Lista4_tri

Questão 1:

library(irtoys)

## Loading required package: sm

## Package 'sm', version 2.2-5.4: type help(sm) for summary information

## Loading required package: ltm

## Loading required package: MASS

## 
## Attaching package: 'MASS'

## The following object is masked from 'package:sm':
## 
##     muscle

## Loading required package: msm

## Loading required package: polycor

library(mirt)

## Loading required package: stats4

## Loading required package: lattice

## 
## Attaching package: 'mirt'

## The following object is masked from 'package:ltm':
## 
##     Science

a.par <- c(2.258,2.019,2.250,2.072,2.654,2.873,3.475,3.465,2.949)
b1.par <- c(-0.347,-0.144,0.615,0.317,0.269,-0.321,-0.289,-0.489,-0.547)
b2.par <- c(0.527,0.708,1.342,1.437,1.138,0.444,0.592,0.303,0.311)
b3.par <- c(1.515,1.560,1.959,1.986,1.940,1.452,1.622,1.210,1.409)
n.itens <- length(a.par)

#########################
### Gera??o dos dados ###
#########################

set.seed(2345) # semente

nr <- 1000 # numero de respondentes

### geracao das proficiencias

theta <- rnorm(nr,0,1)
resp <- matrix(0,nr,n.itens)

### geracao das respostas

mat.prob <- matrix(0,n.itens,4)
for (j in 1:nr) {
  mat.prob <- cbind(rep(0,n.itens),exp(-a.par*(theta[j]-b1.par)),exp(-a.par*(theta[j]-b2.par)),exp(-a.par*(theta[j]-b3.par)))
  mat.prob <- 1/(1+mat.prob)
  mat.prob <- cbind(-t(apply(mat.prob,1,diff)),mat.prob[,4])
  for (i in 1:n.itens)
  resp[j,i] <- sample(4,1,replace=F,mat.prob[i,])
}

### ajuste do modelo de resposta gradual via "mirt" utilizando as respostas simuladas

write(t(resp),file="/Users/Gustavo/Documents/unb/mestrado/TRI/dados.mrg.txt",ncol=n.itens)
resp <- read.table(file="/Users/Gustavo/Documents/unb/mestrado/TRI/dados.mrg.txt")

mrg <- mirt(resp,1,itemtype=c('graded'))

