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 
#>  64  58  84 107  37
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
#>  0.1727 0.08984
#>  0.1816 0.17441
#>  0.1579 0.11196
#>  0.1180 0.16100
#>  0.1086 0.12128
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0     -1.9333
#> λ1      0.1473
#> λ2      0.1668
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1683
#> 0001  0.1329
#> 0010  0.1479
#> 0011  0.2828
#> 0100  0.1229
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 158101 
#> 
#> Posterior Predictive P-value (PPP):
#> M1:   0.5
#> M2:  0.49
#> total scores:  0.6265
a <- summary(output_HMDCM_RT_joint)
a
#> 
#> Model: DINA_HO_RT_joint 
#> 
#> Item Parameters:
#>  ss_EAP  gs_EAP
#>  0.1727 0.08984
#>  0.1816 0.17441
#>  0.1579 0.11196
#>  0.1180 0.16100
#>  0.1086 0.12128
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0     -1.9333
#> λ1      0.1473
#> λ2      0.1668
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1683
#> 0001  0.1329
#> 0010  0.1479
#> 0011  0.2828
#> 0100  0.1229
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 158101 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5008
#> M2:  0.49
#> total scores:  0.6284

a$ss_EAP
#>             [,1]
#>  [1,] 0.17269201
#>  [2,] 0.18155850
#>  [3,] 0.15793767
#>  [4,] 0.11800202
#>  [5,] 0.10859240
#>  [6,] 0.18435075
#>  [7,] 0.13729573
#>  [8,] 0.09056036
#>  [9,] 0.14026537
#> [10,] 0.10935337
#> [11,] 0.10923594
#> [12,] 0.18604417
#> [13,] 0.17070529
#> [14,] 0.09539927
#> [15,] 0.10884538
#> [16,] 0.15417946
#> [17,] 0.18275347
#> [18,] 0.17545294
#> [19,] 0.24954013
#> [20,] 0.13180079
#> [21,] 0.12408538
#> [22,] 0.20856378
#> [23,] 0.13759198
#> [24,] 0.22857160
#> [25,] 0.18071116
#> [26,] 0.23041432
#> [27,] 0.16984212
#> [28,] 0.20469014
#> [29,] 0.17311793
#> [30,] 0.17487781
#> [31,] 0.14947353
#> [32,] 0.12124006
#> [33,] 0.24027107
#> [34,] 0.13190875
#> [35,] 0.17088618
#> [36,] 0.10801057
#> [37,] 0.25565457
#> [38,] 0.12657494
#> [39,] 0.19016467
#> [40,] 0.14297617
#> [41,] 0.12272135
#> [42,] 0.15629315
#> [43,] 0.22920419
#> [44,] 0.24783945
#> [45,] 0.21246526
#> [46,] 0.14814914
#> [47,] 0.28445936
#> [48,] 0.22325116
#> [49,] 0.15027183
#> [50,] 0.16661440
head(a$ss_EAP)
#>           [,1]
#> [1,] 0.1726920
#> [2,] 0.1815585
#> [3,] 0.1579377
#> [4,] 0.1180020
#> [5,] 0.1085924
#> [6,] 0.1843508

(3) Check for parameter estimation accuracy

(cor_thetas <- cor(thetas_true,a$thetas_EAP))
#>           [,1]
#> [1,] 0.7914858
(cor_taus <- cor(taus_true,a$response_times_coefficients$taus_EAP))
#>           [,1]
#> [1,] 0.9869964

(cor_ss <- cor(as.vector(itempars_true[,1]),a$ss_EAP))
#>           [,1]
#> [1,] 0.7527781
(cor_gs <- cor(as.vector(itempars_true[,2]),a$gs_EAP))
#>           [,1]
#> [1,] 0.7282395

AAR_vec <- numeric(L)
for(t in 1:L){
  AAR_vec[t] <- mean(Alphas[,,t]==a$Alphas_est[,,t])
}
AAR_vec
#> [1] 0.9271429 0.9321429 0.9571429 0.9635714 0.9628571

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] 0.7257143 0.7600000 0.8342857 0.8657143 0.8685714

