Introduction

This tutorial is created using R markdown and knitr. It illustrates how to use the GDINA R pacakge (version 2.8.0) to analyze polytomous response data using the sequential models.

Model Estimation

The following code fits the sequential G-DINA model to a set of simulated data, which consist of 20 items (15 polytomous and 5 dichotomous) measuring 5 attributes:

library(GDINA)
## GDINA Package (version 2.8.0; 2020-05-23)
## For tutorials, see https://wenchao-ma.github.io/GDINA
dat <- sim20seqGDINA$simdat
head(dat)
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
## [1,]    0    2    0    2    2    0    0    1    0     0     0     0     3     0
## [2,]    2    0    0    0    0    1    1    1    0     0     1     1     0     2
## [3,]    0    0    2    2    0    0    2    1    0     0     0     0     2     0
## [4,]    0    0    0    0    0    0    0    0    0     0     1     0     0     0
## [5,]    0    2    1    1    1    0    0    1    2     0     2     0     3     0
## [6,]    0    0    2    2    0    0    1    1    1     0     0     3     0     0
##      [,15] [,16] [,17] [,18] [,19] [,20]
## [1,]     2     0     0     1     1     1
## [2,]     0     1     0     0     1     0
## [3,]     2     0     0     0     1     1
## [4,]     0     0     1     0     0     0
## [5,]     1     0     0     1     0     1
## [6,]     2     0     1     0     1     0
Q <- matrix(c(1,    1,  1,  0,  0,  0,  0,
              1,    2,  0,  1,  0,  1,  0,
              2,    1,  1,  0,  1,  0,  0,
              2,    2,  0,  0,  0,  1,  0,
              3,    1,  0,  1,  0,  1,  1,
              3,    2,  1,  0,  0,  0,  0,
              4,    1,  0,  0,  0,  0,  1,
              4,    2,  0,  0,  0,  1,  0,
              5,    1,  0,  0,  1,  0,  0,
              5,    2,  0,  1,  0,  0,  0,
              6,    1,  1,  0,  0,  0,  0,
              6,    2,  0,  1,  1,  0,  0,
              7,    1,  0,  1,  0,  0,  0,
              7,    2,  0,  0,  1,  1,  0,
              8,    1,  0,  0,  0,  1,  0,
              8,    2,  1,  0,  0,  0,  1,
              9,    1,  0,  0,  0,  1,  1,
              9,    2,  0,  0,  1,  0,  0,
              10,   1,  0,  1,  1,  0,  0,
              10,   2,  1,  0,  0,  0,  0,
              11,   1,  1,  1,  0,  0,  0,
              11,   2,  0,  0,  0,  0,  1,
              12,   1,  0,  1,  0,  0,  0,
              12,   2,  0,  0,  0,  1,  0,
              12,   3,  0,  0,  0,  0,  1,
              13,   1,  0,  0,  0,  0,  1,
              13,   2,  0,  0,  0,  1,  0,
              13,   3,  0,  0,  1,  0,  0,
              14,   1,  1,  0,  0,  0,  0,
              14,   2,  0,  1,  0,  0,  0,
              14,   3,  0,  0,  1,  0,  0,
              15,   1,  0,  0,  0,  1,  0,
              15,   2,  0,  0,  0,  0,  1,
              15,   3,  1,  0,  0,  0,  0,
              16,   1,  1,  0,  0,  0,  0,
              17,   1,  0,  1,  0,  0,  0,
              18,   1,  0,  0,  1,  0,  0,
              19,   1,  0,  0,  0,  1,  0,
              20,   1,  0,  0,  0,  0,  1),byrow = TRUE,ncol = 7)

est <- GDINA(dat = dat, Q = Q, sequential = TRUE, model = "GDINA")
## 
Iter = 1  Max. abs. change = 0.53669  Deviance  = 56678.90                                                                                  
Iter = 2  Max. abs. change = 0.04260  Deviance  = 49887.11                                                                                  
Iter = 3  Max. abs. change = 0.00852  Deviance  = 49770.17                                                                                  
Iter = 4  Max. abs. change = 0.00264  Deviance  = 49765.23                                                                                  
Iter = 5  Max. abs. change = 0.00089  Deviance  = 49764.87                                                                                  
Iter = 6  Max. abs. change = 0.00031  Deviance  = 49764.83                                                                                  
Iter = 7  Max. abs. change = 0.00011  Deviance  = 49764.83                                                                                  
Iter = 8  Max. abs. change = 0.00004  Deviance  = 49764.83