## 
Iteration: 1, Log-Lik: -9106.314, Max-Change: 1.73197
Iteration: 2, Log-Lik: -8426.656, Max-Change: 0.94471
Iteration: 3, Log-Lik: -8310.165, Max-Change: 0.47988
Iteration: 4, Log-Lik: -8256.210, Max-Change: 0.25387
Iteration: 5, Log-Lik: -8226.353, Max-Change: 0.18679
Iteration: 6, Log-Lik: -8208.389, Max-Change: 0.11636
Iteration: 7, Log-Lik: -8197.878, Max-Change: 0.16138
Iteration: 8, Log-Lik: -8189.356, Max-Change: 0.11947
Iteration: 9, Log-Lik: -8183.187, Max-Change: 0.07787
Iteration: 10, Log-Lik: -8180.303, Max-Change: 0.07812
Iteration: 11, Log-Lik: -8178.054, Max-Change: 0.05049
Iteration: 12, Log-Lik: -8176.585, Max-Change: 0.04394
Iteration: 13, Log-Lik: -8175.568, Max-Change: 0.04225
Iteration: 14, Log-Lik: -8174.868, Max-Change: 0.03197
Iteration: 15, Log-Lik: -8174.369, Max-Change: 0.02258
Iteration: 16, Log-Lik: -8173.705, Max-Change: 0.01500
Iteration: 17, Log-Lik: -8173.505, Max-Change: 0.01277
Iteration: 18, Log-Lik: -8173.356, Max-Change: 0.00939
Iteration: 19, Log-Lik: -8173.136, Max-Change: 0.00801
Iteration: 20, Log-Lik: -8173.049, Max-Change: 0.00654
Iteration: 21, Log-Lik: -8172.977, Max-Change: 0.00593
Iteration: 22, Log-Lik: -8172.765, Max-Change: 0.00557
Iteration: 23, Log-Lik: -8172.739, Max-Change: 0.00423
Iteration: 24, Log-Lik: -8172.715, Max-Change: 0.00402
Iteration: 25, Log-Lik: -8172.613, Max-Change: 0.00501
Iteration: 26, Log-Lik: -8172.602, Max-Change: 0.00306
Iteration: 27, Log-Lik: -8172.596, Max-Change: 0.00222
Iteration: 28, Log-Lik: -8172.581, Max-Change: 0.00412
Iteration: 29, Log-Lik: -8172.576, Max-Change: 0.00253
Iteration: 30, Log-Lik: -8172.572, Max-Change: 0.00168
Iteration: 31, Log-Lik: -8172.566, Max-Change: 0.00148
Iteration: 32, Log-Lik: -8172.564, Max-Change: 0.00127
Iteration: 33, Log-Lik: -8172.562, Max-Change: 0.00123
Iteration: 34, Log-Lik: -8172.557, Max-Change: 0.00162
Iteration: 35, Log-Lik: -8172.555, Max-Change: 0.00120
Iteration: 36, Log-Lik: -8172.554, Max-Change: 0.00088
Iteration: 37, Log-Lik: -8172.553, Max-Change: 0.00067
Iteration: 38, Log-Lik: -8172.553, Max-Change: 0.00064
Iteration: 39, Log-Lik: -8172.552, Max-Change: 0.00063
Iteration: 40, Log-Lik: -8172.550, Max-Change: 0.00044
Iteration: 41, Log-Lik: -8172.550, Max-Change: 0.00052
Iteration: 42, Log-Lik: -8172.550, Max-Change: 0.00055
Iteration: 43, Log-Lik: -8172.550, Max-Change: 0.00046
Iteration: 44, Log-Lik: -8172.549, Max-Change: 0.00048
Iteration: 45, Log-Lik: -8172.549, Max-Change: 0.00056
Iteration: 46, Log-Lik: -8172.549, Max-Change: 0.00040
Iteration: 47, Log-Lik: -8172.549, Max-Change: 0.00047
Iteration: 48, Log-Lik: -8172.549, Max-Change: 0.00049
Iteration: 49, Log-Lik: -8172.549, Max-Change: 0.00039
Iteration: 50, Log-Lik: -8172.549, Max-Change: 0.00041
Iteration: 51, Log-Lik: -8172.549, Max-Change: 0.00048
Iteration: 52, Log-Lik: -8172.549, Max-Change: 0.00034
Iteration: 53, Log-Lik: -8172.549, Max-Change: 0.00039
Iteration: 54, Log-Lik: -8172.549, Max-Change: 0.00041
Iteration: 55, Log-Lik: -8172.549, Max-Change: 0.00033
Iteration: 56, Log-Lik: -8172.549, Max-Change: 0.00034
Iteration: 57, Log-Lik: -8172.549, Max-Change: 0.00040
Iteration: 58, Log-Lik: -8172.549, Max-Change: 0.00028
Iteration: 59, Log-Lik: -8172.549, Max-Change: 0.00033
Iteration: 60, Log-Lik: -8172.549, Max-Change: 0.00034
Iteration: 61, Log-Lik: -8172.549, Max-Change: 0.00027
Iteration: 62, Log-Lik: -8172.549, Max-Change: 0.00028
Iteration: 63, Log-Lik: -8172.549, Max-Change: 0.00033
Iteration: 64, Log-Lik: -8172.549, Max-Change: 0.00023
Iteration: 65, Log-Lik: -8172.549, Max-Change: 0.00027
Iteration: 66, Log-Lik: -8172.549, Max-Change: 0.00028
Iteration: 67, Log-Lik: -8172.549, Max-Change: 0.00023
Iteration: 68, Log-Lik: -8172.549, Max-Change: 0.00023
Iteration: 69, Log-Lik: -8172.549, Max-Change: 0.00028
Iteration: 70, Log-Lik: -8172.549, Max-Change: 0.00019
Iteration: 71, Log-Lik: -8172.549, Max-Change: 0.00023
Iteration: 72, Log-Lik: -8172.549, Max-Change: 0.00024
Iteration: 73, Log-Lik: -8172.549, Max-Change: 0.00019
Iteration: 74, Log-Lik: -8172.548, Max-Change: 0.00020
Iteration: 75, Log-Lik: -8172.548, Max-Change: 0.00023
Iteration: 76, Log-Lik: -8172.548, Max-Change: 0.00016
Iteration: 77, Log-Lik: -8172.548, Max-Change: 0.00019
Iteration: 78, Log-Lik: -8172.548, Max-Change: 0.00020
Iteration: 79, Log-Lik: -8172.548, Max-Change: 0.00016
Iteration: 80, Log-Lik: -8172.548, Max-Change: 0.00016
Iteration: 81, Log-Lik: -8172.548, Max-Change: 0.00019
Iteration: 82, Log-Lik: -8172.548, Max-Change: 0.00014
Iteration: 83, Log-Lik: -8172.548, Max-Change: 0.00016
Iteration: 84, Log-Lik: -8172.548, Max-Change: 0.00017
Iteration: 85, Log-Lik: -8172.548, Max-Change: 0.00013
Iteration: 86, Log-Lik: -8172.548, Max-Change: 0.00014
Iteration: 87, Log-Lik: -8172.548, Max-Change: 0.00016
Iteration: 88, Log-Lik: -8172.548, Max-Change: 0.00011
Iteration: 89, Log-Lik: -8172.548, Max-Change: 0.00013
Iteration: 90, Log-Lik: -8172.548, Max-Change: 0.00014
Iteration: 91, Log-Lik: -8172.548, Max-Change: 0.00011
Iteration: 92, Log-Lik: -8172.548, Max-Change: 0.00011
Iteration: 93, Log-Lik: -8172.548, Max-Change: 0.00014
Iteration: 94, Log-Lik: -8172.548, Max-Change: 0.00009

