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 
#>  63  55  80 123  29
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
#>  0.1903 0.16441
#>  0.1134 0.08322
#>  0.1505 0.11822
#>  0.1970 0.12669
#>  0.1585 0.09306
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.69340
#> λ1     0.34885
#> λ2     0.04409
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1644
#> 0001  0.1782
#> 0010  0.1719
#> 0011  0.2198
#> 0100  0.2147
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 147146.5 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5228
#> 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
#>  0.1903 0.16441
#>  0.1134 0.08322
#>  0.1505 0.11822
#>  0.1970 0.12669
#>  0.1585 0.09306
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.69340
#> λ1     0.34885
#> λ2     0.04409
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1644
#> 0001  0.1782
#> 0010  0.1719
#> 0011  0.2198
#> 0100  0.2147
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 147146.5 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5084
#> M2:  0.49
#> total scores:  0.6258

a$ss_EAP
#>            [,1]
#>  [1,] 0.1902648
#>  [2,] 0.1134500
#>  [3,] 0.1504925
#>  [4,] 0.1969962
#>  [5,] 0.1584564
#>  [6,] 0.1710181
#>  [7,] 0.1577528
#>  [8,] 0.2121778
#>  [9,] 0.1919964
#> [10,] 0.1601210
#> [11,] 0.2444938
#> [12,] 0.1830459
#> [13,] 0.1930814
#> [14,] 0.1384920
#> [15,] 0.1005799
#> [16,] 0.2426967
#> [17,] 0.1702931
#> [18,] 0.1974680
#> [19,] 0.2610128
#> [20,] 0.1751736
#> [21,] 0.1599600
#> [22,] 0.1294046
#> [23,] 0.1409712
#> [24,] 0.2216429
#> [25,] 0.1991619
#> [26,] 0.1420381
#> [27,] 0.1621668
#> [28,] 0.2168071
#> [29,] 0.1632738
#> [30,] 0.2137082
#> [31,] 0.1908028
#> [32,] 0.1146509
#> [33,] 0.1420352
#> [34,] 0.1391447
#> [35,] 0.2316326
#> [36,] 0.1548822
#> [37,] 0.1935073
#> [38,] 0.1058896
#> [39,] 0.1333153
#> [40,] 0.1563167
#> [41,] 0.1877872
#> [42,] 0.1566914
#> [43,] 0.1938008
#> [44,] 0.2142208
#> [45,] 0.1220436
#> [46,] 0.1857346
#> [47,] 0.1569584
#> [48,] 0.1927644
#> [49,] 0.1677976
#> [50,] 0.2483323
head(a$ss_EAP)
#>           [,1]
#> [1,] 0.1902648
#> [2,] 0.1134500
#> [3,] 0.1504925
#> [4,] 0.1969962
#> [5,] 0.1584564
#> [6,] 0.1710181

(3) Check for parameter estimation accuracy

(cor_thetas <- cor(thetas_true,a$thetas_EAP))
#>           [,1]
#> [1,] 0.8254132
(cor_taus <- cor(taus_true,a$response_times_coefficients$taus_EAP))
#>           [,1]
#> [1,] 0.9895209

(cor_ss <- cor(as.vector(itempars_true[,1]),a$ss_EAP))
#>          [,1]
#> [1,] 0.728508
(cor_gs <- cor(as.vector(itempars_true[,2]),a$gs_EAP))
#>           [,1]
#> [1,] 0.7073186

AAR_vec <- numeric(L)
for(t in 1:L){
  AAR_vec[t] <- mean(Alphas[,,t]==a$Alphas_est[,,t])
}
AAR_vec
#> [1] 0.9407143 0.9521429 0.9564286 0.9657143 0.9600000

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] 0.7971429 0.8257143 0.8428571 0.8685714 0.8514286

