How the estimates are computed

From tract estimates to drive-time areas

The quantity of interest is a characteristic of the people or households in the drive-time area of a site, such as the number of people below the poverty level or the poverty rate. The package starts from ACS estimates for census tracts, each published with a margin of error (MOE), the half-width of its 90 percent confidence interval. Tract boundaries do not follow those of drive-time areas. The package therefore combines the estimates of the tracts that overlap a drive-time area, with weights computed from the areas of the overlaps, and combines their margins of error with the same weights.

cacs_run() does this in five steps, each carried out by a function that can also be called on its own:

1. cacs_acs_prefetch() downloads the ACS 5-year estimates and their margins of error for the census tracts of one state, using the tidycensus package; a download needs a Census API key.
2. cacs_isochrone() gets the drive-time areas around each site, one for each drive time, from a routing service.
3. cacs_intersect_weight() projects the drive-time areas and the tracts to EPSG:5070 (NAD83 / Conus Albers), an equal-area map projection, and measures there, in square meters, the area of each tract and of its part inside each drive-time area. From these areas it computes weights and combines the tract estimates and their margins of error.
4. cacs_propagate_moe() recomputes the margins of error at the chosen confidence level and records the formula used for each row.
5. cacs_derive_rates() computes the five rates listed in cacs_acs_default_rates, such as the poverty rate, each as the ratio of two combined counts, with its margin of error.

Only the first two steps use the network, and the help page of cacs_run() describes how to supply their results instead. This article describes the calculations of steps 3 to 5.

The symbols, the weights, and the estimates

The package computes every result separately for each site and drive time, so the symbols below, the two weights, and the three estimates all describe one site $s$ at one drive time. The three articles named above use the same symbols.

Two weights are computed from these areas. The coverage weight of tract $j$ is the share of the tract’s area that lies inside the drive-time area,

$$w^{cov}_{sj} = \frac{\lvert I_s \cap T_j \rvert}{\lvert T_j \rvert},$$

and the area share of tract $j$ is the area of its overlap divided by the total overlap area of all the tracts,

$$w^{mean}_{sj} = \frac{\lvert I_s \cap T_j \rvert}{\sum_k \lvert I_s \cap T_k \rvert}.$$

Here and below, a sum over tracts runs over the tracts kept for the site, drive time, and variable. A tract is kept if its coverage weight is above the min_weight argument of cacs_intersect_weight() and the ACS data have a row for it and the variable. A tract with no row for the variable is left out without a warning: a count then leaves out that tract’s share, while an average weighted by area shares uses the remaining tracts, whose area shares again sum to one. The column weight_basis of the result records the weight used: "coverage" or "area_mean".

A count is estimated by the coverage-weighted sum of the tract estimates, a median or a per-person value by the average of the tract estimates weighted by area shares, and a rate by the ratio of two coverage-weighted sums:

$$\widehat{Y}_s = \sum_j w^{cov}_{sj}\, Y_j, \qquad \widehat{\bar X}_s = \sum_j w^{mean}_{sj}\, X_j, \qquad \widehat{R}_s = \frac{\widehat{A}_s}{\widehat{B}_s} = \frac{\sum_j w^{cov}_{sj}\, A_j}{\sum_j w^{cov}_{sj}\, B_j}.$$

The two sums of a rate run over the tracts kept for their own variable, so a tract that has a row for one of the two and not the other enters only one of them.

The coverage-weighted sum assumes that whatever a variable counts is spread evenly over each tract’s area. For the number of people below the poverty level, it assumes that those people are spread evenly, not only the population as a whole. The average weighted by area shares stands in for the median or per-person value of the drive-time area. Its weights follow area and ignore how many people live in each tract, so it can be far from that value when the overlapping tracts differ in population density. If any of the tracts has a missing estimate for the variable, the combined estimate and its margin of error are both NA; if only the tract’s margin of error is missing, only the margin of error is NA.

