How to approximate the Standard Error of an estimate over a collection
of tracts

This document provides a detailed and fairly complex set of instructions
for how to compute the standard error for an estimate made by summing
over the published estimates for multiple geographic areas (for example,
tracts). Words in all capital letters represent variables on the
ACS-Census 2000 downloadable datasets.

# Notation

Let P~A~ and P~C~ be the ACS and Census proportions respectively for the
combined grouping of tracts. Let X(i) denote the value of X for tract i.

Num~A~(i) = AVGEST

Den~A~(i) = ACS_UNI

Num~C~(i) = EST_C2K

Den~C~(i) = C2K_UNI

$P_{X} = \frac{\text{Num}_{X}(1) + \ldots + \text{Num}_{X}(n)}{\text{Den}_{X}(1) + \ldots + \text{Den}_{X}(n)} = \frac{\text{Num}_{X}}{\text{Den}_{X}}$
(where X is either A or C)

Goal: We want to find the standard error of the change in the aggregated
grouping of tracts. There will be separate instructions for lines that
are initially counts, totals, and ratios.

# Approximating the Standard Error - Counts

Most of the 364 lines in the profiles are counts which are converted to
proportions before estimating the change. \[These are the lines with the
variable METHOD equal to 'P'.\]

Obtaining SE(P~A~ - P~C~) is essentially a four step process. Separate
methods, with their own special cases, are used to get SE(P~A~) and
SE(P~C~). Then, their values are checked against a maximum value, and
finally SE(P~A~ - P~C~) is computed.

**Note**: ACS_UNI and ACS_UNI_SE are not correctly rounded on your
files. These should be rounded to the nearest unit before proceeding.

1.  First, to calculate SE(P~C~),

> $\text{SE}(P_{C}) = \text{DEFF}\text{00} \times \sqrt{\frac{5}{\text{Den}_{C}} \times P_{C} \times (1 - P_{C})}$
> (1)
>
> If P~C~ \< 0.02 then use P~C~ = 0.02, and if P~C~ \> 0.98 then use
> P~C~ = 0.98. Use these capped values only for SE(P~C~), not when
> calculating P~A~ - P~C~.

DEFF00 is the Census 2000 long form design factor for the appropriate
characteristic. If the tracts you are combining have different values of
DEFF00, then use the largest value of DEFF00 in your calculations.

2.  Second, to calculate SE(P~A~), first calculate

> $\text{SE}(\text{Num}_{A}) = \sqrt{\sum_{i = 1}^{n}\left( \text{SE}\left( \text{Num}_{A}(i) \right) \right)^{2}} = \sqrt{\sum_{i = 1}^{n}\left( \text{AVGSE}(i) \right)^{2}}$
> (2)
>
> $\text{SE}(\text{Den}_{A}) = \sqrt{\sum_{i = 1}^{n}\left( \text{SE}\left( \text{Den}_{A}(i) \right) \right)^{2}} = \sqrt{\sum_{i = 1}^{n}\left( \text{ACS}_{\text{UNI}_{\text{SE}}}(i) \right)^{2}}$
> (3)

Now,

> $\text{SE}(P_{A}) = \frac{1}{\text{Den}_{A}}\sqrt{\left( \text{SE}(\text{Num}_{A}) \right)^{2} - \left( P_{A} \right)^{2} \times \left( \text{SE}(\text{Den}_{A}) \right)^{2}}$
> (4)
>
> There are two special cases. If P~A~ = 1, then use P~A~ = 0 instead.
> If the value under the square root sign is negative, then use
>
> $\text{SE}(P_{A}) = \text{DEFF}\text{00} \times \sqrt{\frac{\text{ACSAVWGT} - 1}{\text{Den}_{A}} \times P_{A} \times (1 - P_{A})}$
> (5)
>
> In this special case *only*, if P~A~ \< 0.02 then use P~A~ = 0.02, and
> if P~A~ \> 0.98 then use P~A~ = 0.98. Use these capped values only for
> SE(P~A~), not when calculating P~A~ - P~C~.

ACSAVWGT is the average of the three single-year county-level average
weights.

3.  Third, before we calculate the standard error of the change, if
    either SE(P~A~) or SE(P~C~) is greater than 0.7, then set that SE to
    0.7.

4.  And finally, fourth,

> $\text{SE}(P_{A} - P_{C})\overset{˙}{=}\sqrt{\left( \text{SE}(P_{A}) \right)^{2} + \left( \text{SE}(P_{C}) \right)^{2}}$
> (6)
>
> SE(P~A~ - P~C~) may be larger than 0.7; it is not capped at that
> value.

# Example - Counts

Let's say we want to aggregate tracts 000200, 000400 and 001100 in Bronx
County, NY (ST=36 & CTY=005), and look at table 1 line 26, the number of
Blacks or African Americans.

