Bayesian estimation method for longitudinal multi-stage multi-level item reaction model parameters

Abstract

The invention discloses a Bayesian estimation method for longitudinal multistage and multilevel item reaction model parameters, and belongs to the technical field of model parameter estimation. The method comprises the following steps: S1, establishing longitudinal multi-level and multi-level project reaction models, including a level 1-level reaction model, a level 2 longitudinal individual growth model and a level 3 multi-level model; s2, based on an improved Bayesian parameter estimation method, estimating all parameters of the project reaction model proposed in the step S1 by introducing auxiliary variables; and S3, verifying the effect of the improved Bayesian parameter estimation method, and comparing the improved Bayesian parameter estimation method with the traditional Bayesian parameter estimation method. According to the Bayesian estimation method for the longitudinal multi-level and multi-level item reaction model parameters, on the basis that estimation precision is guaranteed, the operation time of the QMCMC is better than that of a traditional MCMC, and the operation speed and the estimation efficiency of a longitudinal multi-level and multi-level scoring item reaction theoretical model are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of model parameter estimation, and in particular to a Bayesian estimation method for parameters of a longitudinal multi-level and multi-stage project response model. Background Art

[0002] In the test research of education and psychology, the analysis and exploration of the development trend of latent traits (such as ability) over time has always attracted the attention of researchers. Subjects may have to take multiple tests to enable researchers to effectively evaluate their latent traits and their changing trends. Item response theory is a process of obtaining subjects' responses to test items through tests, and then modeling the responses with the subjects' latent traits for research and analysis. In longitudinal data research, it is usually assumed that students' abilities change over time, which contains richer information than test research at a single time point. If a test consists of several subtests, the latent traits may have a complex structure, and the traditional item response theory model may no longer be applicable to such data. Especially when the model becomes more complex, such as a longitudinal, multi-level, multi-level scoring model, the use of traditional Bayesian algorithms (MCMC) for parameter estimation requires a long computing time and is very time-consuming. The reason is that the Metropolis-Hastings algorithm often continuously accepts and rejects recommended values ​​during the sampling process, and a large number of iterations are performed in each chain to achieve model convergence, especially when hundreds or thousands of data sets are required in simulation. For example, Wollack, Bolt, Cohen, and Lee (2002) set the number of iterations in each chain to 11,000 for estimation of the symmetric nominal reaction model (NRM; Bock, 1972), although their simulation study was relatively small, with 6 simulation conditions (50 replicates per condition) and 16 computers taking more than two months to run the Gibbs sampler. Although today's computing power is much higher than it was more than a decade ago, the time required to conduct a comprehensive simulation study using traditional MCMC methods is still a headache for many methodologists. Therefore, it is necessary to study algorithms that improve efficiency and save time. Summary of the invention

[0003] The purpose of the present invention is to provide a Bayesian estimation method for the parameters of a longitudinal multi-level and multi-level item response model. On the basis of ensuring the estimation accuracy, the running time of QMCMC is better than that of traditional MCMC, which increases the speed by more than 1 / 4, and also greatly improves the estimation efficiency of the longitudinal multi-level and multi-level scoring item response theory model.

[0004] To achieve the above object, the present invention provides a Bayesian estimation method for parameters of a longitudinal multi-level and multi-stage item response model, comprising the following steps:

[0005] S1. Establish a longitudinal multi-level item response model, including level 1 grade response model, level 2 longitudinal individual growth model, and level 3 multi-level model;

[0006] S2, based on the improved Bayesian parameter estimation method, all parameters of the item response model proposed in step S1 are estimated by introducing auxiliary variables;

[0007] S3. Verify the effect of the improved Bayesian parameter estimation method and compare it with the traditional Bayesian parameter estimation method.

[0008] Preferably, in step S1, in establishing the longitudinal multi-level multi-level item response model, for the level 1 grade response model, the probability that the i-th subject scores j points for the k-th item at the t-th time point is:

[0009]

[0010] Among them, Y tik Y represents the response of subject i to the kth item at the tth time point; tik =j means the subject scores j points, 0≤j≤m k , remember m k +1 is the type of reaction, each item has m k +1 response category, the lowest score on the kth item is 0 and the highest score is m k ;

[0011] θ ti represents the potential characteristics of the i-th subject at the t-th time point; ξ tk =(a tk , b tk1 , ..., b tkmk )′ represents the parameter vector composed of project parameters; a tk represents the discrimination parameter of item k at the tth time point; b tkj It represents the difficulty parameter of item k at the jth level at the tth time point.

