Introduction

This tutorial is created using R markdown and knitr. It illustrates how to use the GDINA R pacakge (version 2.7.4) to estimate the G-DINA model.

Model Estimation

The following code estimates the G-DINA model.

library(GDINA)
## GDINA Package [Version 2.7.4; 2019-9-14]
## More information: https://wenchao-ma.github.io/GDINA
## 
Iter = 1  Max. abs. change = 0.40691  Deviance  = 12821.16                                                                                  
Iter = 2  Max. abs. change = 0.05266  Deviance  = 11947.64                                                                                  
Iter = 3  Max. abs. change = 0.02836  Deviance  = 11890.21                                                                                  
Iter = 4  Max. abs. change = 0.01736  Deviance  = 11866.22                                                                                  
Iter = 5  Max. abs. change = 0.01173  Deviance  = 11854.67                                                                                  
Iter = 6  Max. abs. change = 0.00886  Deviance  = 11848.42                                                                                  
Iter = 7  Max. abs. change = 0.00763  Deviance  = 11844.68                                                                                  
Iter = 8  Max. abs. change = 0.00643  Deviance  = 11842.26                                                                                  
Iter = 9  Max. abs. change = 0.00557  Deviance  = 11840.64                                                                                  
Iter = 10  Max. abs. change = 0.00503  Deviance  = 11839.51                                                                                  
Iter = 11  Max. abs. change = 0.00449  Deviance  = 11838.71                                                                                  
Iter = 12  Max. abs. change = 0.00398  Deviance  = 11838.14                                                                                  
Iter = 13  Max. abs. change = 0.00351  Deviance  = 11837.72                                                                                  
Iter = 14  Max. abs. change = 0.00308  Deviance  = 11837.42                                                                                  
Iter = 15  Max. abs. change = 0.00270  Deviance  = 11837.19                                                                                  
Iter = 16  Max. abs. change = 0.00237  Deviance  = 11837.02                                                                                  
Iter = 17  Max. abs. change = 0.00208  Deviance  = 11836.89                                                                                  
Iter = 18  Max. abs. change = 0.00183  Deviance  = 11836.79                                                                                  
Iter = 19  Max. abs. change = 0.00161  Deviance  = 11836.72                                                                                  
Iter = 20  Max. abs. change = 0.00142  Deviance  = 11836.66                                                                                  
Iter = 21  Max. abs. change = 0.00125  Deviance  = 11836.61                                                                                  
Iter = 22  Max. abs. change = 0.00110  Deviance  = 11836.57                                                                                  
Iter = 23  Max. abs. change = 0.00098  Deviance  = 11836.54                                                                                  
Iter = 24  Max. abs. change = 0.00087  Deviance  = 11836.52                                                                                  
Iter = 25  Max. abs. change = 0.00077  Deviance  = 11836.50                                                                                  
Iter = 26  Max. abs. change = 0.00069  Deviance  = 11836.49                                                                                  
Iter = 27  Max. abs. change = 0.00061  Deviance  = 11836.47                                                                                  
Iter = 28  Max. abs. change = 0.00055  Deviance  = 11836.47                                                                                  
Iter = 29  Max. abs. change = 0.00049  Deviance  = 11836.46                                                                                  
Iter = 30  Max. abs. change = 0.00044  Deviance  = 11836.45                                                                                  
Iter = 31  Max. abs. change = 0.00039  Deviance  = 11836.45                                                                                  
Iter = 32  Max. abs. change = 0.00035  Deviance  = 11836.44                                                                                  
Iter = 33  Max. abs. change = 0.00032  Deviance  = 11836.44                                                                                  
Iter = 34  Max. abs. change = 0.00029  Deviance  = 11836.44                                                                                  
Iter = 35  Max. abs. change = 0.00026  Deviance  = 11836.43                                                                                  
Iter = 36  Max. abs. change = 0.00023  Deviance  = 11836.43                                                                                  
Iter = 37  Max. abs. change = 0.00021  Deviance  = 11836.43                                                                                  
Iter = 38  Max. abs. change = 0.00019  Deviance  = 11836.43                                                                                  
Iter = 39  Max. abs. change = 0.00017  Deviance  = 11836.43                                                                                  
Iter = 40  Max. abs. change = 0.00016  Deviance  = 11836.43                                                                                  
Iter = 41  Max. abs. change = 0.00014  Deviance  = 11836.43                                                                                  
Iter = 42  Max. abs. change = 0.00013  Deviance  = 11836.42                                                                                  
Iter = 43  Max. abs. change = 0.00012  Deviance  = 11836.42                                                                                  
Iter = 44  Max. abs. change = 0.00011  Deviance  = 11836.42                                                                                  
Iter = 45  Max. abs. change = 0.00010  Deviance  = 11836.42