prof.est <- fscores(mrg, full.scores=TRUE)
par.est <- coef(mrg,IRTpars=TRUE)

par.est

## $V1
##         a     b1    b2    b3
## par 2.131 -0.333 0.579 1.613
## 
## $V2
##         a     b1   b2    b3
## par 1.995 -0.092 0.76 1.583
## 
## $V3
##         a    b1    b2    b3
## par 2.286 0.676 1.372 1.942
## 
## $V4
##         a    b1    b2    b3
## par 2.069 0.344 1.461 1.947
## 
## $V5
##         a    b1    b2    b3
## par 2.612 0.296 1.234 1.922
## 
## $V6
##         a     b1    b2    b3
## par 2.991 -0.242 0.481 1.469
## 
## $V7
##         a     b1    b2    b3
## par 3.574 -0.245 0.588 1.645
## 
## $V8
##         a     b1    b2    b3
## par 3.641 -0.423 0.293 1.216
## 
## $V9
##         a     b1    b2    b3
## par 2.933 -0.512 0.314 1.475
## 
## $GroupPars
##     MEAN_1 COV_11
## par      0      1

Questão 2: Elabore gráfico com as curvas características de cada item, i.e., \(P_{ik}^+(\theta)\) para as categorias de cada item.

theta.entrada <- c(-4.5,-4,-3.5,-3,-2.5,-2,-1.5,-1,-0.5,0,0.5,1,1.5,2,2.5,3,3.5,4)

for(i in 1:9){
result <- 1/(1+exp(-par.est[[i]][1]*(theta.entrada - par.est[[i]][2])))
plot(theta.entrada,result, xlab = "theta", ylab = "prob", main = paste0("Curva Característica do item ",i))  
lines(theta.entrada,result, type = "l", lty = 1)
}

Questão 3: Você indicaria itens com categorias que poderiam ser aglutinadas?

