Meta analysis method for dichotomous outcome
By introducing HT models of hypergeometric distribution and thick tail distribution in meta-analysis, the bias and instability problems of traditional methods in dealing with double "0" events and high heterogeneity are solved, and a more accurate and reliable risk assessment of drug adverse reactions is achieved.
Patent Information
- Application Number
- CN202510329543.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-03-20
- Publication Date
- 2025-06-20
AI Technical Summary
Traditional meta-analysis methods have bias and instability in dealing with double "0" events and high heterogeneity, making it difficult to accurately evaluate the risk of adverse drug reactions.
A HT model based on hypergeometric distribution and thick tail distribution is proposed. By constructing a random effect model, the OR value of the occurrence risk of the treatment group and the control group is described, and the model parameters are solved through the theoretical proof of mathematical statistics and algorithms.
The HT method can effectively deal with dual "0" events and high heterogeneity problems, improve the accuracy and confidence of analysis results, and provide more reliable support for risk assessment of adverse drug reactions.
Smart Images
Figure CN120183729A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of biomedical technology, and in particular, to a Meta-analysis method for binary outcomes. Background Art
[0002] In recent years, with the wide application of immunotherapy in cancer treatment, especially the use of PD-1 and PD-L1 drugs, although the clinical efficacy is significant and the survival period of patients is extended, the problems of drug-related adverse reactions have gradually attracted attention. The adverse reactions of these drugs not only involve the skin and endocrine systems, but also include various symptoms in the cardiovascular and respiratory systems, such as arrhythmia, dyspnea, pneumonia, and thyroid dysfunction. Therefore, accurately evaluating and analyzing the risk of adverse reactions caused by PD-1 / PD-L1 drugs is of great significance in clinical practice.
[0003] Traditional Meta-analysis methods, such as the Mantel-Haenszel (MH) method, Peto method, and random effects model based on the normal distribution, etc., are widely used in binary data analysis. These methods perform effect combination based on fixed effect or random effect models. However, when there is high heterogeneity between studies, or when double "0" events (i.e., no events are observed in both the treatment group and the control group) are included, these traditional methods will show great limitations. For example, the MH method is prone to bias when dealing with highly heterogeneous data, while the Peto method cannot obtain effective results when dealing with double "0" events and usually requires continuity correction or exclusion of such studies, which will lead to information loss and introduce additional analysis bias.
[0004] In recent years, some new methods have been proposed in Meta-analysis, such as the Binomial-Normal (BN) model and Hypergeometric-Normal (HN) model. These methods attempt to solve the challenges brought by high heterogeneity and double "0" events, but there are still some deficiencies in practical applications. For example, the model has certain requirements for the stability and accuracy of parameter estimation, especially in the case of small sample size or low outcome incidence. In these cases, the assumption based on the normal distribution may no longer be reasonable, resulting in large deviations in the model results. Summary of the Invention
[0005] The present invention proposes a Meta-analysis method for binary outcomes, aiming to solve the bias and instability problems faced by traditional methods in dealing with double "0" events and high heterogeneity, improve the accuracy and confidence of the analysis results, and thus provide more reliable statistical support for the risk assessment of drug adverse reactions.
[0006] The technical solution adopted by the present invention to solve its technical problems is:
[0007] A Meta - analysis method for binary classification outcomes, comprising the following steps:
[0008] S1. Establish an HT model by jointly using the hypergeometric distribution and the heavy - tailed distribution;
[0009] S2. Estimate and statistically infer the parameters of the HT model;
[0010] S3. Use the theoretical proof method of mathematical statistics to study the theoretical properties of the parameter estimation of the HT model;
[0011] S4. Use an algorithm to solve the corresponding model parameters.
[0012] Furthermore, the specific steps of S1 are as follows:
[0013] S11. Process binary data using the hypergeometric distribution;
[0014] S12. Use the heavy - tailed distribution to describe the distribution of high heterogeneity;
[0015] S13. Based on the above - mentioned distributions, construct a random - effects HT model to describe the OR value of the occurrence risk of the concerned outcome in the treatment group and the control group.
[0016] Furthermore, the specific steps of S11 are as follows: For each study, construct a 2×2 contingency table, record the number of events and the sample size in the treatment group and the control group; Calculate the conditional probability of the effect index for each study based on the hypergeometric distribution without performing data continuity correction.