Summary Information

The following code extracts the summary information from GDINA estimates.

## Call:
## GDINA(dat = dat, Q = Q, model = "GDINA")
## 
##   GDINA version 2.7.4 (2019-9-14) 
## ===============================================
## Data
## -----------------------------------------------
## # of individuals    groups    items         
##             1000         1       10
## ===============================================
## Model
## -----------------------------------------------
## Fitted model(s)       = GDINA 
## Attribute structure   = saturated 
## Attribute level       = Dichotomous 
## ===============================================
## Estimation
## -----------------------------------------------
## Number of iterations  = 45 
## 
## For the final iteration:
##   Max abs change in item success prob. = 0.0001 
##   Max abs change in mixing proportions = 0.0000 
##   Change in -2 log-likelihood          = 0.0004 
##   Converged?                           = TRUE 
## 
## Time used             = 0.09674 secs
## 
## Test Fit Statistics
## 
## Loglik = -5918.21 
## 
## AIC    = 11926.42  | penalty [2 * p]  = 90.00 
## BIC    = 12147.27  | penalty [log(n) * p]  = 310.85 
## CAIC   = 12192.27  | penalty [(log(n) + 1) * p]  = 355.85 
## SABIC  = 12004.35  | penalty [log((n + 2)/24) * p]  = 167.93 
## 
## No. of parameters (p)  = 45 
##   No. of estimated item parameters =  38 
##   No. of fixed item parameters =  0 
##   No. of distribution parameters =  7 
## 
## Attribute Prevalence
## 
##    Level0 Level1
## A1 0.5027 0.4973
## A2 0.4974 0.5026
## A3 0.4784 0.5216
AIC(est) #AIC
## [1] 11926.42
BIC(est) #BIC
## [1] 12147.27
logLik(est) #log-likelihood value
## 'log Lik.' -5918.211 (df=45)
deviance(est) # deviance: -2 log-likelihood
## [1] 11836.42
npar(est) # number of parameters
## No. of total parameters = 45 
## No. of population parameters = 7 
## No. of free item parameters = 38 
## No. of fixed item parameters = 0
nobs(est) # number of observations
## [1] 1000

The estimated latent class size can be obtained by

extract(est,"posterior.prob")
##            000       100     010       001       110       101      011
## [1,] 0.1267808 0.1053483 0.11497 0.1201007 0.1312817 0.1451415 0.140898
##           111
## [1,] 0.115479

The tetrachoric correlation between attributes can be calculated by

# psych package needs to be installed
library(psych)
psych::tetrachoric(x = extract(est,"attributepattern"),
                   weight = extract(est,"posterior.prob"))
## Call: psych::tetrachoric(x = extract(est, "attributepattern"), weight = extract(est, 
##     "posterior.prob"))
## tetrachoric correlation 
##    A1    A2    A3   
## A1  1.00            
## A2 -0.02  1.00      
## A3  0.01 -0.04  1.00
## 
##  with tau of 
##      A1      A2      A3 
##  0.0069 -0.0066 -0.0542

You can use extract with argument discrim to extract discrimination indices. The first column gives \(P(1)-P(0)\) and the second column gives the GDINA discrimination index.

