GINI MEAN DIFFERENCE LOG RATIO

GINI MEAN DIFFERENCE LOG RATIO (LET)

Let Subcommand

Compute the log of the ratio of the Gini mean differences for two response variables.

Given a response variable \( X \) with observations \( x_1, x_2, ..., x_n \) sorted from low value to high value, the Gini mean difference is defined as:
\( \begin{array}{lcl} \mbox{GMD} & = & \frac{2}{n(n-1)} \sum_{i=1}^{n} {\sum_{j=1}^{i-1}{|x_{i} - x_{j}|}} \\ & = & \frac{2}{n(n-1)} \sum_{i=1}^{n-1} {i(n-i)x_{i+1} - x_{i}} \end{array} \)

where \( n \) is the sample size.

The Gini mean difference was proposed by Gini (1912) and is the average absolute differences in all pairs of observations. Note that this is a measure of dispersion that does not depend on a measure of location.

This command computes the log of the ratio of the Gini mean differences of two variables. If the Gini mean differences are equal, the ratio of the Gini mean differences is equal to 1 and the log of the ratio is equal to 0. This can be the basis of a robust alternative approach to testing for equal dispersion. Tena (2009) discusses this in more detail.

LET <par> = GINI MEAN DIFFERENCE LOG RATIO <y1> <y2>
                  <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <par> is a parameter where the computed Gini mean difference log ratio is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET A = GINI MEAN DIFFERENCE LOG RATIO Y1 Y2
LET A = GINI MEAN DIFFERENCE LOG RATIO Y1 Y2 SUBSET Y2 > 0

Dataplot statistics can be used in a number of commands. For details, enter
HELP STATISTICS

None

None

GINI MEAN DIFFERENCE	=	Compute the Gini mean differences of a variable.
MEDIAN ABSOLUTE DEVIATION	=	Compute the median absolute deviation of a variable.
AVERAGE ABSOLUTE DEVIATION	=	Compute the average absolute deviation of a variable.
STANDARD DEVIATION	=	Compute the standard deviation of a variable.
VARIANCE	=	Compute the variance of a variable.
RANGE	=	Compute the range of a variable.

Tena (2009), "Test Procedures for Equality of Two Variances in Delta Distributions," Dissertations, 695, https://scholarworks.wmich.edu/dissertations/695.

Gini (1921), "Measurement of Inequality in Incomes," The Economic Journal, 31, pp. 124-126.

Data Analysis

2023/07

 
SKIP 25
READ AUTO83B.DAT Y1 Y2
RETAIN Y2 SUBSET Y2 > 0
.
BOOTSTRAP SAMPLES 10000
TITLE CASE ASIS
LABEL CASE ASIS
CASE ASIS
.
TITLE Bootstrap Plot for Gini Mean Difference Log Ratio
X1LABEL Bootstrap Sample
X2LABEL Dataset: AUTO83B.DAT
Y1LABEL Gini Mean Difference Log Ratio
.
BOOTSTRAP GINI MEAN DIFFERENCE LOG RATIO PLOT Y1 Y2
.
LET LCL = ROUND(B025,3)
LET UCL = ROUND(B975,3)
JUSTIFICATION CENTER
MOVE 50 5
TEXT Lower 95% Confidence Limit: ^LCL, Upper 95% Confidence Limit: ^UCL
LINE DOTTED
LINE COLOR RED
DRAWSDSD 15 ^LCL 85 ^LCL
DRAWSDSD 15 ^UCL 85 ^UCL
    

The following output is generated

 
             Bootstrap Analysis for the GINI MEAN DIFFERENCE LOG RATIO
  
 Response Variable One: Y1
 Response Variable Two: Y2
  
 Number of Bootstrap Samples:                      10000
 Number of Observations:                             249
 Mean of Bootstrap Samples:                      0.03965
 Standard Deviation of Bootstrap Samples:        0.09176
 Median of Bootstrap Samples:                    0.03783
 MAD of Bootstrap Samples:                       0.06284
 Minimum of Bootstrap Samples:                  -0.27461
 Maximum of Bootstrap Samples:                   0.39343
  
  
  
 Percent Points of the Bootstrap Samples
 -----------------------------------
   Percent Point               Value
 -----------------------------------
             0.1    =       -0.23102
             0.5    =       -0.18724
             1.0    =       -0.16822
             2.5    =       -0.13632
             5.0    =       -0.10981
            10.0    =       -0.07637
            20.0    =       -0.03855
            50.0    =        0.03783
            80.0    =        0.11801
            90.0    =        0.15871
            95.0    =        0.19106
            97.5    =        0.22295
            99.0    =        0.26068
            99.5    =        0.28081
            99.9    =        0.32864
  
  
             Percentile Confidence Interval for Statistic
  
 ------------------------------------------
   Confidence          Lower          Upper
  Coefficient          Limit          Limit
 ------------------------------------------
        50.00       -0.02371        0.10227
        75.00       -0.06510        0.14687
        90.00       -0.10981        0.19106
        95.00       -0.13632        0.22295
        99.00       -0.18724        0.28081
        99.90       -0.23955        0.35192
 ------------------------------------------