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 
#>  61  67  78 119  25
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
#>  0.1453 0.16476
#>  0.2235 0.18512
#>  0.2326 0.04447
#>  0.1232 0.15566
#>  0.1367 0.20499
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.87169
#> λ1     0.09275
#> λ2     0.12180
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1069
#> 0001  0.1718
#> 0010  0.1717
#> 0011  0.2851
#> 0100  0.1919
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 154397.3 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5024
#> M2:  0.49
#> total scores:  0.6229
a <- summary(output_HMDCM_RT_joint)
a
#> 
#> Model: DINA_HO_RT_joint 
#> 
#> Item Parameters:
#>  ss_EAP  gs_EAP
#>  0.1453 0.16476
#>  0.2235 0.18512
#>  0.2326 0.04447
#>  0.1232 0.15566
#>  0.1367 0.20499
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.87169
#> λ1     0.09275
#> λ2     0.12180
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1069
#> 0001  0.1718
#> 0010  0.1717
#> 0011  0.2851
#> 0100  0.1919
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 154397.3 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5208
#> M2:  0.49
#> total scores:  0.6268

a$ss_EAP
#>             [,1]
#>  [1,] 0.14527364
#>  [2,] 0.22347089
#>  [3,] 0.23264554
#>  [4,] 0.12324120
#>  [5,] 0.13671703
#>  [6,] 0.15870662
#>  [7,] 0.13474586
#>  [8,] 0.16340068
#>  [9,] 0.23955037
#> [10,] 0.09482755
#> [11,] 0.22594592
#> [12,] 0.21414600
#> [13,] 0.20544524
#> [14,] 0.16827695
#> [15,] 0.18610171
#> [16,] 0.24041734
#> [17,] 0.20221820
#> [18,] 0.19796050
#> [19,] 0.16214569
#> [20,] 0.16190172
#> [21,] 0.16581845
#> [22,] 0.15179118
#> [23,] 0.14965766
#> [24,] 0.22729361
#> [25,] 0.19476440
#> [26,] 0.23051407
#> [27,] 0.15559297
#> [28,] 0.18621418
#> [29,] 0.20944074
#> [30,] 0.10174868
#> [31,] 0.11808534
#> [32,] 0.14596304
#> [33,] 0.18828569
#> [34,] 0.14753469
#> [35,] 0.19077708
#> [36,] 0.23527372
#> [37,] 0.21634465
#> [38,] 0.18328075
#> [39,] 0.20804419
#> [40,] 0.20749658
#> [41,] 0.14474496
#> [42,] 0.25869170
#> [43,] 0.13241310
#> [44,] 0.12952914
#> [45,] 0.21678566
#> [46,] 0.11513858
#> [47,] 0.14294383
#> [48,] 0.23072136
#> [49,] 0.15857348
#> [50,] 0.15212187
head(a$ss_EAP)
#>           [,1]
#> [1,] 0.1452736
#> [2,] 0.2234709
#> [3,] 0.2326455
#> [4,] 0.1232412
#> [5,] 0.1367170
#> [6,] 0.1587066

(3) Check for parameter estimation accuracy

(cor_thetas <- cor(thetas_true,a$thetas_EAP))
#>           [,1]
#> [1,] 0.8061758
(cor_taus <- cor(taus_true,a$response_times_coefficients$taus_EAP))
#>           [,1]
#> [1,] 0.9879757

(cor_ss <- cor(as.vector(itempars_true[,1]),a$ss_EAP))
#>          [,1]
#> [1,] 0.639022
(cor_gs <- cor(as.vector(itempars_true[,2]),a$gs_EAP))
#>           [,1]
#> [1,] 0.5692589

AAR_vec <- numeric(L)
for(t in 1:L){
  AAR_vec[t] <- mean(Alphas[,,t]==a$Alphas_est[,,t])
}
AAR_vec
#> [1] 0.9271429 0.9335714 0.9457143 0.9542857 0.9457143

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] 0.7428571 0.7685714 0.8085714 0.8314286 0.8228571

