Index of qualitative variation

[emanuele fedeli]

Hello,

I want to use an index of qualitative variation (nominal variables) in Stata. Is there a specific command?

Thank you in advance

[Nick Cox]

I think this is one of about 20 or so names in various literatures for a sum of squared proportions. (If you want the complement or reciprocal of that, that's one easy step further.)

See e.g. http://www.statalist.org/forums/foru...index-in-stata

If that doesn't solve the problem please give us an exact definition (equation, not words) and some sample data.

[Rich Goldstein]

never heard of it - could you supply some more info (e.g., a cite)? there are a lot of user-written commands for different measures of variability

[emanuele fedeli]

Hi, my main goal it to measure "variability" for ordinal variables. Below, you have an example.

I wanted to do apply this formula but it does not work:

gen b1 = 1-((p1^2)+(p2^2)+(p3^2)+(p4^2)+(p5^2)+(p6^2))


tab education

*parents' highest education level* | Freq. Percent Cum.
----------------------------------------+-----------------------------------
university or higher | 4,192 31.99 31.99
post-secondary | 3,020 23.05 55.04
upper secondary | 3,106 23.70 78.74
lower secondary | 2,508 19.14 97.88
some primary,lower secondary | 237 1.81 99.69
not applicable | 41 0.31 100.00
----------------------------------------+-----------------------------------
Total | 13,104 100.00

[Nick Cox]

So, this is already answered in the thread linked within #2 with the information that your index is 1 minus what is there called the Herfindahl-Hirschman index.

Here's how to do it in Stata with Mata:

sysuse auto, clear 

tab rep78, matcell(freq) 

mata 
freq = st_matrix("freq") 
1  - sum((freq :/ sum(freq)):^2)
end

In this case Mata shows .7032136106

In your case, it's a matter of naming education as the variable.

[Nick Allum]

I'd like to get bootstrapped standard errors for this statistic. Is there a way of plugging the mata program above into a bootstrap command? Or if not, is there another way of doing it? (I have an experiment where the outcome variable is nominal and I want to compare treatment and control groups on this statistic).

Thanks in advance.
Nick

[Alan Neustadtl]

gen b1 = 1-((p1^2)+(p2^2)+(p3^2)+(p4^2)+(p5^2)+(p6^2))

I have seen this referred to as a diversity index. I believe that the index of qualitative variation is a standardized version of the diversity index that ranges from 0 to 1 so you can compare variability across measures with different numbers of outcome categories. That can be calculated as:

gen b1 = (k/k-1)*1-((p1^2)+(p2^2)+(p3^2)+(p4^2)+(p5^2)+(p6^2)))

Where k is the number of outcome categories.

Best,
Alan

[Mike Lacy]

The IQV, as many sociologists term it, is available in -divcat- (available from SSC) and is described as the "normalized generalized variance." However, since you have an ordinal variable, I suggest you consider the measures of ordinal dispersion implemented in my module -ordvar-, also available at SSC.

To amplify Nick's comment, I might venture that no other statistic has been (re)invented as many times as the IQV.

[Nick Cox]

To answer #6 directly, the answer is surely yes. Here's a minimal code example.

*! 1.0.0 NJC 28 Sept 2016 
program iqv, rclass  
    syntax varname [if] [in] 
    marksample touse, strok 
    tempname freq iqv 
    qui tab `varlist' if `touse', matcell(`freq') 
    mata: freq = st_matrix("`freq'") 
    mata: st_numscalar("`iqv'", 1  - sum((freq :/ sum(freq)):^2)) 
    di _n  "IQV: " %4.3f `iqv'
    return scalar iqv = `iqv' 
end

. sysuse auto
(1978 Automobile Data)

. bootstrap r(iqv) , reps(100) : iqv rep78
(running iqv on estimation sample)

Warning:  Since iqv is not an estimation command or does not set e(sample),
          bootstrap has no way to determine which observations are used in
          calculating the statistics and so assumes that all observations are
          used.  This means no observations will be excluded from the
          resampling because of missing values or other reasons.

          If the assumption is not true, press Break, save the data, and drop
          the observations that are to be excluded.  Be sure that the dataset
          in memory contains only the relevant data.

Bootstrap replications (100)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 
..................................................    50
..................................................   100

Bootstrap results                               Number of obs      =        74
                                                Replications       =       100

      command:  iqv rep78
        _bs_1:  r(iqv)

------------------------------------------------------------------------------
             |   Observed   Bootstrap                         Normal-based
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _bs_1 |   .7032136   .0327004    21.50   0.000      .639122    .7673053
------------------------------------------------------------------------------

. estat bootstrap

Bootstrap results                               Number of obs      =        74
                                                Replications       =       100

      command:  iqv rep78
        _bs_1:  r(iqv)

------------------------------------------------------------------------------
             |    Observed               Bootstrap
             |       Coef.       Bias    Std. Err.  [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _bs_1 |   .70321361  -.0074416   .03270042    .6309599   .7529844  (BC)
------------------------------------------------------------------------------
(BC)   bias-corrected confidence interval

In your case, the approach might need to be extended e.g. to get a CI for the difference of two measures.