# Other Measures

Redistricting is problem with many, many dimensions. This vignette introduces some useful measures related to redistricting, but with smaller categories. See the vignette “Using `redistmetrics`” for the bare-bones of the package.

We first load the `redistmetrics` package and data from New Hampshire. For any function, the `shp` argument can be swapped out for your data, `rvote` and `dvote` for any two party votes, `pop_black` for any group population, `pop` for the total population, and the `plans` argument can be swapped out for your redistricting plans (be it a single plan, a matrix of plans, or a `redist_plans` object).

    library(redistmetrics)
    data(nh)

## Competitiveness

### Talismanic Compactness

This is a measure which offers a balance between competitiveness across the state and competitiveness within individual districts.

Formally, this can be written as:

\[\textrm{Talismanic Competitiveness} = T_p (1 + \alpha T_e)\beta\]

where

\[ T_p = \frac{1}{n_d} * \sum_{k=1}^{n_\textrm{dists}} \big|\frac12 - \textrm{voteshare}_D\big|\] \[ T_e = |\frac{n_\textrm{dists} - Seats_D}{n_\textrm{dists}}-\frac12| \]

Talismanic Compactness can be computed with

    compet_talisman(plans = nh$r_2020, shp = nh, rvote = nrv, dvote = ndv)
    #> [1] 0.04955851 0.04955851

where `nrv` and `ndv` are averages of votes. (In general, you want to compute these scores over many elections and average them.)

## Segregation

### Dissimilarity

Dissimilarity describes how similar the demographic proportions in districts are to the total state population’s demographics.

Formally, this can be written as:

\[ \textrm{Dissimilarity} = \sum_{i = 1}^{n_\textrm{dists}} \frac{(t_d |g_d - G|)}{2T*G(1 - G)}\]

for a group population proportion in district \(d\), \(g_d\), total population in district \(d\), \(t_d\), a group population proportion in a state \(G\), and total population in the state \(T\).

Dissimilarity can be computed with:

    seg_dissim(plans = nh$r_2020, shp = nh, group_pop = pop_black, total_pop = pop)
    #> [1] 0.0273714 0.0273714

## Incumbents

### Incumbent Pairings

We compute incumbent pairings as the number of incumbents who are placed into a district with other incumbents beyond those allowed. Formally, this is:

\[\textrm{Inc. Pairs} = \sum_{d = 1}^{\textrm{ndists}}\max(0, ~n^{(d)}_{inc} - 1)\]

We do not have incumbent data included for New Hampshire. As such, we create fake incumbent data.

    fake_inc <- rep(FALSE, nrow(nh))
    fake_inc[c(1, 2)] <- TRUE

This would indicate that there are only incumbents in the first two rows of the data.

Incumbent pairings can be computed with:

    inc_pairs(plans = nh$r_2020, shp = nh, inc = fake_inc)
    #> [1] 1 1