##         P(1)-P(0)        GDI
## Item 1  0.6934120 0.12020141
## Item 2  0.6405419 0.10257064
## Item 3  0.8198848 0.16773857
## Item 4  0.7797695 0.08455288
## Item 5  0.7173909 0.10215095
## Item 6  0.7406696 0.10203425
## Item 7  0.7168750 0.06522977
## Item 8  0.7893286 0.09124453
## Item 9  0.7062429 0.06232943
## Item 10 0.7253567 0.05496962

Model Fit Evaluation

For test level model data fit, use modelfit function:

## Test-level Model Fit Evaluation
## 
## Relative fit statistics: 
##  -2 log likelihood =  11836.42  ( number of parameters =  45 )
##  AIC  =  11926.42  BIC =  12147.27 
##  CAIC =  12192.27  SABIC =  12004.35 
## 
## Absolute fit statistics: 
##  M2 =  6.8584  df =  10  p =  0.7387 
##  RMSEA2 =  0  with  90 % CI: [ 0 , 0.025 ]
##  SRMSR =  0.0205

itemfit function also provide test level model fit information:

## Summary of Item Fit Analysis
## 
## Call:
## itemfit(GDINA.obj = est)
## 
##                         mean[stats] max[stats] max[z.stats] p-value
## Proportion correct           0.0011     0.0026       0.1683  0.8663
## Transformed correlation      0.0154     0.0630       1.9893  0.0467
## Log odds ratio               0.0687     0.2795       1.9482  0.0514
##                         adj.p-value
## Proportion correct                1
## Transformed correlation           1
## Log odds ratio                    1
## Note: p-value and adj.p-value are associated with max[z.stats].
##       adj.p-values are based on the holm method.

itemfit also provide item-level fit information:

summary(ift)
## 
## Item-level fit statistics
##         z.prop pvalue[z.prop] max[z.r] pvalue.max[z.r] adj.pvalue.max[z.r]
## Item 1  0.0598         0.9523   0.4465          0.6552              1.0000
## Item 2  0.0108         0.9914   0.5339          0.5934              1.0000
## Item 3  0.0063         0.9950   1.4319          0.1522              1.0000
## Item 4  0.0343         0.9726   1.9893          0.0467              0.4200
## Item 5  0.0957         0.9237   1.9893          0.0467              0.4200
## Item 6  0.0372         0.9703   1.4319          0.1522              1.0000
## Item 7  0.1669         0.8674   1.5507          0.1210              1.0000
## Item 8  0.0858         0.9316   1.8360          0.0664              0.5972
## Item 9  0.0721         0.9425   1.8360          0.0664              0.5972
## Item 10 0.1683         0.8663   0.6386          0.5231              1.0000
##         max[z.logOR] pvalue.max[z.logOR] adj.pvalue.max[z.logOR]
## Item 1        0.4503              0.6525                  1.0000
## Item 2        0.5347              0.5929                  1.0000
## Item 3        1.4320              0.1522                  1.0000
## Item 4        1.9482              0.0514                  0.4625
## Item 5        1.9482              0.0514                  0.4625
## Item 6        1.4320              0.1522                  1.0000
## Item 7        1.5333              0.1252                  1.0000
## Item 8        1.8159              0.0694                  0.6244
## Item 9        1.8159              0.0694                  0.6244
## Item 10       0.6214              0.5343                  1.0000

Model Parameters

The following code gives the item probalities of each reduced latent classes. As shown below, the probability of answering item 1 correctly for individuals who do not master the required attribute is 0.2052, and the probability of answering item 1 correctly for individuals who master the required attribute is 0.8986:

coef(est) # item probabilities of success for each reduced latent class
## $`Item 1`
##   P(0)   P(1) 
## 0.2052 0.8986 
## 
## $`Item 2`
##   P(0)   P(1) 
## 0.1391 0.7796 
## 
## $`Item 3`
##   P(0)   P(1) 
## 0.0893 0.9092 
## 
## $`Item 4`
##  P(00)  P(10)  P(01)  P(11) 
## 0.1159 0.2951 0.4741 0.8956 
## 
## $`Item 5`
##  P(00)  P(10)  P(01)  P(11) 
## 0.1057 0.0791 0.0904 0.8231 
## 
## $`Item 6`
##  P(00)  P(10)  P(01)  P(11) 
## 0.1764 0.9029 0.9302 0.9170 
## 
## $`Item 7`
##  P(00)  P(10)  P(01)  P(11) 
## 0.0543 0.4712 0.3922 0.7712 
## 
## $`Item 8`
##  P(00)  P(10)  P(01)  P(11) 
## 0.1107 0.2585 0.2730 0.9000 
## 
## $`Item 9`
##  P(00)  P(10)  P(01)  P(11) 
## 0.0995 0.3746 0.4189 0.8057 
## 
## $`Item 10`
## P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111) 
## 0.1757 0.1326 0.2846 0.3823 0.4928 0.4997 0.6734 0.9011

