HMDCM

library(hmcdm)

Load the spatial rotation data

N = dim(Design_array)[1]
J = nrow(Q_matrix)
K = ncol(Q_matrix)
L = dim(Design_array)[3]

(1) Simulate responses based on the HMDCM model

class_0 <- sample(1:2^K, N, replace = L)
Alphas_0 <- matrix(0,N,K)
for(i in 1:N){
 Alphas_0[i,] <- inv_bijectionvector(K,(class_0[i]-1))
}
thetas_true = rnorm(N)
lambdas_true = c(-1, 1.8, .277, .055)
Alphas <- sim_alphas(model="HO_sep", 
                    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 
#>  33  50  81 147  39
itempars_true <- matrix(runif(J*2,.1,.2), ncol=2)

Y_sim <- sim_hmcdm(model="DINA",Alphas,Q_matrix,Design_array,
                   itempars=itempars_true)

(2) Run the MCMC to sample parameters from the posterior distribution

output_HMDCM = hmcdm(Y_sim,Q_matrix,"DINA_HO",Test_order = Test_order, Test_versions = Test_versions,
                     chain_length=100,burn_in=30,
                     theta_propose = 2,deltas_propose = c(.45,.35,.25,.06))
#> 0

output_HMDCM = hmcdm(Y_sim,Q_matrix,"DINA_HO",Design_array,
                     chain_length=100,burn_in=30,
                     theta_propose = 2,deltas_propose = c(.45,.35,.25,.06))
#> 0

output_HMDCM
#> 
#> Model: DINA_HO 
#> 
#> Sample Size: 350
#> Number of Items: 
#> Number of Time Points: 
#> 
#> Chain Length: 100, burn-in: 30

summary(output_HMDCM)
#> 
#> Model: DINA_HO 
#> 
#> Item Parameters:
#>   ss_EAP gs_EAP
#>  0.13339 0.1740
#>  0.22008 0.2654
#>  0.20151 0.1543
#>  0.08918 0.1968
#>  0.13123 0.1241
#>    ... 45 more items
#> 
#> Transition Parameters:
#>    lambdas_EAP
#> λ0    -1.20013
#> λ1     1.92755
#> λ2     0.24872
#> λ3     0.09357
#> 
#> Class Probabilities:
#>      pis_EAP
#> 0000  0.1941
#> 0001  0.1791
#> 0010  0.2045
#> 0011  0.1750
#> 0100  0.1606
#>    ... 11 more classes
#> 
#> Deviance Information Criterion (DIC): 18884.83 
#> 
#> Posterior Predictive P-value (PPP):
#> M1: 0.5157
#> M2:  0.49
#> total scores:   0.63
a <- summary(output_HMDCM)
a$ss_EAP
#>             [,1]
#>  [1,] 0.13338881
#>  [2,] 0.22007981
#>  [3,] 0.20151405
#>  [4,] 0.08917705
#>  [5,] 0.13123428
#>  [6,] 0.14702905
#>  [7,] 0.16636464
#>  [8,] 0.09941512
#>  [9,] 0.17760779
#> [10,] 0.22205547
#> [11,] 0.20545818
#> [12,] 0.18063000
#> [13,] 0.13404283
#> [14,] 0.18974130
#> [15,] 0.14524659
#> [16,] 0.22102795
#> [17,] 0.17125470
#> [18,] 0.18789287
#> [19,] 0.13420497
#> [20,] 0.17697399
#> [21,] 0.13698542
#> [22,] 0.11493266
#> [23,] 0.10386698
#> [24,] 0.18951550
#> [25,] 0.16333244
#> [26,] 0.11650824
#> [27,] 0.15063358
#> [28,] 0.16053841
#> [29,] 0.16064924
#> [30,] 0.12206793
#> [31,] 0.14986736
#> [32,] 0.13837487
#> [33,] 0.11612465
#> [34,] 0.19796128
#> [35,] 0.17625428
#> [36,] 0.20436015
#> [37,] 0.19814524
#> [38,] 0.11500503
#> [39,] 0.23284451
#> [40,] 0.16604946
#> [41,] 0.13469048
#> [42,] 0.16033371
#> [43,] 0.16202219
#> [44,] 0.07638028
#> [45,] 0.09734455
#> [46,] 0.12763172
#> [47,] 0.11435310
#> [48,] 0.13888442
#> [49,] 0.11363607
#> [50,] 0.11551759
a$lambdas_EAP
#>           [,1]
#> λ0 -1.20013285
#> λ1  1.92755449
#> λ2  0.24871978
#> λ3  0.09357474
mean(a$PPP_total_scores)
#> [1] 0.6295673
mean(upper.tri(a$PPP_item_ORs))
#> [1] 0.49
mean(a$PPP_item_means)
#> [1] 0.5151429