[0012] Preferably, in step S1, in establishing the longitudinal multi-level multi-level item response model, the change of individual ability is represented by the level 2 longitudinal individual growth model, and the growth trajectory of individual ability is linear, as shown below:

[0013] θ ti =π 0i +π 1i d ti +e ti

[0014] Among them, the variable d tiis the time parameter of subject i at the tth moment, π represents the coefficient of the linear function, e ti Represents a random error term, with an expectation of 0 and a variance of σ 2 The normal distribution of .

[0015] Preferably, in step S1, in establishing the longitudinal multi-level multi-level item response model, the individual level model of the level 3 multi-level model is as follows:

[0016] π hi =β h0 +u hi

[0017] Among them, β hs is the fixed effect of the hth parameter, u hi represents the random residual effect of subject i at level 3, vector u=(u 0i ,u 1i ) follows a bivariate normal distribution with zero expectation and variance Ω.

[0018] Preferably, in step S2, Gibbs sampling is performed by introducing auxiliary variables to obey U(0, 1), and the specific process is as follows:

[0019] S21, when the response variable Y tik =2, the indicative function is as follows:

[0020]

[0021] S22, when the response variable Y tik =1, the indicative function is as follows:

[0022]

[0023] S23, when the response variable Y tik =0, the indicative function is as follows:

[0024]

[0025] in,

[0026]

[0027]

[0028] S24, then the joint posterior distribution containing auxiliary variables is as follows:

[0029]

[0030] Preferably, three independent auxiliary variables λ are extracted tik , ψ tik ,tik , given Ψ, Y, λ tik ~[λ tik |Ψ,Y],ψ tik ~[ψ tik |Ψ,Y],υ tik ~[υ tik |Ψ, Y], then the full conditional distribution is as follows:

[0031] When Y tik =2,

[0032] When Y tik =1,

[0033] When Y tik = 0,

[0034] Preferably, extract the item level difficulty parameter b at the tth time point tk , given Ψ, Y, λ, ψ, υ, b tk ~[b tk |Ψ, Y, λ, ψ, υ],

[0035] From 0<λ tik ≤P ★ tik2 , P ★ tik2 ≤ψ tik ≤P ★ tik1 , P ★ tik1 ≤υ tik <1, get b tk1 and b tk2 The full conditional distribution of is as follows:

[0036]

[0037]

[0038] in,

[0039]

[0040] Preferably, extract the item level difficulty parameter a at the tth time point tk , given b, θ, Y, λ, ψ, υ, a tk ~[a tk |b, θ, Y, λ, ψ, υ], from 0<λ tik ≤P ★ tik2 , P★ tik2 ≤ψ tik ≤P ★ tik1 , P ★ tik1 ≤υ tik <1, get a tk The full conditional distribution of is as follows:

[0041]

[0042] When θ>0,

[0043] When θ<0,

[0044]

[0045] When θ>0,

[0046] When θ<0,

[0047] Preferably, the ability parameter θ of the i-th subject at the t-th time point is extracted ti , given ξ, d, π, σ 2 , Y, λ, ψ, υ, θ ti ~[θti|ξ,d,π,σ 2 , Y, λ, ψ, υ], by 0<λ tik ≤P ★ tik2 , P ★ tik2 ≤ψ tik ≤P ★ tik1 , P ★ tik1 ≤υ tik <1, we get θ ti The full conditional distribution of is as follows:

[0048]

[0049] in,

[0050] Preferably, β is extracted. Given π and Ω, β ∼ [β | π, Ω], and the full conditional distribution of β is as follows:

[0051]

[0052]

[0053]

[0054] Therefore, the present invention adopts the above-mentioned Bayesian estimation method for the parameters of the longitudinal multi-level and multi-level item response model. On the basis of ensuring the estimation accuracy, the QMCMC running time is better than the traditional MCMC, the speed is increased by more than 1 / 4, and the estimation efficiency of the longitudinal multi-level and multi-level scoring item response theory model is greatly improved.

