters follow a Jeffreys prior, and the skewness parameter follows a t-distribution, that is:

[0121]

[0122] Where π(λ) k () is a position parameter of 0 and a scale parameter of b. k The degree of freedom is d k The t-distribution t(0, b) k d k Probability density function. μ k σ k , λ k The meaning is the same as before. When At that time, λ k It follows the Jefferys prior distribution; when At that time, λ k It follows a uniform distribution. Let the t-distribution be t(0, b). k d k This can be represented in the following hierarchical form:

[0123]

[0124] Where λ k Follows the pattern with mean 0 and variance of The normal distribution, ω k Obeying shape parameters Scale parameter is The gamma distribution. To ensure μ R >μ P To ensure sensitivity, i.e., to guarantee that the positive control drug is effective compared to the placebo, the joint prior distribution of the parameters can be obtained from equations (1) and (2):

[0125]

[0126] The joint posterior distribution of the parameters satisfying the sensitivity condition can be obtained from Bayes' theorem:

[0127]

[0128] Where μk ,β k η k ω k U k The meaning is the same as before.

[0129] The joint posterior distribution was sampled using the Gibbs sampling method. The sampling process is as follows:

[0130] 1. Set initial values

[0131] 2. After extracting In the (l+1)th iteration, the parameters are fixed: Extract parameters from equation (A.1) Fixed parameters Extract parameters from equation (A.2) Fixed parameters Extract parameters from equation (A.3) Fixed parameters Extract parameters from equation (A.4) Fixed parameters Extract parameters from equation (A.5)

[0132] 3. Repeat the above steps multiple times until the distribution of parameters is stable, i.e., take the stable posterior sample.

[0133] in:

[0134] Equation (A.1) is:

[0135] in, This means that given some conditional parameters And data X k Below, parameters The conditional distribution, i.e. Follow the mean variance It follows a normal distribution.

[0136] Equation (A.2) is:

[0137] in, This means that given some conditional parameters And data X k Below, parameters The conditional distribution, i.e. Follow the mean variance is The truncated normal distribution has its values ​​truncated to the part greater than 0, that is, only the positive values ​​are allowed.

[0138] Equation (A.3) is:

[0139] in, This means that given some conditional parameters And data X k Below, parameters The conditional distribution, i.e. Follow the mean variance is It follows a normal distribution.

[0140] Equation (A.4) is:

[0141] in, This means that given some conditional parameters And data X k Below, parameters The conditional distribution, i.e. Follow the mean variance is It follows a normal distribution.

[0142] Equation (A.5) is:

[0143] in, This means that given some conditional parameters And data X k Below, parameters The conditional distribution, i.e. Obeying shape parameters variance is It follows a normal distribution.

[0144] In equations (A.1)-(A.5) above, l and l+1 represent the iteration number, μ k ,β k η k ω k u kj The meaning is the same as before. This represents the sample drawn in the l-th iteration. After drawing T samples, This can be viewed as derived from the joint posterior distribution p(μ) k ,β k η k ω k U k Extracted from |X). From p(μ) k ,β k η k ω k U k μ extracted from |X) k The value of .

[0145] Within the Bayesian framework, the Bayesian decision criterion for a three-arm non-inferiority test of a new drug relative to a standard drug is:

[0146] P(H1:μ) E -θμ R -(1-θ)μ P >0|μ R -μ P >0,Data)>R NI .

[0147] If the posterior probability value is greater than or equal to the given critical value R NI If the new drug is non-inferior to the standard drug, then it is said that the new drug is not inferior to the standard drug

[0148] Finally, for ease of understanding, the method for testing the non-inferiority of a new drug when efficacy data follows a skewed normal distribution, as proposed in this application, is applied to HIV non-inferiority testing, using the CD4 antigen on the patient's cell surface as an indicator of treatment efficacy. Generally speaking, the higher the CD4 count, the healthier the immune system.

