divintseg 0.7.0

Tools for computing diversity, integration and segregation metrics

divintseg

divintseg is a simple package for computing diversity, integration, and segregation statistics on data sets. Typically, it is used with demographic data such as census data.

Methodology

divintseg uses a straightforward methodology to compute its metrics. It is designed to make mathematical sense and have nice mathematical properties, while at the same time remaining simple enough that it makes sense to non-technical people as well.

Visualizing Diversity, Integration, and Segregation

In order to build up some intuition on what our metrics are trying to model, it's useful to start with some visual illustrations of the concepts the metrics try to capture.

The most basic notion in our methodology is that of a community that consists of members of different non-overlapping groups. In order to build a basic intuition for communities, groups, and the metrics we will compute on them, we will begin with some visual representations.

We'll start with a community that, intuitively, looks both diverse and integrated.

Each small circle represents an individual. The color of the circle represents which one of three groups they belong to. There are equal numbers of blue, green, and orange circles, so we would tend to consider this group to be diverse. Furthermore, the members of the different groups are spread out throughout the community so that every individual has nearby neighbors that are in different groups than they are. This community looks integrated.

In contrast, here is a community that looks diverse but not integrated.

Just like the previous community, this community is diverse. It has an equal number of members of each group. But it is not integrated. Instead, it is segregated. Each of the three groups is concentrated and most individuals do not have nearby neighbors of a different group.

Now let's look at some communities that are less diverse. Here is a non-diverse community. Almost all the individuals are in the blue group.

This is also a segregated community. The few members of the orange and green groups are all together in one corner of the community.

Let's look at another community that is also not diverse, but looks like it might be at least a little more integrated than the last one.

How integrated really is this community? The few individuals in the orange and green groups are scattered around, but there aren't really enough of them to say that the community is integrated. As we will see when we develop the math behind our methodology, a community that is not that diverse cannot really be that integrated either, no matter how the individuals are distributed.

From Visuals to Mathematics

We will introduce our metrics one by one, starting with diversity, then integration, and finally segregation. Informed by the visuals above, we'll try to come up with definitions that make sense and can be translated into mathematical equations and then into code.

Diversity

Let's begin with a working definition of diversity. We say a community is diverse if an average member of the community is likely to encounter people who are not members of their group as they navigate the community. This idea has been proposed multiple times across different fields. It is known as the Gini-Simpson index, the Gibbs-Martin index, the Blau index, and expected heterozygosity in different fields.

Now let's turn that into math. In order to compute the average chance a member of the community encounters someone of a different group, we will first compute, for each group, what the chance that a random person from the entire population comes from a different group. We will then compute the overall average across all groups.

Let's start with the population shown here:

All three groups are the same size. Let's start with the blue group. The chance that a randomly chosen member of the population is a member of the blue group is thus $1/3$, or approximately $0.333$. We'll call this number $p$.

The probability that a member of the blue group encounters someone of one of the other two groups when they encounter a random person from the entire population is $1 - p = 2/3$, or approximately $0.667$.

Since all three groups are of the same size, they all have the same value of $p$. We can summarize this in a table as follows:

Group	Representation $p$	Chance of Encountering a Member of Another Group $= 1 - p$
Blue	$0.333$	$0.667$
Orange	$0.333$	$0.667$
Green	$0.333$	$0.667$

If we define the diversity of the population $D$ to be the average chance of any member of the population encountering a member of another group, then in this example it is

$$D = 0.333(0.667) + 0.333(0.667) + 0.333(0.667) = 0.667.$$

Each of the three terms is for one of the three groups, and for each of them the fraction of the population in the group is $0.333$ and the chance of encountering a member of another group is $0.667$.

Some additional strictly optional mathematical details

(Feel free to skip this section if you like.)