[0055] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 It is a flow chart of a Bayesian estimation method for parameters of a longitudinal multi-level and multi-level item response model of the present invention;

[0057] Figure 2 are the model parameters β, σ of the present invention 2 and the iterative trajectory of the posterior values ​​of τ; where (a) is the iterative trajectory of the posterior value of β0; (b) is the iterative trajectory of the posterior value of β1; (c) is the iterative trajectory of σ 2 The iterative trajectory of the posterior value of ; (d) is τ 00 The iterative trajectory of the posterior value of ; (e) is τ 10 The iterative trajectory of the posterior value of ; (f) is τ 11 Iterative trajectory of the posterior value of ;

[0058] Figure 3 It is a comparison diagram of the capability estimation of MCMC and QMCMC based on real data of the present invention; wherein, (a) is the capability estimation comparison of MCMC and QMCMC based on real data when t=1; (b) is the capability estimation comparison of MCMC and QMCMC based on real data when t=2; (c) is the capability estimation comparison of MCMC and QMCMC based on real data when t=3. DETAILED DESCRIPTION

[0059] In order to make the purpose, technical scheme and advantages disclosed in the embodiments of the present invention more clearly understood, the embodiments of the present invention are 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 embodiments of the present invention and are not intended to limit the embodiments of the present invention. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of this application.

[0060] like Figure 1 As shown, a Bayesian estimation method for parameters of a longitudinal multi-level and multi-level item response model includes the following steps:

[0061] S1. Establish a longitudinal multi-level item response model, including level 1 grade response model, level 2 longitudinal individual growth model, and level 3 multi-level model;

[0062] S2, based on the improved Bayesian parameter estimation method, all parameters of the item response model proposed in step S1 are estimated by introducing auxiliary variables;

[0063] S3. Verify the effect of the improved Bayesian parameter estimation method and compare it with the traditional Bayesian parameter estimation method.

[0064] Example

[0065] S1. Establish a longitudinal multi-level and multi-stage item response model.

[0066] S11, level 1 hierarchical response model;

[0067] Assume that there are K items and T measurement moments for longitudinal measurement evaluation. For level 1, the probability that the i-th subject scores j points on the k-th item at the t-th time point is:

[0068]

[0069] Among them, Y tik Y represents the response of subject i to the kth item at the tth time point; tik =j means the subject scores j points, 0≤j≤m k , remember m k +1 is the type of reaction, each item has m k +1 response category, the lowest score on the kth item is 0 and the highest score is m k ;

[0070] θ ti represents the potential trait (ability) of the i-th (i=1, ..., n) subject at the t-th (t=1, ..., T) time point; ξ tk =(a tk , b tk1 , ..., b tkmk )′ represents the parameter vector composed of project parameters; a tk (k=1, ..., K) represents the discrimination (slope) parameter of item k at the tth time point; b tkj (k=1, ..., K) represents the difficulty (intercept) parameter of item k at the jth level at the tth time point.

[0071] Assume that the difficulty parameter of each item on the level is strictly monotonically increasing: b tkg-1 <btkg tkg+1 , that is, each item has m k The difficulty parameters of each level are b tk1 , ..., b tkmk , for ease of understanding and memory, it is recorded as -∞=b tk0 tk1 <... tkmk+1 =+∞; assuming that P(Y tik =0|θ ti ,ξ tk )=1,P(Y tik =m k +1|θ ti ,ξ tk )=0, the simulation study considers m k =2, which is the case of a third-order reaction.

[0072] S12, Level 2 longitudinal individual growth model;

[0073] The change of individual ability can be represented by a two-level model in many cases. The growth trajectory of individual ability is linear and can be expressed as:

[0074] θ ti =π 0i +π 1i d ti +e ti

[0075] Among them, the variable d ti is the time parameter of subject i at the tth moment, π represents the coefficient of the linear function, e ti Represents a random error term, with an expectation of 0 and a variance of σ 2 The normal distribution of .

