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 
#>  61  62  91 109  27
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
#>  0.2030 0.23168
#>  0.1091 0.12041
#>  0.1770 0.12747
#>  0.1546 0.09689
#>  0.1157 0.11087
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0     -2.3377
#> λ1      0.1963
#> λ2      0.1544
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000 0.09574
#> 0001 0.24318
#> 0010 0.16785
#> 0011 0.21742
#> 0100 0.16746
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 155988.9 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.534
#> M2:  0.49
#> total scores:  0.6292
a <- summary(output_HMDCM_RT_joint)
a
#> 
#> Model: DINA_HO_RT_joint 
#> 
#> Item Parameters:
#>  ss_EAP  gs_EAP
#>  0.2030 0.23168
#>  0.1091 0.12041
#>  0.1770 0.12747
#>  0.1546 0.09689
#>  0.1157 0.11087
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0     -2.3377
#> λ1      0.1963
#> λ2      0.1544
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000 0.09574
#> 0001 0.24318
#> 0010 0.16785
#> 0011 0.21742
#> 0100 0.16746
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 155988.9 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5292
#> M2:  0.49
#> total scores:  0.6272

a$ss_EAP
#>             [,1]
#>  [1,] 0.20301334
#>  [2,] 0.10912259
#>  [3,] 0.17703577
#>  [4,] 0.15462810
#>  [5,] 0.11567108
#>  [6,] 0.12565553
#>  [7,] 0.12469843
#>  [8,] 0.13329129
#>  [9,] 0.16432619
#> [10,] 0.13120551
#> [11,] 0.16416969
#> [12,] 0.19930400
#> [13,] 0.16593883
#> [14,] 0.19189222
#> [15,] 0.27033759
#> [16,] 0.13135951
#> [17,] 0.11547317
#> [18,] 0.09660247
#> [19,] 0.15458020
#> [20,] 0.15971067
#> [21,] 0.13379450
#> [22,] 0.16569405
#> [23,] 0.16890756
#> [24,] 0.15771111
#> [25,] 0.23411613
#> [26,] 0.24103636
#> [27,] 0.17068563
#> [28,] 0.29290814
#> [29,] 0.10543051
#> [30,] 0.17714222
#> [31,] 0.10189851
#> [32,] 0.20225965
#> [33,] 0.27561481
#> [34,] 0.08975443
#> [35,] 0.10310855
#> [36,] 0.17609491
#> [37,] 0.13242350
#> [38,] 0.14521571
#> [39,] 0.18010555
#> [40,] 0.17578248
#> [41,] 0.13574845
#> [42,] 0.15598063
#> [43,] 0.15710907
#> [44,] 0.12691533
#> [45,] 0.10959310
#> [46,] 0.21819883
#> [47,] 0.22386483
#> [48,] 0.13917932
#> [49,] 0.24892180
#> [50,] 0.26665407
head(a$ss_EAP)
#>           [,1]
#> [1,] 0.2030133
#> [2,] 0.1091226
#> [3,] 0.1770358
#> [4,] 0.1546281
#> [5,] 0.1156711
#> [6,] 0.1256555

(3) Check for parameter estimation accuracy

(cor_thetas <- cor(thetas_true,a$thetas_EAP))
#>          [,1]
#> [1,] 0.805015
(cor_taus <- cor(taus_true,a$response_times_coefficients$taus_EAP))
#>           [,1]
#> [1,] 0.9879564

(cor_ss <- cor(as.vector(itempars_true[,1]),a$ss_EAP))
#>           [,1]
#> [1,] 0.7557193
(cor_gs <- cor(as.vector(itempars_true[,2]),a$gs_EAP))
#>           [,1]
#> [1,] 0.6458773

AAR_vec <- numeric(L)
for(t in 1:L){
  AAR_vec[t] <- mean(Alphas[,,t]==a$Alphas_est[,,t])
}
AAR_vec
#> [1] 0.9214286 0.9357143 0.9578571 0.9607143 0.9564286

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] 0.7371429 0.7771429 0.8485714 0.8628571 0.8571429

