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 
#>  70  47  81 124  28
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
#>  0.1127 0.124233
#>  0.2250 0.104211
#>  0.1877 0.009745
#>  0.1335 0.193535
#>  0.1808 0.150073
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.62844
#> λ1     0.21930
#> λ2     0.08968
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1366
#> 0001  0.2043
#> 0010  0.1899
#> 0011  0.1994
#> 0100  0.1981
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 154490 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.4924
#> M2:  0.49
#> total scores:  0.6256
a <- summary(output_HMDCM_RT_joint)
a
#> 
#> Model: DINA_HO_RT_joint 
#> 
#> Item Parameters:
#>  ss_EAP   gs_EAP
#>  0.1127 0.124233
#>  0.2250 0.104211
#>  0.1877 0.009745
#>  0.1335 0.193535
#>  0.1808 0.150073
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.62844
#> λ1     0.21930
#> λ2     0.08968
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1366
#> 0001  0.2043
#> 0010  0.1899
#> 0011  0.1994
#> 0100  0.1981
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 154490 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5004
#> M2:  0.49
#> total scores:  0.6265

a$ss_EAP
#>             [,1]
#>  [1,] 0.11270347
#>  [2,] 0.22499077
#>  [3,] 0.18766803
#>  [4,] 0.13351881
#>  [5,] 0.18084313
#>  [6,] 0.16630679
#>  [7,] 0.21917773
#>  [8,] 0.16926534
#>  [9,] 0.15198695
#> [10,] 0.10504352
#> [11,] 0.18651961
#> [12,] 0.20737220
#> [13,] 0.11106260
#> [14,] 0.12706865
#> [15,] 0.04172292
#> [16,] 0.14512500
#> [17,] 0.12693036
#> [18,] 0.11337881
#> [19,] 0.22802246
#> [20,] 0.16607305
#> [21,] 0.21718210
#> [22,] 0.21294002
#> [23,] 0.13819274
#> [24,] 0.10512822
#> [25,] 0.22038682
#> [26,] 0.18080738
#> [27,] 0.24075552
#> [28,] 0.20718553
#> [29,] 0.15083268
#> [30,] 0.14005061
#> [31,] 0.20968285
#> [32,] 0.19344205
#> [33,] 0.14357316
#> [34,] 0.15476279
#> [35,] 0.14418051
#> [36,] 0.10282458
#> [37,] 0.17717930
#> [38,] 0.20544774
#> [39,] 0.17236065
#> [40,] 0.07957245
#> [41,] 0.17311277
#> [42,] 0.20793720
#> [43,] 0.15947439
#> [44,] 0.10157765
#> [45,] 0.15023576
#> [46,] 0.16526813
#> [47,] 0.21881203
#> [48,] 0.19479450
#> [49,] 0.20101923
#> [50,] 0.09277548
head(a$ss_EAP)
#>           [,1]
#> [1,] 0.1127035
#> [2,] 0.2249908
#> [3,] 0.1876680
#> [4,] 0.1335188
#> [5,] 0.1808431
#> [6,] 0.1663068

(3) Check for parameter estimation accuracy

(cor_thetas <- cor(thetas_true,a$thetas_EAP))
#>           [,1]
#> [1,] 0.8234573
(cor_taus <- cor(taus_true,a$response_times_coefficients$taus_EAP))
#>           [,1]
#> [1,] 0.9857816

(cor_ss <- cor(as.vector(itempars_true[,1]),a$ss_EAP))
#>           [,1]
#> [1,] 0.7852056
(cor_gs <- cor(as.vector(itempars_true[,2]),a$gs_EAP))
#>           [,1]
#> [1,] 0.6492715

AAR_vec <- numeric(L)
for(t in 1:L){
  AAR_vec[t] <- mean(Alphas[,,t]==a$Alphas_est[,,t])
}
AAR_vec
#> [1] 0.9407143 0.9400000 0.9500000 0.9607143 0.9514286

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] 0.7857143 0.7857143 0.8171429 0.8628571 0.8457143