[0076] S13, level 3 multilevel model;

[0077] Assuming that the growth parameters of different subjects are different, and their growth trajectory parameters are determined by individual background covariates (such as family socioeconomic status and gender, etc.), the individual level model is assumed to be:

[0078] π hi =β h0 +u hi

[0079] Among them, β hs is the fixed effect of the hth parameter, u hi (h=0,1) represents the random residual effect of subject i at level 3, and vector u=(u 0i ,u 1i ) follows a bivariate normal distribution with expectation 0 and variance Ω.​​​

[0080] S2. Based on the improved Bayesian parameter estimation method, all parameters of the item response model proposed in step S1 are estimated by introducing auxiliary variables.

[0081] Sampling from a fully conditional distribution is achieved by introducing auxiliary variables. The accelerated Bayesian algorithm (QuickMarkov Chain Monte Carlo; QMCMC) uses three independent auxiliary random variables to make the estimation of all parameters in the three-level item response theory model discussed above be implemented by the Gibbs algorithm, thereby achieving the purpose of improving computational efficiency. When the item response model is a model with more than three rating levels, it is also easy to deduce the corresponding conclusion. Assuming that the three auxiliary variables λ tik , tik and tik All obey U(0,1). The specific implementation process of Gibbs sampling through auxiliary variables is as follows:

[0082] S21, when the response variable Y tik =2, the indicative function is as follows:

[0083]

[0084] S22, when the response variable Y tik =1, the indicative function is as follows:

[0085]

[0086] S23, when the response variable Y tik =0, the indicative function is as follows:

[0087]

[0088] in,

[0089]

[0090]

[0091] S24, then the joint posterior distribution containing auxiliary variables is as follows:

[0092]

[0093] Sampling sequentially from the full conditional distribution:

[0094] (1) Extract three independent auxiliary variables λ tik , ψ tik , tik , given Ψ, Y, λtik ~[λ tik |Ψ,Y],ψ tik ~[ψ tik |Ψ,Y],υ tik ~[υ tik |Ψ, Y], then the full conditional distribution is as follows:

[0095] When Y tik =2,

[0096] When Y tik =1,

[0097] When Y tik = 0,

[0098] (2) Extract the item level difficulty parameter b at the tth time point tk , given Ψ, Y, λ, ψ, υ, b tk ~[b tk |Ψ, Y, λ, ψ, υ], by 0<λ tik ≤P ★ tik2 , P ★ tik2 ≤ψ tik ≤P ★ tik1 , P ★ tik1 ≤υ tik <1, get b tk1 and b tk2 The full conditional distribution of is as follows:

[0099]

[0100]

[0101] in,

[0102]

[0103] (3) Extract the difficulty parameter a of the project level at the tth time point tk , given b, θ, Y, λ, ψ, υ, a tk ~[a tk |b, θ, Y, λ, ψ, υ], from 0<λ tik ≤P ★ tik2 , P ★ tik2 ≤ψ tik ≤P ★tik1 , P ★ tik1 ≤υ tik <1, get a tk The full conditional distribution of is as follows:

[0104]

[0105] When θ>0,

[0106] When θ<0,

[0107]

[0108] When θ>0,

[0109] When θ<0,

[0110] (4) Extract the ability parameter θ of the i-th subject at the t-th time point ti , given ξ, d, π, σ 2 , Y, λ, ψ, υ, θ ti ~[θti|ξ,d,π,σ 2 , Y, λ, ψ, υ], by 0<λ tik ≤P ★ tik2 , P ★ tik2 ≤ψ tik ≤P ★ tik1 , P ★ tik1 ≤υ tik <1, we get θ ti The full conditional distribution of is as follows:

[0111]

[0112] in,

[0113] (5) Extract the random coefficient π of level 2 i =(π 0i , π 1i )′, in θ, σ 2 , β and Ω are given, π i ~[π i |θ i , σ 2 ,β,Ω],θ ti =π0i +π 1i d ti +e ti ,

[0114]

[0115] Simplified:

[0116] θ i =D i π i +e i

[0117] π i is a vector of random regression coefficients, subject to the mean vector And the covariance matrix is Normal distribution, random regression coefficient π i The prior distribution of is a normal distribution with mean vector β and covariance matrix Ω, denoted by β = (β0, β1) ′, π i The full conditional distribution of is:

[0118]

[0119] (6) Extract β. Given π and Ω, β ≈ [β|π, Ω], and the full conditional distribution of β is as follows:

[0120]

[0121]

