Method for testing non-inferior performance of new medicine when curative effect data obeys partial normal distribution

Abstract

The invention discloses a method for testing the non-inferior performance of a new drug when curative effect data obeys partial normal distribution, aiming at the condition that the curative effect data obeys partial normal distribution, the non-inferior performance of the new drug is tested so as to determine the non-inferior performance of the new drug relative to a standard drug, and meanwhile, the reliability of a detection result is ensured. The method comprises the following steps: S1, acquiring drug curative effect data in clinical tests: in a three-arm non-inferior clinical test, based on drug curative effect indexes, respectively acquiring drug curative effect data of three groups of clinical test results of a new drug group, a standard drug control group and a placebo control group, the curative effect data obeying partial normal distribution; s2, probability calculation of first-class errors: in the three-arm non-inferior hypothesis test model, a frequency method or a Bayesian method is used to calculate the drug curative effect data of the three groups of test results to obtain the probability of first-class errors, and the probability of first-class errors is the probability of occurrence of a standby hypothesis H1 when an original hypothesis H0 is established; and S3, non-inferior determination of the new drug: if the probability of making the first type of errors is greater than a given reference value, determining that the new drug has non-inferior relative to the standard drug, and the reference value is a critical value which is given in advance and has clinical significance.

Description

Technical Field

[0001] The embodiments of the present application relate to the technical field of medical drug efficacy detection, and in particular to a method for testing the non-inferiority of a new drug when efficacy data obey a skewed normal distribution. Background Art

[0002] With the continuous development of biomedical technology, in order to meet the needs of human beings to treat various diseases and promote the high-quality development of the country's biomedicine field, researchers must develop new drugs that are cheaper or have fewer side effects. In the development of new drugs, drug safety is of vital importance. The non-inferiority evaluation of new drugs is to verify that the therapeutic effect of new drugs is not inferior to the standard drugs currently being used under a given non-inferiority margin, which can effectively evaluate whether new drugs can replace standard treatment drugs.

[0003] In clinical trials evaluating the non-inferiority of new drugs, designing both the positive drug and the placebo as control groups is called a three-arm non-inferiority clinical trial. By setting up controls in this way, we can not only compare whether the new drug is not inferior to the standard drug, but also compare whether the new drug is superior to the placebo. This can not only ensure the effectiveness of the drug, but also achieve internal and external confirmation of the trial.

[0004] In the current non-inferiority studies of new drugs, the main focus is on cases where the efficacy data follows normal distribution, binomial distribution, and Poisson distribution, and there are fewer cases where the efficacy data follows skewed normal distribution. In view of the characteristics of my country's history, culture, population, and medical resources, in clinical trials, it is common for efficacy data to present a unimodal, asymmetric distribution, which means that the distribution of data has a long tail extending to higher or lower values, that is, the data presents the characteristics of skewed normal distribution. Therefore, in the case where the efficacy data follows a skewed normal distribution, how to test the non-inferiority of new drugs in three-arm non-inferiority clinical trials is an urgent problem to be solved. Summary of the invention

[0005] In order to overcome the above technical defects, the present application provides a method for testing the non-inferiority of a new drug when the efficacy data obeys a skewed normal distribution. In the case where the efficacy data obeys a skewed normal distribution, the non-inferiority of the new drug is tested to determine the non-inferiority of the new drug and ensure the reliability of the test results. The technical solution is as follows:

[0006] This test method is applicable to setting the null hypothesis H0 in the three-arm non-inferiority hypothesis test model to: E -θμ R -(1-θ)μ P ≤0, the alternative hypothesis H1 is set as: μ E -θμ R -(1-θ)μ P >0 in a three-arm non-inferiority clinical trial to test the non-inferiority of a new drug, where μ E , μR , μ P Respectively represent the means of efficacy indicators corresponding to the treatment effects of the new drug group, the standard drug control group, and the placebo control group, θ is the non-inferiority margin, and its value is between [0.5, 1];

[0007] The test method includes:

