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<h1 class="title toc-ignore">Bias of the Gini coefficient from Poisson
sampling</h1>
<h4 class="author">Timothy Daley</h4>
<h4 class="date">7/15/2022</h4>
</div>
<p>The Gini coefficient is commonly used to measure the inequality of a
population, particularly in economic contexts. For example, the Gini
coefficient is commonly used to measure and compare wealth or income
inequality across countries or areas (see <a
href="https://en.wikipedia.org/wiki/Gini_coefficient">Wikipedia</a>).
The Gini coefficent is defined as <span class="math display">\[
\frac{ \sum_i \sum_j | x_i - x_j |}{2 n \sum_j x_j}.
\]</span> The Gini coefficient ranges from 0, when all <span
class="math inline">\(x_i = x_j\)</span> and the population is
completely uniform, to 1, when some <span class="math inline">\(x_k >
0\)</span> and all other <span class="math inline">\(x_{i} = 0\)</span>,
i.e. one person has all the wealth.</p>
<p>The Gini coefficient is used in genomic contexts to compare
inequality of a sampled population, e.g. <a
href="https://genomebiology.biomedcentral.com/articles/10.1186/s13059-016-1010-4"
class="uri">https://genomebiology.biomedcentral.com/articles/10.1186/s13059-016-1010-4</a>
or <a
href="https://genomebiology.biomedcentral.com/articles/10.1186/s13059-015-0843-6"
class="uri">https://genomebiology.biomedcentral.com/articles/10.1186/s13059-015-0843-6</a>.
The latter paper is where our motivation comes from. The Gini
coefficient is used to assess the quality or selection of sequenced
CRISPR screening libraries. Where the usual Gini coefficients assess the
distribution of sampled quantities, e.g. sampling random people in a
population and obtaining the income for each one, our motivating
application involves Poisson or multinomial sampling from a population
to assess the frequency. With the latter, there is a sampling-based bias
that can affect our estimates.</p>
<p>To see this point, consider the extreme situation where the
population being sampled is completely uniform. The true Gini
coefficient is equal to zero, however because the observed sample is not
completely uniform we get a stricly positive estimated Gini coefficient.
Therefore we have some sampling-based bias. How much exactly? We shall
see in this investigation.</p>
<div id="simulation" class="section level1">
<h1>Simulation</h1>
<p>We’ll do some simulations to determine the rate of bias in the
Poisson case.</p>
<pre class="r"><code>gini <- function(x){
if(length(x) == 0){
return(0)
}
if(sum(x) == 0){
return(1)
}
if(!all(x >= 0)){
return(1)
}
x.sorted = sort(x)
l = length(x)
numer = sum((l - 1:l + 1)*x.sorted)
denom = sum(x.sorted)
return((l + 1 - 2*numer/denom)/l)
}
n_rep = 100
n_sgrna = 200000
sampling_depth = n_sgrna*c(1/16, 1/8, 1/4, 1/2, 1, 2, 4, 8, 16, 32)
sampled_gini = data.frame(sampling_depth = rep(sampling_depth, each = n_rep),
gini = rep(0, times = length(sampling_depth)*n_rep))
for(i in 1:dim(sampled_gini)[1]){
lam = sampled_gini$sampling_depth[i]
x = rpois(n_sgrna, lambda = lam)
sampled_gini$gini[i] = gini(x)
}
library(dplyr)
gini_summary = sampled_gini %>% group_by(sampling_depth) %>% summarize(gini = mean(gini))
library(ggplot2)
ggplot(gini_summary, aes(x = sampling_depth, y = gini)) + geom_line() + theme_bw()</code></pre>
<p><img src="gini_files/figure-html/unnamed-chunk-1-1.png" width="672" /></p>
<pre class="r"><code>ggplot(gini_summary, aes(x = sampling_depth, y = gini)) + geom_line() + theme_bw() + scale_y_log10() + scale_x_log10()</code></pre>
<p><img src="gini_files/figure-html/unnamed-chunk-1-2.png" width="672" /></p>
<p>We would naturally think that the convergence rate is some <span
class="math inline">\(O(n^p)\)</span>, e.g. <span
class="math inline">\(O(n^{1/2})\)</span>, as we see from the above
<span class="math inline">\(\log - \log\)</span> plot.</p>
<pre class="r"><code>summary(lm(log(sampling_depth) ~ log(gini), data = gini_summary))</code></pre>
<pre><code>##
## Call:
## lm(formula = log(sampling_depth) ~ log(gini), data = gini_summary)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.340e-04 -1.449e-04 -2.100e-07 1.369e-04 5.385e-04
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.1448271 0.0007031 -1628 <2e-16 ***
## log(gini) -2.0000006 0.0001016 -19686 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.0003198 on 8 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 3.875e+08 on 1 and 8 DF, p-value: < 2.2e-16</code></pre>
<p>We see that the convergence rate of the error is <span
class="math inline">\(O(n^{-2})\)</span> in the uniform Poisson case.
But what if the we’re not in the perfectly uniform case, as we’re
commonly not.</p>
<p>We’ll get to that next time.</p>
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