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

a$DIC
#>              Transition Response_Time Response    Joint    Total
#> D_bar          1959.319      137004.6 14607.29 3549.689 157120.9
#> D(theta_bar)   1704.693      136565.8 14429.81 3440.395 156140.7
#> DIC            2213.944      137443.3 14784.77 3658.983 158101.0
head(a$PPP_total_scores)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.96 0.66 0.80 0.12 0.22
#> [2,] 0.20 0.38 0.08 0.84 0.92
#> [3,] 0.82 0.88 0.32 0.64 0.86
#> [4,] 0.76 0.04 0.96 0.40 0.74
#> [5,] 1.00 0.70 0.32 0.78 0.34
#> [6,] 0.34 0.40 0.78 0.78 0.70
head(a$PPP_total_RTs)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.32 0.56 0.86 0.12 0.20
#> [2,] 0.56 0.84 0.18 0.90 0.02
#> [3,] 0.24 0.46 0.64 0.46 0.78
#> [4,] 0.42 0.64 0.52 0.78 0.28
#> [5,] 0.80 0.26 0.56 0.52 0.66
#> [6,] 0.82 0.00 0.08 0.88 0.32
head(a$PPP_item_means)
#> [1] 0.56 0.60 0.48 0.44 0.48 0.44
head(a$PPP_item_mean_RTs)
#> [1] 0.22 0.52 0.48 0.68 0.14 0.36
head(a$PPP_item_ORs)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
#> [1,]   NA 0.94 0.92 0.18 0.94 0.38 0.82 0.70 0.64  0.98  0.48  0.84  0.34  0.36
#> [2,]   NA   NA 0.82 0.60 0.82 0.92 0.90 0.68 0.54  0.78  0.40  0.28  0.42  0.96
#> [3,]   NA   NA   NA 0.66 0.62 0.42 0.60 0.72 0.78  0.24  0.80  0.82  0.80  0.98
#> [4,]   NA   NA   NA   NA 0.48 0.06 0.16 0.16 0.76  0.16  0.66  0.28  0.90  0.82
#> [5,]   NA   NA   NA   NA   NA 0.30 0.54 0.62 0.46  0.34  0.84  0.36  0.88  0.52
#> [6,]   NA   NA   NA   NA   NA   NA 0.88 0.62 0.58  0.50  0.10  0.78  0.20  0.70
#>      [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
#> [1,]  0.72  0.34  0.90  0.02  0.42  0.04  0.86  0.92  0.70  0.94  0.32  0.48
#> [2,]  0.92  0.90  0.42  0.90  1.00  0.96  0.88  0.92  0.54  0.88  0.62  0.52
#> [3,]  0.84  0.08  0.66  0.56  0.98  0.38  0.12  0.18  0.58  0.46  0.34  0.38
#> [4,]  0.02  0.18  0.32  0.30  0.40  0.28  0.46  0.50  0.20  0.34  0.44  0.34
#> [5,]  0.76  0.50  0.92  0.90  0.84  0.60  0.50  0.98  0.44  0.96  0.46  0.98
#> [6,]  0.30  0.54  0.90  0.72  0.84  0.62  0.10  0.78  0.12  0.76  0.02  0.50
#>      [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
#> [1,]  0.10  0.34  0.58  0.76  0.12  0.66  0.80  0.72  0.94  0.78  0.46  0.94
#> [2,]  0.14  0.78  0.64  0.90  0.46  0.88  0.94  0.92  0.80  0.74  0.90  0.82
#> [3,]  0.84  0.46  0.08  0.68  0.06  0.24  0.36  0.62  0.14  0.60  0.38  0.14
#> [4,]  0.46  0.36  0.48  0.36  0.74  0.34  0.56  0.92  0.58  0.30  0.24  0.26
#> [5,]  0.28  0.24  0.32  0.82  0.40  0.64  0.72  0.72  0.66  0.78  0.62  0.32
#> [6,]  0.02  0.16  0.32  0.48  0.42  0.84  0.66  0.40  0.40  0.54  0.08  0.32
#>      [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
#> [1,]  0.80  0.28  0.50  0.70  0.48  0.80  0.82  0.90  0.84  0.80  0.88  0.52
#> [2,]  0.70  0.76  0.86  0.42  0.30  0.94  0.62  0.76  0.98  0.90  0.78  0.76
#> [3,]  0.50  0.20  0.88  0.56  0.82  0.06  0.78  0.70  0.28  0.68  0.38  0.06
#> [4,]  0.88  0.18  0.36  0.92  0.42  0.94  0.98  0.42  0.80  0.06  0.56  0.12
#> [5,]  0.88  0.26  0.56  0.80  0.24  0.18  0.74  0.26  0.44  0.84  0.64  0.44
#> [6,]  0.18  0.22  0.76  0.12  0.22  0.46  0.52  0.78  0.24  0.34  0.52  0.44