DINA_HO_RT_joint

library(hmcdm)

Load the spatial rotation data

N = length(Test_versions)
J = nrow(Q_matrix)
K = ncol(Q_matrix)
L = nrow(Test_order)

(1) Simulate responses and response times based on the HMDCM model with response times (no covariance between speed and learning ability)

ETAs <- ETAmat(K, J, Q_matrix)
class_0 <- sample(1:2^K, N, replace = L)
Alphas_0 <- matrix(0,N,K)
mu_thetatau = c(0,0)
Sig_thetatau = rbind(c(1.8^2,.4*.5*1.8),c(.4*.5*1.8,.25))
Z = matrix(rnorm(N*2),N,2)
thetatau_true = Z%*%chol(Sig_thetatau)
thetas_true = thetatau_true[,1]
taus_true = thetatau_true[,2]
G_version = 3
phi_true = 0.8
for(i in 1:N){
  Alphas_0[i,] <- inv_bijectionvector(K,(class_0[i]-1))
}
lambdas_true <- c(-2, .4, .055)       # empirical from Wang 2017
Alphas <- sim_alphas(model="HO_joint", 
                    lambdas=lambdas_true, 
                    thetas=thetas_true, 
                    Q_matrix=Q_matrix, 
                    Design_array=Design_array)
table(rowSums(Alphas[,,5]) - rowSums(Alphas[,,1])) # used to see how much transition has taken place
#> 
#>   0   1   2   3   4 
#>  51  57  97 115  30
itempars_true <- matrix(runif(J*2,.1,.2), ncol=2)
RT_itempars_true <- matrix(NA, nrow=J, ncol=2)
RT_itempars_true[,2] <- rnorm(J,3.45,.5)
RT_itempars_true[,1] <- runif(J,1.5,2)

Y_sim <- sim_hmcdm(model="DINA",Alphas,Q_matrix,Design_array,
                   itempars=itempars_true)
L_sim <- sim_RT(Alphas,Q_matrix,Design_array,
                  RT_itempars_true,taus_true,phi_true,G_version)

(2) Run the MCMC to sample parameters from the posterior distribution

output_HMDCM_RT_joint = hmcdm(Y_sim,Q_matrix,"DINA_HO_RT_joint",Design_array,100,30,
                                 Latency_array = L_sim, G_version = G_version,
                                 theta_propose = 2,deltas_propose = c(.45,.25,.06))
#> 0
output_HMDCM_RT_joint
#> 
#> Model: DINA_HO_RT_joint 
#> 
#> Sample Size: 350
#> Number of Items: 
#> Number of Time Points: 
#> 
#> Chain Length: 100, burn-in: 50
summary(output_HMDCM_RT_joint)
#> 
#> Model: DINA_HO_RT_joint 
#> 
#> Item Parameters:
#>   ss_EAP  gs_EAP
#>  0.18173 0.12209
#>  0.20571 0.12054
#>  0.18765 0.04716
#>  0.08832 0.22509
#>  0.11316 0.16415
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.25233
#> λ1     0.24781
#> λ2     0.04834
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1380
#> 0001  0.1780
#> 0010  0.2239
#> 0011  0.2177
#> 0100  0.1477
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 156726.4 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.4964
#> M2:  0.49
#> total scores:  0.6286
a <- summary(output_HMDCM_RT_joint)
a
#> 
#> Model: DINA_HO_RT_joint 
#> 
#> Item Parameters:
#>   ss_EAP  gs_EAP
#>  0.18173 0.12209
#>  0.20571 0.12054
#>  0.18765 0.04716
#>  0.08832 0.22509
#>  0.11316 0.16415
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.25233
#> λ1     0.24781
#> λ2     0.04834
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1380
#> 0001  0.1780
#> 0010  0.2239
#> 0011  0.2177
#> 0100  0.1477
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 156726.4 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5072
#> M2:  0.49
#> total scores:  0.6281

