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 
#>  64  55  83 114  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
#>  0.22532 0.2052
#>  0.21637 0.1363
#>  0.28094 0.1371
#>  0.21710 0.1168
#>  0.09446 0.2245
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0     -3.1008
#> λ1      0.0947
#> λ2      0.2752
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1316
#> 0001  0.2110
#> 0010  0.1564
#> 0011  0.2303
#> 0100  0.1277
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 158202.4 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5076
#> M2:  0.49
#> total scores:  0.6235
a <- summary(output_HMDCM_RT_joint)
a
#> 
#> Model: DINA_HO_RT_joint 
#> 
#> Item Parameters:
#>   ss_EAP gs_EAP
#>  0.22532 0.2052
#>  0.21637 0.1363
#>  0.28094 0.1371
#>  0.21710 0.1168
#>  0.09446 0.2245
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0     -3.1008
#> λ1      0.0947
#> λ2      0.2752
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1316
#> 0001  0.2110
#> 0010  0.1564
#> 0011  0.2303
#> 0100  0.1277
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 158202.4 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5108
#> M2:  0.49
#> total scores:  0.623

a$ss_EAP
#>             [,1]
#>  [1,] 0.22532151
#>  [2,] 0.21637219
#>  [3,] 0.28093710
#>  [4,] 0.21710260
#>  [5,] 0.09446081
#>  [6,] 0.09450295
#>  [7,] 0.15848813
#>  [8,] 0.15109879
#>  [9,] 0.19932304
#> [10,] 0.13248230
#> [11,] 0.14290901
#> [12,] 0.18587316
#> [13,] 0.15098260
#> [14,] 0.17363300
#> [15,] 0.15046872
#> [16,] 0.16897644
#> [17,] 0.17511254
#> [18,] 0.20521302
#> [19,] 0.10588287
#> [20,] 0.10875672
#> [21,] 0.21934760
#> [22,] 0.18824273
#> [23,] 0.13966137
#> [24,] 0.25493274
#> [25,] 0.16736327
#> [26,] 0.21540074
#> [27,] 0.18807268
#> [28,] 0.14223415
#> [29,] 0.14443937
#> [30,] 0.19691753
#> [31,] 0.15981576
#> [32,] 0.20097515
#> [33,] 0.17020442
#> [34,] 0.22851427
#> [35,] 0.14086579
#> [36,] 0.10862603
#> [37,] 0.24408210
#> [38,] 0.23114127
#> [39,] 0.15210578
#> [40,] 0.10940601
#> [41,] 0.18706496
#> [42,] 0.17652787
#> [43,] 0.14491090
#> [44,] 0.15709135
#> [45,] 0.16179697
#> [46,] 0.13527903
#> [47,] 0.14532566
#> [48,] 0.19799098
#> [49,] 0.22146977
#> [50,] 0.20726222
head(a$ss_EAP)
#>            [,1]
#> [1,] 0.22532151
#> [2,] 0.21637219
#> [3,] 0.28093710
#> [4,] 0.21710260
#> [5,] 0.09446081
#> [6,] 0.09450295

(3) Check for parameter estimation accuracy

(cor_thetas <- cor(thetas_true,a$thetas_EAP))
#>           [,1]
#> [1,] 0.8036101
(cor_taus <- cor(taus_true,a$response_times_coefficients$taus_EAP))
#>          [,1]
#> [1,] 0.984953

(cor_ss <- cor(as.vector(itempars_true[,1]),a$ss_EAP))
#>           [,1]
#> [1,] 0.5691691
(cor_gs <- cor(as.vector(itempars_true[,2]),a$gs_EAP))
#>           [,1]
#> [1,] 0.5874792

AAR_vec <- numeric(L)
for(t in 1:L){
  AAR_vec[t] <- mean(Alphas[,,t]==a$Alphas_est[,,t])
}
AAR_vec
#> [1] 0.9100000 0.9185714 0.9378571 0.9428571 0.9500000

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] 0.6857143 0.6971429 0.7857143 0.8085714 0.8400000