coef() can be used to extract various item parameters:

 coef(est) # processing function
## $`Item 1 Cat 1`
##   P(0)   P(1) 
## 0.0989 0.8911 
## 
## $`Item 1 Cat 2`
##  P(00)  P(10)  P(01)  P(11) 
## 0.0824 0.9124 0.1066 0.8890 
## 
## $`Item 2 Cat 1`
##  P(00)  P(10)  P(01)  P(11) 
## 0.1113 0.0982 0.8935 0.9114 
## 
## $`Item 2 Cat 2`
##   P(0)   P(1) 
## 0.0988 0.8861 
## 
## $`Item 3 Cat 1`
## P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111) 
## 0.1214 0.0579 0.1107 0.8900 0.0971 0.8870 0.9201 0.9229 
## 
## $`Item 3 Cat 2`
##   P(0)   P(1) 
## 0.1126 0.8936 
## 
## $`Item 4 Cat 1`
##   P(0)   P(1) 
## 0.1117 0.9224 
## 
## $`Item 4 Cat 2`
##   P(0)   P(1) 
## 0.0865 0.8920 
## 
## $`Item 5 Cat 1`
##   P(0)   P(1) 
## 0.1032 0.8955 
## 
## $`Item 5 Cat 2`
##   P(0)   P(1) 
## 0.1060 0.9171 
## 
## $`Item 6 Cat 1`
##   P(0)   P(1) 
## 0.1161 0.8884 
## 
## $`Item 6 Cat 2`
##  P(00)  P(10)  P(01)  P(11) 
## 0.1019 0.1163 0.0757 0.8831 
## 
## $`Item 7 Cat 1`
##   P(0)   P(1) 
## 0.1107 0.8999 
## 
## $`Item 7 Cat 2`
##  P(00)  P(10)  P(01)  P(11) 
## 0.0984 0.0960 0.1154 0.9025 
## 
## $`Item 8 Cat 1`
##   P(0)   P(1) 
## 0.1071 0.9126 
## 
## $`Item 8 Cat 2`
##  P(00)  P(10)  P(01)  P(11) 
## 0.1006 0.0630 0.0616 0.9113 
## 
## $`Item 9 Cat 1`
##  P(00)  P(10)  P(01)  P(11) 
## 0.1263 0.1119 0.1044 0.8735 
## 
## $`Item 9 Cat 2`
##   P(0)   P(1) 
## 0.0961 0.9040 
## 
## $`Item 10 Cat 1`
##  P(00)  P(10)  P(01)  P(11) 
## 0.1071 0.0979 0.1021 0.8880 
## 
## $`Item 10 Cat 2`
##   P(0)   P(1) 
## 0.1079 0.9084 
## 
## $`Item 11 Cat 1`
##  P(00)  P(10)  P(01)  P(11) 
## 0.1132 0.0936 0.1029 0.8677 
## 
## $`Item 11 Cat 2`
##   P(0)   P(1) 
## 0.0883 0.8997 
## 
## $`Item 12 Cat 1`
##   P(0)   P(1) 
## 0.0840 0.8934 
## 
## $`Item 12 Cat 2`
##   P(0)   P(1) 
## 0.0995 0.9005 
## 
## $`Item 12 Cat 3`
##   P(0)   P(1) 
## 0.0903 0.8579 
## 
## $`Item 13 Cat 1`
##   P(0)   P(1) 
## 0.1017 0.8889 
## 
## $`Item 13 Cat 2`
##   P(0)   P(1) 
## 0.1018 0.9270 
## 
## $`Item 13 Cat 3`
##   P(0)   P(1) 
## 0.0631 0.8967 
## 
## $`Item 14 Cat 1`
##   P(0)   P(1) 
## 0.0973 0.8826 
## 
## $`Item 14 Cat 2`
##   P(0)   P(1) 
## 0.0674 0.9153 
## 
## $`Item 14 Cat 3`
##   P(0)   P(1) 
## 0.0869 0.8951 
## 
## $`Item 15 Cat 1`
##  P(0)  P(1) 
## 0.109 0.892 
## 
## $`Item 15 Cat 2`
##   P(0)   P(1) 
## 0.0931 0.8801 
## 
## $`Item 15 Cat 3`
##   P(0)   P(1) 
## 0.0858 0.9006 
## 
## $`Item 16 Cat 1`
##   P(0)   P(1) 
## 0.1102 0.8781 
## 
## $`Item 17 Cat 1`
##   P(0)   P(1) 
## 0.0984 0.8830 
## 
## $`Item 18 Cat 1`
##   P(0)   P(1) 
## 0.1043 0.8953 
## 
## $`Item 19 Cat 1`
##   P(0)   P(1) 
## 0.1100 0.9068 
## 
## $`Item 20 Cat 1`
##   P(0)   P(1) 
## 0.1003 0.9030
 coef(est,"itemprob") # success probabilities for each item
## $`Item 1`
##       P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Cat 1 0.0908 0.8177 0.0087 0.0884 0.0781 0.7961 0.0110 0.0989
## Cat 2 0.0082 0.0734 0.0903 0.0105 0.8130 0.0950 0.0879 0.7922
## 
## $`Item 2`
##       P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Cat 1 0.1003 0.0885 0.8052 0.0127 0.8213 0.0112 0.1017 0.1038
## Cat 2 0.0110 0.0097 0.0883 0.0986 0.0901 0.0870 0.7918 0.8076
## 
## $`Item 3`
##       P(0000) P(1000) P(0100) P(0010) P(0001) P(1100) P(1010) P(1001) P(0110)