Questão 4: Elabore um gráfico das proficiências estimadas versus proficiências verdadeiras.

plot(theta,prof.est)
abline(0,1, col = "red")

Questão 5: Identifique as respostas dos alunos com proficiência estimada mínima e máxima. Na sua opinião, qual a explicação para a ocorrência de muitos empates nos valores das proficiências estimadas mínimas?

resp[prof.est==min(prof.est),]

##     V1 V2 V3 V4 V5 V6 V7 V8 V9
## 1    1  1  1  1  1  1  1  1  1
## 18   1  1  1  1  1  1  1  1  1
## 20   1  1  1  1  1  1  1  1  1
## 34   1  1  1  1  1  1  1  1  1
## 47   1  1  1  1  1  1  1  1  1
## 50   1  1  1  1  1  1  1  1  1
## 64   1  1  1  1  1  1  1  1  1
## 71   1  1  1  1  1  1  1  1  1
## 73   1  1  1  1  1  1  1  1  1
## 80   1  1  1  1  1  1  1  1  1
## 81   1  1  1  1  1  1  1  1  1
## 95   1  1  1  1  1  1  1  1  1
## 119  1  1  1  1  1  1  1  1  1
## 124  1  1  1  1  1  1  1  1  1
## 142  1  1  1  1  1  1  1  1  1
## 149  1  1  1  1  1  1  1  1  1
## 157  1  1  1  1  1  1  1  1  1
## 169  1  1  1  1  1  1  1  1  1
## 173  1  1  1  1  1  1  1  1  1
## 176  1  1  1  1  1  1  1  1  1
## 183  1  1  1  1  1  1  1  1  1
## 190  1  1  1  1  1  1  1  1  1
## 192  1  1  1  1  1  1  1  1  1
## 193  1  1  1  1  1  1  1  1  1
## 207  1  1  1  1  1  1  1  1  1
## 210  1  1  1  1  1  1  1  1  1
## 217  1  1  1  1  1  1  1  1  1
## 219  1  1  1  1  1  1  1  1  1
## 221  1  1  1  1  1  1  1  1  1
## 225  1  1  1  1  1  1  1  1  1
## 231  1  1  1  1  1  1  1  1  1
## 232  1  1  1  1  1  1  1  1  1
## 241  1  1  1  1  1  1  1  1  1
## 279  1  1  1  1  1  1  1  1  1
## 281  1  1  1  1  1  1  1  1  1
## 286  1  1  1  1  1  1  1  1  1
## 304  1  1  1  1  1  1  1  1  1
## 363  1  1  1  1  1  1  1  1  1
## 366  1  1  1  1  1  1  1  1  1
## 371  1  1  1  1  1  1  1  1  1
## 374  1  1  1  1  1  1  1  1  1
## 380  1  1  1  1  1  1  1  1  1
## 399  1  1  1  1  1  1  1  1  1
## 401  1  1  1  1  1  1  1  1  1
## 402  1  1  1  1  1  1  1  1  1
## 408  1  1  1  1  1  1  1  1  1
## 409  1  1  1  1  1  1  1  1  1
## 431  1  1  1  1  1  1  1  1  1
## 436  1  1  1  1  1  1  1  1  1
## 452  1  1  1  1  1  1  1  1  1
## 454  1  1  1  1  1  1  1  1  1
## 457  1  1  1  1  1  1  1  1  1
## 459  1  1  1  1  1  1  1  1  1
## 468  1  1  1  1  1  1  1  1  1
## 481  1  1  1  1  1  1  1  1  1
## 484  1  1  1  1  1  1  1  1  1
## 485  1  1  1  1  1  1  1  1  1
## 486  1  1  1  1  1  1  1  1  1
## 487  1  1  1  1  1  1  1  1  1
## 489  1  1  1  1  1  1  1  1  1
## 501  1  1  1  1  1  1  1  1  1
## 512  1  