[0017] Furthermore, the specific steps of S12 are as follows: Select an appropriate distribution degree of freedom according to the degree of data heterogeneity to accurately depict the variation between effect values; Determine the distribution degree of freedom in a data - driven manner to maximize the likelihood function value of the model parameter estimation.
[0018] Furthermore, the specific steps of S2 are as follows:
[0019] S21. Take a logarithmic transformation of the constrained parameters in the model to handle the constraint conditions of the constrained parameters;
[0020] S22. Based on the log - likelihood function, use the maximum - likelihood estimation method to determine the model parameters.
[0021] Furthermore, the specific steps of S21 are as follows: Convert the constrained parameters into an unconstrained parameter form for unconstrained optimization calculation; After obtaining the estimated values of the unconstrained parameters, convert them back to the original parameter form.
[0022] Furthermore, the specific steps of S3 are as follows:
[0023] S31. Prove the consistency of the parameter estimation of the HT model in large samples;
[0024] S32. Prove the asymptotic normality of the parameter estimation of the HT model in large samples.
[0025] Furthermore, the specific steps of S4 are as follows:
[0026] S41. Use the Newton-Raphson algorithm to obtain the optimal solution through repeated iterations;
[0027] S42. Adopt a data-driven approach to give an algorithm for selecting the degrees of freedom of the definite distribution.
[0028] Furthermore, in step S41, when using the Newton-Raphson algorithm for iterative calculation, continuously update the parameter estimation value until it converges to the optimal solution; in the iterative process, use the Gaussian-Hermite Quadrature method to calculate the integral operation in the partial derivative to improve the calculation efficiency and accuracy.
[0029] Furthermore, in step S42, under the given conditions, calculate the parameter estimation values and log-likelihood function values of the model under different degrees of freedom; select the degree of freedom that maximizes the log-likelihood function value as the optimal degree of freedom.
[0030] Compared with the prior art, the beneficial effects of the present invention are as follows:
[0031] First, the HT method is based on the hypergeometric distribution, can solve the Meta-analysis of binary outcomes containing "0" events, does not require data continuity correction, and can make full use of data information. Therefore, different from the T-type regression model, whose research outcome index is a directly observable variable, the HT model solves the effect index of each study by maximizing the likelihood function with parameters, that is, θ i The unobservable situation.
[0032] Second, the HT method can handle the Meta-analysis problem with high inter-study heterogeneity. When there is high heterogeneity, the existing methods have large estimation biases and cannot obtain accurate estimates. The HT method is based on the assumption of a thick-tailed t-distribution and determines the appropriate degrees of freedom based on a data-driven approach to solve the problem of high heterogeneity.
[0033] Third, the HT method has a certain robustness and can be used for small sample Meta-analysis research. Compared with the normal distribution, the thick-tailed characteristic of the t-distribution enables it to weaken or reduce the interference of outliers, and can accurately grasp the data characteristics even in small samples. Description of the Drawings
[0034] Figure 1Graph for comparing average bias and median bias results under different simulation settings;
[0035] Figure 2 Graph for comparing coverage rate results under different simulation settings;
[0036] Figure 3 Graph for comparing RMSE results under different simulation settings. Specific implementation manners
[0037] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention. The present invention will be further described in conjunction with the drawings and embodiments:
[0038] A Meta-analysis method for binary outcomes, the core principle of which is to construct a new Meta-analysis model, namely the HT model, by combining the hypergeometric distribution and the t-distribution for analyzing the combined effect of binary outcomes, including the following steps:
[0039] S1. Establish the HT model by jointly using the hypergeometric distribution and the heavy-tailed distribution;
[0040] S2. Estimate and statistically infer the parameters of the HT model;
[0041] S3. Use the theoretical proof method of mathematical statistics to study the theoretical properties of the parameter estimation of the HT model;
[0042] S4. Use an algorithm to solve the corresponding model parameters.
[0043] Furthermore, the specific steps of S1 are as follows:
[0044] S11. Use the hypergeometric distribution to process binary data; the hypergeometric distribution is used to process the binary data of each study, including the statistics of the number of events and the sample size. In this way, when calculating the probability, no data continuity correction is required, and the model can handle even if there are double "0" events.