More formally, what we computed is

$$D = \sum p(1 - p)$$

The Gini-Simpson index formulation of diversity is normally written as the equivalent expression

$$D = 1 - \sum p^2.$$

The two are equivalent because

$$D = \sum p(1 - p) = \sum p - \sum p^2 = 1 - \sum p^2$$

The last step works because the $p$ values are probabilities for each group, so they add up to $1$, i.e. $\sum p = 1$.

But in our discussion we stick to the earlier formulation because we think it more clearly expresses what we are computing and why, especially for small examples like the ones we are considering here.

Now let's look at another example. It is one of the communities we looked at above.

In this example, each of the groups is also exactly one third of the population of the community, so we have the exact same numbers as before:

Group	Representation $p$	Chance of Encountering a Member of Another Group $= 1 - p$
Blue	$0.333$	$0.667$
Orange	$0.333$	$0.667$
Green	$0.333$	$0.667$

And again,

$$D = 0.333(0.667) + 0.333(0.667) + 0.333(0.667) = 0.667.$$

So this community has the same exact diversity as the last one, though it is clearly more integrated. We'll return to that later.

Finally, let's look at a less diverse community. Again, this is one we looked at before.

Let's compute the $p$ for each of the three groups, and then compute $D$. We'll add a column to our table where we will compute $p(1-p)$ for each group, and then we will sum these up at the bottom of the table to get $D$.

Group	$p$	$1 - p$	Weighted Representation $p(1-p)$
Blue	$0.963$	$0.037$	$0.036$
Orange	$0.022$	$0.978$	$0.022$
Green	$0.015$	$0.985$	$0.015$
Weighted Sum			$0.073$

So this community's diversity is $0.073$. As we would expect from visual inspection, it is much lower than the diversity of the previous two communities ($0.667$).

Integration

As we saw above, two communities can have the exact same diversity, but to the eye appear to be very different when it comes to integration. Integration is all about whether the members of a diverse community actually do interact, as the definition of diversity assumes they do, by randomly encountering one another, or if, on the contrary, they live in segregated neighborhoods within the community in which they rarely encounter members of other groups.

In order to make this notion of integration a little more formal, in a way that we can then write math and code to compute it, we will say that a community is integrated if the average member of the community is likely to encounter people who are not members of their group as they navigate their local neighborhood within the community. Another way of putting this is that if the neighborhoods within a community are diverse, then the community is integrated. If the community as a whole is diverse, but none of the neighborhoods in the community are themselves diverse, then the community is not integrated. Mathematically speaking, integration is the population-weighted average of neighborhood diversity in the community.

Let's look at an example of a community consisting of three neighborhoods of equal population.

We can compute the diversity withing each of the three neighborhoods. Since each neighborhood has exactly equal numbers of members of each group, each neighborhood has diversity $D = 0.667$. We get this number by doing the exact same kind of calculation we did for the diversity of the community with equal members of each group above. We just repeat it three times, once for each neighborhood.

Now, let's define $r$ for each neighborhood to be the fraction of the total population of the community that lives in the neighborhood. For our current example, $r = 1/3$ for each of the three neighborhoods since they are of equal size.

Knowing $r$ and $D$ for each neighborhood, we can compute the integration of the community by multiplying the $r$ and $D$ values together for each neighborhood and summing them up. We do this in the following table.

Neighborhood	$r$	$D$	Weighted Diversity $rD$
A	$0.333$	$0.667$	$0.222$
B	$0.333$	$0.667$	$0.222$
C	$0.333$	$0.667$	$0.222$
Weighted Sum			$0.667$

So the integration of our community is $I = 0.667$. This is exactly the same as the overall diversity of the community.

We won't go into the details here, but one of the consequences of the way we set up our mathematical definitions of diversity and integration is that the value of $I$ for a community can never be more that the value of $D$. That is, $I \le D$ in all cases. More generally, $I$ and $D$ are also both between $0$ and $1$, so $0 \le I \le D \le 1$ no matter how our community and the neighborhoods within it are constructed. No matter how big the community is, how big the neighborhoods are, whether the neighborhoods are all the same size or not, or how many groups there are, the fundamental relationship

$$0 \le I \le D \le 1$$

will always hold true.