The following code gives the item probalities of each reduced latent classes with standard errors.

coef(est, withSE = TRUE) # item probabilities of success & standard errors
## $`Item 1`
##        P(0)   P(1)
## Est. 0.2052 0.8986
## S.E. 0.0258 0.0224
## 
## $`Item 2`
##        P(0)   P(1)
## Est. 0.1391 0.7796
## S.E. 0.0221 0.0247
## 
## $`Item 3`
##        P(0)   P(1)
## Est. 0.0893 0.9092
## S.E. 0.0213 0.0199
## 
## $`Item 4`
##       P(00)  P(10)  P(01)  P(11)
## Est. 0.1159 0.2951 0.4741 0.8956
## S.E. 0.0279 0.0369 0.0379 0.0282
## 
## $`Item 5`
##       P(00)  P(10)  P(01)  P(11)
## Est. 0.1057 0.0791 0.0904 0.8231
## S.E. 0.0249 0.0252 0.0256 0.0335
## 
## $`Item 6`
##       P(00)  P(10)  P(01) P(11)
## Est. 0.1764 0.9029 0.9302 0.917
## S.E. 0.0412 0.0314 0.0280 0.023
## 
## $`Item 7`
##       P(00)  P(10)  P(01)  P(11)
## Est. 0.0543 0.4712 0.3922 0.7712
## S.E. 0.0238 0.0393 0.0369 0.0322
## 
## $`Item 8`
##       P(00)  P(10)  P(01)  P(11)
## Est. 0.1107 0.2585 0.2730 0.9000
## S.E. 0.0262 0.0382 0.0386 0.0349
## 
## $`Item 9`
##       P(00)  P(10)  P(01)  P(11)
## Est. 0.0995 0.3746 0.4189 0.8057
## S.E. 0.0294 0.0372 0.0362 0.0288
## 
## $`Item 10`
##      P(000) P(100) P(010) P(001) P(110) P(101) P(011) P(111)
## Est. 0.1757 0.1326 0.2846 0.3823 0.4928 0.4997 0.6734 0.9011
## S.E. 0.0449 0.0570 0.0563 0.0554 0.0567 0.0517 0.0479 0.0391

The following code gives delta parameters.

coef(est, what = "delta") # delta parameters
## $`Item 1`
##     d0     d1 
## 0.2052 0.6934 
## 
## $`Item 2`
##     d0     d1 
## 0.1391 0.6405 
## 
## $`Item 3`
##     d0     d1 
## 0.0894 0.8199 
## 
## $`Item 4`
##     d0     d1     d2    d12 
## 0.1159 0.1792 0.3582 0.2423 
## 
## $`Item 5`
##      d0      d1      d2     d12 
##  0.1057 -0.0267 -0.0153  0.7594 
## 
## $`Item 6`
##      d0      d1      d2     d12 
##  0.1764  0.7265  0.7539 -0.7397 
## 
## $`Item 7`
##      d0      d1      d2     d12 
##  0.0543  0.4169  0.3379 -0.0380 
## 
## $`Item 8`
##     d0     d1     d2    d12 
## 0.1107 0.1478 0.1623 0.4792 
## 
## $`Item 9`
##     d0     d1     d2    d12 
## 0.0995 0.2751 0.3194 0.1117 
## 
## $`Item 10`
##      d0      d1      d2      d3     d12     d13     d23    d123 
##  0.1757 -0.0431  0.1089  0.2066  0.2513  0.1604  0.1822 -0.1409

The following code gives delta parameters with standard errors.