[0008] S1. Acquisition of drug efficacy data in clinical trials: In a three-arm non-inferiority clinical trial, based on the drug efficacy index, obtain the drug efficacy data of the three groups of clinical trial results of the new drug group, the standard drug control group and the placebo control group, where the efficacy data follow a skewed normal distribution;

[0009] S2. Calculation of the probability of making a type I error: In the three-arm non-inferiority hypothesis test model, the frequency method or Bayesian method is used to calculate the drug efficacy data of the three groups of test results to obtain the probability of making a type I error, where the probability of making a type I error is the probability of the alternative hypothesis H1 occurring when the null hypothesis H0 is established;

[0010] In the absence of historical data on drug efficacy, the frequency method is used to construct an approximate test statistic when the efficacy data obeys a skewed normal distribution using the drug efficacy data obtained in the current trial. Given the moment estimator of the unknown parameter, the Bootstrap sampling algorithm is used to perform sample sampling and construct the Bootstrap test statistic, and then the probability of making a first-type error is calculated. In the presence of historical data on drug efficacy, the Bayesian method is used to obtain the posterior distribution of the parameter by combining the historical data on drug efficacy and the drug efficacy data obtained in the current trial, and the Gibbs sampling algorithm is used to sample from the posterior distribution. According to the Bayesian decision criterion, the probability that the alternative hypothesis is true is used as the probability of making a first-type error.

[0011] S3. Determination of non-inferiority of new drugs: 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, where the reference value is a pre-given clinically significant critical value.

[0012] Optionally, in a possible implementation of the test method, the location parameter, scale parameter, and skewness parameter corresponding to the drug efficacy data obeying the skewed normal distribution are used as unknown parameters, and the specific steps of using the frequency method to calculate the probability of making a first type error include:

[0013] S2.1. Use the efficacy data of the three groups of test results in the current trial to calculate the first parameter moment estimator corresponding to the location parameter, scale parameter and skewness parameter when the drug efficacy data obeys the skewed normal distribution;

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

[0015] S2.3, sampling the first parameter moment estimator calculated in S2.1 by the Bootstrap sampling algorithm to obtain Bootstrap sample data, and calculating the moment estimates of the corresponding location parameter, scale parameter and skewness parameter in the Bootstrap sample data;

[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 a possible implementation of the test method, the location parameter, scale parameter, and skewness parameter corresponding to the drug efficacy data obeying the skewed normal distribution are used as test parameters, and the specific steps of using the Bayesian method to calculate the probability of making a first type error include:

[0019] S2.6. Construct the conditional posterior distribution of the location parameter, scale parameter and skewness parameter based on the historical drug efficacy data and the drug efficacy data obtained in the current trial;

[0020] S2.7, use the Gibbs sampling algorithm to sample from the conditional posterior distribution of the parameter to obtain the posterior samples of the location parameters corresponding to the three groups of test results;

[0021] S2.8. Calculate the probability that the alternative hypothesis is true based on the posterior samples of the location parameters corresponding to the three sets of test results and use it as the probability of committing a type I error.

[0022] Optionally, in a possible implementation of the test method, the specific steps of constructing the posterior distribution include:

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

[0024] S2.6.2. Observe the drug efficacy data obtained in the current trial to establish the likelihood function of the data;

[0025] S2.6.3. Combining historical drug efficacy data with drug efficacy data obtained in the current trial, use Bayes’ theorem to combine prior distribution and likelihood function to construct the joint posterior distribution of parameters;

[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 a possible implementation of the inspection method, the process of using the Gibbs sampling algorithm to perform sampling calculation is as follows:

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

[0029] S2.7.2. For each parameter, fix the remaining 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 several times until the conditional posterior distribution of the parameter is stable, that is, take the stable posterior sample.