Now let's look at a community where $I < D$, meaning that integration is less than diversity in the community.

If we repeat our calculation of $D$ for each neighborhood and use that to calculate $I$ again, we get

Neighborhood	$r$	$D$	Weighted Diversity $rD$
A	$0.333$	$0.667$	$0.222$
B	$0.333$	$0.444$	$0.148$
C	$0.333$	$0.444$	$0.148$
Weighted Sum			$0.519$

So $I = 0.519$ for this community. Looking at this community vs. the previous one, it does appear to be less integrated. Neighborhood A is diverse, with equal numbers of each of the three groups, but the other two neighborhoods are less diverse. Each of them is completely lacking one of the three groups and has unequal numbers of the other two.

So, as far as out math working out to produce $I = 0.519 < D = 0.667$ for this community, things make sense. The way the people in the community are divided up into neighborhoods results in integration being less than the diversity of the community as a whole. This is in contrast to the previous example where each neighborhood was as diverse as the whole community, and as a result, $I$ was equal to $D$.

Now let's look at a third example, one in which the diversity of the community as a whole was already low, and even the limited diversity that exists is not shared among the neighborhoods. This should result in a value of $I$ even lower than the already low value of $D$.

If we do our calculation of $D$ as we did above when we looked at this community without the neighborhood boundaries, $D = 0.073$. Now let's calculate $I$.

Neighborhood	$r$	$D$	Weighted Diversity $rD$
A	$0.333$	$0.000$	$0.000$
B	$0.333$	$0.000$	$0.000$
C	$0.333$	$0.204$	$0.068$
Weighted Sum			$0.068$

Two of the neighborhoods (A and B) have no diversity at all. Neighborhood C has a little bit. The overall integration of the community is $I = 0.068$, which is less than the diversity of $D = 0.073$ as we expected.

Finally, just to drive home the point that diversity and integration are different concepts, let's look at a community with high diversity but no integration at all.

Overall diversity of the community is $D = 0.667$, but if we calculate $I$ we get

Neighborhood	$r$	$D$	Weighted Diversity $rD$
A	$0.333$	$0.000$	$0.000$
B	$0.333$	$0.000$	$0.000$
C	$0.333$	$0.000$	$0.000$
Weighted Sum			$0.000$

$I = 0$. Despite the community being diverse, it is not integrated at all.

Segregation

Segregation is the opposite of integration. Since we know that for all communities, $0 \le I \le 1$ we simply define segregation as $S = 1 - I$. We don't generally use $S$ as often as we use $D$ and $I$, since it is so related to $I$, but for completeness, the divintseg library can compute it.

Code Examples

Now that we have gone through the methodology behind divintseg at length, let's look at some examples of how to use the code itself.

In most cases, data we will want to analyze with divintseg will exist in Pandas DataFrames, or in some other format that is easy to convert to a DataFrame. We'll use them in our examples.

Diversity

We begin with some diversity computations.

First, let's start with a very simple example consisting of a single-row DataFrame with a column for each group. The numbers in the columns represent the number of people in the community that belong to each group. The first community we looked at had 108 members of each group. So we could construct it in code and compute its diversity as follows:

import pandas as pd

from divintseg import diversity

df = pd.DataFrame(
    [[108, 108, 108]],
    columns=['blue', 'green', 'orange']
)

print(diversity(df))

This will print

0    0.666667
Name: diversity, dtype: float64

The return value of the call to diversity(df) is a pandas Series with a single element, the diversity of the single row of df. And as we would expect, it got the same number we calculated manually above.