coef(est, what = "delta", withSE = TRUE) # delta parameters
## $`Item 1`
##          d0     d1
## Est. 0.2052 0.6934
## S.E. 0.0258 0.0383
## 
## $`Item 2`
##          d0     d1
## Est. 0.1391 0.6405
## S.E. 0.0221 0.0364
## 
## $`Item 3`
##          d0     d1
## Est. 0.0894 0.8199
## S.E. 0.0213 0.0317
## 
## $`Item 4`
##          d0     d1     d2    d12
## Est. 0.1159 0.1792 0.3582 0.2423
## S.E. 0.0279 0.0492 0.0494 0.0738
## 
## $`Item 5`
##          d0      d1      d2    d12
## Est. 0.1057 -0.0267 -0.0153 0.7594
## S.E. 0.0249  0.0399  0.0373 0.0618
## 
## $`Item 6`
##          d0     d1     d2     d12
## Est. 0.1764 0.7265 0.7539 -0.7397
## S.E. 0.0412 0.0563 0.0542  0.0740
## 
## $`Item 7`
##          d0     d1     d2     d12
## Est. 0.0543 0.4169 0.3379 -0.0380
## S.E. 0.0238 0.0490 0.0459  0.0749
## 
## $`Item 8`
##          d0     d1     d2    d12
## Est. 0.1107 0.1478 0.1623 0.4792
## S.E. 0.0262 0.0487 0.0496 0.0792
## 
## $`Item 9`
##          d0     d1     d2    d12
## Est. 0.0995 0.2751 0.3194 0.1117
## S.E. 0.0294 0.0515 0.0497 0.0736
## 
## $`Item 10`
##          d0      d1     d2     d3    d12    d13    d23    d123
## Est. 0.1757 -0.0431 0.1089 0.2066 0.2513 0.1604 0.1822 -0.1409
## S.E. 0.0449  0.0765 0.0759 0.0767 0.1215 0.1176 0.1117  0.1676

The following code gives \(P(0)\) and 1-\(P(0)\), which is guessing and slipping parameters.

coef(est, what = "gs") # guessing and slip parameters
##         guessing   slip
## Item 1    0.2052 0.1014
## Item 2    0.1391 0.2204
## Item 3    0.0893 0.0908
## Item 4    0.1159 0.1044
## Item 5    0.1057 0.1769
## Item 6    0.1764 0.0830
## Item 7    0.0543 0.2288
## Item 8    0.1107 0.1000
## Item 9    0.0995 0.1943
## Item 10   0.1757 0.0989

The following code gives guessing and slipping parameters with standard errors.

coef(est, what = "gs", withSE = TRUE) # guessing and slip parameters & standard errors
##         guessing   slip SE[guessing] SE[slip]
## Item 1    0.2052 0.1014       0.0258   0.0224
## Item 2    0.1391 0.2204       0.0221   0.0247
## Item 3    0.0893 0.0908       0.0213   0.0199
## Item 4    0.1159 0.1044       0.0279   0.0282
## Item 5    0.1057 0.1769       0.0249   0.0335
## Item 6    0.1764 0.0830       0.0412   0.0230
## Item 7    0.0543 0.2288       0.0238   0.0322
## Item 8    0.1107 0.1000       0.0262   0.0349
## Item 9    0.0995 0.1943       0.0294   0.0288
## Item 10   0.1757 0.0989       0.0449   0.0391

The following code gives the proportions of each latent classes. As you can see, the estimated proportion of individuals in the population who do not master any attribute (i.e., 000) is 0.1268:

coef(est,"lambda")
## p(000) p(100) p(010) p(001) p(110) p(101) p(011) p(111) 
## 0.1268 0.1054 0.1149 0.1201 0.1313 0.1452 0.1409 0.1155

