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 
#> 68 62 83 99 38
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
#>  0.1557 0.1143
#>  0.2245 0.1741
#>  0.2263 0.1156
#>  0.1533 0.2161
#>  0.1638 0.1624
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0     -2.0626
#> λ1      0.1431
#> λ2      0.2529
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.0859
#> 0001  0.1950
#> 0010  0.1360
#> 0011  0.2987
#> 0100  0.1423
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 154707 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5164
#> M2:  0.49
#> total scores:  0.6248
a <- summary(output_HMDCM_RT_joint)
a
#> 
#> Model: DINA_HO_RT_joint 
#> 
#> Item Parameters:
#>  ss_EAP gs_EAP
#>  0.1557 0.1143
#>  0.2245 0.1741
#>  0.2263 0.1156
#>  0.1533 0.2161
#>  0.1638 0.1624
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0     -2.0626
#> λ1      0.1431
#> λ2      0.2529
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.0859
#> 0001  0.1950
#> 0010  0.1360
#> 0011  0.2987
#> 0100  0.1423
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 154707 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5172
#> M2:  0.49
#> total scores:  0.6257

a$ss_EAP
#>            [,1]
#>  [1,] 0.1557205
#>  [2,] 0.2244812
#>  [3,] 0.2262660
#>  [4,] 0.1532767
#>  [5,] 0.1638374
#>  [6,] 0.1507844
#>  [7,] 0.1952788
#>  [8,] 0.1364254
#>  [9,] 0.1975456
#> [10,] 0.1644454
#> [11,] 0.1298776
#> [12,] 0.2212982
#> [13,] 0.2286232
#> [14,] 0.1627503
#> [15,] 0.1541261
#> [16,] 0.2147110
#> [17,] 0.2043280
#> [18,] 0.1620837
#> [19,] 0.2023271
#> [20,] 0.1051622
#> [21,] 0.1243876
#> [22,] 0.1804976
#> [23,] 0.2746697
#> [24,] 0.1359099
#> [25,] 0.2426224
#> [26,] 0.1727554
#> [27,] 0.2772798
#> [28,] 0.2448241
#> [29,] 0.2045482
#> [30,] 0.1153920
#> [31,] 0.2095491
#> [32,] 0.2167588
#> [33,] 0.1888326
#> [34,] 0.2458929
#> [35,] 0.2068952
#> [36,] 0.1455357
#> [37,] 0.2163113
#> [38,] 0.1978453
#> [39,] 0.2248851
#> [40,] 0.1874084
#> [41,] 0.1766877
#> [42,] 0.1930233
#> [43,] 0.1351152
#> [44,] 0.1264922
#> [45,] 0.2971824
#> [46,] 0.1334946
#> [47,] 0.1585078
#> [48,] 0.1526695
#> [49,] 0.2295854
#> [50,] 0.2177479
head(a$ss_EAP)
#>           [,1]
#> [1,] 0.1557205
#> [2,] 0.2244812
#> [3,] 0.2262660
#> [4,] 0.1532767
#> [5,] 0.1638374
#> [6,] 0.1507844

(3) Check for parameter estimation accuracy

(cor_thetas <- cor(thetas_true,a$thetas_EAP))
#>           [,1]
#> [1,] 0.8165708
(cor_taus <- cor(taus_true,a$response_times_coefficients$taus_EAP))
#>           [,1]
#> [1,] 0.9868757

(cor_ss <- cor(as.vector(itempars_true[,1]),a$ss_EAP))
#>           [,1]
#> [1,] 0.5570496
(cor_gs <- cor(as.vector(itempars_true[,2]),a$gs_EAP))
#>          [,1]
#> [1,] 0.526791

AAR_vec <- numeric(L)
for(t in 1:L){
  AAR_vec[t] <- mean(Alphas[,,t]==a$Alphas_est[,,t])
}
AAR_vec
#> [1] 0.8842857 0.8978571 0.9171429 0.9328571 0.9335714

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] 0.6142857 0.6685714 0.7257143 0.7714286 0.7885714