## Cat 1  0.1077  0.0129  0.0514  0.0983  0.7898  0.0062  0.0118  0.0947  0.0862
## Cat 2  0.0137  0.1085  0.0065  0.0125  0.1002  0.0518  0.0989  0.7953  0.0109
##       P(0101) P(0011) P(1110) P(1101) P(1011) P(0111) P(1111)
## Cat 1  0.7871  0.8165  0.0103  0.0944  0.0979  0.8190  0.0982
## Cat 2  0.0999  0.1036  0.0868  0.7926  0.8222  0.1039  0.8247
## 
## $`Item 4`
##        P(00)  P(10)  P(01)  P(11)
## Cat 1 0.1021 0.0121 0.8426 0.0996
## Cat 2 0.0097 0.0997 0.0798 0.8228
## 
## $`Item 5`
##        P(00)  P(10)  P(01)  P(11)
## Cat 1 0.0923 0.0086 0.8006 0.0742
## Cat 2 0.0109 0.0947 0.0949 0.8212
## 
## $`Item 6`
##       P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Cat 1 0.1042 0.7979 0.1026 0.1073 0.7851 0.8211 0.0136 0.1038
## Cat 2 0.0118 0.0905 0.0135 0.0088 0.1033 0.0672 0.1025 0.7846
## 
## $`Item 7`
##       P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Cat 1 0.0998 0.8114 0.1001 0.0979 0.8136 0.7961 0.0108 0.0877
## Cat 2 0.0109 0.0886 0.0106 0.0128 0.0864 0.1039 0.0999 0.8122
## 
## $`Item 8`
##       P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Cat 1 0.0963 0.1003 0.8209 0.1005 0.8551 0.0095 0.8564 0.0810
## Cat 2 0.0108 0.0068 0.0918 0.0066 0.0575 0.0976 0.0562 0.8317
## 
## $`Item 9`
##       P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Cat 1 0.1141 0.0121 0.1012 0.0944 0.0107 0.0100 0.7896 0.0839
## Cat 2 0.0121 0.1142 0.0108 0.0100 0.1012 0.0944 0.0839 0.7896
## 
## $`Item 10`
##       P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Cat 1 0.0956 0.0098 0.0874 0.0911  0.009 0.0094 0.7922 0.0814
## Cat 2 0.0116 0.0973 0.0106 0.0110  0.089 0.0927 0.0958 0.8067
## 
## $`Item 11`
##       P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Cat 1 0.1032 0.0854 0.0938 0.0113 0.7911 0.0094 0.0103 0.0870
## Cat 2 0.0100 0.0083 0.0091 0.1018 0.0766 0.0842 0.0926 0.7807
## 
## $`Item 12`
##       P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Cat 1 0.0756 0.8046 0.0084 0.0756 0.0889 0.8046 0.0084 0.0889
## Cat 2 0.0076 0.0808 0.0688 0.0012 0.7319 0.0126 0.0107 0.1143
## Cat 3 0.0008 0.0080 0.0068 0.0072 0.0726 0.0762 0.0649 0.6903
## 
## $`Item 13`
##       P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Cat 1 0.0914 0.0914 0.0074 0.7985 0.0074 0.7985 0.0648 0.0648
## Cat 2 0.0097 0.0011 0.0883 0.0847 0.0097 0.0093 0.7721 0.0851
## Cat 3 0.0007 0.0093 0.0059 0.0057 0.0845 0.0811 0.0520 0.7389
## 
## $`Item 14`
##       P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Cat 1 0.0907 0.8231 0.0082 0.0907 0.0747 0.8231 0.0082 0.0747
## Cat 2 0.0060 0.0543 0.0813 0.0007 0.7377 0.0062 0.0093 0.0848
## Cat 3 0.0006 0.0052 0.0077 0.0059 0.0702 0.0532 0.0797 0.7231
## 
## $`Item 15`
##       P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Cat 1 0.0989 0.0989 0.8090 0.0131 0.8090 0.0131 0.1070 0.1070
## Cat 2 0.0093 0.0010 0.0759 0.0877 0.0083 0.0095 0.7176 0.0781
## Cat 3 0.0009 0.0091 0.0071 0.0082 0.0748 0.0864 0.0674 0.7069
## 
## $`Item 16`
##         P(0)   P(1)
## Cat 1 0.1102 0.8781
## 
## $`Item 17`
##         P(0)  P(1)
## Cat 1 0.0984 0.883
## 
## $`Item 18`
##         P(0)   P(1)
## Cat 1 0.1043 0.8953
## 
## $`Item 19`
##       P(0)   P(1)
## Cat 1 0.11 0.9068
## 
## $`Item 20`
##         P(0)  P(1)
## Cat 1 0.1003 0.903

