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I have two distributions and I want to test whether there's an inequality of variances. They're non-normal, so Levene's test is appropriate. The scipy implementation offers three options: test on the mean, the median, and the trimmed mean. The trimmed mean is appropriate for heavy tailed distributions.

My question is, how do I know if my distribution is heavy-tailed? My understanding is that it's heavy-tailed if it's not exponentially bounded. I've tried to check this but I'm not sure if my method is correct. Here's what I did:

  1. Converted my data into z-scores so as to standardise it and plotted it.

enter image description here

  1. Plotted the exponential distribution function across the range of my data, plotted it.

  2. Compared the two visually.

enter image description here

EDIT: I'm here adding some descriptive statistics about my two distributions in response to the comments. This is the raw data, not the z scores.

Distribution 1: 

count    38.000000
mean      1.160140
std       1.281058
min       0.220619
25%       0.451241
50%       0.623582
75%       1.478313
kurtosis  7.57
max       6.719054

Distribution 2: 

count    40.000000
mean      0.887812
std       0.720215
min       0.252508
25%       0.408433
50%       0.617842
75%       1.120488
kurtosis  6.27
max       3.939130

My conclusion is that my distribution is not heavy tailed, but I'm not confident about it. Can anyone advise?

EDIT 2: This is my full data, long form:

Group   entropy
1   6.71905356
1   0.56407487
1   0.738029138
1   0.630035416
1   3.017076375
1   2.510090903
1   0.254787047
1   0.376719953
1   0.456298101
1   0.328258469
1   0.767253283
1   0.641643213
1   2.905235741
1   3.615227362
1   0.244727319
1   2.317604878
1   1.504713298
1   0.999700392
1   0.669730607
1   0.766398132
1   0.449555621
1   0.360902977
1   0.297898424
1   1.399111031
1   0.67895411
1   0.56984134
1   0.536010552
1   2.226602414
1   1.998649951
1   0.220619041
1   0.547186366
1   0.446506256
1   0.495662791
1   0.458900635
0   1.699580285
0   1.017646859
0   0.618058775
0   0.740520854
0   0.558418925
0   0.264262271
0   1.4136416
0   0.538862166
0   2.089605078
0   2.206855803
0   0.494698728
0   0.36284015
0   0.947420619
0   1.515928283
0   0.682302263
0   0.515864165
0   0.400418084
0   0.401584527
0   1.195820577
0   0.544921866
0   0.284516915
0   1.902155181
0   1.095376897
0   0.263003363
0   0.674095659
0   3.939129819
0   0.617625765
0   0.364223021
0   0.355701427
0   0.887284165
0   0.312722361
0   0.570313528
0   0.4107156
0   0.453855313
0   1.441497841
1   1.720034593
1   0.590291826
1   0.444819008
0   0.252508237
0   1.226010557
0   0.526118886
0   1.046928619
0   0.679454156
1   0.617128565
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    $\begingroup$ A few comments: 1) Bartlett's test is not appropriate for non-normally distributed data (go to the Wikipedia page); 2) you can quantify tailedness with the kurtosis statistic; 3) your data may well be exponentially distributed (if so, only one parameter, lambda, is required to fit a probability density distribution and the mean and variance will be 1/lambda and 1/lambda^2). Personally, instead of implementing an out-of-the-box test for homogeneity of variance, I would see if the two random variables are exponentially distributed. If so, implement a permutation test for the parameter lambda. $\endgroup$ Jan 10, 2021 at 11:37
  • $\begingroup$ Please back up here. It seems that your two distributions (your graph shows only one) are entropy measures, so they are bounded above and below, which won't necessarily bite. Very different variances is usually a sign that you should work on a transformed scale. (Taking z-scores does some small harm here, as it obscures the real range of the data; otherwise it is irrelevant.) $\endgroup$
    – Nick Cox
    Jan 10, 2021 at 11:38
  • $\begingroup$ I think there is some confusion here between (a) an exponential distribution (b) how the right tail behaves. $\endgroup$
    – Nick Cox
    Jan 10, 2021 at 11:40
  • $\begingroup$ @Ventrilocus: (1) Sorry, should have said Levene's test, which is what I used. Edited to fix. (2) I'll try this, thanks. (3) Not sure how I'd do this, but will look into it! $\endgroup$
    – Lodore66
    Jan 10, 2021 at 12:08
  • $\begingroup$ @NickCox: How are they bounded? I appreciate entropy can't go below zero, but what's the upper bound? I'm not using metric entropy here, which is constrained between 0 and 1. Or have I misunderstood? $\endgroup$
    – Lodore66
    Jan 10, 2021 at 12:09

1 Answer 1

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This isn't a complete answer, but the graphs it shows can't be included easily in a comment.

Your summary statistics allow box plots to be drawn and transformations to be tried in exploratory fashion.

Here I note that distribution 1 does have higher variability than distribution 2, as the standard deviations imply. I am not a great fan of tests comparing variances, but they exist. Perhaps a better procedure would be to get a confidence interval for the variance ratio by bootstrapping and see if it contains 1.

I suggest considering working on logarithmic scale.

Either way your sample sizes (38 and 40; Python's default of adding 6 decimal places is curious, but I've seen the same dopeyness in my own favourite software) are small enough that you could post the entire dataset here. Particularly interesting to me is how far the higher variability of distribution 1 is attributable to just a few high values in its tail.

enter image description here

EDIT: The full data allow more to be said. First, as a generic method to get a firmer idea of the variance ratio, I used Stata for bootstrapping. The variance ratio is 3.164 or so, which looks a fair bit bigger than 1, but the 95% confidence intervals all include 1. The normal-based confidence interval doesn't "know" that the variance ratio can't be zero or negative, but set that aside.

. estat bootstrap, all

Bootstrap results                               Number of obs     =         78
                                                Replications      =      10000

      command:  var_ratio entropy
        _bs_1:  r(var_ratio)

------------------------------------------------------------------------------
             |    Observed               Bootstrap
             |       Coef.       Bias    Std. Err.  [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _bs_1 |   3.1638341   .8053801   3.1112753   -2.934153   9.261822   (N)
             |                                       .7221368   12.66759   (P)
             |                                       .7664097   13.44563  (BC)
------------------------------------------------------------------------------
(N)    normal confidence interval
(P)    percentile confidence interval
(BC)   bias-corrected confidence interval

Naturally, other statistics could be used ad hoc to compare variability such as a ratio of SDs or IQRs.

More detailed graphs, a quantile-box plot and a Lego-style histogram, suggest a possible pattern: the two groups have very similar lower values, but group 1 has a longer tail. It's not a matter of a simple additive or multiplicative shift.

enter image description here

enter image description here

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  • $\begingroup$ Wow! This is really generous of you to do this. You're right that my data are small and I can easily enough post the full dataset here. I'll edit the main post to do this. Thank you! Some great insights in what you've given. $\endgroup$
    – Lodore66
    Jan 10, 2021 at 20:24

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