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

a$DIC
#>              Transition Response_Time Response    Joint    Total
#> D_bar          1644.209      132763.2 15429.68 3737.795 153574.9
#> D(theta_bar)   1430.446      132328.9 15053.73 3629.588 152442.7
#> DIC            1857.971      133197.4 15805.63 3846.002 154707.0
head(a$PPP_total_scores)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.96 0.36 0.54 0.92 0.58
#> [2,] 0.34 0.88 0.52 0.10 0.96
#> [3,] 1.00 0.84 0.70 0.80 0.54
#> [4,] 0.86 0.40 0.30 0.70 0.88
#> [5,] 0.88 0.82 1.00 0.90 0.84
#> [6,] 0.54 0.90 0.34 0.08 0.80
head(a$PPP_total_RTs)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.78 0.56 0.32 0.52 0.52
#> [2,] 0.38 0.18 0.18 0.96 0.86
#> [3,] 0.54 0.10 0.38 0.66 0.68
#> [4,] 0.58 0.54 0.46 0.56 0.40
#> [5,] 0.86 0.30 0.12 0.32 0.88
#> [6,] 0.40 0.76 0.70 0.56 0.50
head(a$PPP_item_means)
#> [1] 0.56 0.56 0.62 0.44 0.46 0.46
head(a$PPP_item_mean_RTs)
#> [1] 0.40 0.58 0.54 0.32 0.58 0.56
head(a$PPP_item_ORs)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
#> [1,]   NA 0.46 0.94 0.96 0.86 0.98 0.58 0.50 0.90  0.48  0.22  0.84  0.82  0.92
#> [2,]   NA   NA 0.52 0.96 0.68 0.82 0.70 0.52 0.14  0.46  0.76  0.82  0.46  0.66
#> [3,]   NA   NA   NA 0.94 0.98 0.90 1.00 0.76 0.90  0.96  0.08  0.80  0.94  0.88
#> [4,]   NA   NA   NA   NA 1.00 0.84 0.90 0.82 0.90  0.86  0.68  0.96  1.00  0.94
#> [5,]   NA   NA   NA   NA   NA 0.90 0.74 0.56 0.92  0.62  0.74  0.94  0.86  0.80
#> [6,]   NA   NA   NA   NA   NA   NA 0.78 0.88 0.32  0.46  0.74  0.98  0.00  0.82
#>      [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
#> [1,]  0.66  0.54  0.88  0.52  0.86  0.82  0.76  1.00  0.42  1.00  0.24  0.56
#> [2,]  0.58  0.94  0.52  0.78  0.94  0.28  0.58  0.22  0.82  0.62  0.32  0.60
#> [3,]  0.66  0.32  0.90  0.88  0.70  0.80  0.56  0.86  0.42  0.36  0.78  0.88
#> [4,]  0.98  0.52  0.94  0.92  0.86  0.98  0.72  0.94  0.94  0.72  1.00  0.66
#> [5,]  0.98  0.50  1.00  0.90  0.92  0.90  0.86  0.96  0.98  1.00  0.56  0.92
#> [6,]  0.80  0.90  0.92  0.92  0.94  0.78  0.30  0.60  0.72  0.44  0.28  0.14
#>      [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
#> [1,]  0.92  0.10  0.84  0.70  0.42  0.40  0.32  0.68  0.56  0.22  0.64  0.46
#> [2,]  0.10  0.02  0.52  0.34  0.82  0.64  0.52  0.82  0.74  0.16  0.40  0.76
#> [3,]  1.00  0.50  0.64  0.88  0.96  0.96  0.68  0.92  0.56  0.20  0.54  0.72
#> [4,]  0.84  0.94  0.82  0.98  1.00  1.00  0.92  0.64  0.98  0.98  0.50  1.00
#> [5,]  0.74  0.22  0.70  1.00  0.86  0.56  0.54  0.64  0.36  1.00  0.76  0.90
#> [6,]  0.42  0.06  0.48  0.20  0.28  0.12  0.34  0.26  0.06  0.44  0.06  0.92
#>      [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
#> [1,]  0.88  0.56  0.96  0.72  0.88  0.54  0.96  0.86  0.32  0.68  0.82  0.86
#> [2,]  0.84  0.66  0.94  0.72  0.60  0.64  0.96  0.60  0.68  0.86  0.90  0.78
#> [3,]  0.90  0.80  0.76  0.76  0.92  0.58  0.64  1.00  0.38  0.92  0.90  0.98
#> [4,]  0.78  0.98  0.84  0.78  0.94  0.96  0.94  1.00  0.86  0.90  0.92  0.68
#> [5,]  0.54  0.66  0.96  0.86  0.86  0.62  0.58  0.90  0.66  0.76  0.30  0.94
#> [6,]  0.36  0.02  0.44  0.48  0.44  0.92  0.88  0.88  0.38  0.92  0.76  0.64