# Sim3PL

##### Simulated Data for fitting a 3PL and 3PLu

Simulated responses of 10000 persons to 10 dichotomous items under two different simulation conditions.

- Keywords
- datasets

##### Usage

`data("Sim3PL", package = "psychotools")`

##### Details

Data were simulated under the 3PL (`resp`

) and 3PLu (`resp2`

) (see
`plmodel`

). For the 3PL scenario, the random number generator's
seed was set to 277. For the 3PLu scenario, the random number generator's seed
was set to 167. Person parameters \(\theta_{i}\) of 10000 persons were drawn
from the standard normal distribution. Item difficulties \(b_{j}\) of 10
items (under the classical IRT parametrization) were drawn from the standard
normal distribution. Item discrimination parameters \(a_{j}\) were drawn
from a log-normal distribution with a mean of \(0\) and a variance of
\(0.0625\) on the log scale. For the 3PL, guessing parameters
\(g_{j}\) were drawn from a uniform distribution with a lower limit of
\(0.1\) and an upper limit of \(0.2\). For the 3PLu, upper asymptote
parameters \(u_{j}\) were drawn from a uniform distribution with a lower
limit of \(0.8\) and an upper limit of \(0.9\). In both scenarios, a
\(10000\) x \(10\) matrix based on realizations of a uniform distribution
with a lower limit of \(0\) and an upper limit of \(1\) was generated and
compared to a \(10000\) x \(10\) matrix based on the probability function
under the respective model. If the probability of person \(i\) solving item
\(j\) exceeded the corresponding realization of the uniform distribution,
this cell of the matrix was set to \(1\), e.g., person \(i\) solved item
\(j\).

##### Format

A data frame containing 10000 observations on 2 variables.

- resp
Item response matrix with 10 items (see details below).

- resp2
Item response matrix with 10 items (see details below).

##### See Also

##### Examples

```
# NOT RUN {
## overview
data("Sim3PL", package = "psychotools")
str(Sim3PL)
## data generation
M <- 10000
N <- 10
## 3PL scenario
set.seed(277)
theta <- rnorm(M, 0, 1)
a <- rlnorm(N, 0, 0.25)
b <- rnorm(N, 0, 1)
g <- runif(N, 0.1, 0.2)
u <- rep(1, N)
probs <- matrix(g, M, N, byrow = TRUE) + matrix(u - g, M, N, byrow = TRUE) *
plogis(matrix(a, M, N, byrow = TRUE) * outer(theta, b, "-"))
resp <- (probs > matrix(runif(M * N, 0, 1), M, N)) + 0
all.equal(resp, Sim3PL$resp, check.attributes = FALSE)
## 3PLu scenario
set.seed(167)
theta <- rnorm(M, 0, 1)
a <- rlnorm(N, 0, 0.25)
b <- rnorm(N, 0, 1)
g <- rep(0, N)
u <- runif(N, 0.8, 0.9)
probs <- matrix(g, M, N, byrow = TRUE) + matrix(u - g, M, N, byrow = TRUE) *
plogis(matrix(a, M, N, byrow = TRUE) * outer(theta, b, "-"))
resp2 <- (probs > matrix(runif(M * N, 0, 1), M, N)) + 0
all.equal(resp2, Sim3PL$resp2, check.attributes = FALSE)
# }
```

*Documentation reproduced from package psychotools, version 0.5-1, License: GPL-2 | GPL-3*