[0149] The specific steps of the inspection are as follows:

[0150] First, we compare the logCD4 levels of patients in the three drug groups. The data are measurements from patients in groups 7 and 8. The total sample size of the dataset is n = 501 cases, with n patients in each group. E =n R =n P =167. zA400d was used as the new drug (experimental group xExp). zA400dA400n was used as the standard drug (control group xRef), and zA225z was used as the placebo (placebo group xPla). Data description is as follows: The overall parameter values ​​for the three groups are unknown, from... Figure 4 The frequency histogram with probability density curves shows that the skewness of the three sets of data is not obvious. Figure 5 The characteristics of three sets of data are presented. The means and variances of the three sets of data are very similar. The skewness data obtained by the skewness formula show that all three sets of data have negative skewness.

[0151] Secondly, to understand the skewness of the test data, the Shapiro-Wilk test, Kolmogorov-Smirnov test, and Cramer-von Mises test were first used. The p-values ​​for the normality tests were 0.000391639, 0.000791679, and 0.003596725, respectively. This indicates that at the 5% significance level, the experimental group is not normally distributed. Furthermore, the chi-square goodness-of-fit test was used, assuming the original data were skewed normally distributed. The values ​​of the test statistics are then determined. Therefore, the null hypothesis will not be rejected at the 5% significance level. Thus, the distribution of the experimental group can be considered approximately normally distributed.

[0152] Similarly, conversely, we obtained the following p-values ​​for the normality test of the control group: 0.005087814, 0.009443264, and 0.01567969; and the following p-values ​​for the normality test of the placebo: 0.000335353, 0.001029581, and 0.004230334. Therefore, neither the control group nor the placebo is normally distributed. The test statistic for the control group is... Test statistic for placebo Both the control group and the placebo can be considered to be approximately positively skewed.

[0153] Finally, the three groups of drugs in the case data were tested using experimental methods.

[0154] H0:μ E -θμ R -(1-θ)μ P ≤0 vs. H1: μ E -θμ R -(1-θ)μ P >0,

[0155] The p-value of the test is 0.48, which is greater than the pre-specified significance level of 0.05, therefore the null hypothesis can be accepted.

[0156] Next, we analyze the case data within the Bayesian framework. Under the hypothesis testing problem described above, we calculate that the probability of committing a Type I error is 0.065, therefore we accept the null hypothesis.

[0157] Compared with the standard drug zA400dA400n, the efficacy of the new drug zA225z does not meet the non-inferiority requirement.

[0158] Obviously, those skilled in the art should understand that the modules or steps of the embodiments of this application described above can be implemented using general-purpose computer devices. They can be centralized on a single computer device or distributed across a network of multiple computer devices. Optionally, they can be implemented using computer-executable program code, thereby storing them in a storage device for execution by a computer device. In some cases, the steps shown or described can be performed in a different order than those presented here, or they can be fabricated as separate integrated circuit modules, or multiple modules or steps can be fabricated as a single integrated circuit module. Thus, the embodiments of this application are not limited to any particular combination of hardware and software.

[0159] It should be noted that the above are merely preferred embodiments of this application and do not limit the scope of patent protection of this application. Any equivalent structural or procedural changes made using the content of this application's specification and drawings, or direct or indirect applications in other related technical fields, are similarly included within the scope of patent protection of this application.

Claims

1. A method for testing the non-inferiority of a new drug when efficacy data follow a skewed normal distribution, characterized in that, The aforementioned testing method is applicable when the null hypothesis H0 in the three-arm non-inferiority hypothesis testing model is set to: μ E -θμ R -(1-θ)μ P ≤0, alternative hypothesis H1 is set as: μ E -θμ R -(1-θ)μ P In a three-arm non-inferiority clinical trial where μ > 0 is used to test the non-inferiority of a new drug, the hypothesis is that μ < 0. E μ R μ P θ represents the mean efficacy index corresponding to the treatment effect of the new drug group, the standard drug control group, and the placebo control group, respectively, and θ is the non-inferiority margin, which takes values ​​between [0.5, 1]. The testing method includes: S1. Obtaining drug efficacy data in clinical trials: In a three-arm non-inferiority clinical trial, drug efficacy data were obtained from three groups of clinical trial results, namely the new drug group, the standard drug control group, and the placebo control group, based on drug efficacy indicators. The efficacy data followed a skewed normal distribution. S2. Calculation of the probability of committing a Type I error: In the three-arm non-inferiority hypothesis testing model, the probability of committing a Type I error is calculated by using the frequency method or Bayesian method on the drug efficacy data of the three sets of experimental results. The probability of committing a Type I error is the probability of the alternative hypothesis H1 occurring when the null hypothesis H0 is true. In the absence of historical drug efficacy data, a frequentist approach is used to construct an approximate test statistic for the efficacy data following a skewed normal distribution using the drug efficacy data obtained in the current trial. Given the moment estimator of the unknown parameters, a Bootstrap sampling algorithm is used to sample and construct the Bootstrap test statistic, thereby calculating the probability of committing a Type I error. In the presence of historical drug efficacy data, a Bayesian approach is used to obtain the posterior distribution of the parameters by combining the historical and current drug efficacy data. The Gibbs sampling algorithm is used to sample from the posterior distribution, and according to the Bayesian decision criterion, the probability of the alternative hypothesis being true is taken as the probability of committing a Type I error. S3. Non-inferiority assessment of new drugs: If the probability of committing a Type I error is greater than a given reference value, then the new drug is determined to be non-inferior to the standard drug, where the reference value is a pre-given clinically significant threshold.

