## Introduction

## Scalability coefficients

### Estimating scalability coefficients

### Hypothesis tests and confidence intervals for scalability coefficients

## Automated item selection procedure

Criterion | AISP | ||
---|---|---|---|

Null hypothesis | Hypothesis matches criterion | Accepts criterion if | |

1: \(H_{ij}>0\) | \(H_{ij}\le 0\) | \(\checkmark\) | \(\Delta _{ij} \geq z_\mathrm{crit}\) |

2: \(H_{i}\ge c\) | \(H_{i}\le 0\) | – | \(\Delta _{i} \geq z_\mathrm{crit}\) and \({\widehat{H}}_i \ge c\) |

Criterion | T-AISP | ||
---|---|---|---|

Null hypothesis | Hypothesis matches criterion | Accepts criterion if | |

1: \(H_{ij}>0\) | \(H_{ij}\le 0\) | \(\checkmark\) | \(z_{ij} \geq z_\mathrm{crit}\) |

2: \(H_{i} > c\) | \(H_{i}\le c\) | \(\checkmark\) | \(z_{i} \geq z_\mathrm{crit}\) |

## Solving the two MSA issues

### Point estimates , standard errors , and statistical tests of scalability coefficients for clustered data

### Test-guided automated item selection procedure

## A two-step, test-guided MSA, for nonclustered and clustered data

## Real-data example

Item | M | SD | |
---|---|---|---|

1 | The teachers usually know how I feel | 2.84 | 0.89 |

2 | I can talk about problems with the teachers | 3.18 | 0.92 |

3 | If I feel unhappy, I can talk to the teachers about it | 3.03 | 0.98 |

4 | I feel at ease with the teachers | 3.52 | 0.77 |

5 | The teachers understand me | 3.23 | 0.81 |

6 | I have good contact with the teachers | 3.34 | 0.83 |

7 | I would prefer to have other teachers* | 3.22 | 0.85 |

8 | I have a lot of contact with my classmates | 4.06 | 0.76 |

9 | I would prefer to be in another class* | 3.89 | 1.08 |

10 | We have a nice class | 3.89 | 0.96 |

11 | I get along well with my classmates | 4.01 | 0.73 |

12 | I sometimes feel alone in the class* | 4.12 | 0.92 |

13 | I enjoy hanging out with my classmates | 4.00 | 0.74 |

Total scale | 3.57 | 0.53 |

### Method

### Results

Item | c | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.00 | 0.05 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.50 | 0.55 | |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |

2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 |

3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |

5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |

6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |

7 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

8 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 3 | 0 |

9 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 |

10 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 |

11 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 0 |

12 | 1 | 1 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

13 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 |

SWMD | SWMK | ||||||||
---|---|---|---|---|---|---|---|---|---|

Item | \({\widehat{H}}\) | SE | 95% CI | ICC | Item | \({\widehat{H}}\) | SE | 95% CI | ICC |

1 | 0.609 | 0.033 | [0.545; 0.674] | 0.120 | 8 | 0.547 | 0.036 | [0.477; 0.617] | 0.077 |

2 | 0.641 | 0.026 | [0.589; 0.693] | 0.111 | 9 | 0.551 | 0.036 | [0.480; 0.621] | 0.103 |

3 | 0.619 | 0.029 | [0.562; 0.676] | 0.103 | 10 | 0.644 | 0.025 | [0.595; 0.693] | 0.196 |

4 | 0.634 | 0.033 | [0.568; 0.699] | 0.142 | 11 | 0.594 | 0.031 | [0.532; 0.655] | 0.111 |

5 | 0.650 | 0.028 | [0.594; 0.705] | 0.082 | 12 | – | – | – | – |

6 | 0.566 | 0.031 | [0.506; 0.626] | 0.129 | 13 | 0.627 | 0.028 | [0.572; 0.682] | 0.097 |

7 | – | – | – | – | |||||

Total | 0.620 | 0.026 | [0.570; 0.670] | 0.169 | Total | 0.592 | 0.025 | [0.543; 0.642] | 0.183 |