Q-matrix validation

The Qval() function is used for Q-matrix validation. By default, it implements de la Torre and Chiu’s (2016) algorithm. The following example use the stepwise method (Ma & de la Torre, 2019) instead.

Qv <- Qval(est, method = "Wald")
Qv
## 
## Q-matrix validation based on Stepwise Wald test 
## 
## Suggested Q-matrix: 
## 
##    X1 X2 A1 A2 A3 A4 A5
## 1   1 1  1  0  0  0  0 
## 2   1 2  0  1  0  0* 0 
## 3   2 1  0* 0  1  0  0 
## 4   2 2  0  0  0  1  0 
## 5   3 1  0  0* 0  0* 1 
## 6   3 2  1  0  0  0  0 
## 7   4 1  0  0  0  0  1 
## 8   4 2  0  0  0  1  0 
## 9   5 1  0  0  1  0  0 
## 10  5 2  0  1  0  0  0 
## 11  6 1  1  0  0  0  0 
## 12  6 2  0  1  1  0  0 
## 13  7 1  0  1  0  0  0 
## 14  7 2  0  0  1  1  0 
## 15  8 1  0  0  0  1  0 
## 16  8 2  1  0  0  0  1 
## 17  9 1  0  0  0  1  1 
## 18  9 2  0  0  1  0  0 
## 19 10 1  0  1  1  0  0 
## 20 10 2  1  0  0  0  0 
## 21 11 1  1  1  0  0  0 
## 22 11 2  0  0  0  0  1 
## 23 12 1  0  1  0  0  0 
## 24 12 2  0  0  1* 1  0 
## 25 12 3  0  0  0  0  1 
## 26 13 1  0  0  0  0  1 
## 27 13 2  0  0  0  1  0 
## 28 13 3  0  0  1  0  0 
## 29 14 1  1  0  0  0  0 
## 30 14 2  0  1  0  0  0 
## 31 14 3  0  0  1  0  0 
## 32 15 1  0  0  0  1  0 
## 33 15 2  0  0  0  0  1 
## 34 15 3  1  0  0  0  0 
## 35 16 1  1  0  0  0  0 
## 36 17 1  0  1  0  0  0 
## 37 18 1  0  0  1  0  0 
## 38 19 1  0  0  0  1  0 
## 39 20 1  0  0  0  0  1 
## Note: * denotes a modified element.