[0122] (7) Extraction level 2 residual variance σ 2 , given θ, π, v, ω, σ 2 ~[σ 2 |θ,π,v,ω], its prior distribution is Inverse-Gamma(v,ω). σ 2 The full conditional distribution of is:

[0123]

[0124] remember Then there is

[0125] σ 2 |θ (r) , π (r) , v, ω ~ Inverse-Gamma (v+nT / 2, ω+F)

[0126] (8) The covariance matrix of the random error at level 3 is Ω. Given π, β, λ, Ξ, Ω ≈ [Ω|π, β, λ, Ξ], let the prior distribution of the given Ω be the Inverse-Wishart (λ, Ξ) distribution. The full conditional distribution of Ω is:

[0127]

[0128] remember Then there is

[0129] Ω|π (r) ,B (r) ,λ,Ξ~Inverse-Wishart(λ+n,F1+Ξ).

[0130] S3. Verify the effect of the improved Bayesian parameter estimation method and compare it with the traditional Bayesian parameter estimation method.

[0131] S31, simulation design;

[0132] In this example, the simulation study demonstrates the effect of MCMC sampling algorithm and QMCMC algorithm on the estimation and correction of longitudinal multi-level model parameters, and compares them. Consider three time points (ie, t = 1, 2, 3). In the process of estimating model parameters, 30% of the items at each time point are set as anchor items, where the anchor items are assumed to be known or pre-known. The simulation conditions are set as follows:

[0133] (a) The number of items tested at each time point is: K = 10, K = 20, or K = 30, that is, there are 3, 6, or 9 anchor items at each time point;

[0134] (b) The number of subjects (sample size) is: SS = 500, 1000 or 2000. There are 9 simulation designs in total (3 test lengths × 3 sample sizes of subjects).

[0135] The response data in the simulation is measured by a longitudinal three-level hierarchical item response model. Assuming that there are K items and T measurement moments for longitudinal measurement and evaluation, the model is expressed as:

[0136]

[0137] For level 1, the probability that the i-th subject scores j on the k-th item at time point t is p tikj ; Y tik Y represents the response of subject i to the kth item at the tth time point; tik=j means the subject scored j points, 0≤j≤2, there are three response categories, each item has three response categories. The score on the kth item can be 0, 1 or 2 points.

[0138] θ ti represents the potential trait (ability) of the i-th (i=1, ..., n) subject at the t-th (t=1, ..., T) time point; ξ tk =(a tk , b tk1 , ..., b tkmk )′ represents the parameter vector composed of project parameters; a tk (k=1, ..., K) represents the discrimination (slope) parameter of item k at the tth time point; b tkj (k=1, ..., K) represents the difficulty (intercept) parameter of item k at the jth level at the tth time point.

[0139] Assume that the difficulty parameter of each item on the level is strictly monotonically increasing: b tk1 tk2 , that is, each project has two level difficulty parameters b tk1 and b tk2 ; Assume that P(Y tik =0|θ ti ,ξ tk )=1,P(Y tik =3|θ ti ,ξ tk )=0, the simulation study considers m k =2, which is the case of a third-order reaction.

[0140] The linear growth model is used to describe the level 2 personal ability growth curve, and the corresponding structural model is expressed as:

[0141]

[0142] Among them, e ti ~N(0,σ 2 ), t = 1, 2, 3, d ti is a variable representing time;

[0143]

[0144] The true value a of the discrimination parameter of the item at different time points tk It is generated by the following distribution log(N(exp(1), 0.15)), t = 1, 2, 3. The true value b of the discrimination parameter of the item tk At three time points, they are (-1, 1), (-0.75, 1.25), and (-0.5, 1.5), and the subject's ability parameter θ​i The generation distribution is N(A i π i , σ 2 I T×T ), where the true value of the level 2 residual variance is set to 0.24 (ie, σ 2 =0.24), D i represents the time-based loading matrix of subject i, and the random regression coefficient π in level 2 i It obeys the multivariate normal distribution, with the mean vector β and the covariance matrix Ω. Therefore, π i To generate , we need to first know the true values ​​of the fixed effect β and the covariance matrix Ω, where the fixed effect parameter β = (0; 0.15) and the covariance matrix is ​​set to Ω = (0.20; 00.1).

