DINA_HO_RT

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 
#>  70  52  82 112  34
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
#>  0.1848 0.1899
#>  0.1763 0.1353
#>  0.2425 0.1403
#>  0.1301 0.1541
#>  0.1835 0.1727
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.83978
#> λ1     0.27932
#> λ2     0.08818
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.2150
#> 0001  0.1711
#> 0010  0.1758
#> 0011  0.2148
#> 0100  0.1582
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 159862.2 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5128
#> 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
#>  0.1848 0.1899
#>  0.1763 0.1353
#>  0.2425 0.1403
#>  0.1301 0.1541
#>  0.1835 0.1727
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.83978
#> λ1     0.27932
#> λ2     0.08818
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.2150
#> 0001  0.1711
#> 0010  0.1758
#> 0011  0.2148
#> 0100  0.1582
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 159862.2 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5088
#> M2:  0.49
#> total scores:  0.627

a$ss_EAP
#>             [,1]
#>  [1,] 0.18475319
#>  [2,] 0.17634185
#>  [3,] 0.24252807
#>  [4,] 0.13008456
#>  [5,] 0.18350636
#>  [6,] 0.17726515
#>  [7,] 0.17971682
#>  [8,] 0.03657850
#>  [9,] 0.14209030
#> [10,] 0.15709071
#> [11,] 0.12357039
#> [12,] 0.16007177
#> [13,] 0.15575961
#> [14,] 0.10842622
#> [15,] 0.19206181
#> [16,] 0.08060138
#> [17,] 0.14232637
#> [18,] 0.13765846
#> [19,] 0.20461871
#> [20,] 0.08367335
#> [21,] 0.12844598
#> [22,] 0.23548715
#> [23,] 0.10666963
#> [24,] 0.22388528
#> [25,] 0.20969528
#> [26,] 0.15023088
#> [27,] 0.18736555
#> [28,] 0.20654446
#> [29,] 0.23591350
#> [30,] 0.12132564
#> [31,] 0.23082009
#> [32,] 0.10703288
#> [33,] 0.15391558
#> [34,] 0.13209995
#> [35,] 0.21967517
#> [36,] 0.15076908
#> [37,] 0.25882885
#> [38,] 0.12800171
#> [39,] 0.14424402
#> [40,] 0.11538197
#> [41,] 0.09829498
#> [42,] 0.20673604
#> [43,] 0.15251383
#> [44,] 0.16045880
#> [45,] 0.12444373
#> [46,] 0.20167966
#> [47,] 0.15814662
#> [48,] 0.13969299
#> [49,] 0.12800321
#> [50,] 0.08266470
head(a$ss_EAP)
#>           [,1]
#> [1,] 0.1847532
#> [2,] 0.1763418
#> [3,] 0.2425281
#> [4,] 0.1300846
#> [5,] 0.1835064
#> [6,] 0.1772652

(3) Check for parameter estimation accuracy

(cor_thetas <- cor(thetas_true,a$thetas_EAP))
#>           [,1]
#> [1,] 0.8115132
(cor_taus <- cor(taus_true,a$response_times_coefficients$taus_EAP))
#>           [,1]
#> [1,] 0.9887804

(cor_ss <- cor(as.vector(itempars_true[,1]),a$ss_EAP))
#>           [,1]
#> [1,] 0.7823194
(cor_gs <- cor(as.vector(itempars_true[,2]),a$gs_EAP))
#>           [,1]
#> [1,] 0.5586448

AAR_vec <- numeric(L)
for(t in 1:L){
  AAR_vec[t] <- mean(Alphas[,,t]==a$Alphas_est[,,t])
}
AAR_vec
#> [1] 0.9528571 0.9478571 0.9571429 0.9657143 0.9685714

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] 0.8228571 0.8114286 0.8428571 0.8714286 0.8914286