[0031] The technical scheme used in the embodiment of the present application can achieve the following technical effects: when the efficacy data obeys the skewed normal distribution, a frequency method and a Bayesian method are provided to calculate the probability of making a first-type error, specifically: in the absence of historical drug efficacy data, the frequency method is used to construct an approximate test statistic when the efficacy data obeys the skewed normal distribution using the drug efficacy data obtained in the current trial, and the moment estimator of the unknown parameter is given, and the Bootstrap sampling algorithm is used to perform sample sampling and construct the Bootstrap test statistic, and then the probability of making a first-type error is calculated. In the case of the existence of historical drug efficacy data, the Bayesian method is used to obtain the posterior distribution of the parameter by combining the historical drug efficacy data and the drug efficacy data obtained in the current trial, and the posterior probability obtained by sampling and calculating from the posterior distribution by the Gibbs sampling algorithm is used as the probability of making a first-type error; thus, the non-inferiority of the new drug is judged based on the probability of making a first-type error, thereby solving the problem of judging whether the new drug is non-inferior to the standard drug when the efficacy data obeys the skewed normal distribution. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] The accompanying drawings exemplarily illustrate the embodiments and constitute a part of the specification, and together with the text description of the specification, are used to explain the exemplary implementation of the embodiments. The embodiments shown are for illustrative purposes only and do not limit the scope of the claims. In all drawings, the same reference numerals refer to similar but not necessarily identical elements.

[0033] Figure 1 A flowchart of a method for testing the non-inferiority of a new drug when the efficacy data in this application obeys a skewed normal distribution;

[0034] Figure 2A flowchart for calculating the probability of making a Type I error using the frequency method in this application;

[0035] Figure 3 A flowchart for calculating the probability of making a Type I error using the Bayesian method in this application;

[0036] Figure 4 is a frequency histogram with a probability density curve;

[0037] Figure 5 Characteristics of the three data sets: new drug, standard drug, and placebo. DETAILED DESCRIPTION

[0038] In order to make the purpose, technical solutions and advantages of the present application more clearly understood, the present application is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not intended to limit the present application. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in the field without making creative work are within the scope of protection of the present application.

[0039] It should be noted that the descriptions involving "first", "second", etc. in the embodiments of the present application are only for descriptive purposes and cannot be understood as indicating or implying their relative importance or implicitly indicating the number of technical features indicated. Therefore, the features defined as "first" and "second" may explicitly or implicitly include at least one of the features. In addition, the technical solutions between the various embodiments can be combined with each other, but they must be based on the ability of ordinary technicians in the field to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be deemed that such combination of technical solutions does not exist and is not within the scope of protection required by this application.

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

[0041] First, the following terms are explained:

[0042] Bootstrap sampling algorithm: The Bootstrap method is a numerical simulation technique that estimates the distribution of sample statistics through repeated sampling. Its core idea is to use the original data set for multiple random samplings, and each time the sample is drawn to simulate the sample distribution of the population, so as to obtain the estimated value of the sample statistic and the corresponding confidence interval.

[0043] Gibbs sampling algorithm: is a special Markov chain Monte Carlo (MCMC) method for sampling from a multivariate joint distribution, especially for sampling from the posterior distribution. When direct sampling is difficult, Gibbs sampling updates the parameters one by one in an iterative manner and uses the conditional distribution for sampling.

[0044] Test statistic: A test statistic is a statistic used to calculate hypothesis testing. Under the null hypothesis, this statistic follows a given probability distribution, but not under another hypothesis. Therefore, if the value of the test statistic falls outside the critical value of the above distribution, it can be considered that the above null hypothesis may not be correct. The main function of the test statistic is to play a key 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 the population parameter, thereby making statistical inferences.

[0045] The probability of making a type I error refers to the probability of incorrectly rejecting the null hypothesis in a hypothesis test, that is, the probability that the test results incorrectly support the alternative hypothesis when the null hypothesis is true.

[0046] As described in the background technology, in view of the actual situation in my country, this application designs a technical solution for testing the non-inferiority of new drugs in three-arm non-inferiority clinical trials when the efficacy data obey a skewed normal distribution. It should be noted that the non-inferiority of new drugs described in this article is an evaluation of the biological non-inferiority of new drugs, which can also be referred to as biological non-inferiority, or non-inferiority, etc., which have the same meaning and are all expressed as new drug non-inferiority in the following text.