[0145] S32, prior distribution;

[0146] Assume that the prior distributions of discrimination and difficulty parameters are a tk ~logN(0,1) and b tk ~Uniform(-2,2), the fixed effect β follows a normal prior distribution N(0,100). The prior distribution of the residual variance at level 2 is assumed to follow an inverse gamma distribution with shape parameter v=0.001 and scale parameter ω=0.001, and the prior for the covariance matrix Ω at level 3 is set to an inverse Wishart distribution with a small degree of freedom λ=3 and a parameter matrix Ξ.

[0147] S33, parameter correction;

[0148] Regarding the accuracy analysis of parameter estimation, the average deviation and average root mean square error are used for evaluation, ie, bias (Bias) and root mean square error (RMSE), and M = 100 data sets are selected for simulation.

[0149] S34, convergence diagnostics;

[0150] In the convergence diagnosis, 90 test items and 500 test subjects at three time points are used as an example to illustrate. The convergence diagnosis method is used to diagnose the convergence of MCMC sampling, and is verified by detecting the sampling trajectory curve of the parameters under 10,000 iterations. Here, the first 5,000 iteration values ​​are discarded as the burning period. Figure 2 As shown, the fixed effect vector β and the variance parameter σ at level 2 are shown. 2 , the trajectory of the level 3 variance and covariance parameter τ. Figure 1 You can see the curve trajectory diagrams of the six iterations, all of which verify the convergence of MCMC sampling.

[0151] S34, simulation results;

[0152] Under the above nine simulation designs, the average values ​​of Bias and RMSE of the item discrimination parameters at each time point are shown in Table 1; the average values ​​of Bias and RMSE of the difficulty parameters at each time point are shown in Tables 2 and 3. The following conclusions can be drawn from Tables 1, 2 and 3:

[0153] Under the condition of a fixed number of subjects, when the test length at each time point increased from 10 to 30, the Bias and RMSE of the discrimination parameters and difficulty parameters of the items at each time point were mostly significantly reduced.

[0154] As shown in Table 1, the average Bias and RMSE of the discrimination parameter at the first time point, when the number of subjects is 500, when the test length increases from 10 to 30, the Bias drops from -0.0383 to -0.0081 and the RMSE drops from 0.1703 to 0.1338 under MCMC estimation, and the Bias drops from -0.438 to -0.0197 and the RMSE drops from 0.1892 to 0.1443 under QMCMC estimation. The trends at other time points are similar.

[0155] Table 1 Estimation of item discrimination parameters

[0156]

[0157] As shown in Table 2, the average Bias and RMSE of the difficulty parameter of the first level at the first time point, when the number of subjects was 500 and the test length increased from 10 to 30, the RMSE decreased from 0.0943 to 0.0935 under MCMC estimation, and the RMSE decreased from 0.1014 to 0.0942 under QMCMC estimation. The trends at other time points were similar.

[0158] Table 2 Item level difficulty parameter b ·1· Estimates

[0159]

[0160] As shown in Table 3, the average Bias and RMSE of the difficulty parameter of the second level at the first time point, when the number of subjects was 500 and the test length increased from 10 to 30, the RMSE decreased from 0.0986 to 0.0936 under MCMC estimation, and the RMSE decreased from 0.1054 to 0.0956 under QMCMC estimation. The trends at other time points were similar.

[0161] Table 3 Item level difficulty parameter b ·2· Estimates

[0162]

[0163] Under the condition of fixed test length at each time point, when the number of subjects increased from 500 to 2000, the average Bias and average RMSE of the discrimination parameters of all items decreased significantly.

[0164] As shown in Table 1, when the trial length at each time point is 10 items and the number of subjects increases from 500 to 2000, the estimation of the discrimination parameter at the first time point, Bias under MCMC estimation decreases from -0.0383 to -0.0179, RMSE decreases from 0.1703 to 0.0817, and Bias under QMCMC estimation decreases from -0.438 to -0.0232, RMSE decreases from 0.1892 to 0.1101. The trends at the other time points are almost similar;

[0165] As shown in Table 2, the average RMSE of the first level difficulty parameter at the first time point, when the number of subjects increased from 500 to 2000, the RMSE decreased from 0.0943 to 0.0472 under MCMC estimation, and from 0.1014 to 0.0522 under QMCMC estimation; the trends at other time points were similar;

[0166] As shown in Table 3, the average RMSE of the difficulty parameter of the second level at the first time point, when the number of subjects increased from 500 to 2000, the RMSE under MCMC estimation decreased from 0.0986 to 0.0487, and the RMSE under QMCMC estimation decreased from 0.1054 to 0.0511. The trends at other time points were similar. The simulation results of the above two estimation methods, Bias and RMSE, both show that the estimated results of the discrimination parameter and difficulty parameter of the item parameters are closer to the true value.

