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 
#>  67  49  79 129  26
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
#>  0.10572 0.1025
#>  0.14765 0.1882
#>  0.17772 0.1061
#>  0.20545 0.2311
#>  0.07481 0.1794
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.74342
#> λ1     0.34487
#> λ2     0.08736
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1876
#> 0001  0.2051
#> 0010  0.1607
#> 0011  0.1679
#> 0100  0.1941
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 155443 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5076
#> M2:  0.49
#> total scores:  0.6251
a <- summary(output_HMDCM_RT_joint)
a
#> 
#> Model: DINA_HO_RT_joint 
#> 
#> Item Parameters:
#>   ss_EAP gs_EAP
#>  0.10572 0.1025
#>  0.14765 0.1882
#>  0.17772 0.1061
#>  0.20545 0.2311
#>  0.07481 0.1794
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.74342
#> λ1     0.34487
#> λ2     0.08736
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1876
#> 0001  0.2051
#> 0010  0.1607
#> 0011  0.1679
#> 0100  0.1941
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 155443 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5076
#> M2:  0.49
#> total scores:  0.625

a$ss_EAP
#>             [,1]
#>  [1,] 0.10572029
#>  [2,] 0.14764775
#>  [3,] 0.17771958
#>  [4,] 0.20545010
#>  [5,] 0.07480516
#>  [6,] 0.11027512
#>  [7,] 0.15432273
#>  [8,] 0.19726466
#>  [9,] 0.13006040
#> [10,] 0.24271729
#> [11,] 0.22655114
#> [12,] 0.15568252
#> [13,] 0.12144776
#> [14,] 0.12885847
#> [15,] 0.11748241
#> [16,] 0.14170231
#> [17,] 0.19091638
#> [18,] 0.14665189
#> [19,] 0.18829297
#> [20,] 0.15596921
#> [21,] 0.13360584
#> [22,] 0.15646014
#> [23,] 0.22354200
#> [24,] 0.19549286
#> [25,] 0.17574291
#> [26,] 0.08578716
#> [27,] 0.20557775
#> [28,] 0.15168265
#> [29,] 0.16038929
#> [30,] 0.21557818
#> [31,] 0.12571759
#> [32,] 0.11854224
#> [33,] 0.16347358
#> [34,] 0.17515197
#> [35,] 0.18085111
#> [36,] 0.18890819
#> [37,] 0.17286079
#> [38,] 0.19008726
#> [39,] 0.17487369
#> [40,] 0.18983465
#> [41,] 0.17567377
#> [42,] 0.17678561
#> [43,] 0.17394711
#> [44,] 0.21814667
#> [45,] 0.22859169
#> [46,] 0.21411009
#> [47,] 0.13424969
#> [48,] 0.16103405
#> [49,] 0.22249557
#> [50,] 0.23098546
head(a$ss_EAP)
#>            [,1]
#> [1,] 0.10572029
#> [2,] 0.14764775
#> [3,] 0.17771958
#> [4,] 0.20545010
#> [5,] 0.07480516
#> [6,] 0.11027512

(3) Check for parameter estimation accuracy

(cor_thetas <- cor(thetas_true,a$thetas_EAP))
#>           [,1]
#> [1,] 0.8040642
(cor_taus <- cor(taus_true,a$response_times_coefficients$taus_EAP))
#>           [,1]
#> [1,] 0.9877751

(cor_ss <- cor(as.vector(itempars_true[,1]),a$ss_EAP))
#>           [,1]
#> [1,] 0.6348623
(cor_gs <- cor(as.vector(itempars_true[,2]),a$gs_EAP))
#>           [,1]
#> [1,] 0.7432809

AAR_vec <- numeric(L)
for(t in 1:L){
  AAR_vec[t] <- mean(Alphas[,,t]==a$Alphas_est[,,t])
}
AAR_vec
#> [1] 0.9371429 0.9385714 0.9521429 0.9628571 0.9635714

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] 0.7714286 0.7857143 0.8342857 0.8628571 0.8742857