Now let's try something a little more advanced, with three neighborhoods in a community like in our examples above.

import pandas as pd

from divintseg import diversity

df = pd.DataFrame(
    [
        ['A', 36, 36, 36],
        ['B', 36, 36, 36],
        ['C', 36, 36, 36],
    ],
    columns=['neighborhood', 'blue', 'green', 'orange']
)

df.set_index('neighborhood', inplace=True)

print(diversity(df))

This time the output is

neighborhood
A    0.666667
B    0.666667
C    0.666667
Name: diversity, dtype: float64

diversity(df) calculated the diversity of each row independently. Again we reproduced some of the same results as we got manually above.

Now let's try another example with some different diversity in different neighborhoods.

import pandas as pd

from divintseg import diversity

df = pd.DataFrame(
    [
        ['A', 36, 36, 36],
        ['B', 72, 0, 36],
        ['C', 0, 72, 36],
    ],
    columns=['neighborhood', 'blue', 'green', 'orange'],
)

df.set_index('neighborhood', inplace=True)

print(diversity(df))

Now the output is

neighborhood
A    0.666667
B    0.444444
C    0.444444
Name: diversity, dtype: float64

just as we would expect.

Integration

Now let's move on to integration. The API is almost as simple as for diversity, but we have to specify what column or index represents the neighborhood.

More generally, since we might not actually be working with neighborhoods, but with various other kinds of nested geographic areas. For example, if we are working with US Census data, we might be interested in integration at the block group level computed over the diversity of the different blocks in the block group. But we might also want to skip a level in the census hierarchy as compute the integration of census tracts (groups of multiple block groups) over diversity down at the block level. The integration API gives us the flexibility to choose how we do this.

Here is an example where we put two communities in the same DataFrame. The first, community "X" has equally diverse neighborhoods. The second, community "Y" has unequally diverse neighborhoods.

import pandas as pd

from divintseg import integration

df = pd.DataFrame(
    [
        ['X', 'A', 36, 36, 36],
        ['X', 'B', 36, 36, 36],
        ['X', 'C', 36, 36, 36],
        
        ['Y', 'A', 36, 36, 36],
        ['Y', 'B', 72, 0, 36],
        ['Y', 'C', 0, 72, 36],
    ],
    columns=['community', 'neighborhood', 'blue', 'green', 'orange'],
)

print(integration(df, by='community', over='neighborhood'))

The two keyword arguments are important. The first by="community, tells the API that we want our results by community. There are two unique communities in the data, "X", and "Y", so we should get two results. The second keyword, over='neigborhood' tells us what column to use to represent the inner level of geography at which to compute the diversity numbers that we then aggregate up to the level specified by the by= argument.

The result is

           integration
community             
X             0.666667
Y             0.518519

This again matches the results we computed manually for these example communities and neighborhoods.

Diversity, Integration (and Segregation) All at Once

More often than not, we want to compute diversity and integration for the same communities at the same time. We can do that with a single API divintseg.di. It can also optionally tell us segregation too. Here is how to use it.

import pandas as pd

from divintseg import di

df = pd.DataFrame(
    [
        ['W', 'A', 108, 0, 0],
        ['W', 'B', 0, 108, 0],
        ['W', 'C', 0, 0, 108],
        
        ['X', 'A', 36, 36, 36],
        ['X', 'B', 36, 36, 36],
        ['X', 'C', 36, 36, 36],
        
        ['Y', 'A', 36, 36, 36],
        ['Y', 'B', 72, 0, 36],
        ['Y', 'C', 0, 72, 36],
        
        ['Z', 'A', 108, 0, 0],
        ['Z', 'B', 108, 0, 0],
        ['Z', 'C', 96, 5, 7],
    ],
    columns=['community', 'neighborhood', 'blue', 'green', 'orange'],
)

print(di(df, by='community', over='neighborhood', add_segregation=True))

This gives us everything we would want to know about diversity, integration, and segregation in these communities in one output DataFrame.

           diversity  integration  segregation
community      
W           0.666667     0.000000     1.000000                               
X           0.666667     0.666667     0.333333
Y           0.666667     0.518519     0.481481
Z           0.071997     0.067844     0.932156