The following code gives item success probabilities for all latent classes,

coef(est,"LCprob")
##            000    100    010    001    110    101    011    111
## Item 1  0.2052 0.8986 0.2052 0.2052 0.8986 0.8986 0.2052 0.8986
## Item 2  0.1391 0.1391 0.7796 0.1391 0.7796 0.1391 0.7796 0.7796
## Item 3  0.0893 0.0893 0.0893 0.9092 0.0893 0.9092 0.9092 0.9092
## Item 4  0.1159 0.2951 0.1159 0.4741 0.2951 0.8956 0.4741 0.8956
## Item 5  0.1057 0.1057 0.0791 0.0904 0.0791 0.0904 0.8231 0.8231
## Item 6  0.1764 0.9029 0.9302 0.1764 0.9170 0.9029 0.9302 0.9170
## Item 7  0.0543 0.4712 0.0543 0.3922 0.4712 0.7712 0.3922 0.7712
## Item 8  0.1107 0.2585 0.2730 0.1107 0.9000 0.2585 0.2730 0.9000
## Item 9  0.0995 0.0995 0.3746 0.4189 0.3746 0.4189 0.8057 0.8057
## Item 10 0.1757 0.1326 0.2846 0.3823 0.4928 0.4997 0.6734 0.9011

The following code returns EAP estimates of attribute patterns (for the first six individuals). As you can see, the EAP estimate of attribute profile for the first individual is (1, 0, 1):

head(personparm(est)) # EAP estimates of attribute profiles
##      A1 A2 A3
## [1,]  1  0  1
## [2,]  1  1  1
## [3,]  0  1  1
## [4,]  1  1  1
## [5,]  0  0  1
## [6,]  1  0  0

By specifying what argument, the following code gives MAP estimates of attribute patterns (for the first six individuals).

head(personparm(est, what = "MAP")) # MAP estimates of attribute profiles
##   A1 A2 A3 multimodes
## 1  1  0  1      FALSE
## 2  1  1  1      FALSE
## 3  0  1  1      FALSE
## 4  1  1  1      FALSE
## 5  0  0  1      FALSE
## 6  0  0  0      FALSE

The following code extracts MLE estimates of attribute patterns (for the first six individuals).

head(personparm(est, what = "MLE")) # MLE estimates of attribute profiles
##   A1 A2 A3 multimodes
## 1  1  0  1      FALSE
## 2  1  1  1      FALSE
## 3  0  1  1      FALSE
## 4  1  1  1      FALSE
## 5  0  0  1      FALSE
## 6  0  0  0      FALSE

Some Plots

The following code gives item response functions of item 10.

plot(est, item = 10)

The following code gives item response functions of item 10 with error bars.

plot(est, item = 10, withSE = TRUE) # with error bars

The following code plots mastery probabilities of three attributes for individuals 1,20 and 50.

#plot mastery probability for individuals 1, 20 and 50
plot(est, what = "mp", person = c(1, 20, 50))