1  1  1  1  1  1  1  1
## 514  1  1  1  1  1  1  1  1  1
## 520  1  1  1  1  1  1  1  1  1
## 546  1  1  1  1  1  1  1  1  1
## 551  1  1  1  1  1  1  1  1  1
## 557  1  1  1  1  1  1  1  1  1
## 588  1  1  1  1  1  1  1  1  1
## 594  1  1  1  1  1  1  1  1  1
## 603  1  1  1  1  1  1  1  1  1
## 605  1  1  1  1  1  1  1  1  1
## 606  1  1  1  1  1  1  1  1  1
## 611  1  1  1  1  1  1  1  1  1
## 612  1  1  1  1  1  1  1  1  1
## 615  1  1  1  1  1  1  1  1  1
## 632  1  1  1  1  1  1  1  1  1
## 636  1  1  1  1  1  1  1  1  1
## 646  1  1  1  1  1  1  1  1  1
## 647  1  1  1  1  1  1  1  1  1
## 654  1  1  1  1  1  1  1  1  1
## 665  1  1  1  1  1  1  1  1  1
## 667  1  1  1  1  1  1  1  1  1
## 672  1  1  1  1  1  1  1  1  1
## 686  1  1  1  1  1  1  1  1  1
## 698  1  1  1  1  1  1  1  1  1
## 699  1  1  1  1  1  1  1  1  1
## 718  1  1  1  1  1  1  1  1  1
## 732  1  1  1  1  1  1  1  1  1
## 734  1  1  1  1  1  1  1  1  1
## 738  1  1  1  1  1  1  1  1  1
## 741  1  1  1  1  1  1  1  1  1
## 743  1  1  1  1  1  1  1  1  1
## 746  1  1  1  1  1  1  1  1  1
## 752  1  1  1  1  1  1  1  1  1
## 767  1  1  1  1  1  1  1  1  1
## 772  1  1  1  1  1  1  1  1  1
## 773  1  1  1  1  1  1  1  1  1
## 774  1  1  1  1  1  1  1  1  1
## 791  1  1  1  1  1  1  1  1  1
## 796  1  1  1  1  1  1  1  1  1
## 804  1  1  1  1  1  1  1  1  1
## 810  1  1  1  1  1  1  1  1  1
## 818  1  1  1  1  1  1  1  1  1
## 822  1  1  1  1  1  1  1  1  1
## 832  1  1  1  1  1  1  1  1  1
## 840  1  1  1  1  1  1  1  1  1
## 845  1  1  1  1  1  1  1  1  1
## 853  1  1  1  1  1  1  1  1  1
## 854  1  1  1  1  1  1  1  1  1
## 859  1  1  1  1  1  1  1  1  1
## 862  1  1  1  1  1  1  1  1  1
## 871  1  1  1  1  1  1  1  1  1
## 880  1  1  1  1  1  1  1  1  1
## 885  1  1  1  1  1  1  1  1  1
## 895  1  1  1  1  1  1  1  1  1
## 898  1  1  1  1  1  1  1  1  1
## 907  1  1  1  1  1  1  1  1  1
## 914  1  1  1  1  1  1  1  1  1
## 915  1  1  1  1  1  1  1  1  1
## 923  1  1  1  1  1  1  1  1  1
## 937  1  1  1  1  1  1  1  1  1
## 939  1  1  1  1  1  1  1  1  1
## 944  1  1  1  1  1  1  1  1  1
## 945  1  1  1  1  1  1  1  1  1
## 950  1  1  1  1  1  1  1  1  1
## 952  1  1  1  1  1  1  1  1  1
## 961  1  1  1  1  1  1  1  1  1
## 964  1  1  1  1  1  1  1  1  1
## 966  1  1  1  1  1  1  1  1  1
## 970  1  1  1  1  1  1  1  1  1
## 973  1  1  1  1  1  1  1  1  1
## 981  1  1  1  1  1  1  1  1  1
## 982  1  1  1  1  1  1  1  1  1
## 986  1  1  1  1  1  1  1  1  1

resp[prof.est==max(prof.est),]