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

a$DIC
#>              Transition Response_Time Response    Joint    Total
#> D_bar          2106.370      133261.0 14862.97 3199.193 153429.5
#> D(theta_bar)   1834.658      132833.9 14656.24 3044.158 152369.0
#> DIC            2378.082      133688.0 15069.69 3354.228 154490.0
head(a$PPP_total_scores)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.52 0.92 0.52 0.90 0.56
#> [2,] 0.74 0.92 0.14 0.98 0.22
#> [3,] 0.28 0.96 0.26 0.72 0.08
#> [4,] 0.52 0.38 0.82 0.64 0.32
#> [5,] 0.90 1.00 0.80 1.00 0.56
#> [6,] 0.82 0.74 0.90 0.32 0.48
head(a$PPP_total_RTs)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.70 0.52 0.78 0.66 0.28
#> [2,] 0.28 0.82 0.46 0.42 0.86
#> [3,] 0.18 0.56 0.32 0.86 0.62
#> [4,] 0.88 0.12 0.08 0.78 0.56
#> [5,] 0.14 0.60 0.48 0.36 0.60
#> [6,] 0.74 0.68 0.92 0.12 0.56
head(a$PPP_item_means)
#> [1] 0.64 0.44 0.40 0.50 0.52 0.56
head(a$PPP_item_mean_RTs)
#> [1] 0.66 0.34 0.60 0.48 0.64 0.60
head(a$PPP_item_ORs)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
#> [1,]   NA 0.24 0.86 0.20 0.76 0.46 0.78 0.30 0.70  1.00  0.30  0.78  0.84  0.06
#> [2,]   NA   NA 0.60 0.38 0.08 0.84 0.46 0.08 0.64  0.74  0.90  0.54  0.88  0.44
#> [3,]   NA   NA   NA 0.30 0.78 0.92 0.96 0.82 0.28  0.98  0.76  0.78  0.56  0.54
#> [4,]   NA   NA   NA   NA 0.00 0.06 0.02 0.36 0.62  0.20  0.98  0.10  0.14  0.20
#> [5,]   NA   NA   NA   NA   NA 0.18 0.70 0.26 0.86  0.34  0.60  0.44  0.68  0.10
#> [6,]   NA   NA   NA   NA   NA   NA 0.28 0.28 0.52  0.66  0.70  1.00  0.48  0.32
#>      [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
#> [1,]  0.56  0.70  0.36  0.18  0.74  0.34  0.38  0.98  0.78  0.80  0.02  0.60
#> [2,]  0.68  0.54  0.32  0.24  0.50  0.36  0.68  0.92  0.08  0.24  0.12  0.30
#> [3,]  0.88  0.72  0.68  0.06  0.58  0.80  0.52  0.26  0.60  1.00  0.78  0.90
#> [4,]  0.00  0.26  0.14  0.32  0.70  0.32  0.04  0.06  0.34  0.10  0.26  0.40
#> [5,]  0.04  0.74  0.02  0.12  0.44  0.46  1.00  1.00  0.84  0.92  0.74  0.82
#> [6,]  0.96  1.00  0.94  0.10  0.74  0.76  0.42  0.96  0.08  0.90  0.30  0.90
#>      [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
#> [1,]  0.68  0.96  0.10  0.50  0.40  0.04  0.46  0.88  0.16  0.48  0.10  0.40
#> [2,]  0.90  0.72  0.16  0.22  0.14  0.36  0.92  0.84  0.24  0.44  0.92  0.10
#> [3,]  0.68  0.92  0.98  0.44  0.90  0.44  0.78  0.84  0.84  0.18  0.54  0.66
#> [4,]  0.40  0.12  0.02  0.14  0.62  0.48  0.50  0.16  0.20  0.32  0.18  0.24
#> [5,]  0.98  0.98  0.62  0.38  0.68  0.64  0.40  0.82  0.98  0.60  0.56  0.70
#> [6,]  0.92  0.92  0.94  0.34  0.56  0.26  0.74  0.64  0.22  0.14  0.44  0.72
#>      [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
#> [1,]  0.68  0.12  0.26  0.56  0.56  0.94  0.88  0.24  0.86  0.08  0.94  0.44
#> [2,]  0.60  0.12  0.46  0.64  0.84  0.88  0.90  0.42  0.50  0.52  0.70  0.66
#> [3,]  0.74  0.64  0.04  0.82  0.46  0.92  0.76  0.86  0.98  0.98  0.88  0.48
#> [4,]  0.30  0.34  0.20  1.00  0.32  0.64  0.60  0.30  0.32  0.68  0.54  0.50
#> [5,]  1.00  0.94  0.54  0.62  0.22  0.46  0.68  0.84  0.78  0.56  0.78  0.34
#> [6,]  0.74  0.16  0.66  0.30  0.50  0.82  0.36  0.12  0.58  0.58  0.48  0.02