  -------- --------- --------- --------- --------- -------- -------- ------------
  TR       AVGEST    ACS_UNI   EST_C2K   C2K_UNI   DEFF00   AVGSE    ACS_UNI_SE

  000200   620.67    3029.33   970       3590      2.2      215.10   473.98

  000400   757.33    3515.67   900       3250      2.2      243.49   536.10

  001100   50.00     104.33    80        400       1.8      66.28    73.56

  Total    1428.00   6649.33   1950      7240                        
  -------- --------- --------- --------- --------- -------- -------- ------------

$$\begin{array}{r}
P_{A} = \frac{\text{1428}\text{.}\text{00}}{\text{6649}\text{.}\text{33}} = 0\text{.}\text{2148} \\
P_{C} = \frac{\text{1950}}{\text{7240}} = 0\text{.}\text{2693} \\
P_{A} - P_{C} = 0\text{.}\text{2148} - 0\text{.}\text{2693} = - 0\text{.}\text{0545}
\end{array}$$

The three tracts have different DEFF00 values, so we'll use the largest
one, 2.2.

$$\text{SE}(P_{C}) = 2\text{.}2 \times \sqrt{\frac{5}{\text{7240}} \times 0\text{.}\text{2693} \times (1 - 0\text{.}\text{2693})} = 0\text{.}\text{0256}$$

$$\begin{array}{r}
\text{SE}(\text{Num}_{A}) = \sqrt{\text{215}\text{.}\text{10}^{2} + \text{243}\text{.}\text{49}^{2} + \text{66}\text{.}\text{28}^{2}} = \text{331}\text{.}\text{58} \\
\text{SE}(\text{Den}_{A}) = \sqrt{\text{473}\text{.}\text{98}^{2} + \text{536}\text{.}\text{10}^{2} + \text{73}\text{.}\text{56}^{2}} = \text{719}\text{.}\text{35}
\end{array}$$

$$\text{SE}(P_{A}) = \frac{1}{\text{6649}\text{.}\text{33}}\sqrt{\left( \text{331}\text{.}\text{58} \right)^{2} - \left( 0\text{.}\text{2148} \right)^{2} \times \left( \text{719}\text{.}\text{35} \right)^{2}} = 0\text{.}\text{0441}$$

$$\text{SE}(P_{A} - P_{C}) = \sqrt{0\text{.}\text{0441}^{2} + 0\text{.}\text{0256}^{2}} = 0\text{.}\text{0510}$$

# Approximating the Standard Error - Totals

Some characteristics, such as the total number of persons or housing
units, are not converted to proportions to compute the change. \[Lines
with METHOD = 'T'.\]

Simply,

$$\text{Est}_{A} = \sum_{i = 1}^{n}\text{AVGEST}_{i}$$

$$\text{Est}_{C} = \sum_{i = 1}^{n}\text{EST}_{}$$

$$\text{SE}(\text{Est}_{A}) = \sqrt{\sum_{i = 1}^{n}\text{AVGSE}_{i}^{2}}$$

$$\text{SE}(\text{Est}_{C}) = \sqrt{\sum_{i = 1}^{n}{\text{NSE}\text{00}_{}^{2}}}$$

> $$\text{SE}(\text{Est}_{A} - \text{Est}_{C})\overset{˙}{=}\sqrt{\left( \text{SE}(\text{Est}_{A}) \right)^{2} + \left( \text{SE}(\text{Est}_{C}) \right)^{2}}$$

# Approximating the Standard Error -- Ratios

Unfortunately, standard errors for most ratio lines (including means and
percents) can't be calculated for aggregations of tracts given the
information in the output files. However, standard errors for six of the
ratio lines can be computed, using several different methods.

Group 1: lines 1-73 (average household size) and 3-6 (percent
unemployed).

The standard errors of these ratios can be computed because both the
numerator and denominator are separate lines in the profile. For line
1-73, the numerator is line 1-1 and the denominator is line 1-63. For
line 3-6, the numerator is line 3-5 and the denominator is line 3-3.

Num~A~(i) = AVGEST from the numerator line

Den~A~(i) = AVGEST from the denominator line

Num~C~(i) = EST_C2K from the numerator line

Den~C~(i) = EST_C2K from the denominator line

$$\text{SE}(\text{Num}_{A}) = \sqrt{\sum_{i = 1}^{n}\left( \text{AVGSE}_{\text{num}\ \text{line}}(i) \right)^{2}}$$

$$\text{SE}(\text{Den}_{A}) = \sqrt{\sum_{i = 1}^{n}\left( \text{AVGSE}_{\text{den}\ \text{line}}(i) \right)^{2}}$$

$$\text{SE}(P_{A}) = \frac{1}{\text{Den}_{A}}\sqrt{\left( \text{SE}(\text{Num}_{A}) \right)^{2} + \left( P_{A} \right)^{2} \times \left( \text{SE}(\text{Den}_{A}) \right)^{2}}$$

Note the use of "+" instead of "-" in the formula for SE(P~A~). If
P~A~=1, then use P~A~=0 when computing SE(P~A~).