(3) Evaluate the accuracy of estimated parameters

Attribute-wise agreement rate between true and estimated alphas

AAR_vec <- numeric(L)
for(t in 1:L){
  AAR_vec[t] <- mean(Alphas[,,t]==a$Alphas_est[,,t])
}
AAR_vec
#> [1] 0.9364286 0.9564286 0.9678571 0.9714286 0.9657143

Pattern-wise agreement rate between true and estimated alphas

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] 0.7971429 0.8285714 0.8771429 0.8914286 0.8771429

(4) Evaluate the fit of the model to the observed response

a$DIC
#>              Transition Response_Time Response    Joint    Total
#> D_bar          2180.716            NA 14554.27 1290.278 18025.27
#> D(theta_bar)   1893.860            NA 14030.95 1240.900 17165.71
#> DIC            2467.572            NA 15077.60 1339.657 18884.83

head(a$PPP_total_scores)
#>           [,1]      [,2]      [,3]      [,4]      [,5]
#> [1,] 0.7571429 0.7571429 0.1857143 0.3142857 0.2000000
#> [2,] 0.5428571 0.4428571 0.3571429 0.4000000 0.8285714
#> [3,] 0.9571429 0.4428571 0.2714286 0.7571429 0.8428571
#> [4,] 0.7857143 0.4285714 0.7428571 0.8714286 0.6857143
#> [5,] 0.5285714 0.7857143 0.6285714 0.7285714 0.2000000
#> [6,] 0.6000000 0.6857143 0.8714286 0.2857143 0.8142857
head(a$PPP_item_means)
#> [1] 0.5428571 0.4428571 0.4142857 0.4857143 0.6428571 0.4571429
head(a$PPP_item_ORs)
#>      [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]      [,8]
#> [1,]   NA 0.9428571 0.5714286 0.6142857 0.6571429 0.7000000 0.4714286 0.7857143
#> [2,]   NA        NA 0.6714286 0.6571429 0.9571429 0.8428571 0.9714286 0.4428571
#> [3,]   NA        NA        NA 0.4857143 0.7571429 0.7714286 0.9285714 0.8000000
#> [4,]   NA        NA        NA        NA 0.8285714 0.1428571 0.8714286 0.7714286
#> [5,]   NA        NA        NA        NA        NA 0.5142857 0.5000000 0.6857143
#> [6,]   NA        NA        NA        NA        NA        NA 0.6428571 0.8285714
#>           [,9]     [,10]     [,11]     [,12]      [,13]     [,14]     [,15]
#> [1,] 0.3571429 0.4285714 0.9714286 0.6285714 0.07142857 0.6714286 0.7714286
#> [2,] 0.7571429 0.9285714 0.8714286 0.4571429 0.75714286 0.6428571 0.7285714
#> [3,] 0.3571429 0.7857143 0.8000000 0.6857143 0.51428571 0.6857143 0.8428571
#> [4,] 0.6000000 0.6428571 0.8857143 0.4571429 0.65714286 0.3714286 0.2571429
#> [5,] 0.5285714 0.2714286 0.9857143 0.3285714 0.57142857 0.5285714 0.5142857
#> [6,] 0.4857143 0.7142857 0.9571429 0.4714286 0.40000000 0.8142857 0.4428571
#>          [,16]      [,17]      [,18]     [,19]     [,20]      [,21]     [,22]
#> [1,] 0.2857143 0.80000000 0.50000000 0.7000000 0.4428571 0.85714286 0.3714286
#> [2,] 0.7714286 0.28571429 0.22857143 0.3571429 0.9571429 0.78571429 0.1714286
#> [3,] 0.4428571 0.55714286 0.31428571 0.7000000 0.1285714 0.87142857 0.9000000
#> [4,] 0.1714286 0.55714286 0.55714286 0.4857143 0.4857143 0.65714286 0.5571429
#> [5,] 0.5428571 0.07142857 0.48571429 0.3571429 0.9428571 0.67142857 0.1714286
#> [6,] 0.9428571 0.54285714 0.02857143 0.7857143 0.8857143 0.07142857 0.2857143
#>          [,23]     [,24]     [,25]      [,26]     [,27]     [,28]     [,29]
#> [1,] 0.8142857 0.7142857 0.7857143 0.92857143 0.4285714 0.2714286 1.0000000
#> [2,] 0.9714286 0.7714286 0.7285714 0.30000000 0.8428571 0.6142857 0.6857143
#> [3,] 0.1285714 0.3857143 0.9714286 0.37142857 0.5714286 0.4714286 0.9714286
#> [4,] 0.5285714 0.2571429 0.5285714 0.61428571 0.8142857 0.3142857 0.5000000
#> [5,] 0.5428571 0.2428571 0.3571429 0.21428571 0.4285714 0.6714286 0.4428571
#> [6,] 0.6714286 0.2428571 0.9285714 0.07142857 0.4714286 0.4428571 0.5714286
#>          [,30]     [,31]     [,32]     [,33]     [,34]      [,35]     [,36]
#> [1,] 0.7000000 1.0000000 0.9285714 0.7428571 0.2000000 0.95714286 0.8714286
#> [2,] 0.8000000 1.0000000 1.0000000 0.2285714 0.4714286 0.42857143 0.3428571
#> [3,] 0.5142857 0.9571429 0.6142857 0.3000000 0.2142857 0.80000000 0.2142857
#> [4,] 0.6428571 0.9000000 0.2571429 0.7000000 0.6714286 0.05714286 0.4571429
#> [5,] 0.1714286 0.9000000 0.8000000 0.9000000 0.7428571 0.27142857 0.6857143
#> [6,] 0.5285714 0.7571429 0.3571429 0.8000000 0.6285714 0.22857143 0.1285714
#>          [,37]     [,38]      [,39]     [,40]     [,41]     [,42]     [,43]
#> [1,] 0.6000000 0.8142857 0.72857143 0.9571429 0.8714286 0.9285714 0.6000000
#> [2,] 0.9571429 0.7428571 0.78571429 0.8142857 0.6857143 0.9571429 0.6857143
#> [3,] 1.0000000 0.7857143 0.72857143 0.4571429 0.2000000 0.4571429 0.2857143
#> [4,] 0.7428571 0.3714286 0.01428571 0.8714286 0.3428571 0.4714286 0.9571429
#> [5,] 0.7428571 0.4714286 0.87142857 0.8714286 0.5285714 0.4428571 0.1857143
#> [6,] 0.9142857 0.5000000 0.87142857 0.5142857 0.7000000 0.1285714 0.3714286
#>          [,44]     [,45]     [,46]      [,47]     [,48]     [,49]     [,50]
#> [1,] 0.9857143 0.6428571 0.8714286 0.72857143 0.9142857 0.9857143 0.7714286
#> [2,] 0.9428571 0.6285714 1.0000000 1.00000000 0.9714286 0.9714286 0.9142857
#> [3,] 0.7142857 0.5714286 0.2285714 0.02857143 0.8571429 0.9857143 0.2285714
#> [4,] 0.8714286 0.3142857 0.7285714 0.44285714 0.4714286 0.7142857 0.1142857
#> [5,] 0.9857143 0.5714286 0.6142857 0.54285714 0.3285714 0.9142857 0.4285714
#> [6,] 0.9857143 0.3000000 0.6285714 0.41428571 0.9142857 0.7142857 0.9285714
library(bayesplot)
pp_check(output_HMDCM)

