entropies | R Documentation |

Entropy `I_1`

is a scalar measure of how much information is required to predict
the outcome of a choice number 1 exactly, and consequently is a measure of item effectiveness suitable for multiple choice tests and rating scales.
Joint entropy `J_{1,2}`

is a scalar measure of the cross-product of multinomial
vectors 1 and 2. Mutual entropy `I_{1,2} = I_1 + I_2 - J_{1,2}`

is a measure
of the co-dependency of items 1 and 2, and thus the analogue of the negative
log of a squared correlation `R^2`

. this function computes all four types
of entropies for two specificed items.

```
entropies(index, m, n, chcemat, noption)
```

`index` |
A vector of length N containing score index values for each test taker. |

`m` |
The index of the first choice. |

`n` |
The index of the second choice. |

`chcemat` |
The data matrix containing the indices of choisen options for each test taker. |

`noption` |
A vector containing the number of options for all items. |

A named list object containing objects produced from analyzing the simulations, one set for each simulation:

`I_m: ` |
The entropy of item m. |

`I_n: ` |
The entropy of item n. |

`J_nm: ` |
The joint entropy of items m and n. |

`I_nm: ` |
The mutual entropy of items m and n. |

Juan Li and James Ramsay

Ramsay, J. O., Li J. and Wiberg, M. (2020) Full information optimal scoring. Journal of Educational and Behavioral Statistics, 45, 297-315.

Ramsay, J. O., Li J. and Wiberg, M. (2020) Better rating scale scores with information-based psychometrics. Psych, 2, 347-360.

`Entropy_plot`

```
# Load needed objects
chcemat <- Quant_13B_problem_dataList$chcemat
index <- Quant_13B_problem_parmList$index
noption <- matrix(5,24,1)
# compute mutual entropies for all pairs of the first 6 items
Mvec <- 1:6
Mlen <- length(Mvec)
Hmutual <- matrix(0,Mlen,Mlen)
for (i1 in 1:Mlen) {
for (i2 in 1:i1) {
Result <- entropies(index, Mvec[i1], Mvec[i2], chcemat, noption)
Hmutual[i1,i2] = Result$Hmutual
Hmutual[i2,i1] = Result$Hmutual
}
}
print("Matrix of mutual entries (off-digagonal) and self-entropies (diagonal)")
print(round(Hmutual,3))
```

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