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

a$DIC
#>              Transition Response_Time Response    Joint    Total
#> D_bar          1990.836      133126.8 15066.03 3277.902 153461.6
#> D(theta_bar)   1723.832      132699.9 14961.74 3140.410 152525.9
#> DIC            2257.839      133553.8 15170.31 3415.393 154397.3
head(a$PPP_total_scores)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.46 0.10 0.30 0.82 0.62
#> [2,] 0.84 0.00 0.24 0.24 0.06
#> [3,] 0.40 0.32 0.28 0.44 0.06
#> [4,] 0.88 0.92 0.84 0.24 0.98
#> [5,] 0.74 0.88 0.46 0.52 0.04
#> [6,] 0.42 0.80 0.24 0.50 0.88
head(a$PPP_total_RTs)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.38 0.14 0.22 0.96 0.72
#> [2,] 0.04 0.96 0.56 0.96 0.32
#> [3,] 0.80 0.40 0.40 0.32 0.50
#> [4,] 0.96 0.08 0.84 0.72 0.12
#> [5,] 0.36 0.54 0.98 0.22 0.84
#> [6,] 0.42 0.24 0.74 0.32 0.86
head(a$PPP_item_means)
#> [1] 0.64 0.44 0.52 0.50 0.46 0.56
head(a$PPP_item_mean_RTs)
#> [1] 0.42 0.72 0.52 0.66 0.62 0.48
head(a$PPP_item_ORs)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
#> [1,]   NA 0.86 0.58 0.40 0.44 0.60 0.60 0.74 0.08  0.74  0.36  0.94  0.78  0.76
#> [2,]   NA   NA 0.98 0.88 0.80 0.48 0.82 0.88 0.64  0.62  0.78  0.80  0.80  0.46
#> [3,]   NA   NA   NA 0.84 1.00 0.98 0.98 0.76 0.80  0.92  0.02  0.54  0.42  0.56
#> [4,]   NA   NA   NA   NA 0.96 0.94 0.82 0.94 0.80  0.96  0.94  0.82  0.90  0.96
#> [5,]   NA   NA   NA   NA   NA 0.78 0.50 0.30 0.12  0.36  0.56  0.08  0.96  0.32
#> [6,]   NA   NA   NA   NA   NA   NA 0.84 0.74 0.74  0.82  0.40  0.94  1.00  0.26
#>      [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
#> [1,]  0.62  0.14  0.62  0.56  0.56  0.28  0.40  0.22  0.48  0.16  0.94  0.14
#> [2,]  0.16  0.60  0.28  0.24  0.38  0.72  0.76  0.92  0.96  0.84  0.58  0.74
#> [3,]  0.88  0.18  0.58  0.74  0.88  0.44  0.90  0.58  0.62  0.96  0.44  0.32
#> [4,]  0.68  0.70  0.66  0.82  0.80  0.80  0.82  0.98  0.54  0.88  0.80  0.90
#> [5,]  0.10  0.14  0.02  0.30  0.44  0.20  0.34  0.92  0.44  0.16  0.88  0.20
#> [6,]  0.48  0.06  0.12  0.36  0.84  0.14  0.64  0.98  0.38  0.16  0.54  0.56
#>      [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
#> [1,]  0.32  0.86  0.82  0.68  1.00  0.92  0.04  0.60  1.00  0.64  0.90  0.82
#> [2,]  0.70  0.90  0.62  0.98  0.68  0.78  0.86  0.64  0.94  0.44  0.98  0.76
#> [3,]  0.90  0.48  0.38  0.88  0.64  0.84  1.00  0.74  0.66  0.60  1.00  0.74
#> [4,]  0.74  0.84  0.92  0.42  0.60  0.74  0.50  0.82  0.94  0.38  0.72  1.00
#> [5,]  0.72  0.30  0.44  0.70  0.26  0.72  0.88  0.98  0.90  0.44  0.96  0.14
#> [6,]  0.88  0.62  0.64  0.56  0.90  0.16  0.56  0.80  0.82  0.14  0.72  0.40
#>      [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
#> [1,]  0.00  0.42  0.44  0.74  0.10  0.96  0.04  0.14  0.30  0.18  0.96  0.76
#> [2,]  0.68  0.78  0.58  1.00  0.40  0.42  0.82  0.34  0.10  0.64  0.42  0.82
#> [3,]  0.50  0.86  0.68  1.00  1.00  0.78  0.82  0.92  0.98  1.00  1.00  0.96
#> [4,]  0.56  0.92  0.76  0.52  1.00  0.96  0.38  0.94  0.70  0.84  0.38  0.50
#> [5,]  0.62  0.38  0.96  0.68  0.66  0.54  0.66  0.80  0.36  0.84  0.52  0.80
#> [6,]  0.26  0.78  0.90  0.94  0.92  0.84  0.82  0.46  0.50  0.82  0.48  1.00