ACS margins of error are at the 90 percent level, where the margin of error $M$ of an estimate is 1.645 times its standard error (SE) (U.S. Census Bureau 2020, chap. 7):

$$M = 1.645\,\mathrm{SE}, \qquad \mathrm{SE} = \frac{M}{1.645}.$$

The package divides each tract margin of error by 1.645 and combines the resulting standard errors into the standard error of a combined estimate, written $\mathrm{SE}(\widehat{Y}_s)$ for a count and in the same way for the other estimates. It reports the margin of error $M(\widehat{Y}_s) = z\,\mathrm{SE}(\widehat{Y}_s)$, where $z = 1.645$ at the default level of 90 percent. vignette("theory-moe-propagation", package = "catchmentACS") describes other levels, which are set with the level argument of cacs_propagate_moe().

Kinds of estimate and their weights

The column estimand_family of the results names the kind of estimate in each row. The kind decides which weights combine the tract estimates, as recorded in weight_basis, and which formula gives the margin of error (see the section “Margins of error”):

If whatever a count counts is spread evenly over the tract, as the coverage-weighted sum assumes, the part of a tract inside the drive-time area holds the same share of that count as of the tract’s area. The count is therefore split by the coverage weight: a tract half inside the drive-time area adds half of its count. The assumption is made for each variable on its own, and it asks more of a subgroup than of the population as a whole. A tract whose residents below the poverty level all live in one corner can have an evenly spread population and an unevenly spread poverty count. A median or a per-person value does not scale with the part of the tract that is inside. The package averages these values with area shares instead. A rate divides two counts, each added up with coverage weights.

As the help page of cacs_intersect_weight() describes, the kind of estimate is read from the ACS code alone. Codes of tables B19013 (median household income) and B25077 (median value of owner-occupied housing units) are taken as medians, and codes of table B19301 (per capita income) as per-person values. Any other code of the form B, five digits, an underscore, and three digits, such as B17001_002, is taken as a count. Median gross rent (B25064_001), for instance, comes from another table and is summed with coverage weights as if it were a count; the package gives no warning. A code of any other form, such as one from a table whose name starts with C or S or ends in a letter (B17001A), is not combined: its row has the kind "metadata_only" and NA values. cacs_propagate_moe() stops with an error on such rows, and so does cacs_run().

A worked example with both weights

This example follows two variables through the 10-minute drive-time area of site AL_SITE_19: the number of people below the poverty level (B17001_002), a count, and per capita income (B19301_001), a per-person value. The call below sets keep_tract_audit = TRUE, so the result also lists the areas from which the weights are computed:

# The drive-time areas and ACS data described at the top of this article
iso <- readRDS(system.file(
  "extdata", "legacy_2025_isochrones.rds", package = "catchmentACS"
))
acs <- readRDS(system.file(
  "extdata", "sample_alabama_subset.rds", package = "catchmentACS"
))

site_id    <- "AL_SITE_19"
drive_time <- 10L

iso_one <- iso[
  iso$site_id == site_id & iso$drive_time_min == drive_time, ,
  drop = FALSE
]

weighted <- cacs_intersect_weight(
  iso_sf           = iso_one,
  acs_sf           = acs,
  weight_method    = "area",
  keep_tract_audit = TRUE,
  verbose          = FALSE
)

A coverage weight divides int_area_m2, the area of the part of a tract inside the drive-time area, by tract_area_m2, the area of the tract. An area share divides int_area_m2 by its total over the tracts. The code below computes both from the cacs_tract_audit attribute of the result:

tracts <- attr(weighted, "cacs_tract_audit") |>
  transmute(
    GEOID,
    int_area_m2,
    tract_area_m2,
    w_cov  = int_area_m2 / tract_area_m2,            # coverage weight
    w_mean = int_area_m2 / sum(int_area_m2)          # area share
  ) |>
  arrange(desc(int_area_m2))

tracts

#> # A tibble: 3 × 5
#>   GEOID       int_area_m2 tract_area_m2 w_cov w_mean
#>   <chr>             <dbl>         <dbl> <dbl>  <dbl>
#> 1 01027001201    9182624.      9182624. 1.000  0.788
#> 2 01029001201    1238064.      9182624. 0.135  0.106
#> 3 01025001201    1238026.      9182624. 0.135  0.106

The tract that contains the site lies wholly inside the drive-time area and has 79 percent of the overlap area. About 13 percent of each of the other two tracts lies inside.