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

a$DIC
#>              Transition Response_Time Response    Joint    Total
#> D_bar          1985.595      135047.9 14476.15 3394.077 154903.7
#> D(theta_bar)   1673.223      134622.6 14239.71 3282.907 153818.5
#> DIC            2297.967      135473.1 14712.59 3505.248 155988.9
head(a$PPP_total_scores)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.14 0.64 0.08 0.58 0.50
#> [2,] 0.82 0.36 0.64 0.04 0.28
#> [3,] 0.72 0.48 0.66 0.74 0.72
#> [4,] 0.78 0.86 0.88 0.90 0.94
#> [5,] 0.86 0.40 0.92 0.82 0.94
#> [6,] 0.36 0.94 0.38 0.14 0.82
head(a$PPP_total_RTs)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.34 0.90 0.00 0.34 0.82
#> [2,] 0.96 0.22 0.60 0.72 0.44
#> [3,] 1.00 0.38 0.80 0.24 0.02
#> [4,] 0.26 0.50 0.22 0.44 0.90
#> [5,] 0.10 0.96 0.70 0.08 0.94
#> [6,] 0.30 0.90 0.22 0.38 0.38
head(a$PPP_item_means)
#> [1] 0.54 0.62 0.70 0.50 0.38 0.44
head(a$PPP_item_mean_RTs)
#> [1] 0.42 0.52 0.48 0.32 0.44 0.36
head(a$PPP_item_ORs)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
#> [1,]   NA 0.56 0.82 0.38 0.74 0.78 0.50 0.38 0.24  0.24  0.88  0.50  0.90  0.44
#> [2,]   NA   NA 0.98 0.52 0.44 0.28 0.50 0.70 0.78  0.46  0.58  0.36  0.20  0.22
#> [3,]   NA   NA   NA 0.86 0.74 1.00 0.92 0.88 0.88  0.96  0.48  0.98  0.20  1.00
#> [4,]   NA   NA   NA   NA 0.78 0.42 1.00 0.86 0.92  0.64  0.30  0.88  0.70  0.90
#> [5,]   NA   NA   NA   NA   NA 0.32 0.62 0.72 0.68  0.50  0.44  0.30  0.40  0.80
#> [6,]   NA   NA   NA   NA   NA   NA 0.62 0.86 0.72  0.60  0.32  1.00  0.48  0.40
#>      [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
#> [1,]  0.80  0.68  0.20  1.00  0.94  0.86  0.38  0.82  0.98  0.10  0.80  0.82
#> [2,]  0.80  0.86  0.10  1.00  0.92  0.46  0.72  0.94  1.00  0.12  0.86  0.96
#> [3,]  0.96  1.00  0.74  0.94  0.92  0.78  0.76  0.98  0.40  0.58  0.82  0.22
#> [4,]  0.66  0.90  0.56  0.76  0.98  0.70  0.94  0.48  0.20  0.64  0.92  0.88
#> [5,]  0.92  0.80  0.20  0.88  0.94  0.36  0.36  1.00  0.54  0.32  1.00  0.96
#> [6,]  0.62  0.12  0.08  0.64  0.98  0.92  0.32  0.72  0.58  0.46  0.74  0.62
#>      [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
#> [1,]  1.00  0.50  0.02  0.10  0.32  0.54  0.50  0.38  0.50  0.22  0.86  0.38
#> [2,]  0.92  0.74  0.76  0.84  0.76  0.70  0.76  0.74  0.16  0.28  0.44  0.52
#> [3,]  0.66  0.68  0.76  0.44  0.80  0.82  0.92  0.66  0.50  0.96  0.74  1.00
#> [4,]  0.60  0.68  0.86  0.82  0.52  0.48  0.58  0.76  0.32  0.20  0.48  0.54
#> [5,]  1.00  0.58  0.32  0.58  0.74  0.76  0.38  0.98  0.80  0.36  0.52  0.58
#> [6,]  0.90  0.14  0.52  0.70  0.52  0.74  0.02  0.92  0.44  0.08  0.62  0.44
#>      [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
#> [1,]  0.28  0.10  0.30  0.56  0.88  0.46  1.00  0.80  0.06  0.48  0.30  0.62
#> [2,]  0.30  0.12  0.08  0.90  0.48  0.92  0.96  1.00  0.60  0.50  0.76  0.48
#> [3,]  0.48  0.58  0.38  0.70  0.78  0.54  0.94  0.68  0.06  0.68  0.42  0.76
#> [4,]  0.94  0.10  0.44  0.84  0.56  0.86  0.80  0.38  0.28  0.34  0.52  0.68
#> [5,]  0.28  0.36  0.82  0.86  0.74  0.82  1.00  0.90  0.46  0.18  0.94  0.48
#> [6,]  0.26  0.06  0.72  0.76  0.44  0.06  0.90  0.86  0.04  0.40  0.46  0.18