##     V1 V2 V3 V4 V5 V6 V7 V8 V9
## 213  4  4  4  4  4  4  4  4  4
## 236  4  4  4  4  4  4  4  4  4
## 267  4  4  4  4  4  4  4  4  4
## 375  4  4  4  4  4  4  4  4  4
## 474  4  4  4  4  4  4  4  4  4
## 666  4  4  4  4  4  4  4  4  4

Questão 6: Compare graficamente as estimativas dos parâmetros dos 9 itens obtidas a partir das respostas simuladas com aquelas fornecidas originalmente.

mat.est <- matrix(ncol = 4, nrow = 9)

for(i in 1:9) mat.est[i,1] <- par.est[[i]][1] ## a
for(i in 1:9) mat.est[i,2] <- par.est[[i]][2] ## b1
for(i in 1:9) mat.est[i,3] <- par.est[[i]][3] ## b2
for(i in 1:9) mat.est[i,4] <- par.est[[i]][4] ## b3 
colnames(mat.est) <- c("a", "b1", "b2", "b3")

data.par <- cbind(a.par,b1.par,b2.par,b3.par)
colnames(data.par) <- c("a", "b1", "b2", "b3")

mat.est <- data.frame(mat.est)
data.par <- data.frame(data.par)

plot(data.par$a, mat.est$a, xlab = "Parâmetro 'a' original", ylab = "'a' estimado", 
     main = "'a' real vs. 'a' estimado")
abline(0,1, col = "red")

plot(data.par$b1, mat.est$b1, xlab = "Parâmetro 'b1' original", ylab = "'b1' estimado", 
     main = "'b1' real vs. 'b1' estimado")
abline(0,1, col = "red")

plot(data.par$b2, mat.est$b2, xlab = "Parâmetro 'b2' original", ylab = "'b2' estimado", 
     main = "'b2' real vs. 'b2' estimado")
abline(0,1, col = "red")

plot(data.par$b3, mat.est$b3, xlab = "Parâmetro 'b3' original", ylab = "'b3' estimado", 
     main = "'b3' real vs. 'b3' estimado")
abline(0,1, col = "red")

Questão 7: Gere valores de parâmetros para 50 itens, gere as respostas para estes itens para 1000 respondentes, ajuste o modelo de resposta gradual aos dados e elabore o gráfico das proficiências estimadas versus proficiências verdadeiras. Compare este gráfico com aquele para as estimativas obtidas a partir dos 9 itens originais.

## geracao artificial de um numero maior de itens

n.it <- 50 # numero de itens simulados
a.par.sim <- runif(n.it,0.5,3)
b.par.sim <- matrix(rnorm(3*n.it,0,1),n.it,3)
b.par.sim <- t(apply(b.par.sim,1,sort))
b1.par.sim <- b.par.sim[,1]
b2.par.sim <- b.par.sim[,2]
b3.par.sim <- b.par.sim[,3]

theta.sim <- rnorm(nr,0,1)
resp.sim <- matrix(0,nr,n.it)

### geracao das respostas

mat.prob.sim <- matrix(0,n.it,4)
for (j in 1:nr) {
  mat.prob.sim <- cbind(rep(0,n.it),exp(-a.par.sim*(theta.sim[j]-b1.par.sim)),
    exp(-a.par.sim*(theta.sim[j]-b2.par.sim)),exp(-a.par.sim*(theta.sim[j]-b3.par.sim)))
  mat.prob.sim <- 1/(1+mat.prob.sim)
  mat.prob.sim <- cbind(-t(apply(mat.prob.sim,1,diff)),mat.prob.sim[,4])
  for (i in 1:n.it)
  resp.sim[j,i] <- sample(4,1,replace=F,mat.prob.sim[i,])
}

resp.sim <- read.table(file="/Users/Gustavo/Documents/unb/mestrado/dados.mrg.txt")

mrg.sim <- mirt(resp.sim,1,itemtype=c('graded'))