The package’s rows for the two variables show the kind of each estimate and the weights used:

two_rows <- weighted |>
  filter(variable %in% c("B17001_002", "B19301_001")) |>
  select(variable, estimate, moe, weight_sum, estimand_family, weight_basis)

two_rows

#> # A tibble: 2 × 6
#>   variable   estimate    moe weight_sum estimand_family            weight_basis
#>   <chr>         <dbl>  <dbl>      <dbl> <chr>                      <chr>       
#> 1 B17001_002    1146.   87.3       1.27 spatial_total              coverage    
#> 2 B19301_001   32737. 2340.        1.27 area_weighted_scalar_proxy area_mean

The count uses the coverage weights. Its estimate, about 1,146 people, is the count of the tract that contains the site plus about 13 percent of the count of each of the other two tracts. On both rows, weight_sum is the sum of the coverage weights computed above, although per capita income is averaged with the area shares.

All three tracts have a row for per capita income in the ACS data, so the area shares computed above are the ones that the package uses for it. The code below applies both kinds of weights to the three tract values:

pc_income <- acs |>
  sf::st_drop_geometry() |>
  filter(GEOID %in% tracts$GEOID, variable == "B19301_001") |>
  select(GEOID, X = estimate)

demo <- tracts |>
  select(GEOID, w_cov, w_mean) |>
  left_join(pc_income, by = "GEOID")

demo

#> # A tibble: 3 × 4
#>   GEOID       w_cov w_mean     X
#>   <chr>       <dbl>  <dbl> <dbl>
#> 1 01027001201 1.000  0.788 34144
#> 2 01029001201 0.135  0.106 15508
#> 3 01025001201 0.135  0.106 39531

pc_sum <- sum(demo$w_cov  * demo$X)   # coverage weights, as for a count
pc_avg <- sum(demo$w_mean * demo$X)   # area shares, as the package does

c(coverage_weighted_sum = pc_sum, area_share_average = pc_avg)

#> coverage_weighted_sum    area_share_average 
#>              41564.57              32737.04

The average weighted by area shares, 32,737 dollars, is the estimate that the package reports for B19301_001 above. It lies between the smallest and the largest tract values, 15,508 and 39,531 dollars. vignette("theory-spatial-aggregation", package = "catchmentACS") shows with two tracts how far this average can be from the per capita income of the people in the drive-time area.

The same tract values weighted by coverage weights add up to 41,565 dollars, more than the per capita income of any of the three tracts. This sum is not a per capita income: it grows with the number of tracts in the drive-time area and with the part of each that lies inside.

Each of the two rows also has a margin of error, computed with the same weights as its estimate; the next section gives the formulas.

Margins of error

The margins of error of counts, medians, and per-person values come from cacs_intersect_weight() at the 90 percent level and from cacs_propagate_moe() at the chosen level, and those of the rates from cacs_derive_rates(). Each margin of error is computed from the published tract margins of error $M_j$ and the weights of the estimate, with the weights treated as fixed numbers. A count is a sum of weighted tract estimates, and its margin of error comes from the formula that the Census Bureau’s handbook for ACS data users gives for a sum (U.S. Census Bureau 2020, chap. 8):

$$M(\widehat{Y}_s) = z \sqrt{\sum_j \left(\frac{w^{cov}_{sj}\, M_j}{1.645}\right)^2}.$$

With the default level of 90 percent ($z = 1.645$), this reduces to $\sqrt{\sum_j (w^{cov}_{sj} M_j)^2}$. A median or a per-person value gets the same expression with the area shares $w^{mean}_{sj}$ in place of the coverage weights. For these values the margin of error is that of an area-weighted average of tract values, an approximation that the package makes and the handbook does not give.