[0167] The modified effects of structural parameters are shown in Tables 4, 5 and 6, and the following conclusions can be drawn:

[0168] When the sample size SS of the subjects is given, the RMSE of the fixed effect parameter (βs) decreases as the length of the experiment at each time point increases from 10 to 30; the variance parameter (σ 2 ) also decreases as the trial length per time point increases from 10 to 30; the RMSE of the covariance parameter (τ) at level 3 decreases as the trial length per time point increases from 10 to 30.

[0169] When the test length at each time point is given, the RMSE of the fixed effect parameter (βs) in the parameters of the structural model decreases as the sample size SS of the subjects increases from 500 to 2000; the variance parameter (σ 2) also decreases as the sample size SS of the subjects increases from 500 to 2000; the RMSE of the covariance parameter (τ) at level 3 also decreases as the sample size SS of the subjects increases from 500 to 2000.

[0170] Table 4 Estimation of fixed and random effect parameters when the sample size is 2000

[0171]

[0172] Table 5 Estimation of fixed and random effect parameters when the sample size is 1000

[0173]

[0174] Table 6 Estimation of fixed and random effect parameters when the sample size is 500

[0175]

[0176] The simulation results of the above two estimation methods show that the estimated values ​​of the structural parameters are closer to the true values ​​of the structural parameters.

[0177] In summary, based on the above indicators, it is clear that the Bayesian sampling algorithm provides relatively accurate estimates of the parameters, and both parameter estimation algorithms MCMC and QMCMC are relatively stable. When comparing the estimation accuracy of the two algorithms, the accuracy of the project parameters, fixed effect parameters, and random effect parameters is very small. At the same time, the correlation coefficient between the estimated values ​​of the ability parameters obtained by the two estimation methods is almost always 1, such as Figure 3 shown.

[0178] However, from the perspective of computational speed and efficiency, under simulation conditions, the running time of the two algorithms, MCMC and QMCMC, was compared. It was found that the traditional MCMC took about 370 seconds to estimate the model parameters corresponding to a data set after 10,000 iterations, while the QMCMC took about 276 seconds to estimate the model parameters corresponding to a data set after 10,000 iterations. That is, the time used by the traditional MCMC to estimate 100 data sets can be used by QMCMC to estimate 137 data sets.

[0179] Therefore, the present invention adopts the above-mentioned Bayesian estimation method for the parameters of the longitudinal multi-level and multi-level item response model. On the basis of ensuring the estimation accuracy, the QMCMC running time is better than the traditional MCMC, the speed is increased by more than 1 / 4, and the estimation efficiency of the longitudinal multi-level and multi-level scoring item response theory model is greatly improved.

[0180] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A Bayesian estimation method for parameters of a longitudinal multi-level item response model, characterized in that: The following steps are involved: S1. Establish a longitudinal multi-level item response model, including level 1 grade response model, level 2 longitudinal individual growth model, and level 3 multi-level model; S2, based on the improved Bayesian parameter estimation method, all parameters of the item response model proposed in step S1 are estimated by introducing auxiliary variables; S3. Verify the effect of the improved Bayesian parameter estimation method and compare it with the traditional Bayesian parameter estimation method.

2. A Bayesian estimation method for parameters of a longitudinal multi-level item response model according to claim 1, characterized in that: In step S1, in establishing the longitudinal multi-level multi-level item response model, for the level 1 grade response model, the probability that the i-th subject scores j points for the k-th item at the t-th time point is: Among them, Y tik Y represents the response of subject i to the kth item at the tth time point; tik =j means the subject scores j points, 0≤j≤m k , remember m k +1 is the type of reaction, each item has m k +1 response category, the lowest score on the kth item is 0 and the highest score is m k ; θ ti represents the potential characteristics of the i-th subject at the t-th time point; ξ tk =(a tk , b tk1 , ..., b tkmk )′ represents the parameter vector composed of project parameters; a tk represents the discrimination parameter of item k at the tth time point; b tkj It represents the difficulty parameter of item k at the jth level at the tth time point.