To further examine the q-vectors, you can draw the mesa plots (de la Torre & Ma, 2016):

plot(Qv, item = 2) # the 2nd row in the Q-matrix - not item 2

We can also examine whether the G-DINA model with the suggested Q had better relative fit:

sugQ <- extract(Qv, what = "sug.Q")
est.sugQ <- GDINA(dat, sugQ, sequential = TRUE, verbose = 0)
anova(est,est.sugQ)
## 
## Information Criteria and Likelihood Ratio Test
## 
##          #par    logLik Deviance      AIC      BIC     CAIC    SABIC chisq df
## est       131 -24882.42 49764.83 50026.83 50760.55 50891.55 50344.36         
## est.sugQ  123 -24883.89 49767.78 50013.78 50702.69 50825.69 50311.91  2.95  8
##          p-value
## est             
## est.sugQ    0.94

Item-level model comparison

Based on the suggested Q-matrix, we perform item level model comparison using the Wald test (see de la Torre, 2011; de la Torre & Lee, 2013; Ma, Iaconangelo & de la Torre, 2016) to check whether any reduced CDMs can be used. Note that score test and likelihood ratio test (Sorrel, Abad, Olea, de la Torre, and Barrada, 2017; Sorrel, de la Torre, Abad, & Olea, 2017; Ma & de la Torre, 2018) may also be used.

mc <- modelcomp(est.sugQ)
mc
## 
## Item-level model selection:
## 
## test statistic: Wald 
## Decision rule: simpler model + largest p value rule at 0.05 alpha level.
## Adjusted p values were based on holm correction.
## 
##               models pvalues adj.pvalues
## Item 1 Cat 1   GDINA                    
## Item 1 Cat 2   GDINA                    
## Item 2 Cat 1   GDINA                    
## Item 2 Cat 2   GDINA                    
## Item 3 Cat 1   GDINA                    
## Item 3 Cat 2   GDINA                    
## Item 4 Cat 1   GDINA                    
## Item 4 Cat 2   GDINA                    
## Item 5 Cat 1   GDINA                    
## Item 5 Cat 2   GDINA                    
## Item 6 Cat 1   GDINA                    
## Item 6 Cat 2    DINA  0.3378           1
## Item 7 Cat 1   GDINA                    
## Item 7 Cat 2    DINA  0.7968           1
## Item 8 Cat 1   GDINA                    
## Item 8 Cat 2    DINA  0.2278           1
## Item 9 Cat 1    DINA  0.5768           1
## Item 9 Cat 2   GDINA                    
## Item 10 Cat 1   DINA  0.9124           1
## Item 10 Cat 2  GDINA                    
## Item 11 Cat 1   DINA  0.6323           1
## Item 11 Cat 2  GDINA                    
## Item 12 Cat 1  GDINA                    
## Item 12 Cat 2  GDINA                    
## Item 12 Cat 3  GDINA                    
## Item 13 Cat 1  GDINA                    
## Item 13 Cat 2  GDINA                    
## Item 13 Cat 3  GDINA                    
## Item 14 Cat 1  GDINA                    
## Item 14 Cat 2  GDINA                    
## Item 14 Cat 3  GDINA                    
## Item 15 Cat 1  GDINA                    
## Item 15 Cat 2  GDINA                    
## Item 15 Cat 3  GDINA                    
## Item 16 Cat 1  GDINA                    
## Item 17 Cat 1  GDINA                    
## Item 18 Cat 1  GDINA                    
## Item 19 Cat 1  GDINA                    
## Item 20 Cat 1  GDINA

We can fit the models suggested by the Wald test based on the rule in Ma, Iaconangelo and de la Torre (2016) and compare the combinations of CDMs with the G-DINA model:

est.wald <- GDINA(dat, sugQ, model = extract(mc,"selected.model")$models, sequential = TRUE, verbose = 0)
anova(est.sugQ,est.wald)
## 
## Information Criteria and Likelihood Ratio Test
## 
##          #par    logLik Deviance      AIC      BIC     CAIC    SABIC chisq df
## est.sugQ  123 -24883.89 49767.78 50013.78 50702.69 50825.69 50311.91         
## est.wald  111 -24888.01 49776.02 49998.02 50619.72 50730.72 50267.07  8.24 12
##          p-value
## est.sugQ        
## est.wald    0.77

Absolute fit evaluation

The test level absolute fit include M2 statistic, RMSEA and SRMSR (Maydeu-Olivares, 3013; Liu, Tian, & Xin, 2016; Hansen, Cai, Monroe, & Li, 2016; Ma, 2019) and the item level absolute fit include log odds and transformed correlation (Chen, de la Torre, & Zhang, 2013), as well as heat plot for item pairs.