For a rate $\widehat{R}_s = \widehat{A}_s / \widehat{B}_s$, the handbook distinguishes a proportion, whose numerator is part of its denominator, from other ratios and gives a formula for each. In terms of the margins of error of the two counts,

$$M(\widehat{R}_s) = \frac{\sqrt{M(\widehat{A}_s)^2 \mp \widehat{R}_s^2\, M(\widehat{B}_s)^2}}{\widehat{B}_s},$$

where the minus sign gives the proportion formula and the plus sign the ratio formula. The column moe_formula_effective records the formula used in each row: "weighted_sum" for counts, "weighted_mean" for medians and per-person values, and "proportion_subset" or "general_ratio_conservative" for rates.

All five rates are proportions: the numerator of each is part of its denominator. For a proportion, the handbook uses the ratio formula only in place of a proportion formula that fails (see below). Unless the argument formula_dispatch is changed, cacs_derive_rates() and cacs_run() nevertheless give all five rates the ratio formula, whose margin of error is never the narrower of the two. vignette("theory-derived-rates", package = "catchmentACS") lists the formula of every rate for each value of formula_dispatch and gives the reason recorded for unemp_rate.

The proportion formula fails when the value under its square root is negative. The package then computes that row with the ratio formula, as the handbook advises for this case, and records the substitution in the row. vignette("theory-moe-propagation", package = "catchmentACS") describes the columns that record it and how zero denominators and missing values are handled.

Limitations

Counts and rates rest on the assumption, stated with the notation above, that whatever a variable counts is spread evenly over each tract’s area. Comber and Zeng (2019, 8) describe area weighting as the best-known method for transferring counts between two sets of areas. They note that its assumption rarely holds, but that the method is reasonable when no other data on where people live are available. That method moves counts, and the averages this package forms for medians and per-person values are its own. They weight each tract by the area it shares with the drive-time area and ignore how many people live in it. They therefore match the drive-time area’s own per-person value only when the overlapping tracts hold the same number of residents per unit area. For a median, not even that is enough, because the median of a combined distribution is not an average of the tract medians. vignette("theory-spatial-aggregation", package = "catchmentACS") works through an example. The package has no population weighting yet: weight_method = "population" gives an error.

All the margins of error treat the estimates of different tracts as independent. The handbook’s approximation formulas leave out the covariance between estimates, so they can overstate or understate a margin of error (U.S. Census Bureau 2020, chap. 8). They also depart further from the standard error computed from the ACS microdata as more estimates are combined. If the tract estimates in a drive-time area are positively correlated, the reported margins of error of counts, medians, and per-person values are too small. The margins of error also leave out error from area weighting and uncertainty in the drive-time areas. catchmentACS does not use the Census Bureau’s variance replicate tables, which reflect this covariance (U.S. Census Bureau 2020, chap. 8) and performed best among the three approaches compared by Folch et al. (2023). vignette("theory-moe-propagation", package = "catchmentACS") discusses each assumption and what it means for rates.

The margins of error describe each site and drive time separately. Each drive-time area of a site contains its areas for shorter drive times, and the areas of nearby sites can overlap the same tracts, so their estimates are computed in part from the same tract estimates and are not independent. The package does not compute the covariance between them. A comparison that treats two such estimates as independent, such as the Census Bureau’s test for comparing two estimates (U.S. Census Bureau 2020, chap. 7), leaves this covariance out. For two counts, the shared tracts make the estimates positively correlated, so the test finds fewer differences than it should. For two rates, the sign of the correlation also depends on how the errors of the numerator and the denominator in each shared tract are related.

cacs_derive_rates() computes the five rates in cacs_acs_default_rates and gives an error for any other list of rates. A ratio of two other counts can be formed from the weighted counts and their margins of error in the result, with the formulas above, as vignette("theory-derived-rates", package = "catchmentACS") describes.

Data from outside a box drawn around the contiguous United States and the District of Columbia, such as data for Alaska, Hawaii, or Puerto Rico, stop cacs_intersect_weight() with an error. vignette("theory-spatial-aggregation", package = "catchmentACS") describes the projection on which the areas are measured. When the ACS data are the tracts of one state, as cacs_acs_prefetch() downloads them, the part of a drive-time area that extends into another state adds nothing to the estimates.

References