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

[0048] Embodiment 1

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

[0050] See also Figure 1 In the embodiments of the present application, the test method for non-inferiority of a new drug when the efficacy data obeys a skewed normal distribution includes:

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

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

[0053] The efficacy data is determined based on the drug efficacy indicators corresponding to the new drug. 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 has two control groups, namely 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 going to replace.

[0055] In the three-arm non-inferiority hypothesis test model designed for a three-arm non-inferiority clinical trial, the null hypothesis H0 is set to: μ E -θμ R -(1-θ)μ P ≤0, the alternative hypothesis H1 is set as: μ E -θμ R -(1-θ)μ P >0,μ E , μ R , μ P They represent the mean values ​​of the efficacy indicators of 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 making the first type of error:

[0057] In the three-arm non-inferiority hypothesis test model, the frequency method or Bayesian method is used to calculate the drug efficacy data of the three groups of test results to obtain the probability of making a first-type error, where the probability of making a first-type error is the probability of the alternative hypothesis H1 occurring when the null hypothesis H0 is established.

[0058] Generally speaking, before the non-inferiority test of a new drug, the technicians have known the historical data related to the efficacy of the new drug in advance, that is, there are a large number of data samples at this time, but there are also situations where the historical data related to the efficacy of the new drug has not been tested before, that is, the data sample size is small at this time. Therefore, for the above two situations, the frequency method or the Bayesian method is selected for probability calculation.

[0059] Specifically, in the absence of historical data on drug efficacy, the frequency method is used to construct an approximate test statistic when the efficacy data obeys a skewed normal distribution using the drug efficacy data obtained in the current trial, and the moment estimator of the unknown parameter is given. The Bootstrap sampling algorithm is used to perform sample sampling and construct the Bootstrap test statistic, and then the probability of making a first-type error is calculated. In the presence of historical data on drug efficacy, the Bayesian method is used to obtain the posterior distribution of the parameter by combining the historical data on drug efficacy and the drug efficacy data obtained in the current trial, and the posterior probability calculated by sampling from the posterior distribution using the Gibbs sampling algorithm is used as the probability of making a first-type error.

[0060] S3. Determination of non-inferiority of new drugs:

[0061] 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, where the reference value is a pre-given clinically significant critical value.

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

[0063] In the embodiment of the present application, for the case where the efficacy data obeys the skewed normal distribution, a frequency method and a Bayesian method are provided to calculate the probability of making a first type error, specifically: in the absence of historical drug efficacy data, the frequency method is used to construct an approximate test statistic when the efficacy data obeys the skewed normal distribution using the drug efficacy data obtained in the current trial, and the moment estimator of the unknown parameter is given, and the Bootstrap sampling algorithm is used to perform sample sampling and construct the Bootstrap test statistic, and then the probability of making a first type error is calculated. In the case where there is historical drug efficacy data, the Bayesian method is used to obtain the posterior distribution of the parameter by combining the historical drug efficacy data and the drug efficacy data obtained in the current trial, and the Gibbs sampling algorithm is used to sample from the posterior distribution. According to the Bayesian decision criterion, the probability that the alternative hypothesis is true is used as the probability of making a first type error; thus, the non-inferiority of the new drug is judged based on the probability of making a first type error, thereby solving the problem of judging whether the new drug is non-inferior to the standard drug when the efficacy data obeys the skewed normal distribution.

[0064] In order to facilitate a more detailed explanation of the technical solutions of the frequency method and the Bayesian method below, the drug efficacy data in the three-arm non-inferiority hypothesis test model in this application is first explained.

[0065] The three-arm non-inferiority test includes three drug groups, the experimental drug (i.e., 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 represents the therapeutic effect value of the jth individual in the kth group, SN(μ,σ 2 ,λ) means the mean is μ and the variance is σ 2 Skewed normal distribution with skewness λ, μ k represents the efficacy value of group k, n k represents the sample size of the kth group, represents the variance of the kth group, λ k represents the skewness of the kth group, k = E, R, P; j = 1, 2, ..., n k Without loss of generality, we assume μ k The larger the value of , the better the treatment effect of the kth group.

