catchmentACS computes estimates for drive-time areas from American
Community Survey (ACS) estimates for census tracts. This article gives
the whole calculation in one place: the weights used for each kind of
quantity, the formulas for the margins of error, and what the
calculations assume. Three other articles derive what is stated here,
one for the weights
(vignette("theory-spatial-aggregation", package = "catchmentACS")),
one for the margins of error
(vignette("theory-moe-propagation", package = "catchmentACS")),
and one for the rates
(vignette("theory-derived-rates", package = "catchmentACS")).
The examples run without network access on data bundled with the package: drive-time areas that are circles with a radius of 1 km per minute of drive time, and random ACS values for small squares standing in for census tracts. The numbers in this article therefore show how the calculations work and describe no real place.
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:
-
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. -
cacs_isochrone()gets the drive-time areas around each site, one for each drive time, from a routing service. -
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. -
cacs_propagate_moe()recomputes the margins of error at the chosen confidence level and records the formula used for each row. -
cacs_derive_rates()computes the five rates listed incacs_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.
| Symbol | Meaning |
|---|---|
| I_s | the drive-time area of site s: the area reachable from the site within the drive time, which contains the areas for shorter drive times |
| T_j | census tract j |
| I_s \cap T_j | the part of tract j inside the drive-time area |
| \lvert\,\cdot\,\rvert | area, in square meters on EPSG:5070 |
| Y_j | the ACS estimate of a count for tract j, such as the number of people below the poverty level |
| X_j | the ACS estimate of a median or a per-person value for tract j, such as median household income or per capita income |
| A_j, B_j | the ACS estimates of the numerator and the denominator of a rate for tract j |
| M_j | the margin of error published with a tract estimate, at the 90 percent confidence level |
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”):
estimand_family |
Kind of estimate | Estimate | Weights (weight_basis) |
|---|---|---|---|
"spatial_total" |
a count, such as the number of people below the poverty level
(B17001_002) |
\widehat{Y}_s | coverage weights w^{cov}_{sj}
("coverage") |
"median_proxy" |
a median, such as median household income
(B19013_001) |
\widehat{\bar X}_s | area shares w^{mean}_{sj}
("area_mean") |
"area_weighted_scalar_proxy" |
a per-person value: per capita income (B19301_001) |
\widehat{\bar X}_s | area shares w^{mean}_{sj}
("area_mean") |
"derived_rate" |
one of the five rates that cacs_derive_rates() adds,
such as the poverty rate |
\widehat{R}_s | coverage weights, for the numerator and the denominator
("coverage") |
"metadata_only" |
an ACS code that is not combined (see below) | none: estimate and moe are
NA
|
none ("none") |
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. 1.00 0.788
#> 2 01029001201 1238064. 9182624. 0.135 0.106
#> 3 01025001201 1238026. 9182624. 0.135 0.106The 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_meanThe 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$GEOID, variable == "B19301_001") |>
select(GEOID, X = estimate)
demo <- tracts |>
select(GEOID, w_cov, w_mean) |>
left_join(pc_income, by = "GEOID")
demo#> # A tibble: 3 × 4
#> GEOID w_cov w_mean X
#> <chr> <dbl> <dbl> <dbl>
#> 1 01027001201 1.00 0.788 34144
#> 2 01029001201 0.135 0.106 15508
#> 3 01025001201 0.135 0.106 39531
pc_sum <- sum(demo$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 32737.04The 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.