# test level absolute fit
mft <- modelfit(est.wald)
mft
## Test-level Model Fit Evaluation
## 
## Relative fit statistics: 
##  -2 log likelihood =  49776.02  ( number of parameters =  111 )
##  AIC  =  49998.02  BIC =  50619.72 
##  CAIC =  50730.72  SABIC =  50267.07 
## 
## Absolute fit statistics: 
##  Mord =  102.438  df =  99  p =  0.3864 
##  RMSEA2 =  0.0042  with  90 % CI: [ 0 , 0.0126 ]
##  SRMSR =  0.0169

The estimated latent class size can be obtained by

extract(est.wald,"posterior.prob")
##           00000      10000      01000      00100      00010      00001
## [1,] 0.03453115 0.03420516 0.02623529 0.03614817 0.03982501 0.02987453
##           11000      10100      10010      10001      01100      01010
## [1,] 0.02615635 0.02437048 0.02880361 0.03777212 0.03203646 0.03275289
##           01001      00110      00101      00011      11100      11010
## [1,] 0.03259152 0.02838396 0.03133634 0.03331199 0.02743016 0.03471439
##           11001      10110      10101      10011      01110      01101
## [1,] 0.03439462 0.02887284 0.03184059 0.03495408 0.03261572 0.03154267
##           01011      00111      11110      11101      11011      10111
## [1,] 0.03092711 0.02990523 0.02316411 0.02415087 0.02351595 0.03482464
##           01111      11111
## [1,] 0.03273822 0.03607377

The tetrachoric correlation between attributes can be calculated by

# psych package needs to be installed
library(psych)
psych::tetrachoric(x = extract(est.wald,"attributepattern"),
                   weight = extract(est.wald,"posterior.prob"))
## Call: psych::tetrachoric(x = extract(est.wald, "attributepattern"), 
##     weight = extract(est.wald, "posterior.prob"))
## tetrachoric correlation 
##    A1    A2    A3    A4    A5   
## A1  1.00                        
## A2 -0.02  1.00                  
## A3 -0.03  0.04  1.00            
## A4  0.00  0.02  0.01  1.00      
## A5  0.06  0.00  0.03 -0.01  1.00
## 
##  with tau of 
##     A1     A2     A3     A4     A5 
##  0.037  0.048  0.037 -0.013 -0.024

Classification Accuracy

The following code calculates the test-, pattern- and attribute-level classification accuracy indices based on GDINA estimates using approaches in Iaconangelo (2017) and Wang, Song, Chen, Meng, and Ding (2015).

CA(est.wald)
## Classification Accuracy 
## 
## Test level accuracy =  0.9367 
## 
## Pattern level accuracy: 
## 
##  00000  10000  01000  00100  00010  00001  11000  10100  10010  10001  01100 
## 0.8589 0.8960 0.8928 0.9479 0.9135 0.8715 0.9469 0.8843 0.8556 0.9234 0.9528 
##  01010  01001  00110  00101  00011  11100  11010  11001  10110  10101  10011 
## 0.9035 0.9572 0.8843 0.9119 0.9454 0.9587 0.9576 0.9898 0.9404 0.9463 0.9832 
##  01110  01101  01011  00111  11110  11101  11011  10111  01111  11111 
## 0.9741 0.9016 0.9440 0.9531 0.9690 0.9615 0.9935 0.9664 0.9859 0.9934 
## 
## Attribute level accuracy: 
## 
##     A1     A2     A3     A4     A5 
## 0.9871 0.9879 0.9822 0.9876 0.9897

References

Chen, J., de la Torre, J., & Zhang, Z. (2013). Relative and Absolute Fit Evaluation in Cognitive Diagnosis Modeling. Journal of Educational Measurement, 50, 123-140.

de la Torre, J., & Lee, Y. S. (2013). Evaluating the wald test for item-level comparison of saturated and reduced models in cognitive diagnosis. Journal of Educational Measurement, 50, 355-373.

de la Torre, J., & Ma, W. (2016, August). Cognitive diagnosis modeling: A general framework approach and its implementation in R. A short course at the fourth conference on the statistical methods in Psychometrics, Columbia University, New York.

