How to explain and use Stephen Jenkins' J indice

[Chen Samulsion]

Dear Stata users,

Professor Stephen Jenkins developed command named -ineqord- (SSC, Distribution-Date: 20200314) for comparing distributions of ordinal data, especially subjective well-being (SWB) scales data. He propose an indice called J, including J downward-looking status measure index (J_d) & J upward-looking status measure index (J_u).

Each index is equal to the area between the generalized Lorenz (GL) curve for the relevant status distribution and the GL curve for the distribution with no status inequality [in which case the GL curve is a straight line between the origin and point (1; 1)], divided by the total area beneath the perfect equality curve (= 0:5). (Jenkins, 2020, The Stata Journal, st0606, P509)

Here is some literature source: (1) https://journals.sagepub.com/doi/pdf...36867X20953565 (2) https://www.tandfonline.com/doi/full...4.2019.1697729
I want to know how to explain J indice in empirical studies, and is it comparable to Gini index? Thank you very much.

[Stephen Jenkins]

Chen: the paragraph you cite provides a discussion of how one might relate the J index to areas between generalised Lorenz curves (and thence sort of analogous to generalised poverty gap indices which are also thus). Gini coefficients relate to areas between Lorenz curves. Your intended audience may well be unfamiliar with such fine points and the detail 'wasted' on them. And, after all, how the heck would you explain to that audience what an Apouey et al. index (for example)? All in all, why not simply say that bigger numbers for J mean more inequality? Good luck

[Chen Samulsion]

Thank you vey much professor. Best regards.

[Neeraj Kumar]

Prof. Stephen Jenkins, I am using the -ineqord- command to measure inequality in Life Satisfaction. I am Just struggling with interpretation of the J indices. How to interpret these J_d (downward-looking status); J_u (upward-looking status) . Kindly help me with interpretation of these two values.

[Jared Greathouse]

Hey Neeraj Kumar, please rephrase your question as we ask you to in the FAQ. Please, give context to your question and report your results in

Code

Delimiters, that way it'll be easier to read your output so anyone can help you.

[Neeraj Kumar]

Sir, Since my variable is Ordinal (Life Satisfaction), I am Using the ineqord command to measure the inequality developed by Prof. Stephen Jenkins. I am more interested to know the interpretation of J indices, which is also developed by Prof. Stephen Jenkins. I am sharing the result of J_d (downward-looking status); J_u (upward-looking status). This J indices formulated on the basis Cowell-Flachaire inequality indices to measure the inequality in Ordinal variable. I want to know how to interpret the J_d (downward-looking status) and J_u (upward-looking status).

 . ineqord A170
  
Inequality indices: Cowell-Flachaire, downward-looking status

----------------------------------------------------------
  All obs |       I(0)      I(.25)       I(.5)      I(.75)
----------+-----------------------------------------------
          |    0.78068     0.86695     1.11179     1.94073
----------------------------------------------------------
  
Inequality indices: Cowell-Flachaire, upward-looking status

----------------------------------------------------------
  All obs |       I(0)      I(.25)       I(.5)      I(.75)
----------+-----------------------------------------------
          |    0.75024     0.85072     1.10354     1.93693
----------------------------------------------------------
 
Inequality indices: J_d (downward-looking status); J_u (upward-looking status)

----------------------------------
  All obs |         Jd          Ju
----------+-----------------------
          |    0.59213     0.59411
----------------------------------

[William Lisowski]

I think Stephen Jenkins makes a good point in post #2 - "why not simply say that bigger numbers for J mean more inequality?".

The upward-looking status measures how unequal the distribution is when individuals are compared to the highest possible value; the downward-looking status measures how unequal the distribution is when individuals are compared to the lowest possible value.

[Neeraj Kumar]

Thank you so much Mr. Lisowski for reply. "why not simply say that bigger numbers for J mean more inequality?". In this bigger number of J implies what? is it Upwhave ard-looking status (J_u) or downward looking status (J_u)?

Actually I have to generate one variable called inequality for my further analysis ( I am having 14 states and 4 years for each state and each year there will be one inequality value ) . That's why I am asking which value I have to considered for measuring the inequality.
once again thank you so much Mr. William Lisowski for you help.

[William Lisowski]

Both imply more inequality, the difference is that a bigger J_u implies more inequality when individuals are compared to those between them and the highest possible value, and J_d implies more inequality when individuals are compared to those between them and the lowest possible value.

The papers cited in post #1 discusses this in some detail, but they are fairly technical. This seminar paper by Cowell & Flachaire tells us, for their measures, for downward-looking status inequality is higher when the distribution is skewed towards the higher categories; for upward- looking status inequality is higher when the distribution is skewed towards the lower categories.

If you want a single measure, you have to choose which one to use.

What we lack is a tutorial in place of the academic papers cited, to explain it to us with more approachable examples.

[Neeraj Kumar]

Thank you so much Mr. William Lisowski for your guidance.

[Neeraj Kumar]

Sorry for bothering you Mr. William Lisowski. But I am having one question related to J_u (upward-looking status). I have gone through the paper you have mentioned in post #9. I got to know whether I have to use Upward-looking status or downward looking status. According to Cowell-Flachaire inequality indice the value of upward or downward looking status should be fall between 0 and 1. In the case of J indices also the value should fall between 0 and 1 (Having bit doubt). Since I am using Upward looking status, the value value I am getting is more than 1. I am sharing the result. Just let me know whether I am doing any mistake or it's fine.

Inequality indices: J_d (downward-looking status); J_u (upward-looking status)

----------------------------------
  All obs |         Jd          Ju
----------+-----------------------
          |    0.93011     1.04966
----------------------------------

[William Lisowski]

In the case of J indices also the value should fall between 0 and 1 (Having bit doubt)

In the paper by Stephen Jenkins Inequality comparisons with ordinal data found at https://doi.org/10.1111/roiw.12489 there is mention of a lower bound of zero but no mention of an upper bound.

[Stephen Jenkins]

Like the more familiar Gini index, my "J" index is a ratio of areas, as described in the article cited by William Lisowski (thank you for your replies on this topic; I've been away). The maximum J value arises when the area under the generalised Lorenz curve is at maximum. (Cf. how values of one for the standard Gini coefficient arise.)

I'm being a little coy because ascertaining what distributions of respondents across the categories is associated with the maximum value of an inequality index for ordinal data, at least those indices constructed in the Cowell-Flachaire tradition, is not necessarily straightforward. (Hence also little discussion of this in my RIW paper.) For discussion of this issue, with reference to Cowell-Flachaire indices per se, see: Frank A. Cowell and Emmanuel Flachaire,(2022), "Maximum Inequality: The Case of Categorical Data", Chapter 5 in Research on Economic Inequality: Poverty, Inequality and Shocks, Volume 29, 95–103

[Neeraj Kumar]

Thank you so much Mr. William Lisowski for your prompt reply and consistent guidance. Thank you Prof. Jenkins for your reply.