pp_check(output_HMDCM, plotfun="dens_overlay", type="item_mean")

pp_check(output_HMDCM, plotfun="hist", type="item_OR")
#> Note: in most cases the default test statistic 'mean' is too weak to detect anything of interest.
#> `stat_bin()` using `bins = 30`. Pick better value `binwidth`.

pp_check(output_HMDCM, plotfun="stat_2d", type="item_mean")
#> Note: in most cases the default test statistic 'mean' is too weak to detect anything of interest.

pp_check(output_HMDCM, plotfun="scatter_avg", type="total_score")

pp_check(output_HMDCM, plotfun="error_scatter_avg", type="total_score")

Convergence checking

Checking convergence of the two independent MCMC chains with different initial values using coda package.

# output_HMDCM1 = hmcdm(Y_sim, Q_matrix, "DINA_HO", Design_array,
#                      chain_length=100, burn_in=30,
#                      theta_propose = 2, deltas_propose = c(.45,.35,.25,.06))
# output_HMDCM2 = hmcdm(Y_sim, Q_matrix, "DINA_HO", Design_array,
#                      chain_length=100, burn_in=30,
#                      theta_propose = 2, deltas_propose = c(.45,.35,.25,.06))
# 
# library(coda)
# 
# x <- mcmc.list(mcmc(t(rbind(output_HMDCM1$ss, output_HMDCM1$gs, output_HMDCM1$lambdas))),
#                mcmc(t(rbind(output_HMDCM2$ss, output_HMDCM2$gs, output_HMDCM2$lambdas))))
# 
# gelman.diag(x, autoburnin=F)