Abul Naga and Yalcin Index

[SimonMolinsky]

Abul Naga and Yalcin indices

Create a function to calculate Abul Naga and Yalcin indices from ordinal data. The function definition is:

def any_index(ds: Iterable, a_param: Optional[int] = 1, b_param: Optional[int] = 1):
    pass

where:

• ds: sorted or unsorted dataset, an array with responses on an ordinal scale,
• a_param: $\alpha$ parameter, must be positive and greater or equal to 1, the default value is 1,
• b_param: $\beta$ parameter, must be positive and greater or equal to 1; the default value is 1.

Parameters a_param and b_param are designed to put different weights to inequalities below and above the median (see example below).

Steps

The algorithm has five steps:

1. Calculate the median value $m$ from ds. It must be an integer!
2. Get the number of categories $n$.
3. Build cumulative distribution for each category $P{i}$, where the last category $P{n}$ should be equal to 1.
4. Calculate the index based on the equation:

I_{\alpha , \beta} = \frac{\sum_{i < m}{P_{i}^{\alpha}} - \sum_{i \ge m}P_{i}^{\beta} + (n+1-m)}{k_{\alpha , \beta} + (n+1-m)}

where:

• $\alpha \geq 1$
• $\beta \geq 1$

and:

$$k_{\alpha , \beta}=(m-1)(\frac{1}{2})^{\alpha}-[1-(n-m)(\frac{1}{2})^{\beta}]$$

With $(\alpha=\beta)$ inequality is at a minimum when everyone is in the same category and at a maximum when half of the population lies in the lowest category and half in the highest category. Different calibrations of the parameters $\alpha$ and $\beta$ allow the researcher to give different weights to inequalities above and below the median of the responsiveness distribution - for higher values of $\alpha$ ($\beta$), less weight is given to inequalities below (above) the median. However, by design, the index can only be used to compare distributions with the same median category [1].

Returns

Abul Naga and Yalcin index: float

Bibliography

• [1] Abul Naga RH, Yalcin T. Inequality measurement for ordered response health data. J Health Econ. 2008 Dec;27(6):1614-25. doi: 10.1016/j.jhealeco.2008.07.015. Epub 2008 Aug 19. PMID: 18838185.

• [2] Andrew M. Jones, Nigel Rice, Silvana Robone, Pedro Rosa Dias. Inequality and Polarisation in Health Systems’ Responsiveness: A Cross- Country Analysis. HEDG Working Paper 10/27, October 2010. URL: https://www.york.ac.uk/media/economics/documents/herc/wp/10_27.pdf

[SimonMolinsky]

@Arturstrag, how is it going? Would you like some help with this task?

[SimonMolinsky]

@Arturstrag I'd like to take this issue from you :) If this is okay, then on Monday (5th of Feb), I'll change the assignment of the issue.

[Arturstrag]

Ok