[0067] Embodiment 2

[0068] Next, the technical solution for calculating the probability of making a type I error using the frequency method is explained for the case where there is no historical data on the efficacy of a new drug, as follows:

[0069] like Figure 2 As shown in the figure, the location parameter, scale parameter and skewness parameter corresponding to the drug efficacy data obeying the skewed normal distribution are used as the test parameters. The specific steps of using the frequency method to calculate the probability of making the first type of error include:

[0070] S2.1. Use the efficacy data of the three groups of test results in the current trial to calculate the parameter moment estimators corresponding to the location parameter, scale parameter and skewness parameter when the drug efficacy data obeys the skewed normal distribution.

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

[0072] like Then the parameter The moment estimator of 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 Respectively represent Xk The sample mean, sample second-order center distance and sample third-order center distance of .

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

[0076] Specifically, the calculation formula of the test statistic constructed by the central limit theorem is as follows:

[0077]

[0078] S2.3. The parameter distance estimator calculated in S2.1 is sampled by the Bootstrap sampling algorithm to obtain Bootstrap sample data, and the moment estimates of the corresponding location parameter, scale parameter and skewness parameter in the Bootstrap sample data are calculated.

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

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

[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 calculation formula for 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, the parameter estimates can be obtained from the efficacy data and the Bootstrap sampling samples, and the corresponding test statistics T and T B The corresponding test statistic can be obtained from the calculation of T B The probability of ≥T can be used to get the P value of the test.

[0087] The probability value obtained in this application is expressed as: p = P (T B ≥T), that is, the probability when the bootstrap test statistic is greater than or equal to the test statistic of the non-inferiority hypothesis, that is, sample T B The frequency of occurrence of ≥T, and then the probability of making a type I error is determined based on this probability. If p < α, the null hypothesis H0 is rejected at the significance level α, that is, the experimental treatment is considered to be non-inferior to the positive control treatment.

[0088] In the technical solution of the embodiment of the present application, when the efficacy data obeys a skewed normal distribution, the test statistic constructed according to the central limit theorem does not obey a specific distribution function form when the sample size is small, so the probability of the occurrence of the alternative hypothesis cannot be directly calculated by the probability distribution function form of the test statistic. Through the Bootstrap sampling method, the frequency of sample occurrence can be used instead of the probability of the occurrence of the alternative hypothesis to judge the biological non-inferiority of the new drug. Therefore, the approximate test statistic based on the Bootstrap method can more accurately judge the non-inferiority of the new drug relative to the standard drug.

[0089] Embodiment 3

[0090] The Bayesian method is a statistical method that updates our understanding of an unknown parameter by combining prior knowledge with observed data. The core 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, the technical solution for calculating the probability of making a type I error using the Bayesian method is explained for the case where the new drug has historical data on drug efficacy, as follows:

[0092] like Figure 3 As shown in the figure, the location parameter, scale parameter and skewness parameter corresponding to the drug efficacy data obeying the skewed normal distribution are taken as unknown parameters. The specific steps of using the Bayesian method to calculate the probability of making the first type of error include:

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

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

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

[0096] Prior distribution is our assumption about the parameters before observing the data.

[0097] S2.6.2. Observe the drug efficacy data obtained in the current trial and establish the likelihood function of the data.

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

[0099] S2.6.3. The historical data on the efficacy of the combination and the drug efficacy data obtained in the current trial are used to construct the joint posterior distribution of the parameters using the Bayesian theorem combined with the prior distribution and the likelihood function.

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

[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 from the conditional posterior distribution of the parameter to obtain the posterior samples of the location parameters corresponding to the three sets of test results.

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

[0104] S2.7.1 Give initial values ​​of location parameters, scale parameters and skewness parameters, i.e., 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 parameter is stable, that is, take the stable posterior sample.

[0107] S2.8. Calculate the probability that the alternative hypothesis is true based on the posterior samples of the location parameters corresponding to the three sets of test results and use it as the probability of committing a type I error.

