Given a set of item parameters and a set of abilities, it is possible to simulate a set of item responses.

Users can also impose a factor structure that maps items to specific dimensions, and impose various missing data structures.

Users can also permute the discrimination of items to impose misfit on the simulated data.

The item parameters for a simple item are represented as:

```
library(conquestr)
<- matrix(c(0, 0, 0, 0, 1, 1, 0, 1), ncol = 4, byrow = TRUE)
myItem colnames(myItem) <- c("k", "d", "t", "a")
print(myItem)
#> k d t a
#> [1,] 0 0 0 0
#> [2,] 1 1 0 1
plotModelCCC(myItem)
```

where `k`

is the category score, `d`

is the
delta dot (\(\dot{\delta}\)),
`t`

is the tau, and `a`

is the discrimination
parameter. By convention, the first row of the item matrix is zeros.

items are stored in lists, and a list of items can be visualised as the test characteristic curve which illustrates the expected raw score give an ability, theta (\(\Theta\)).

```
<- list(myItem, myItem)
myItems 2]][2, 2] <- -1 # make the second item delta equal to -1
myItems[[print(plotModelExp(myItems))
```

In the unidminsaional case, abilities are a simple vector of abilities (for example, EAPs):

`<- rnorm(100) myAbils `

Given fixed item parameters and abilities, it I possible then to simulate item responses for each case. By default, there is no missing data and item response fit the item response model implied by the item parameters. The first 10 responses are shown, along with the person ability (column 3):

```
<- genResponses(myAbils, myItems)
myResponses print(cbind(myResponses[1:10, 1:2], myAbils[1:10]))
#> [,1] [,2] [,3]
#> [1,] 0 1 0.7965057
#> [2,] 0 0 -0.8527630
#> [3,] 1 1 0.8427809
#> [4,] 0 0 -0.7379118
#> [5,] 0 0 -0.4384575
#> [6,] 0 1 -1.5821600
#> [7,] 1 1 1.1906216
#> [8,] 1 1 0.2683473
#> [9,] 0 1 -0.2535734
#> [10,] 1 1 0.5555316
```

A polytomous item can be described by including an item matrix with
more than two rows. An item with its own discrimination parameter can be
described by including the value in the `a`

column. An item
with non-integer scoring can be described by varying the value in the
`k`

column. Note that for identification purposes, the values
in the `t`

column should sum to zero, though this is not
enforced to allow for sophisticated designs.

```
<- matrix(
myItemPoly c(
0, 0.0, 0.0, 0,
1, 0.5, -1.5, 1,
2, 0.5, 1.5, 1
),ncol = 4, byrow = TRUE
)
<- matrix(
myItem2Pl c(
0, 0, 0.00, 0.00,
1, -2, -1.25, 0.85,
2, -2, -0.25, 0.85,
3, -2, 0.30, 0.85,
4, -2, 1.20, 0.85
),ncol = 4, byrow = TRUE
)
<- matrix(
myItemNonIntScore c(
0, 0, 0.00, 0,
1.2, 0, -0.35, 1,
1.8, 0, 0.15, 1,
3.8, 0, 0.20, 1
),ncol = 4, byrow = TRUE
)
<- append(myItems, list(myItemPoly, myItem2Pl, myItemNonIntScore))
myItems plotModelCCC(myItemPoly)
```

`plotModelExp(myItems)`

The multidimensional structure of the model is defined by allocating
items to dimensions by constructing a B-matrix (sometimes called a
scoring matrix – see Wilson, M., Zheng, X., & McGuire, L. (2012).
Formulating latent growth using an explanatory item response model
approach. Journal of Applied Measurement, 13(1), 1–22.). The B matrix
maps items (rows) to dimensions (columns). Note that this is a
simplified B matrix, in that it is strictly binary – any aspect of
increasing coefficients (for example over time) should be captured in
the scoring column, `k`

, within the item matrix.

Note that now that the model is multidimensional, the abilities provided should also be a matrix, rather than a vector: cases (rows) by dimensions (columns). Any covariance structure amongst the dimensions is encoded within the matrix of abilities.

Note that one of the items does not use integer scoring. If the software to analyse the data assumes integer scoring, then secondary processing is required to yield traditional categorical data, and the scoring assumptions should be otherwise input into the software.

```
<- matrix(
myBMatrix c(
1, 0,
1, 0,
0, 1,
0, 1,
0, 1
),byrow = TRUE,
ncol = 2
)
<- rnorm(length(myAbils))
myAbils2 <- cbind(myAbils, myAbils2) # assumes expected correlation = 0
myAbilsMat
<- genResponses(myAbilsMat, myItems, BMatrix = myBMatrix)
myResponses print(cbind(myResponses[1:10, 1:5], myAbilsMat[1:10, 1:2]))
#> myAbils myAbils2
#> [1,] 0 1 2 4 3.8 0.7965057 1.3364925
#> [2,] 0 1 0 3 1.2 -0.8527630 -1.2347370
#> [3,] 1 1 1 4 3.8 0.8427809 0.6101917
#> [4,] 0 1 1 4 3.8 -0.7379118 2.2117249
#> [5,] 0 1 1 1 1.2 -0.4384575 -0.7333140
#> [6,] 0 1 2 3 3.8 -1.5821600 0.2933033
#> [7,] 0 0 1 4 3.8 1.1906216 0.8467243
#> [8,] 0 0 1 3 0.0 0.2683473 -1.3508211
#> [9,] 0 0 1 4 3.8 -0.2535734 0.1860145
#> [10,] 0 1 1 4 0.0 0.5555316 0.1555404
```