a$ss_EAP
#>             [,1]
#>  [1,] 0.18172984
#>  [2,] 0.20571084
#>  [3,] 0.18764730
#>  [4,] 0.08831541
#>  [5,] 0.11316258
#>  [6,] 0.12749734
#>  [7,] 0.16526013
#>  [8,] 0.19290705
#>  [9,] 0.19752326
#> [10,] 0.11672841
#> [11,] 0.10735308
#> [12,] 0.18651972
#> [13,] 0.23939671
#> [14,] 0.23659833
#> [15,] 0.12225033
#> [16,] 0.22750307
#> [17,] 0.17374229
#> [18,] 0.16720903
#> [19,] 0.16760024
#> [20,] 0.18417524
#> [21,] 0.13489969
#> [22,] 0.14625810
#> [23,] 0.20436702
#> [24,] 0.18245568
#> [25,] 0.22044639
#> [26,] 0.14727817
#> [27,] 0.16521516
#> [28,] 0.17076557
#> [29,] 0.13856993
#> [30,] 0.13478178
#> [31,] 0.21243626
#> [32,] 0.24888085
#> [33,] 0.14633973
#> [34,] 0.23837025
#> [35,] 0.16449767
#> [36,] 0.17209176
#> [37,] 0.10378234
#> [38,] 0.14154425
#> [39,] 0.20597578
#> [40,] 0.08946400
#> [41,] 0.14822972
#> [42,] 0.18334679
#> [43,] 0.16799209
#> [44,] 0.05833630
#> [45,] 0.22830662
#> [46,] 0.15549188
#> [47,] 0.16248934
#> [48,] 0.14696982
#> [49,] 0.09206677
#> [50,] 0.19339051
head(a$ss_EAP)
#>            [,1]
#> [1,] 0.18172984
#> [2,] 0.20571084
#> [3,] 0.18764730
#> [4,] 0.08831541
#> [5,] 0.11316258
#> [6,] 0.12749734

(3) Check for parameter estimation accuracy

(cor_thetas <- cor(thetas_true,a$thetas_EAP))
#>           [,1]
#> [1,] 0.7823287
(cor_taus <- cor(taus_true,a$response_times_coefficients$taus_EAP))
#>           [,1]
#> [1,] 0.9863389

(cor_ss <- cor(as.vector(itempars_true[,1]),a$ss_EAP))
#>           [,1]
#> [1,] 0.7497122
(cor_gs <- cor(as.vector(itempars_true[,2]),a$gs_EAP))
#>           [,1]
#> [1,] 0.5792237

AAR_vec <- numeric(L)
for(t in 1:L){
  AAR_vec[t] <- mean(Alphas[,,t]==a$Alphas_est[,,t])
}
AAR_vec
#> [1] 0.9335714 0.9442857 0.9521429 0.9557143 0.9614286

PAR_vec <- numeric(L)
for(t in 1:L){
  PAR_vec[t] <- mean(rowSums((Alphas[,,t]-a$Alphas_est[,,t])^2)==0)
}
PAR_vec
#> [1] 0.7600000 0.7914286 0.8285714 0.8428571 0.8685714

(4) Evaluate the fit of the model to the observed response and response times data (here, Y_sim and R_sim)

