# Selection of the first items

### Description

This command selects the first items of the adaptive test, either randomly or on the basis of their difficulty level (for dichotomous items only).

### Usage

1 2 3 4 |

### Arguments

`itemBank` |
numeric: a suitable matrix of item parameters. See |

`model` |
either |

`fixItems` |
either a vector of integer values or |

`seed` |
either a numeric value, |

`nrItems` |
numeric: the number of starting items to be randomly selected (default is 1). Can be equal to zero to avoid initial selection of items (see |

`theta` |
numeric: a vector of the initial ability levels for selecting the first items (default is the single value 0). Ignored if either |

`D` |
numeric: the metric constant. Default is |

`randomesque` |
integer: the number of 'randomesque' items to be picked up optimally for each value of |

`random.seed` |
either |

`startSelect` |
character: the criterion for selecting the first items. Possible values are |

`nAvailable` |
either a boolean vector indicating which items (denoted by 1's) are available at the start of the test and which (denoted by 0's) are not, or |

### Details

This function permits to select the first item(s) of the test. It works with both dichotomous and polytomous item banks.

Dichotomous IRT models are considered whenever `model`

is set to `NULL`

(default value). In this case, `it`

must be a matrix with one row per item and four columns, with the values of the discrimination, the difficulty, the pseudo-guessing and the inattention parameters (in this order). These are the parameters of the four-parameter logistic (4PL) model
(Barton and Lord, 1981).

Polytomous IRT models are specified by their respective acronym: `"GRM"`

for Graded Response Model, `"MGRM"`

for Modified Graded Response Model, `"PCM"`

for Partical Credit Model, `"GPCM"`

for Generalized Partial Credit Model, `"RSM"`

for Rating Scale Model and `"NRM"`

for Nominal Response Model. The `it`

still holds one row per item, end the number of columns and their content depends on the model. See `genPolyMatrix`

for further information and illustrative examples of suitable polytomous item banks.

The number of starting items is given by the length of `fixItems`

argument, the `nrItems`

argument (in case of random selection) or by the length of `theta`

argument (in case of optimal selection), with default value 1 in all cases. It can be set to zero; in this case, only `NULL`

values are returned in the output list and the CAT process will start without starting items.

The first item(s) of the adaptive test can be selected by one of the following methods.

By specifying the item(s) to be administered. The argument

`fixItems`

then holds the item number(s) as listed in the item bank. Setting`fixItems`

to`NULL`

(default value) disables this method.By selecting it (them) randomly into the item bank. The argument

`seed`

permits to fix the random selection by specifying the random seed number and the number of selected items is fixed by the`nrItems`

argument. Setting`seed`

to`NA`

disables the random seed (though items are still picked up randomly in the bank); in other words, successive runs of`startItems`

with`seed=NA`

may lead to different item(s) selection. Setting`seed`

to`NULL`

(default value) disables this selection method.By selecting the item(s) according to an initial sequence of ability values set by the

`theta`

argument. In this case, five criteria can be used, specified through the`startSelect`

argument:`"MFI"`

(default): one selects the most informative item(s) for the given initial ability value(s);`"bOpt"`

: one selects the item(s) whose difficulty level is as close as possible to the inital ability value(s);`"thOpt"`

: one selects the item(s) with the ability value where they get their maximum Fisher information is as close as possible to the inital ability value(s) (see Magis, 2013, for further details);`"progressive"`

for the*progressive*method (see`nextItem`

);`"proportional"`

for the*proportional*method (see`nextItem`

).

If the

`"progressive"`

or`"proportional"`

methods are selected, this will force the values of`fixItems`

to`NULL`

,`seed`

to`NULL`

, and`nrItems`

to 1. Thus, a single item will be selected randomly.

The third method above will be used if and only if both `fixItems`

and `seed`

arguments are fixed to `NULL`

.
Otherwise, one of the first two methods will be used (see also `testList`

for details about debugging
misspecifications of the starting arguments).

The sequence of initial ability estimates is specified by the argument `theta`

. For each component of `theta`

one item will be picked up optimally according to the chosen `startSelect`

argument. however, it is possible to perform randomesque selection at this stage by setting the `randomesque`

argument to an integer value larger than one. In this case, the `randomesque`

most optimal items are chosen per value of `theta`

, and the final starting item is chosen randomly among the `randomesque`

items. By default, only one item is picked up per ability level (and is therefore *the* most optimal one).

Only part of the full item bank can be made available for the selection of the first item(s), while others can be dropped out from this first step. This is fixed by the `nAvailable`

argument, which is a vector with as many components as items in the bank and with zeros and ones only. Values 1 code for available items, values 0 for non-available items. By default, `nAvailable`

is `NULL`

and all items are available. Note that `nrItems`

should never be larger than the number of available items (i.e. `sum(nAvailable)`

). Otherwise an error message is returned.

### Value

A list with four arguments:

`items` |
the selected items (identified by their number in the item bank) or |

`par` |
the matrix of item parameters of the selected items (one row per item) or |

`thStart` |
the sequence of starting ability values used for selecting the items or |

`startSelect` |
the value of the |

### Note

Currently only the `"MFI"`

value for `startSelect`

is allowed for polytomous IRT models (i.e., when `model`

is not `NULL`

). Otherwise an error message is returned.