2. The testing method according to claim 1, characterized in that, When drug efficacy data follows a skewed normal distribution, the location, scale, and skewness parameters are treated as unknown parameters. The specific steps for calculating the probability of committing a Type I error using the frequency method include: S2.1 Using the efficacy data from the three groups of trials in the current trial, calculate the first parameter moment estimators for the location parameter, scale parameter, and skewness parameter corresponding to the drug efficacy data when they follow a partially normal distribution; S2.2 Substitute the moment estimators of the location parameter, scale parameter, and skewness parameter into the approximate test statistic constructed based on the central limit theorem to obtain the test statistic for the non-inferiority hypothesis; S2.

3. The Bootstrap sample data is obtained by sampling the first parameter moment estimate calculated in S2.1 using the Bootstrap sampling algorithm, and the moment estimates of the corresponding position parameter, scale parameter and skewness parameter in the Bootstrap sample data are calculated. S2.4 Substitute the moment estimates of the location parameter, scale parameter, and skewness parameter calculated based on the Bootstrap sample data in S2.3 into the calculation formula of the test statistic for the non-inferiority hypothesis in S2.2 to obtain the bootstrap test statistic. S2.

5. Obtain the probability of committing a Type I error based on the test statistic of the non-inferiority hypothesis and the bootstrap test statistic.

3. The testing method according to claim 1, characterized in that, When drug efficacy data follows a skewed normal distribution, the location, scale, and skewness parameters are treated as unknown parameters. The specific steps for calculating the probability of committing a Type I error using Bayesian methods include: S2.6 Construct conditional posterior distributions for location parameters, scale parameters, and skewness parameters based on historical drug efficacy data and drug efficacy data obtained in the current trial. S2.

7. Use the Gibbs sampling algorithm to sample the posterior samples of the location parameters corresponding to the three sets of experimental results from the conditional posterior distribution of the parameters. S2.8 Calculate the probability of the alternative hypothesis being true based on the posterior samples of the position parameters corresponding to the three sets of experimental results, and use it as the probability of committing a Type I error.

4. The testing method according to claim 3, characterized in that, The specific steps for constructing the posterior distribution include: S2.6.1 Determine the prior distributions of location parameters, scale parameters, and skewness parameters based on historical drug efficacy data; S2.6.2 Establish the likelihood function of the drug efficacy data obtained in the current experiment; S2.6.

3. Combine historical data on the efficacy of the compound with drug efficacy data obtained in the current trial, and use Bayes' theorem to construct the joint posterior distribution of the parameters by combining prior distribution and likelihood function; S2.6.4 Based on the joint posterior distribution of the parameters, construct the conditional posterior distributions of the location parameter, scale parameter, and skewness parameter respectively.

5. The testing method according to claim 3 or 4, characterized in that, The process of performing sampling calculations using the Gibbs sampling algorithm is as follows: S2.7.

1. Given the initial values ​​of the position parameter, scale parameter, and skewness parameter, i.e., the initialization parameters; S2.7.2 For each parameter, fix the other parameters and only update the current parameter. That is, the parameter update needs to be sampled from the corresponding conditional posterior distribution. S2.7.3 Repeat the above steps multiple times until the conditional posterior distribution of the parameters is stable, i.e., take the stable posterior sample.

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