a$DIC
#>              Transition Response_Time Response    Joint    Total
#> D_bar          2375.810      135684.9 14760.43 2994.890 155816.0
#> D(theta_bar)   2127.038      135258.0 14687.85 2832.824 154905.7
#> DIC            2624.583      136111.9 14833.01 3156.955 156726.4
head(a$PPP_total_scores)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.14 0.82 0.56 0.30 0.96
#> [2,] 0.90 0.96 1.00 0.10 0.78
#> [3,] 0.76 0.28 0.46 0.22 0.50
#> [4,] 0.94 0.22 0.98 0.88 0.68
#> [5,] 0.92 0.36 0.76 0.70 0.54
#> [6,] 0.52 0.30 0.96 0.38 0.84
head(a$PPP_total_RTs)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.76 0.56 0.40 0.86 0.60
#> [2,] 0.42 0.54 0.06 0.94 0.64
#> [3,] 0.38 0.46 0.68 0.46 0.44
#> [4,] 0.68 0.40 0.98 0.58 0.50
#> [5,] 0.70 0.02 0.88 0.42 1.00
#> [6,] 0.06 0.78 0.38 0.96 0.14
head(a$PPP_item_means)
#> [1] 0.54 0.48 0.56 0.48 0.60 0.50
head(a$PPP_item_mean_RTs)
#> [1] 0.34 0.46 0.42 0.22 0.60 0.60
head(a$PPP_item_ORs)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
#> [1,]   NA 0.68 0.56 0.86 0.88 0.82 0.74 0.76 0.08  0.70  0.42  0.34  0.74  0.34
#> [2,]   NA   NA 0.08 0.30 0.26 0.74 0.42 0.64 0.94  0.32  0.98  0.90  0.96  0.78
#> [3,]   NA   NA   NA 0.36 0.32 0.44 0.66 0.14 0.80  0.30  0.42  0.64  0.54  0.58
#> [4,]   NA   NA   NA   NA 0.94 0.74 0.84 0.52 0.88  0.86  0.90  0.44  0.82  0.44
#> [5,]   NA   NA   NA   NA   NA 0.46 0.54 0.64 0.78  0.46  0.76  0.46  0.84  0.94
#> [6,]   NA   NA   NA   NA   NA   NA 0.30 0.82 0.62  0.38  0.72  0.68  1.00  0.44
#>      [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
#> [1,]  0.60  0.14  0.56  0.22  0.48  0.64  0.32  0.54  0.06  0.30  0.40  0.50
#> [2,]  0.38  0.84  0.60  1.00  0.82  0.14  0.54  0.16  0.26  0.58  0.62  0.20
#> [3,]  0.22  0.18  0.84  0.30  0.40  0.74  0.86  0.68  0.14  0.80  0.06  0.50
#> [4,]  0.86  0.66  0.70  0.90  0.74  0.70  0.28  0.44  0.78  0.58  0.74  0.74
#> [5,]  0.58  0.10  0.62  0.36  0.30  0.98  0.68  0.52  0.20  0.46  0.82  0.72
#> [6,]  0.48  0.60  0.46  0.18  0.18  0.24  0.28  0.62  0.08  0.66  0.78  0.06
#>      [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
#> [1,]  0.04  0.34  0.62  0.08  0.38  0.20  0.42  0.72  0.64  0.18  0.22  0.42
#> [2,]  0.16  0.00  0.04  0.42  0.66  0.32  0.70  0.66  0.52  0.52  0.92  0.20
#> [3,]  0.54  0.42  0.48  0.40  0.16  0.04  0.84  0.66  0.18  0.18  0.00  0.02
#> [4,]  0.40  0.84  0.92  0.10  1.00  0.52  0.76  0.80  0.90  0.72  0.92  0.88
#> [5,]  0.06  0.60  1.00  0.70  0.64  0.08  0.76  0.58  0.66  0.34  0.46  0.66
#> [6,]  0.82  0.14  0.42  0.02  0.26  0.16  0.70  0.62  0.30  0.46  0.18  0.32
#>      [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
#> [1,]  0.00  0.22  0.88  0.90  0.90  0.92  0.52  0.94  0.70  0.62  0.98  0.52
#> [2,]  0.04  0.84  0.06  0.90  0.70  0.90  0.38  0.94  0.74  0.92  0.98  0.86
#> [3,]  0.66  0.12  0.56  0.14  0.28  1.00  0.34  0.76  0.12  0.44  0.80  0.84
#> [4,]  0.84  1.00  0.74  0.84  0.90  1.00  0.62  0.82  0.36  0.36  0.58  0.86
#> [5,]  0.00  0.60  0.90  0.44  0.42  0.96  0.86  0.50  0.90  0.78  0.70  0.68
#> [6,]  0.10  0.78  0.48  0.90  0.34  0.14  0.68  0.32  0.22  0.60  0.36  0.60