[0045] S12. Use the heavy-tailed distribution to describe the distribution of high heterogeneity; in order to handle the heterogeneity between different studies, the HT model uses the t-distribution to characterize the variation between effect values. Due to its heavy-tailed property, the t-distribution can avoid estimation bias in high-heterogeneity data, thereby maintaining the reliability of the confidence interval.
[0046] S13. Based on the above distribution, construct a random-effect HT model to describe the OR value of the occurrence risk of the concerned outcome in the treatment group and the control group.
[0047] Scheme refinement. The specific steps of S11 are as follows: For each study, construct a 2×2 contingency table, and record the number of events and sample size in the treatment group and the control group; Calculate the conditional probability of the effect index for each study based on the hypergeometric distribution without performing continuity correction on the data.
[0048] Scheme refinement. The specific steps of S12 are as follows: Select an appropriate distribution degree of freedom according to the heterogeneity degree of the data to accurately describe the variation between effect values; Determine the distribution degree of freedom in a data-driven manner to maximize the likelihood function value of model parameter estimation.
[0049] Specifically, when conducting a Meta-analysis, first collect the 2×2 contingency table data of each study. Assume that a total of n studies are included in the Meta-analysis, y 1i and y 0i respectively represent the number of events in the treatment group and the control group of the i-th study, n 1i and n 0i respectively represent the number of people in the treatment group and the control group of the i-th study. Then the 2×2 contingency table of each study is as follows:
[0050] Table 1 2×2 contingency table of events occurring in the treatment group and the control group
[0051]
[0052] Specifically, let x i represent the observed data of the i-th study, that is, x i =(y 1i ,y 0i ,n 1i ,n 0i ), θ i represent the log OR value of the occurrence risk of the outcome of interest in the treatment group and the control group in the i-th study, θ represents the combined effect value. Consider the following random effects HT model:
[0053] θ i =θ + σ * Z i , σ * >0, Z i ~t(v).
[0054] Furthermore, the specific steps of S2 are as follows:
[0055] S21. Take a logarithmic transformation of the constrained parameters in the model to handle the constraint conditions of the constrained parameters;
[0056] S22. Based on the log-likelihood function, use the maximum likelihood estimation method to determine the model parameters.
[0057] The solution is refined, and the specific steps of S21 are as follows: Convert the constrained parameters into an unconstrained parameter form for unconstrained optimization calculation; after obtaining the estimated value of the unconstrained parameters, convert them back to the original parameter form.
[0058] Specifically, the present invention first gives the conditional probability of the observed data under the condition of a given θi based on the hypergeometric distribution model, that is, the likelihood function containing the parameter θ i
[0059]
[0060] where M L = max(0, Y i - n 1i ), M U = min(Y i , n 1i ).
[0061] Furthermore, since the value range of σ * in the HT model is σ * > 0, the value range of the parameters is restricted during the model solution process. To avoid directly calculating the constrained optimization problem, the following transformation is made for σ * , that is, let σ = log(σ * ). Denote β = (θ, σ) T , and then the likelihood function and log-likelihood function of the model are derived as follows:
[0062]
[0063] According to the idea of maximum likelihood estimation, find the model parameters that make the log-likelihood function reach the maximum value, that is
[0064]
[0065] Then it is called the maximum likelihood estimate of the model parameter β.
[0066] Furthermore, the specific steps of S3 are as follows:
[0067] S31. Prove the consistency of the HT model parameter estimation in large samples;
[0068] S32. Prove the asymptotic normality of the HT model parameter estimation in large samples.
[0069] The following are the asymptotic theoretical properties of the HT model parameter estimation in large samples to illustrate the theoretical correctness of the model used in the present invention.
[0070] Specifically, let x1, x2,..., xn is a set of independent observation samples from n studies, satisfying that p(x; β) is identifiable with respect to the parameter β, and at the true parameter β0, there is:
[0071]
[0072] The following Lemma (1) and Lemma (2) provide assistance for the proof of theoretical properties.
[0073] Lemma (1). lnp(x; β) is differentiable with respect to the parameter β.
[0074] Lemma (2). In the neighborhood of the true parameter β0, exists for all x.
[0075] Furthermore, the large-sample theoretical properties of the parameter estimation of the present invention are as described in the following Theorem (1) and Theorem (2), which respectively show the consistency and asymptotic normality of the parameters.