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

a$DIC
#>              Transition Response_Time Response    Joint    Total
#> D_bar           2367.07      126098.6 14779.49 2987.256 146232.4
#> D(theta_bar)    2126.19      125667.3 14706.49 2818.264 145318.3
#> DIC             2607.95      126529.8 14852.48 3156.247 147146.5
head(a$PPP_total_scores)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.80 0.28 0.14 0.96 0.38
#> [2,] 0.62 0.52 0.26 0.60 0.96
#> [3,] 0.84 0.84 0.76 0.74 1.00
#> [4,] 0.86 0.62 0.48 0.64 0.80
#> [5,] 0.44 0.38 0.52 0.46 0.12
#> [6,] 0.38 0.48 0.58 0.54 0.74
head(a$PPP_total_RTs)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.08 0.56 0.42 0.48 0.38
#> [2,] 0.52 0.26 0.74 0.96 0.68
#> [3,] 0.14 0.18 0.98 0.36 0.34
#> [4,] 0.28 0.86 0.58 0.00 0.84
#> [5,] 0.22 0.60 0.96 0.26 0.34
#> [6,] 0.38 0.94 0.72 0.40 0.06
head(a$PPP_item_means)
#> [1] 0.58 0.48 0.34 0.58 0.52 0.50
head(a$PPP_item_mean_RTs)
#> [1] 0.68 0.40 0.92 0.44 0.66 0.62
head(a$PPP_item_ORs)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
#> [1,]   NA 0.66 0.72 0.28 0.50 0.98 0.36 0.42 0.22  0.62  0.72  1.00  0.96  0.52
#> [2,]   NA   NA 0.06 0.44 0.76 0.96 0.34 0.72 0.96  0.60  0.78  0.90  0.94  0.88
#> [3,]   NA   NA   NA 0.68 0.60 0.10 0.26 0.80 0.38  0.22  0.02  0.40  0.56  0.34
#> [4,]   NA   NA   NA   NA 0.24 0.66 0.30 0.14 0.72  0.74  0.56  0.18  0.68  0.14
#> [5,]   NA   NA   NA   NA   NA 0.58 0.50 0.66 0.58  0.40  0.46  0.82  0.40  0.54
#> [6,]   NA   NA   NA   NA   NA   NA 0.42 0.36 0.70  0.36  0.90  0.44  0.64  0.76
#>      [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
#> [1,]  0.70  0.84  1.00  0.92  0.90  0.54  0.96  0.62  0.50  0.74  0.90  0.94
#> [2,]  0.86  0.28  0.94  0.90  1.00  0.86  0.26  0.02  0.86  0.80  0.76  0.16
#> [3,]  0.54  0.34  0.20  0.62  0.98  0.20  0.40  0.74  0.18  0.42  0.62  0.96
#> [4,]  0.42  0.68  0.34  0.52  0.98  0.56  0.24  0.38  0.66  0.18  0.44  0.48
#> [5,]  0.20  0.34  0.24  0.50  0.96  0.10  0.48  0.14  0.68  0.10  1.00  0.86
#> [6,]  0.06  0.66  0.74  0.82  0.84  0.62  0.74  0.38  0.48  0.30  0.88  0.30
#>      [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
#> [1,]  0.86  0.62  0.94  0.96  0.66  0.44  0.88  0.94  0.76  0.74  0.54  0.84
#> [2,]  0.22  0.84  0.24  0.78  0.10  0.94  0.58  0.68  0.22  0.98  0.64  0.02
#> [3,]  0.80  0.76  0.48  0.12  0.66  0.90  0.62  0.52  0.76  0.26  0.62  0.46
#> [4,]  0.84  0.34  0.48  0.82  0.92  0.80  0.66  0.90  0.32  0.74  0.24  0.20
#> [5,]  0.26  0.58  0.48  0.32  0.22  0.66  0.70  0.32  0.66  0.42  0.12  0.28
#> [6,]  0.78  0.82  0.28  0.44  0.38  0.54  0.56  0.64  0.22  0.80  0.24  0.10
#>      [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
#> [1,]  0.54  0.74  0.96  0.18  0.52  0.94  0.90  0.48  0.42  0.78  0.44   0.4
#> [2,]  0.30  0.74  0.28  0.48  0.12  0.98  0.60  0.10  0.04  0.74  0.30   0.5
#> [3,]  0.86  0.42  0.76  0.42  0.10  0.60  0.68  0.92  0.64  0.30  0.88   0.9
#> [4,]  0.84  0.18  0.40  0.68  0.60  0.86  0.68  0.68  1.00  0.92  0.96   0.9
#> [5,]  0.36  0.78  0.10  0.38  0.18  0.78  0.32  0.44  0.20  0.44  0.44   0.2
#> [6,]  0.60  0.74  0.16  0.18  0.02  0.70  0.48  0.00  0.02  0.14  0.32   0.3