INTERDECILE RATIO
 Dataplot Vol 2 Vol 1

# INTERDECILE RATIO

Name:
INTERDECILE RATIO (LET)
Type:
Let Subcommand
Purpose:
Compute the interdecile ratio of a variable.
Description:
Given a univariate data set and two quantiles of the data, Q1 and Q2 (where Q1 < Q2), the interdecile ratio computes the ratio of the sum of the data that is greater than or equal to the Q2 quantile of the data to the sum of the data is less than or equal to the Q1 quantile. That is,

$\mbox{IDR} = \frac{\sum_{i=N_{q2}}^{N}{Y_{i}}} {\sum_{i=1}^{N_{q1}}{Y_{i}}}$

with Nq1 and Nq2 denoting the points corresponding to the Q1 and Q2 quantiles, respectively.

This statistic has been proposed as measure of income inequality. For example, if Q1 = 0.2 and Q2 = 0.8, then this ratio compares the income of the bottom 20% relative to the income of the upper 20% percent.

The economist Palma has specifically suggested the Palma index which is based on Q1 = 0.4 and Q2 = 0.9. That is, the income of the bottom 40% is compared to the income of the top 10%. This is based on empirical studies which indicate the stability of incomes of the middle 50% (from roughly the 40th to the 90th decile) across countries. This measure has been proposed as an alternative to the Gini index. The Gini index indicates difference in income inequality, but is not particularly informative about where those differences occur. The Palma index is sensitive to the differences in the tails (i.e., the bottom 40% and the top 10%). The Cobham and Sumner article gives a number of arguments for preferring the Palma index to the Gini index.

The desired quantiles can be set with the commands

LET XQNUM = <value>
LET XQDENOM = <value>

Note that Dataplot will take the larger value for the numerator and the smaller value for the denominator. If these values are not given, Dataplot will use 0.9 for the numerator and 0.4 for the denominator (i.e., the Palma index).

Currently this command is restricted to non-negative data. If any negative numbers are encountered in the data, an error will be reported and the statistic will not be computed.

Syntax:
LET <par> = INTERDECILE RATIO <var>
<SUBSET/EXCEPT/FOR qualification>
where <par> is a parameter where the computed interdecile ratio is stored;
<var> is a response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
LET A = INTERDECILE RATIO Y

LET XQNUM = 0.8
LET XQDENOM = 0.2
LET A = INTERDECILE RATIO Y

Note:
There are various methods for computing quantiles of a data set. The algorithm used for this command is based on the interpolation of order statistics. Specifically, if q is the desired quantile, N is the sample size, and Y is the sorted data set

1. ANI=q*(N+1)
2. NI = integer part of ANI
3. R = ANI - NI
4. NIP1 = NI + 1
5. If NI is less than 1, set it to 1. If NIP1 is greater than N, set it to N.
6. The q-th quantile is then (1 -R)*Y(NI) + R*Y(NI1P)

In the context of this command, we are actually interested in the NI and the NI1P. For the denominator, we sum the points from 1 to NI and add REM*Y*N(NI1P). For the numerator, we sum the points from NI to N and subtract REM*Y(NI1P).

If a different algorithm is used to computed the desired quantiles, a slightly different result could be returned for the interdecile ratio statistic.

Note:
Dataplot statistics can be used in a number of commands. For details, enter

Default:
None
Synonyms:
DECILE RATIO is an synonym for INTERDECILE RATIO
Related Commands:
 QUANTILE = Compute a specified quantile of a variable. INTERQUARTILE RANGE = = Compute the interquartile range of a variable. Q QUANTILE RANGE = Compute the q quantile range of a variable.
Reference:
Cobham and Sumner (2014), "Is Ineqaulity All About the Tails", Significance, Vol. 11, No. 1, pp. 10-13.

Palma (2011), "Homogeneous Middles vs. Hererogeneous Tails, and the End of the 'Inverted-U': It's All About the Share of the Rich", Development and Change, 42(1), pp. 87-153.

Applications:
Data Analysis
Implementation Date:
2015/02
Program:

. Step 1:   Read the Data
.
.           The first data set follows a normal distribution
.           while the second follows a lognormal distribution
.
skip 25
.
. Step 2:   Compute the statistic for both data sets
.
let xqnum = 0.9
let xqdenom = 0.4
let decrat1 = interdecile ratio y1
let decrat2 = interdecile ratio y2
.
let xqnum = 0.8
let xqdenom = 0.2
let decrat3 = interdecile ratio y1
let decrat4 = interdecile ratio y2
.
. Step 3:   Print the results
.
let decrat1 = round(decrat1,2)
let decrat2 = round(decrat2,2)
let decrat3 = round(decrat3,2)
let decrat4 = round(decrat4,2)
print "num: 0.9, den: 0.4"
print "decrat1:  ^decrat1"
print "decrat2:  ^decrat2"
print "num: 0.8, den: 0.2"
print "decrat3:  ^decrat3"
print "decrat4:  ^decrat4"

The following output is generated

num: 0.9, den: 0.4
decrat1:  0.25
decrat2:  0.31
num: 0.8, den: 0.2
decrat3:  1.01
decrat4:  1.26


NIST is an agency of the U.S. Commerce Department.

Date created: 02/09/2015
Last updated: 02/09/2015