Advanced Topics

##             000        100        010        001        110        101
## [1,] -15.108780  -8.336519 -13.759121  -9.046286  -7.458953  -3.508211
## [2,] -19.066682 -12.294420 -14.950767 -13.177555  -8.650599  -7.639480
## [3,] -15.608150 -10.694877 -11.529638 -10.837671 -10.460689  -9.061469
## [4,] -17.901173 -10.796473 -15.804861 -11.202255 -10.397619  -6.142558
## [5,]  -8.584137  -9.816582 -13.093769  -3.587529 -14.211994  -6.864976
## [6,]  -6.356676  -7.258905  -9.138395  -9.267932  -6.554173 -11.407240
##             011        111
## [1,] -10.010643  -5.751538
## [2,]  -7.209424  -2.950319
## [3,]  -5.427149  -8.084197
## [4,]  -7.773934  -5.000620
## [5,] -10.990620 -14.742772
## [6,] -14.363668 -13.823401
head(indlogPost(est)) # individual log-posterior
##              000       100        010          001         110        101
## [1,] -11.8422008 -5.254964 -10.590807  -5.83405663  -4.1578241 -0.1064772
## [2,] -16.0548514 -9.467614 -12.037202 -10.22007449  -5.6042195 -4.4924951
## [3,] -10.3817971 -5.653549  -6.401550  -5.66566755  -5.1997868 -3.6999613
## [4,] -13.2039637 -6.284289 -11.205917  -6.55939618  -5.6658616 -1.3101943
## [5,]  -4.9960740 -6.413543  -9.603971  -0.05381585 -10.5893821 -3.1417586
## [6,]  -0.8340671 -1.921320  -3.714052  -3.79967324  -0.9970151 -5.7494767
##              011          111
## [1,] -6.63885140  -2.57867661
## [2,] -4.09238208  -0.03220729
## [3,] -0.09558416  -2.95156165
## [4,] -2.97151364  -0.39712955
## [5,] -7.29734498 -11.24842717
## [6,] -8.73584656  -8.39451030
extract(est,"designmatrix") #design matrix
## [[1]]
##      [,1] [,2]
## [1,]    1    0
## [2,]    1    1
## 
## [[2]]
##      [,1] [,2]
## [1,]    1    0
## [2,]    1    1
## 
## [[3]]
##      [,1] [,2]
## [1,]    1    0
## [2,]    1    1
## 
## [[4]]
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    1    1    0    0
## [3,]    1    0    1    0
## [4,]    1    1    1    1
## 
## [[5]]
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    1    1    0    0
## [3,]    1    0    1    0
## [4,]    1    1    1    1
## 
## [[6]]
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    1    1    0    0
## [3,]    1    0    1    0
## [4,]    1    1    1    1
## 
## [[7]]
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    1    1    0    0
## [3,]    1    0    1    0
## [4,]    1    1    1    1
## 
## [[8]]
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    1    1    0    0
## [3,]    1    0    1    0
## [4,]    1    1    1    1
## 
## [[9]]
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    1    1    0    0
## [3,]    1    0    1    0
## [4,]    1    1    1    1
## 
## [[10]]
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## [1,]    1    0    0    0    0    0    0    0
## [2,]    1    1    0    0    0    0    0    0
## [3,]    1    0    1    0    0    0    0    0
## [4,]    1    0    0    1    0    0    0    0
## [5,]    1    1    1    0    1    0    0    0
## [6,]    1    1    0    1    0    1    0    0
## [7,]    1    0    1    1    0    0    1    0
## [8,]    1    1    1    1    1    1    1    1
extract(est,"linkfunc") #link functions
##  [1] "identity" "identity" "identity" "identity" "identity" "identity"
##  [7] "identity" "identity" "identity" "identity"
## R version 3.6.1 (2019-07-05)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 18362)
## 
## 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_1.8.12 GDINA_2.7.4 
## 
## loaded via a namespace (and not attached):
##  [1] tidyselect_0.2.5     xfun_0.9             purrr_0.3.2         
##  [4] lattice_0.20-38      colorspace_1.4-1     htmltools_0.3.6     
##  [7] yaml_2.2.0           rlang_0.4.0          pkgdown_1.4.0       
## [10] later_0.8.0          pillar_1.4.2         nloptr_1.2.1        
## [13] foreign_0.8-71       glue_1.3.1           stringr_1.4.0       
## [16] munsell_0.5.0        gtable_0.3.0         memoise_1.1.0       
## [19] evaluate_0.14        labeling_0.3         knitr_1.24          
## [22] httpuv_1.5.1         parallel_3.6.1       Rcpp_1.0.2          
## [25] xtable_1.8-4         promises_1.0.1       scales_1.0.0        
## [28] backports_1.1.4      desc_1.2.0           truncnorm_1.0-8     
## [31] alabama_2015.3-1     mime_0.7             fs_1.3.1            
## [34] mnormt_1.5-5         ggplot2_3.2.1        digest_0.6.20       
## [37] stringi_1.4.3        dplyr_0.8.3          shiny_1.3.2         
## [40] numDeriv_2016.8-1.1  grid_3.6.1           rprojroot_1.3-2     
## [43] tools_3.6.1          magrittr_1.5         lazyeval_0.2.2      
## [46] Rsolnp_1.16          tibble_2.1.3         crayon_1.3.4        
## [49] pkgconfig_2.0.2      MASS_7.3-51.4        shinydashboard_0.7.1
## [52] assertthat_0.2.1     rmarkdown_1.15       rstudioapi_0.10     
## [55] R6_2.4.0             nlme_3.1-140         compiler_3.6.1