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

a$DIC
#>              Transition Response_Time Response    Joint    Total
#> D_bar          2265.045      134281.0 14789.52 3171.785 154507.4
#> D(theta_bar)   2002.054      133848.1 14710.93 3010.690 153571.7
#> DIC            2528.035      134714.0 14868.10 3332.881 155443.0
head(a$PPP_total_scores)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.74 0.36 0.88 0.56 0.78
#> [2,] 0.64 0.68 0.42 0.82 0.50
#> [3,] 0.48 1.00 0.36 0.80 1.00
#> [4,] 0.82 0.58 1.00 0.82 0.16
#> [5,] 0.40 0.48 0.58 0.98 0.12
#> [6,] 0.98 0.58 0.56 0.56 0.58
head(a$PPP_total_RTs)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.72 0.10 0.66 0.46 0.88
#> [2,] 0.74 0.52 0.34 0.70 0.70
#> [3,] 0.46 0.86 0.46 0.04 0.28
#> [4,] 0.30 0.20 0.26 0.72 0.78
#> [5,] 0.16 0.30 0.68 0.64 1.00
#> [6,] 0.76 0.30 0.64 0.42 0.32
head(a$PPP_item_means)
#> [1] 0.58 0.50 0.46 0.46 0.50 0.56
head(a$PPP_item_mean_RTs)
#> [1] 0.42 0.60 0.40 0.58 0.58 0.46
head(a$PPP_item_ORs)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
#> [1,]   NA 0.54 0.62 0.54 0.54 0.68 0.64 0.54 0.52  0.70  0.28  0.38  0.10  0.42
#> [2,]   NA   NA 0.90 0.90 0.36 0.80 0.54 0.34 0.66  0.20  0.16  0.68  0.34  0.34
#> [3,]   NA   NA   NA 0.78 0.44 0.26 0.92 0.72 0.46  0.56  0.36  0.28  0.50  0.90
#> [4,]   NA   NA   NA   NA 0.32 0.68 0.22 0.42 0.52  0.68  0.60  0.62  0.94  0.80
#> [5,]   NA   NA   NA   NA   NA 0.52 0.44 0.66 0.26  0.68  0.18  0.96  0.82  0.68
#> [6,]   NA   NA   NA   NA   NA   NA 0.58 0.80 0.58  0.48  0.56  0.98  0.46  0.86
#>      [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
#> [1,]  0.38  0.50  0.16  0.62  0.00  0.20  0.98  0.38  0.76  0.96  0.24  0.58
#> [2,]  0.48  0.48  0.82  0.66  0.70  0.66  0.50  0.56  0.80  0.48  0.82  0.86
#> [3,]  0.44  0.66  0.54  0.72  0.32  0.92  0.52  0.52  0.74  0.76  0.70  0.94
#> [4,]  0.96  0.76  0.56  0.88  0.60  0.82  0.82  0.88  0.82  0.16  0.58  0.46
#> [5,]  0.58  0.88  0.54  0.46  0.26  0.08  0.66  0.68  0.36  0.96  0.10  0.46
#> [6,]  0.58  0.66  0.92  0.66  0.78  0.44  0.88  0.60  0.72  0.60  0.46  0.50
#>      [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
#> [1,]  0.04  0.62  0.84  0.22  0.64  0.50  0.70  0.10  0.58  0.96  0.10  0.74
#> [2,]  0.74  0.76  0.60  0.88  0.48  0.94  0.14  0.08  0.48  0.60  0.96  0.68
#> [3,]  0.24  0.92  0.90  0.98  0.32  0.86  0.64  0.78  0.76  0.24  0.94  0.52
#> [4,]  0.64  0.96  0.60  0.96  0.60  0.94  0.64  0.62  0.44  0.72  0.48  0.74
#> [5,]  0.24  0.50  0.92  0.50  0.26  0.22  0.58  0.62  0.52  0.86  0.92  0.48
#> [6,]  0.44  0.60  0.98  0.78  0.66  0.76  0.64  0.30  0.90  0.92  1.00  0.42
#>      [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
#> [1,]  0.44  0.62  0.60  0.88  0.58  0.84  0.50  0.82  1.00  0.78  0.62  0.78
#> [2,]  0.80  0.82  0.06  0.64  0.76  0.90  0.20  0.82  0.58  0.14  0.36  0.28
#> [3,]  0.52  0.98  0.46  0.16  0.48  0.56  0.14  0.62  0.14  0.80  0.02  0.96
#> [4,]  0.78  0.96  0.34  0.22  0.56  0.48  0.46  0.98  0.76  0.22  0.14  0.10
#> [5,]  0.28  0.30  0.20  0.24  0.60  0.40  0.72  0.52  0.84  0.12  0.06  0.82
#> [6,]  0.34  0.88  0.10  0.30  0.74  0.94  0.06  0.54  0.86  0.32  0.52  0.16