3. The Bayesian estimation method for parameters of a longitudinal multi-level item response model according to claim 1, characterized in that: In step S1, in which the longitudinal multi-level item response model is established, the change of individual ability is represented by the level 2 longitudinal individual growth model, and the growth trajectory of individual ability is linear, as shown below: i ti =π 0i +p 1i d ti +e ti Among them, the variable d ti is the time parameter of subject i at the tth moment, π represents the coefficient of the linear function, e ti Represents a random error term, with an expectation of 0 and a variance of σ 2 The normal distribution of .

4. A Bayesian estimation method for parameters of a longitudinal multi-level item response model according to claim 1, characterized in that: In step S1, in establishing the longitudinal multilevel item response model, the individual level model of the level 3 multilevel model is as follows: π hi ＝β h0 +u hi Among them, β hs is the fixed effect of the hth parameter, u hi represents the random residual effect of subject i at level 3, vector u=(u 0i ,u 1i ) follows a bivariate normal distribution with zero expectation and variance Ω.

5. The Bayesian estimation method for parameters of a longitudinal multi-level item response model according to claim 1, characterized in that: In step S2, Gibbs sampling is performed by introducing auxiliary variables to obey U(0, 1), and the specific process is as follows: S21, when the response variable Y tik =2, the indicative function is as follows: S22, when the response variable Y tik =1, the indicative function is as follows: S23, when the response variable Y tik =0, the indicative function is as follows: in, S24, then the joint posterior distribution containing auxiliary variables is as follows:

6. A Bayesian estimation method for parameters of a longitudinal multi-level item response model according to claim 5, characterized in that: Extract three independent auxiliary variables λ tik , ψ tik , tik , given Ψ, Y, λ tik ~[λ tik |Ψ,Y],ψ tik ~[ψ tik |Ψ,Y],υ tik ~[υ tik |Ψ, Y], then the full conditional distribution is as follows: When Y tik =2, When Y tik =1, When Y tik = 0, 7. A Bayesian estimation method for parameters of a longitudinal multi-level item response model according to claim 5, characterized in that: Extract the item level difficulty parameter b at the tth time point tk , given Ψ, Y, λ, ψ, υ, b tk ~[b tk |Ψ, Y, λ, ψ, υ], by 0<λ tik ≤P ★ tik2 , P ★ tik2 ≤ψ tik ≤P ★ tik1 , P ★ tik1 ≤υ tik <1, get b tk1 and b tk2 The full conditional distribution of is as follows: in, 8. A Bayesian estimation method for parameters of a longitudinal multi-level item response model according to claim 5, characterized in that: Extract the difficulty parameter a of the item level at the tth time point tk , given b, θ, Y, λ, ψ, υ, a tk ~[a tk |b, θ, Y, λ, ψ, υ], from 0<λ tik ≤P ★ tik2 , P ★ tik2 ≤ψ tik ≤P ★ tik1 , P ★ tik1 ≤υ tik <1, get a tk The full conditional distribution of is as follows: Among them, when θ>0, When θ<0, When θ>0, When θ<0, 9. A Bayesian estimation method for parameters of a longitudinal multi-level item response model according to claim 5, characterized in that: Extract the ability parameter θ of the i-th subject at the t-th time point ti , given ξ, d, π, σ 2 , Y, λ, ψ, υ, θ ti ~[θti|ξ,d,π,σ 2 , Y, λ, ψ, υ], by 0<λ tik ≤P ★ tik2 , P ★ tik2 ≤ψ tik ≤P ★ tik1 , P ★ tik1 ≤υ tik <1, we get θ ti The full conditional distribution of is as follows: in, 10. A Bayesian estimation method for parameters of a longitudinal multi-level item response model according to claim 5, characterized in that: Extract β. Given π and Ω, β~[β|π,Ω], the full conditional distribution of β is as follows:

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