# Probability of response vectors in conquestr

#### Dan Cloney

#### 2023-03-12

# Introduction

In some cases users might like to return the probability of a response on a given item. For example, given a fixed set of item parameters, return the probabilities at varying levels of theta to produce custom probability plots.

# Example

## dichotomous case

The probability of responding correctly to a dichotomous item under Rasch-like models (e.g., 1PL models) is often expressed as:

\[\begin{equation} p(x_{ni} = 1)=\frac{exp(\theta_{n} - \delta_{i})}{1 + (\theta_{n} - \delta_{i})} (\#eq:slm) \end{equation}\]

Imagine the item parameters of a single item represented as:

    library(conquestr)
    myItem <- matrix(c(0, 0, 0, 1, 1, 0), ncol =3, byrow=TRUE)
    colnames(myItem)<- c("k", "d", "t")
    print(myItem)
    #>      k d t
    #> [1,] 0 0 0
    #> [2,] 1 1 0

Then the probability of scoring 0 and 1 on this item, at = 0.5:

    myProbs <- simplep(0.5, myItem)
    #> assuming item discrimination is constant and equal to 1
    print(myProbs)
    #>           [,1]
    #> [1,] 0.6224593
    #> [2,] 0.3775407

A simple ICC can be drawn:

    myProbsList <- list()
    myThetaRange <- seq(-4, 4, by = 0.1)
    for (i in seq(myThetaRange)) {
      myProbsList[[i]] <- pX(x = 1, probs = simplep(myThetaRange[i], myItem))
    }
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
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    #> assuming item discrimination is constant and equal to 1
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    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    plot(unlist(myProbsList))

## polytomous case

In the case of polytomously scored items, the probability model can be generalised:

\[\begin{equation} p(X_{ni} = x)=\frac{exp\sum\limits_{k=0}^{x}(\theta_{n} - (\delta_{i} + \tau_{ik}))}{\sum\limits_{j=0}^{m}exp(\sum\limits_{k=0}^{j} (\theta_{n} - (\delta_{i} + \tau_{ik})))} (\#eq:pcm) \end{equation}\]

An item can them be represented such that:

    library(conquestr)
    myItem <- matrix(c(0, 0, 0, 1, 1, -0.2, 2, 1, 0.2), ncol =3, byrow=TRUE)
    colnames(myItem)<- c("k", "d", "t")
    print(myItem)
    #>      k d    t
    #> [1,] 0 0  0.0
    #> [2,] 1 1 -0.2
    #> [3,] 2 1  0.2

Then the probability of scoring 0, 1 and 2 on this item, at = 0.5:

    myProbs <- simplep(0.5, myItem)
    #> assuming item discrimination is constant and equal to 1
    print(myProbs)
    #>           [,1]
    #> [1,] 0.4742264
    #> [2,] 0.3513155
    #> [3,] 0.1744581

A simple ICC can be drawn:

    myProbsList <- list()
    myThetaRange <- seq(-4, 4, by = 0.1)
    for (i in seq(myThetaRange)) {
      myProbsList[[i]] <- simplep(myThetaRange[i], myItem)
    }
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
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    #> assuming item discrimination is constant and equal to 1
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    #> assuming item discrimination is constant and equal to 1
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    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    myProbs <- (matrix(unlist(myProbsList), ncol = 3, byrow = TRUE))
    plot(myThetaRange, myProbs[,1])
    points(myThetaRange, myProbs[,2])
    points(myThetaRange, myProbs[,3])
    abline(v = c(myItem[2, 2], sum(myItem[2, 2:3]), sum(myItem[3, 2:3])))

## Expected scores

The expected score for the an item can be calculated at a given value of theta. Taking an aribitary set of items, it is possible therefor to calculate the test expected score.

    library(conquestr)
    myItems <- list()
    myItems[[1]] <- matrix(c(0, 0, 0, 1, 1, -0.2, 2, 1, 0.2), ncol =3, byrow=TRUE)
    myItems[[2]] <- matrix(c(0, 0, 0, 1, -1, -0.4, 2, -1, 0.4), ncol =3, byrow=TRUE)
    myItems[[3]] <- matrix(c(0, 0, 0, 1, 1.25, -0.6, 2, 1.25, 0.6), ncol =3, byrow=TRUE)
    myItems[[4]] <- matrix(c(0, 0, 0, 1, 2, 0.2, 2, 2, -0.2), ncol =3, byrow=TRUE)
    myItems[[5]] <- matrix(c(0, 0, 0, 1, -2.5, -0.2, 2, -2.5, 0.2), ncol =3, byrow=TRUE)
    for (i in seq(myItems)) {
      colnames(myItems[[i]])<- c("k", "d", "t")
    }
    print(myItems)
    #> [[1]]
    #>      k d    t
    #> [1,] 0 0  0.0
    #> [2,] 1 1 -0.2
    #> [3,] 2 1  0.2
    #>
    #> [[2]]
    #>      k  d    t
    #> [1,] 0  0  0.0
    #> [2,] 1 -1 -0.4
    #> [3,] 2 -1  0.4
    #>
    #> [[3]]
    #>      k    d    t
    #> [1,] 0 0.00  0.0
    #> [2,] 1 1.25 -0.6
    #> [3,] 2 1.25  0.6
    #>
    #> [[4]]
    #>      k d    t
    #> [1,] 0 0  0.0
    #> [2,] 1 2  0.2
    #> [3,] 2 2 -0.2
    #>
    #> [[5]]
    #>      k    d    t
    #> [1,] 0  0.0  0.0
    #> [2,] 1 -2.5 -0.2
    #> [3,] 2 -2.5  0.2

    expectedRes <- list()
    for (i in seq(myThetaRange)) {
      tmpExp <- 0
      for (j in seq(myItems)) {
        tmpE <- simplef(myThetaRange[i], myItems[[j]])
        tmpExp <- tmpExp + tmpE
      }
      expectedRes[[i]] <- tmpExp
    }
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
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    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1
    #> assuming item discrimination is constant and equal to 1

    plot(myThetaRange, unlist(expectedRes))