[0076] Theorem (1). (Consistency) Let x1, x2,..., x n be a set of samples from p(x; β), and lnp(x; β) satisfies Lemma 1. Then, when n → ∞, the likelihood equation has a solution with probability 1, and this solution is consistent with respect to β0, that is n → ∞.
[0077] Theorem (2). (Asymptotic Normality) Assume that Θ is an open region, the probability density function is p(x; β), β ∈ Θ and satisfies Lemma 2. Then there is
[0078] Furthermore, the specific steps of S4 are as follows:
[0079] S41. Use the Newton-Raphson algorithm to obtain the optimal solution through repeated iterations;
[0080] Specifically, to calculate the maximum likelihood estimate of the HT model parameters is equivalent to solving
[0081]
[0082] When solving, use the Newton-Raphson algorithm to obtain the optimal solution through repeated iterations. The iteration formula is as follows:
[0083]
[0084] where m = 0, 1, 2,....
[0085] Since the partial derivative calculation involves integral operations, Gaussian-Hermite Quadrature is used for the calculation. In numerical calculations, Gaussian-Hermite Quadrature is used to approximate the integral values of the following types:
[0086]
[0087] The basic idea is
[0088]
[0089] where t s is the root of the Hermite polynomial H s (x) (s = 1, 2, …, S)
[126] .
[0090] The iterative algorithm is shown in Algorithm 1 in detail.
[0091]
[0092] S42. A selection algorithm for determining the degrees of freedom of the distribution is given in a data-driven manner.
[0093] Specifically, in step S310, an algorithm for solving the model parameter β is given, but this is an algorithm under the condition that the degrees of freedom v is given. In fact, the degrees of freedom v of the t-distribution can be adjusted and is one of the adjustable parameters of the model.
[0094] When v is small, the t-distribution shows the characteristic of heavy tails. When v is large, the t-distribution approximates the normal distribution. Theoretically, by adjusting v, the switching between the BN method (non-approximate method) and the HT method can be achieved.
[0095] This step will give an algorithm for determining the selection of the degrees of freedom v of the t-distribution. The degrees of freedom of the t-distribution are selected in a data-driven manner, that is, under the condition that v is given, the model parameter β is calculated, and then the value of the log-likelihood function l n (β; x) is solved.
[0096] The HT model algorithm uses the idea of maximum likelihood estimation, so a set of parameter results that can make the likelihood function reach the maximum value is finally selected. The algorithm for data-driven selection of the adjustable parameter v is shown in Algorithm 2.
[0097]
[0098]
[0099] Scheme refinement. In step S41, when using the Newton-Raphson algorithm for iterative calculation, continuously update the parameter estimate until it converges to the optimal solution; during the iteration process, use the Gaussian-Hermite Quadrature method to calculate the integral operation in the partial derivative to improve the calculation efficiency and accuracy.
[0100] Scheme refinement. In step S42, under given conditions, calculate the model parameter estimates and log-likelihood function values at different degrees of freedom; select the degree of freedom that maximizes the log-likelihood function value as the optimal degree of freedom.
[0101] The present invention is further illustrated by the following numerical simulation examples, and the performance of the HT model is evaluated through horizontal comparison.
[0102] Specifically, the generation of simulation data depends on the random effects model. Select the MH, Peto, and BN models for comparison with the HT model proposed in the present invention, and compare from the bias of model parameter estimation (Bias), confidence interval coverage rate (coverage probability, CP), and estimation error.
[0103] In terms of the parameter settings of the numerical simulation, first assume that the number of people in the treatment group and the control group is 1:1. Under the given conditions of θ0 and σ0, use
[0104] θ i = θ0 + σ0Z i , Z i ~t(v)
[0105] to randomly generate the θ i values of each group. Subsequently, under the given number of people in the treatment group and the disease incidence rate, use the hypergeometric distribution to randomly generate the number of diseased people in the experimental group, and calculate the total number of diseased people minus the number of diseased people in the experimental group according to the incidence rate and the number of people in each group to obtain the number of diseased people in the control group.
[0106] There are a total of 6 parameters that vary in the simulation study, including the number N of studies included in the Meta-analysis, the number n i of people included in each study, the effect value OR (i.e., exp(θ0)), the heterogeneity parameter σ0, the degree of freedom v of the model, and the incidence rate p of the disease in the experimental group. The specific settings are shown in Table 2.