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

a$DIC
#>              Transition Response_Time Response    Joint    Total
#> D_bar          1914.792      136395.3 15164.48 3631.777 157106.4
#> D(theta_bar)   1645.803      135961.0 14901.81 3501.764 156010.4
#> DIC            2183.781      136829.6 15427.15 3761.790 158202.4
head(a$PPP_total_scores)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.42 0.76 0.20 0.60 0.62
#> [2,] 0.90 0.46 0.86 0.84 0.88
#> [3,] 0.94 0.06 0.92 0.48 0.86
#> [4,] 0.72 0.44 0.58 0.50 0.76
#> [5,] 0.86 0.70 0.18 0.72 0.16
#> [6,] 0.94 0.44 0.32 0.88 0.32
head(a$PPP_total_RTs)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.42 0.14 0.84 0.12 0.30
#> [2,] 0.62 0.64 0.68 0.34 0.16
#> [3,] 0.14 0.58 0.66 0.70 0.34
#> [4,] 0.16 0.20 0.72 0.70 0.26
#> [5,] 0.94 0.16 0.76 0.26 0.92
#> [6,] 0.90 0.26 0.64 0.00 0.26
head(a$PPP_item_means)
#> [1] 0.40 0.48 0.56 0.48 0.46 0.68
head(a$PPP_item_mean_RTs)
#> [1] 0.58 0.42 0.24 0.30 0.42 0.38
head(a$PPP_item_ORs)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
#> [1,]   NA 0.86 0.96 0.76 0.88 0.74 0.28 0.58 0.70  0.74  0.98  0.74  0.32  0.98
#> [2,]   NA   NA 1.00 0.72 0.58 0.80 0.54 0.04 0.40  0.84  0.98  0.78  0.50  1.00
#> [3,]   NA   NA   NA 0.44 0.82 0.56 0.94 0.46 0.98  0.90  0.44  0.80  0.64  0.98
#> [4,]   NA   NA   NA   NA 0.94 0.72 0.98 0.86 0.86  0.94  0.88  0.54  0.82  0.24
#> [5,]   NA   NA   NA   NA   NA 0.52 0.54 0.58 0.34  0.14  0.96  0.60  0.22  0.36
#> [6,]   NA   NA   NA   NA   NA   NA 0.72 0.60 0.74  0.94  0.50  0.42  0.60  0.82
#>      [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
#> [1,]  0.88  0.64  0.90  0.84  0.84  0.86  0.58  0.44  0.46  0.84  0.34  0.18
#> [2,]  0.70  0.94  0.62  0.84  0.98  0.88  0.34  0.26  0.30  0.38  0.70  0.72
#> [3,]  0.64  0.16  0.52  0.48  0.90  0.76  0.94  0.58  0.18  0.88  0.48  0.78
#> [4,]  0.90  0.62  1.00  0.72  0.92  0.88  0.56  0.26  0.68  0.66  0.42  0.50
#> [5,]  0.14  0.20  0.54  0.84  0.86  0.48  0.50  0.46  0.46  0.40  0.78  0.16
#> [6,]  0.28  0.28  0.64  0.28  0.64  0.66  0.06  0.92  0.52  0.58  0.42  0.00
#>      [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
#> [1,]  0.00  0.24  0.22  0.36  0.02  0.66  0.74  0.80  0.86  0.40  0.50  0.84
#> [2,]  0.72  0.34  0.46  1.00  0.02  0.08  0.66  0.68  0.28  0.20  0.42  0.24
#> [3,]  0.32  0.68  0.92  0.70  0.66  0.20  0.16  0.50  0.68  0.56  0.58  0.86
#> [4,]  0.44  0.06  0.96  0.16  0.70  0.64  0.12  0.30  0.78  0.62  0.92  0.56
#> [5,]  0.64  0.20  0.24  0.22  0.76  0.40  0.96  0.56  0.82  0.00  0.46  0.40
#> [6,]  0.54  0.18  0.32  0.58  0.00  0.26  0.48  0.58  0.16  0.08  0.64  0.28
#>      [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
#> [1,]  0.26  0.48  0.92  0.94  0.66  0.92  0.94  0.58  0.80  1.00  1.00  0.12
#> [2,]  0.34  0.28  0.34  0.98  0.20  0.28  0.42  0.08  0.62  0.58  0.38  0.04
#> [3,]  0.44  0.80  0.46  0.20  0.72  0.94  0.52  0.70  0.84  0.68  1.00  0.04
#> [4,]  0.56  0.68  0.02  1.00  0.70  0.78  0.54  0.36  0.68  0.92  0.80  0.92
#> [5,]  0.60  0.70  0.22  0.96  0.10  0.40  0.74  0.20  0.74  0.46  0.90  0.14
#> [6,]  0.06  0.76  0.68  0.98  0.42  0.36  0.60  0.02  0.48  0.46  0.56  0.28