[0108] In the Bayesian framework, the Bayesian decision criterion for the 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 , then the new drug is said to be non-inferior to the standard drug.

[0110] In the technical solution of the embodiment of the present application, the Bayesian method allows the integration of historical data information in the probability calculation. This means that if there is already some information or assumptions about the problem, this knowledge can be integrated into the analysis in the form of a priori probability distribution. This is particularly useful for situations where sample data is sparse or incomplete, and sample information can be updated to provide a more comprehensive analysis.

[0111] Exemplary, for a three-arm non-inferiority hypothesis test 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 They represent the mean efficacy of the experimental group, positive control group and placebo group respectively, and θ is a given constant between [0.5,1].

[0114] This application provides the test process of this hypothesis test as follows:

[0115] According to the random representation of skewed normal distribution introduced by Azzalini (1986), the above three-arm non-inferiority hypothesis test model is rewritten as a stratified model in the following form:

[0116]

[0117] in Indicates that the mean is μ k +σ k δ k U kj , the variance is Normal distribution. TN(0,1)I{u kj >0} is a normal distribution with a mean of 0 and a variance of 1, but its value is truncated at the part greater than 0, that is, only positive values ​​are allowed. kj is a hidden 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 be rewritten as

[0119]

[0120] In Bayesian estimation, the choice of prior distribution is crucial to determine the posterior distribution. In this case, we can consider that the location parameter and scale parameter follow the Jeffreys prior, which is an uninformative prior used to express parameter uncertainty when prior information is lacking. The prior distribution of the skewness parameter follows the t distribution because it has a 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 parameter and scale parameter follow the Jeffreys prior, the prior distribution of the skewness parameter follows the t distribution, that is:

[0121]

[0122] Where π(λ k ) is the position parameter is 0 and the scale parameter is b k , the degrees of freedom are d k The t distribution of t(0, b k , d k ) probability density function. k , σ k , k The meaning is the same as before. When k Follow Jefferys prior distribution; when When k Obey uniform distribution. k , d k ) is expressed in the following hierarchical form:

[0123]

[0124] where λ k The mean is 0 and the variance is The normal distribution of k The shape parameter is The scale parameter is Gamma distribution. To ensure μ R >μ P The sensitivity is satisfied, that is, it is guaranteed that the positive control drug is an effective drug compared with the placebo. From formulas (1) and (2), we can know that the joint prior distribution of the parameters is:

[0125]

[0126] The Bayesian formula can be used to obtain the joint posterior distribution of parameters when the sensitivity condition is met:

[0127]

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

[0129] The Gibbs sampling method is used to sample the joint posterior distribution. The sampling process is as follows:

[0130] 1. Set the initial value

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

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

[0133] in:

[0134] (A.1) is:

[0135] in, This means that given some conditional parameters and data X k Next, parameters The conditional distribution of The mean is variance The normal distribution of .

[0136] (A.2) is:

[0137] in, This means that given some conditional parameters and data X k Next, parameters The conditional distribution of The mean is The variance is The truncated normal distribution is truncated at the part greater than 0, that is, only the positive part is allowed.

[0138] (A.3) is:

[0139] in, This means that given some conditional parameters and data X k Next, parameters The conditional distribution of The mean is The variance is The normal distribution of .

[0140] (A.4) is:

[0141] in, This means that given some conditional parameters and data X k Next, parameters The conditional distribution of The mean is The variance is The normal distribution of .

[0142] (A.5) is:

[0143] in, This means that given some conditional parameters and data X k Next, parameters The conditional distribution of The shape parameter is The variance is The normal distribution of .

[0144] In the above formulas (A.1)-(A.5), l and l+1 represent the number of iterations, μ k , β k , η k ,ω k ,u kj The meaning is the same as before, represents the sample drawn at the lth iteration. After T samples are drawn, It can be viewed as the joint posterior distribution p(μ k , β k , η k ,ω k , U k |X) extracted. is from p(μ k , β k , η k ,ω k , U k |X) extracted from k The value of .

[0145] In the Bayesian framework, the Bayesian decision criterion for the 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 , then the new drug is said to be non-inferior to the standard drug.