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

a$DIC
#>              Transition Response_Time Response    Joint    Total
#> D_bar          2280.300      139060.8 14479.59 3121.008 158941.6
#> D(theta_bar)   1977.589      138631.7 14425.66 2986.123 158021.1
#> DIC            2583.011      139489.8 14533.51 3255.892 159862.2
head(a$PPP_total_scores)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.74 0.96 0.70 0.72 0.60
#> [2,] 0.58 0.66 1.00 0.38 0.96
#> [3,] 0.54 0.52 0.84 0.96 0.70
#> [4,] 0.24 0.50 0.30 0.80 0.40
#> [5,] 0.52 0.74 0.76 0.16 0.64
#> [6,] 0.80 0.62 0.78 0.52 1.00
head(a$PPP_total_RTs)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.26 0.74 0.20 0.16 0.56
#> [2,] 0.26 0.50 0.28 0.32 0.50
#> [3,] 0.20 0.28 0.42 0.90 0.96
#> [4,] 0.24 0.60 0.86 0.18 0.20
#> [5,] 0.52 0.52 0.22 0.52 0.30
#> [6,] 0.20 0.40 0.44 0.56 0.70
head(a$PPP_item_means)
#> [1] 0.44 0.50 0.56 0.62 0.54 0.52
head(a$PPP_item_mean_RTs)
#> [1] 0.54 0.36 0.66 0.42 0.56 0.58
head(a$PPP_item_ORs)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
#> [1,]   NA 0.42 0.72 0.50 0.52 0.80 0.10 1.00 0.96  1.00  0.88  0.02  1.00  0.84
#> [2,]   NA   NA 0.40 0.18 0.70 0.60 0.94 0.68 0.90  0.52  0.02  0.84  0.44  0.60
#> [3,]   NA   NA   NA 0.30 0.20 0.10 0.14 0.24 0.92  0.08  0.72  0.58  0.84  0.98
#> [4,]   NA   NA   NA   NA 0.12 0.12 0.42 0.14 0.82  0.50  0.56  0.24  0.58  0.60
#> [5,]   NA   NA   NA   NA   NA 0.80 0.56 0.50 0.82  0.86  0.20  0.34  0.74  0.30
#> [6,]   NA   NA   NA   NA   NA   NA 0.70 0.42 0.84  0.74  0.62  0.02  0.62  0.24
#>      [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
#> [1,]  0.34  0.22  0.72  0.86  0.94  1.00  0.32  0.06  0.78  0.26  0.26  0.80
#> [2,]  0.04  0.40  0.74  0.72  0.24  0.62  0.30  0.70  0.80  0.48  0.14  0.84
#> [3,]  0.94  0.36  0.86  0.94  0.92  0.22  0.62  0.44  0.12  0.02  0.02  0.48
#> [4,]  0.62  0.32  0.08  0.42  0.56  0.86  0.70  0.20  0.22  0.32  0.04  0.76
#> [5,]  0.04  0.58  0.54  0.92  0.54  0.80  0.46  0.28  0.96  0.26  0.16  0.62
#> [6,]  0.26  0.66  0.10  0.96  0.30  0.54  0.64  0.42  0.68  0.46  0.78  0.76
#>      [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
#> [1,]  0.66  0.50  0.52  0.26  0.36  0.10  0.90  0.12  0.74  0.28  0.42  0.04
#> [2,]  0.34  0.28  0.84  0.98  0.46  0.26  0.72  0.84  0.98  0.54  0.30  0.24
#> [3,]  0.48  0.08  0.22  0.32  0.32  0.94  0.72  0.70  0.98  0.16  0.10  0.70
#> [4,]  0.68  0.76  0.22  0.14  0.12  0.48  0.82  0.42  0.42  0.62  0.88  0.72
#> [5,]  0.86  0.26  0.52  0.56  0.98  0.42  0.58  0.30  0.30  0.44  0.30  0.24
#> [6,]  0.84  0.94  0.98  1.00  0.18  0.66  0.98  0.12  0.90  0.68  0.88  0.34
#>      [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
#> [1,]  0.34  0.64  0.84  0.98  0.18  0.46  0.78  0.56  0.50  0.28  0.30  0.28
#> [2,]  0.86  0.46  0.70  0.30  0.12  0.38  0.84  0.98  0.14  0.92  0.40  0.04
#> [3,]  0.62  0.28  0.34  0.18  0.12  0.92  0.80  0.40  0.40  0.56  0.62  0.06
#> [4,]  0.20  0.40  0.88  0.36  0.68  0.34  0.26  0.76  0.62  0.60  0.96  0.78
#> [5,]  0.14  0.54  0.36  0.68  0.38  0.20  0.72  0.60  0.46  0.58  0.70  0.14
#> [6,]  0.46  0.76  0.74  0.40  0.08  0.48  0.86  0.12  0.32  0.74  0.40  0.08