[0107] Table 2 Parameter settings in the simulation
[0108]
[0109] The results of bias, confidence interval coverage rate, and error in this example are shown below in sequence.
[0110] (1) Bias:
[0111] Figure 1 A shows the average bias results of the effect values estimated by four simulations under different simulation settings. Figure 1 B shows the results of the median bias. The HT method performs well under various simulation settings and has the lowest estimated average bias. Among the simulation results with an incidence rate not less than 0.01, i.e., p = 0.01, 0.05, 0.1, regardless of the magnitudes of the heterogeneity parameters σ0 and v, the HT method exhibits the smallest average bias.
[0112] At a relatively high level of heterogeneity, the performances of the HT method and the BN method are relatively close. However, as the incidence rate of adverse reactions in the simulation settings increases, the overall performance of the HT method is better than that of the BN method.
[0113] The HT method performs well under various simulation settings. The MH method performs the worst under all simulation settings and has the highest estimated average bias. Among the simulation results with an incidence rate not less than 0.01, i.e., p = 0.01, 0.05, 0.1, regardless of the magnitudes of the heterogeneity parameters σ0 and v, the HT method exhibits the smallest average bias.
[0114] (2) Coverage rate:
[0115] In this embodiment, the CP value reflects the probability that the obtained 95% confidence interval contains the true value. The closer the CP value is to 95%, the more reliable the estimated statistical inference result is.
[0116] Figure 2 The coverage rate results for all simulation settings are shown. The HT method is the method with the coverage rate closest to 95% when the heterogeneity is relatively high, but it is affected by many factors. First, in the cases with a relatively high incidence rate, i.e., p = 0.1 and 0.05, the coverage rate results obtained by the HT method are better. However, when the sample size is small (i.e., the sample size included in the Meta-analysis is small, n = 15) and the sample heterogeneity is low, the performance of the HT method is inferior to that of the other three methods.
[0117] The HT method performs relatively well in the case of a low incidence rate but relatively high heterogeneity. The BN method is slightly better than the HT method. When the heterogeneity and the incidence rate are relatively low, the CP value of the HT method deviates relatively far from 95%.
[0118] (3) Error:
[0119] Figure 3 The results of the Root Mean Square Error (RMSE) are shown. Under all simulation settings, the RMSE of all models decreases significantly as the incidence rate increases; while as the variance between samples increases, the RMSE of the models increases.
[0120] Under the conditions of within - study heterogeneity and between - study heterogeneity in a given simulation, the RMSE results of the HT method and the BN method are relatively close, and for the HT method, the RMSE is slightly smaller than that of the BN method, indicating that the results of the HT method have better stability.
[0121] The present invention is modeled based on the conditional hypergeometric distribution and the t - distribution, and is used to estimate the combined effect index (log OR value) of binary outcomes, and its performance is superior to that of traditional models.
[0122] Specifically, by comparing the HT method with traditional Meta - analysis methods, the following advantages of the present invention are found:
[0123] On the one hand, when dealing with binary data, the HT method can well handle the situation of single "0" or double "0" events, which is a problem that traditional methods cannot effectively handle. When the traditional inverse - variance method deals with "0" events, the variance is 0 and the weight tends to ∞, so continuity correction is usually required, and the same is true for the MH method. The Peto method can only handle single "0" problems, and double "0" studies will be excluded, which has been proven to lose some information. The method of the present study uses the conditional hypergeometric distribution, so that even when "0" exists, the calculation of conditional probability is not affected, and thus the likelihood function of the model can be accurately obtained for estimating the effect index;
[0124] On the other hand, the HT method can better handle the problem of high heterogeneity. Traditional methods mostly consider the assumption of normal distribution when combining between - study indicators, and consider using a random - effects model to handle heterogeneity when it exists, which can solve the heterogeneity problem to a certain extent. However, when there is high heterogeneity in the data, the fitting result of the normal distribution has a large deviation. Different from the normal distribution, the t - distribution used by the HT method has the characteristics of thick tails and long tails, which can give certain calculation weights to the tail data in the calculation and has better stability. When there is large heterogeneity, the long - tailed distribution performs better than the normal distribution.