[0148] Finally, for ease of understanding, the test method for non-inferiority of new drugs when the efficacy data proposed in this application obeys a skewed normal distribution is applied to HIV non-inferiority testing, with the CD4 antigen on the surface of patient cells as an indicator for judging the therapeutic effect. Generally speaking, the higher the CD4 count, the healthier the immune system.

[0149] The inspection steps are as follows:

[0150] First, we compare the logCD4 of the three drug groups. The data are the measurement results of patients in groups 7 and 8. The total sample size of the data set is n = 501 cases, and the number of patients in each group is n E =n R =n P =167. zA400d is the new drug (experimental group xExp). zA400dA400n is the standard drug (control group xRef), and zA225z is the placebo (placebo group xPla). The data are described as follows: The overall parameter values ​​of the three groups of data are unknown. Figure 4 From the frequency histogram with probability density curve, we can see that the skewness characteristics of the three sets of data are not obvious. Figure 5 The characteristics of the three sets of data are given. The means of the three sets of data are very similar, and the variances of the three sets of data are also very similar. The skewness data obtained from the skewness formula shows that the three sets of data are all negatively skewed.

[0151] Secondly, in order to understand the skewed normality of the test data, the Shapiro-Wilk test, Kolmogorov-Smirnov test and Cramer-von Mises test were first used. The p-values ​​of the normality test were 0.000391639, 0.000791679 and 0.003596725 respectively. We can get that at the 5% significance level, the experimental group is not normally distributed. Further, using the chi-square goodness of fit test, it is assumed that the original data is skewed normally distributed. The value of the test statistic is So the null hypothesis will not be rejected at the 5% significance level. Therefore, the distribution of the experimental group can be considered to be approximately skewed normal.

[0152] Again, in the same way, in turn, we get the normality test p values ​​of the control group are 0.005087814, 0.009443264, 0.01567969, and the order p values ​​of the normality test of the placebo are 0.000335353, 0.001029581, 0.004230334, so the control group and the placebo are not normally distributed, and the test statistic of the control group is Placebo test statistic It can be considered that both the control group and the placebo are approximately skewed normal.

[0153] Finally, the test method is used to test the three groups of drugs in the case data.

[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, so the null hypothesis can be accepted.

[0156] Next, we analyze the case data under the Bayesian framework and calculate the probability of making a type I error under the above hypothesis testing problem to be 0.065, so we accept the null hypothesis.

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

[0158] Obviously, those skilled in the art should understand that the modules or steps of the above-mentioned embodiments of the present application can be implemented by general-purpose computer devices, they can be concentrated on a single computer device, or distributed on a network composed of multiple computer devices, optionally, they can be implemented by executable program codes of computer devices, so that they can be stored in a storage device and executed by the computer device, and in some cases, the steps shown or described can be executed in a different order from that herein, or they can be made into individual integrated circuit modules, or multiple modules or steps therein can be made into a single integrated circuit module for implementation. In this way, the embodiments of the present application are not limited to any specific combination of hardware and software.

[0159] It should be noted that the above are only preferred embodiments of the present application, and the patent protection scope of the present application is not limited thereto. Any equivalent structure or equivalent process transformation made using the contents of the specification and drawings of the present application, or directly or indirectly applied in other related technical fields, are also included in the patent protection scope of the present application.