## Item re-scored so that all values are within a distance of 1

## 
Iteration: 1, Log-Lik: -50116.631, Max-Change: 1.77786
Iteration: 2, Log-Lik: -49077.065, Max-Change: 0.45459
Iteration: 3, Log-Lik: -48800.185, Max-Change: 0.28616
Iteration: 4, Log-Lik: -48692.168, Max-Change: 0.28517
Iteration: 5, Log-Lik: -48670.122, Max-Change: 0.09794
Iteration: 6, Log-Lik: -48634.695, Max-Change: 0.08362
Iteration: 7, Log-Lik: -48608.903, Max-Change: 0.05986
Iteration: 8, Log-Lik: -48592.898, Max-Change: 0.06550
Iteration: 9, Log-Lik: -48576.843, Max-Change: 0.06070
Iteration: 10, Log-Lik: -48566.967, Max-Change: 0.05229
Iteration: 11, Log-Lik: -48556.452, Max-Change: 0.06055
Iteration: 12, Log-Lik: -48549.316, Max-Change: 0.03654
Iteration: 13, Log-Lik: -48545.593, Max-Change: 0.03258
Iteration: 14, Log-Lik: -48541.740, Max-Change: 0.03131
Iteration: 15, Log-Lik: -48538.543, Max-Change: 0.03048
Iteration: 16, Log-Lik: -48535.543, Max-Change: 0.02714
Iteration: 17, Log-Lik: -48532.885, Max-Change: 0.02089
Iteration: 18, Log-Lik: -48531.955, Max-Change: 0.02663
Iteration: 19, Log-Lik: -48530.235, Max-Change: 0.02537
Iteration: 20, Log-Lik: -48529.106, Max-Change: 0.01383
Iteration: 21, Log-Lik: -48528.657, Max-Change: 0.00979
Iteration: 22, Log-Lik: -48528.064, Max-Change: 0.01367
Iteration: 23, Log-Lik: -48527.730, Max-Change: 0.00815
Iteration: 24, Log-Lik: -48527.450, Max-Change: 0.01300
Iteration: 25, Log-Lik: -48526.856, Max-Change: 0.00523
Iteration: 26, Log-Lik: -48526.711, Max-Change: 0.00591
Iteration: 27, Log-Lik: -48526.592, Max-Change: 0.00632
Iteration: 28, Log-Lik: -48526.370, Max-Change: 0.00438
Iteration: 29, Log-Lik: -48526.295, Max-Change: 0.00276
Iteration: 30, Log-Lik: -48526.269, Max-Change: 0.00477
Iteration: 31, Log-Lik: -48526.184, Max-Change: 0.00138
Iteration: 32, Log-Lik: -48526.183, Max-Change: 0.00337
Iteration: 33, Log-Lik: -48526.138, Max-Change: 0.00369
Iteration: 34, Log-Lik: -48526.079, Max-Change: 0.00289
Iteration: 35, Log-Lik: -48526.057, Max-Change: 0.00080
Iteration: 36, Log-Lik: -48526.055, Max-Change: 0.00162
Iteration: 37, Log-Lik: -48526.045, Max-Change: 0.00074
Iteration: 38, Log-Lik: -48526.042, Max-Change: 0.00017
Iteration: 39, Log-Lik: -48526.041, Max-Change: 0.00027
Iteration: 40, Log-Lik: -48526.041, Max-Change: 0.00238
Iteration: 41, Log-Lik: -48526.027, Max-Change: 0.00013
Iteration: 42, Log-Lik: -48526.026, Max-Change: 0.00006

prof.est.sim <- fscores(mrg.sim, full.scores=TRUE)
par.est.sim <- coef(mrg.sim,IRTpars=TRUE)

plot(theta.sim,prof.est.sim)
abline(0,1, col= "red")