### Author(s)

David Magis

Department of Education, University of Liege, Belgium

david.magis@ulg.ac.be

Juan Ramon Barrada

Department of Psychology and Sociology, Universidad Zaragoza, Spain

barrada@unizar.es

### References

Barton, M.A., and Lord, F.M. (1981). *An upper asymptote for the three-parameter logistic item-response model*. Research Bulletin 81-20. Princeton, NJ: Educational Testing Service.

Haley, D.C. (1952). *Estimation of the dosage mortality relationship when the dose is subject to error.*
Technical report no 15. Palo Alto, CA: Applied Mathematics and Statistics Laboratory, Stanford University.

Magis, D; (2013). A note on the item information function of the four-parameter logistic model. *Applied Psychological Measurement*, *37*, 304-315. doi: 10.1177/0146621613475471

Magis, D., and Raiche, G. (2012). Random Generation of Response Patterns under Computerized Adaptive Testing with the R
Package *catR*. *Journal of Statistical Software*, *48 (8)*, 1-31. URL http://www.jstatsoft.org/v48/i08/

### See Also

`testList`

, `genPolyMatrix`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | ```
## Dichotomous models ##
# Loading the 'tcals' parameters
data(tcals)
# Item bank creation with 'tcals' item parameters
bank <- as.matrix(tcals[,1:4])
# Random selection of 4 starting items
startItems(bank, seed = 1, nrItems = 4)
# Random selection of 4 starting items without fixing the seed
startItems(bank, seed = NA, nrItems = 4)
startItems(bank, seed = NA, nrItems = 4) # may provide a different result!
# Selection of the first 5 starting items
startItems(bank, fixItems = 1:5)
# Selecting 1 starting item, initial ability estimate is 0
startItems(bank)
# Selecting 3 starting items for ability levels -1, 0 and 2
startItems(bank, theta = c(-1, 0, 2))
# Same with 5 randomesque items per theta value
startItems(bank, theta = c(-1, 0, 2), randomesque = 5)
# 5 randomesque items per theta value, with fixed random seed number
startItems(bank, theta = c(-1, 0, 2), randomesque = 5, random.seed = 1)
# Idem but with 'bOpt' criterion
startItems(bank, theta = c(-1, 0, 2), startSelect = "bOpt")
# Selecting only the first 10 items as available items
avail <- c(rep(1, 10), rep(0, nrow(bank)-10))
startItems(bank, theta = c(-1, 0, 2), nAvailable = avail)
## Not run:
# Selecting too many items among available ones
startItems(bank, nrItems = 11, theta = 2, halfRange = 3, nAvailable = avail)
## End(Not run)
## Polytomous models ##
# Generation of an item bank under GRM with 100 items and at most 4 categories
m.GRM <- genPolyMatrix(100, 4, "GRM")
m.GRM <- as.matrix(m.GRM)
# Random selection of 4 starting items
startItems(m.GRM, model = "GRM", seed = 1, nrItems = 4)
# Selection of the first 5 starting items
startItems(m.GRM, model = "GRM", fixItems = 1:5)
# Selecting 3 starting items for theta values -1, 0 and 2
startItems(m.GRM, model = "GRM", theta = c(-1, 0, 2))
## Not run:
# Idem but with 'bOpt' criterion
startItems(m.GRM, model = "GRM", nrItems = 3, theta = 1, halfRange = 2,
startSelect = "bOpt")
## End(Not run)
# Selecting only the first 10 items as available items
avail <- c(rep(1, 10), rep(0, nrow(m.GRM)-10))
startItems(m.GRM, model = "GRM", theta = c(-1, 0, 2),
nAvailable = avail)
## Not run:
# Selecting too many items among available ones
startItems(m.GRM, model = "GRM", theta = seq(from = -2, to = 2, length = 11),
nAvailable = avail)
## End(Not run)
# Generation of an item bank under PCM with 20 items and 4 categories
m.PCM <- genPolyMatrix(20, 4, "PCM", same.nrCat = TRUE)
m.PCM <- as.matrix(m.PCM)
# Random selection of 4 starting items
startItems(m.PCM, model = "PCM", seed = 1, nrItems = 4)
# Selection of the first 5 starting items
startItems(m.PCM, model = "PCM", fixItems = 1:5)
# Selecting 3 starting items for theta values -1, 0 and 2
startItems(m.PCM, model = "PCM", theta = c(-1, 0, 2))
## Not run:
# Idem but with 'bOpt' criterion
startItems(m.PCM, model = "PCM", theta = c(-1, 0, 2), startSelect = "bOpt")
## End(Not run)
# Selecting only the first 10 items as available items
avail <- c(rep(1, 10), rep(0, nrow(m.PCM)-10))
startItems(m.PCM, model = "PCM", theta = c(-1, 0, 2), nAvailable = avail)
## Not run:
# Selecting too many items among available ones
startItems(m.PCM, model = "PCM", theta = seq(from = -2, to = 2, length = 11),
nAvailable = avail)
## End(Not run)
``` |