Claims

1. A method for testing the non-inferiority of a new drug when the efficacy data obeys a skewed normal distribution, characterized in that: The test method is applicable to setting the null hypothesis H0 in the three-arm non-inferiority hypothesis test model to: E -θμ R -(1-θ)μ P ≤0, the alternative hypothesis H1 is set as: μ E -θμ R -(1-θ)μ P >0 in a three-arm non-inferiority clinical trial to test the non-inferiority of a new drug, where μ E , μ R , μ P Respectively represent the means of efficacy indicators corresponding to the treatment effects of the new drug group, the standard drug control group, and the placebo control group, θ is the non-inferiority margin, and its value is between [0.5, 1]; The inspection method comprises: S1. Acquisition of drug efficacy data in clinical trials: In a three-arm non-inferiority clinical trial, based on the drug efficacy index, the drug efficacy data of the three groups of clinical trial results of the new drug group, the standard drug control group and the placebo control group were obtained respectively, and the efficacy data followed a skewed normal distribution; S2. Calculation of the probability of making the first type of error: In the three-arm non-inferiority hypothesis test model, the frequency method or Bayesian method is used to calculate the drug efficacy data of the three groups of test results to obtain the probability of making a first-type error, where the probability of making a first-type error is the probability of the alternative hypothesis H1 occurring when the null hypothesis H0 is established; In the absence of historical data on drug efficacy, the frequency method is used to construct an approximate test statistic when the efficacy data obeys a skewed normal distribution using the drug efficacy data obtained in the current trial. Given the moment estimator of the unknown parameter, the Bootstrap sampling algorithm is used to perform sample sampling and construct the Bootstrap test statistic, and then the probability of making a first-type error is calculated. In the presence of historical data on drug efficacy, the Bayesian method is used to obtain the posterior distribution of the parameter by combining the historical data on drug efficacy and the drug efficacy data obtained in the current trial, and the Gibbs sampling algorithm is used to sample from the posterior distribution. According to the Bayesian decision criterion, the probability that the alternative hypothesis is true is used as the probability of making a first-type error. S3. Determination of non-inferiority of new drugs: 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, where the reference value is a pre-given clinically significant critical value.

2. The inspection method according to claim 1, characterized in that: When the drug efficacy data obeys the skewed normal distribution, the corresponding location parameter, scale parameter and skewness parameter are taken as unknown parameters. The specific steps of using the frequency method to calculate the probability of making the first type of error include: S2.

1. Use the efficacy data of the three groups of test results in the current trial to calculate the first parameter moment estimator corresponding to the location parameter, scale parameter and skewness parameter when the drug efficacy data obeys the skewed normal distribution; S2.2, Substitute the moment estimators of the location parameter, scale parameter, and skewness parameter into the approximate test statistic constructed according to the central limit theorem to obtain the test statistic for the non-inferiority hypothesis; S2.3, sampling the first parameter moment estimator calculated in S2.1 by the Bootstrap sampling algorithm to obtain Bootstrap sample data, and calculating the moment estimates of the corresponding location parameter, scale parameter and skewness parameter in the Bootstrap sample data; 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 inspection method according to claim 1, characterized in that: When the drug efficacy data obeys the skewed normal distribution, the corresponding location parameter, scale parameter and skewness parameter are used as unknown parameters. The specific steps of using the Bayesian method to calculate the probability of making the first type of error include: S2.

6. Construct the conditional posterior distribution of the location parameter, scale parameter and skewness parameter based on the historical drug efficacy data and the drug efficacy data obtained in the current trial; S2.7, use the Gibbs sampling algorithm to sample from the conditional posterior distribution of the parameter to obtain the posterior samples of the location parameters corresponding to the three groups of test results; S2.

8. Calculate the probability that the alternative hypothesis is true based on the posterior samples of the location parameters corresponding to the three sets of test results and use it as the probability of committing a type I error.

4. The inspection method according to claim 3, characterized in that: The specific steps of constructing the posterior distribution include: S2.6.

1. Determine the prior distribution of location parameter, scale parameter and skewness parameter based on historical data of drug efficacy; S2.6.

2. Observe the drug efficacy data obtained in the current trial to establish the likelihood function of the data; S2.6.

3. The historical efficacy data of the combination and the efficacy data of the drug obtained in the current trial are used to construct the joint posterior distribution of the parameters using the Bayesian theorem combined with the prior distribution and the 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 inspection method according to claim 3 or 4, characterized in that: The process of sampling calculation using the Gibbs sampling algorithm is as follows: S2.7.1, give the initial values ​​of the location parameter, scale parameter and skewness parameter, i.e., initialization parameters; S2.7.

2. For each parameter, fix the remaining 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 several times until the conditional posterior distribution of the parameter is stable, that is, take the stable posterior sample.

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