Hansen, M., Cai, L., Monroe, S., & Li, Z. (2016). Limited-information goodness-of-fit testing of diagnostic classification item response models. British Journal of Mathematical and Statistical Psychology. 69, 225–252.

Iaconangelo, C.(2017). Uses of Classification Error Probabilities in the Three-Step Approach to Estimating Cognitive Diagnosis Models. (Unpublished doctoral dissertation). New Brunswick, NJ: Rutgers University.

Liu, Y., Tian, W., & Xin, T. (2016). An Application of M2 Statistic to Evaluate the Fit of Cognitive Diagnostic Models. Journal of Educational and Behavioral Statistics, 41, 3-26.

Ma, W. (2019). Evaluating the fit of sequential G-DINA model using limited-information measures. Applied Psychological Measurement.

Ma, W. & de la Torre, J. (2018). Category-level model selection for the sequential G-DINA model. Journal of Educational and Behavorial Statistics.

Ma,W., & de la Torre, J. (2019). An empirical Q-matrix validation method for the sequential G-DINA model. British Journal of Mathematical and Statistical Psychology.

Ma, W., Iaconangelo, C., & de la Torre, J. (2016). Model similarity, model selection and attribute classification. Applied Psychological Measurement, 40, 200-217.

Maydeu-Olivares, A. (2013). Goodness-of-Fit Assessment of Item Response Theory Models. Measurement, 11, 71-101.

Sorrel, M. A., Abad, F. J., Olea, J., de la Torre, J., & Barrada, J. R. (2017). Inferential Item-Fit Evaluation in Cognitive Diagnosis Modeling. Applied Psychological Measurement, 41, 614-631.

Sorrel, M. A., de la Torre, J., Abad, F. J., & Olea, J. (2017). Two-Step Likelihood Ratio Test for Item-Level Model Comparison in Cognitive Diagnosis Models. Methodology, 13, 39-47.

Wang, W., Song, L., Chen, P., Meng, Y., & Ding, S. (2015). Attribute-Level and Pattern-Level Classification Consistency and Accuracy Indices for Cognitive Diagnostic Assessment. Journal of Educational Measurement, 52 , 457-476.

## R version 4.0.0 (2020-04-24)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 17763)
## 
## Matrix products: default
## 
## locale:
## [1] LC_COLLATE=English_United States.1252 
## [2] LC_CTYPE=English_United States.1252   
## [3] LC_MONETARY=English_United States.1252
## [4] LC_NUMERIC=C                          
## [5] LC_TIME=English_United States.1252    
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] psych_1.9.12.31 GDINA_2.8.0    
## 
## loaded via a namespace (and not attached):
##  [1] xfun_0.13            lattice_0.20-41      colorspace_1.4-1    
##  [4] vctrs_0.2.4          htmltools_0.4.0      yaml_2.2.1          
##  [7] rlang_0.4.5          pkgdown_1.5.1        later_1.0.0         
## [10] pillar_1.4.3         nloptr_1.2.2.1       glue_1.4.0          
## [13] lifecycle_0.2.0      stringr_1.4.0        munsell_0.5.0       
## [16] gtable_0.3.0         memoise_1.1.0        evaluate_0.14       
## [19] knitr_1.28           fastmap_1.0.1        httpuv_1.5.2        
## [22] parallel_4.0.0       Rcpp_1.0.4.6         xtable_1.8-4        
## [25] promises_1.1.0       backports_1.1.6      scales_1.1.0        
## [28] desc_1.2.0           truncnorm_1.0-8      alabama_2015.3-1    
## [31] mime_0.9             fs_1.4.1             mnormt_1.5-7        
## [34] ggplot2_3.3.0        digest_0.6.25        stringi_1.4.6       
## [37] shiny_1.4.0.2        numDeriv_2016.8-1.1  grid_4.0.0          
## [40] rprojroot_1.3-2      tools_4.0.0          magrittr_1.5        
## [43] Rsolnp_1.16          tibble_3.0.1         crayon_1.3.4        
## [46] pkgconfig_2.0.3      MASS_7.3-51.5        ellipsis_0.3.0      
## [49] shinydashboard_0.7.1 assertthat_0.2.1     rmarkdown_2.1       
## [52] rstudioapi_0.11      R6_2.4.1             nlme_3.1-147        
## [55] compiler_4.0.0