For line 1-73:

$$\text{SE}(\text{Num}_{C}) = \sqrt{\sum_{i = 1}^{n}\left( \text{NSE}\text{00}_{}(i) \right)^{2}}$$

$$\text{SE}(\text{Den}_{C}) = \sqrt{\sum_{i = 1}^{n}\left( \text{NSE}\text{00}_{}(i) \right)^{2}}$$

$$\text{SE}(P_{C}) = \frac{1}{\text{Den}_{C}}\sqrt{\left( \text{SE}(\text{Num}_{C}) \right)^{2} + \left( P_{C} \right)^{2} \times \left( \text{SE}(\text{Den}_{C}) \right)^{2}}$$

For line 3-6:

$\text{SE}(P_{C}) = \text{DEFF}\text{00} \times \sqrt{\frac{5}{\text{Den}_{C}} \times P_{C} \times (1 - P_{C})}$

If P~C~ \< 0.02 then use P~C~ = 0.02, and if P~C~ \> 0.98 then use P~C~
= 0.98. If DEFF00 differs among the tracts being combined, use the
largest value of DEFF00.

$$\text{SE}(P_{A} - P_{C})\overset{˙}{=}\sqrt{\left( \text{SE}(P_{A}) \right)^{2} + \left( \text{SE}(P_{C}) \right)^{2}}$$

Do *not* cap SE(PA) or SE(PC) at 0.7.

Group 2: lines 2-15 (percent high school graduate or higher), 2-16
(percent bachelor's degree or higher), 2-33 (percent employed, 21-64,
with a disability), and 2-35 (percent employed, 21-64, with no
disability).

For these four percents, the denominator is a separate line in the
profile: 2-7, 2-7, 2-32 and 2-34, respectively.

Den~A~(i) = AVGEST from the denominator line

SE(Den~A~(i)) = AVGSE from the denominator line

P~A~(i) = AVGEST/100 from the percent line

Num~A~(i) = Den~A~(i) × P~A~(i)

SE(P~A~(i)) = AVGSE/100 from the percent line

Den~C~(i) = EST_C2K from the denominator line

P~C~(i) = EST_C2K/100 from the percent line

Num~C~(i) = Den~C~(i) × P~C~(i)

$$\text{SE}(\text{Num}_{A}(i)) = \sqrt{\left( \text{Den}_{A}(i) \times \text{SE}(P_{A}(i)) \right)^{2} - \left( P_{A}(i) \times \text{SE}(\text{Den}_{A}(i)) \right)^{2}}$$

If Den~A~(i) = 0, then set SE(Num~A~(i)) = SE(Den~A~(i)). We use a
special approximation for the standard error when an estimate is zero.
If the denominator is zero, the numerator is zero also, and their
standard errors are the same. The above equation will not work in this
case.

There is a chance that the value under the square root sign in the
equation above will be negative. This could be due to rounding (it would
not be negative if unrounded values were used) or other reasons. If that
is the case for a tract, then SE(P~A~ -- P~C~) can't be computed for the
grouping that contains that tract.

$$\text{SE}(\text{Num}_{A}) = \sqrt{\sum_{i = 1}^{n}\left( \text{SE}(\text{Num}_{A}(i) \right)^{2}}$$

$$\text{SE}(\text{Den}_{A}) = \sqrt{\sum_{i = 1}^{n}\left( \text{AVGSE}_{\text{den}\ \text{line}}(i) \right)^{2}}$$

$$\text{SE}(P_{A}) = \frac{1}{\text{Den}_{A}}\sqrt{\left( \text{SE}(\text{Num}_{A}) \right)^{2} + \left( P_{A} \right)^{2} \times \left( \text{SE}(\text{Den}_{A}) \right)^{2}}$$

For the census estimate,

$\text{SE}(P_{C}) = \text{DEFF}\text{00} \times \sqrt{\frac{5}{\text{Den}_{C}} \times P_{C} \times (1 - P_{C})}$

If P~C~ \< 0.02 then use P~C~ = 0.02, and if P~C~ \> 0.98 then use P~C~
= 0.98. If DEFF00 differs among the tracts being combined, use the
largest value of DEFF00.

$$\text{SE}(P_{A} - P_{C})\overset{˙}{=}\sqrt{\left( \text{SE}(P_{A}) \right)^{2} + \left( \text{SE}(P_{C}) \right)^{2}}$$

Do *not* cap SE(P~A~) or SE(P~C~) at 0.7. You may multiply P~A~ -- P~C~
and SE(P~A~ -- P~C~) by 100% to convert them to percent estimates.

# Approximating the Standard Error -- Medians

Neither estimates nor standard errors of aggregations of tracts for
median lines can be calculated with the data available.