[0125] The simulation study of the embodiment shows that as the heterogeneity parameter σ increases, the bias of the simulation results of the HT method is the smallest among all study models, and the CP value is closer to 95%, showing better statistical inference accuracy.
[0126] Previous case studies have proved that when the normal assumption does not hold, the random - effects model will get completely opposite and wrong conclusions. On the contrary, some flexible distributions such as the skewed t - distribution, asymmetric Subbotin distribution, etc. are more suitable.
[0127] Applying the HT model to analyze the collected data can effectively solve problems such as double "0" events, high heterogeneity, and low incidence. In the Meta-analysis of drug adverse reactions, this model can provide more accurate combined effect estimates and give reliable confidence intervals. Finally, the output of the model includes the combined estimation results of the effect values of each study (such as odds ratio OR), the effect size of each study, and the overall effect estimation of all studies. Through statistical inference, the risk level of drug adverse reactions can be determined, providing more reliable evidence for clinical practice.
[0128] The above shows and describes the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments. The above embodiments and the descriptions in the specification only illustrate the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements all fall within the scope of protection required by the present invention. The scope of protection required by the present invention is defined by the appended claims and their equivalents.
Claims
1. A meta-analysis method for binary outcomes, characterized in that: The following steps are involved: S1. Combine hypergeometric distribution and fat-tailed distribution to establish HT model; S2, estimate and statistically infer the HT model parameters; S3. Use the theoretical proof method of mathematical statistics to study the theoretical properties of HT model parameter estimation; S4. Use the algorithm to solve the corresponding model parameters.
2. The meta-analysis method for binary outcomes according to claim 1, characterized in that: The specific steps of S1 are as follows: S11. Use hypergeometric distribution to process binary data; S12. Use fat-tailed distributions to describe distributions with high heterogeneity; S13. A random effects HT model was constructed based on the distribution to describe the OR value of the risk of occurrence of the outcome of interest in the treatment group and the control group.
3. The meta-analysis method for binary outcomes according to claim 2, characterized in that: The specific steps of S11 are as follows: construct a 2×2 contingency table for each study to record the number of events and sample size in the treatment group and the control group; calculate the conditional probability of the effect indicator of each study based on the hypergeometric distribution without the need for continuity correction of the data.
4. The meta-analysis method for binary outcomes according to claim 2 or 3, characterized in that: The specific steps of S12 are as follows: selecting appropriate distribution degrees of freedom according to the degree of heterogeneity of the data to accurately characterize the changes between effect values; determining the distribution degrees of freedom in a data-driven manner to maximize the likelihood function value of the model parameter estimation.
5. The meta-analysis method for binary outcomes according to claim 1, characterized in that: The specific steps of S2 are as follows: S21, taking logarithmic transformation of the constrained parameters in the model to deal with the restriction conditions of the constrained parameters; S22. Based on the log-likelihood function, the model parameters are determined using the maximum likelihood estimation method.
6. The meta-analysis method for binary outcomes according to claim 5, characterized in that: The specific steps of S21 are as follows: converting the constrained parameters into unconstrained parameter form to perform unconstrained optimization calculation; after obtaining the estimated value of the unconstrained parameter, converting it back to the original parameter form.
7. The meta-analysis method for binary outcomes according to claim 1, characterized in that: The specific steps of S3 are as follows: S31. Prove the consistency of HT model parameter estimates under large samples; S32. Prove the asymptotic normality of HT model parameter estimates under large samples.
8. The meta-analysis method for binary outcomes according to claim 1, characterized in that: The specific steps of S4 are as follows: S41, use Newton-Raphson algorithm to obtain the optimal solution through repeated iterations; S42. A data-driven approach is used to provide a selection algorithm for determining the distribution degrees of freedom.
9. The meta-analysis method for binary outcomes according to claim 8, characterized in that: In step S41, when the Newton-Raphson algorithm is used for iterative calculation, the parameter estimation value is continuously updated until convergence to the optimal solution; during the iteration process, the Gaussian-Hermite Quadrature method is used to calculate the integral operation in the partial derivative to improve the calculation efficiency and accuracy.
10. The meta-analysis method for binary outcomes according to claim 8 or 9, characterized in that: In step S42, under given conditions, the model parameter estimates and log-likelihood function values under different degrees of freedom are calculated; the degree of freedom that maximizes the log-likelihood function value is selected as the optimal degree of freedom.