MOEs for derived averages

I need some statistical advice. Paging !

When calculating MOEs for derived estimates -- specifically, averages -- should I use the formula for ratio, or proportion?

The two formulas shown in the ACS handbook chapter 8 (pages 64 and 65) are nearly identical. except that the proportion formula uses a minus operator under the radix whereas the ratio formula uses a plus.

I'm finding that using the proportion formula sometimes results in an error from trying to take the square root of a negative number.

Here's an example from the ACS 2022 1-year data for the nation:

B25065_E001 (aggregate gross rent) = $63,086,890,700

B25065_M001 (MOE for above) = ±$234,476,359

B25063_E002 (count of cash renters) = 42,971,061

B25063_M002 (MOE for above) = ±162,515

I'm trying to derive average gross rent as (aggregate gross rent / count of cash renters), or about $1,468.13 for the USA. Seems about right. But plugging the numbers into the proportion formula leads to madness:

MOE(P-hat) = sqrt(B25065_M0012 - ((B25065_E001 / B25063_E002)2 * B25063_M0022)) / B25063_E002

MOE(P-hat) = sqrt(5.49e+16 - (2,155,391 * 1.85e+15)) / 42,971,061

MOE(P-hat) = sqrt(-3.98e+21) / 42,971,061 Confounded

I feel like I'm missing something. Is it because the source estimates and MOEs are counting different things? Is there a different formula for calculating MOEs for derived averages?

Thanks for any guidance.

Parents
  • To possibly answer my own question and ask a new one: Could I use formula (9) to do this?

    Calculating Measures of Error for the Product of Two Estimates

    Since dividing x / y is the same as the product x * 1/y, could this work?

    EDIT: trying this out....

    MOE(X-hat * 1/Y-hat) = sqrt((X-hat**2 * (1/MOE[Y-hat])**2) + ((1/Y-hat)**2 * MOE[X-hat]**2))

    = sqrt((B25065_E001**2 * (1/B25063_M002)**2) + ((1/B25063_E002)**2 * B25065_M001**2))

    = sqrt((63086890700**2 * (1/162515)**2) + ((1/42971061)**2 * 234476359**2))

    = sqrt(2.3886483 +  29.7746023)

    = $5.67

    ...which seems low, but MOEs should be pretty low for the nation as a whole. 

    Better, worse, or just completely wrong? Thanks

  • My comments refer to this document

    https://www.census.gov/content/dam/Census/library/publications/2020/acs/acs_general_handbook_2020_ch08.pdf

    The 2 formulas are (6) and (7) in this document.

    Formula (6) has a - sign under the square root.

    The caveat for that formula is that

    "Users should note that if the value under the square root is negative, then substitute a “plus” for the “minus” sign
    under the square root in formula (6). This modified formula is the same as the formula for the MOE of a ratio,
    which will be discussed in the next section."  This occurs when either the proportion/ratio  is large or the MoE or the denominator is large when compared to the numerator MoE

    Formula (7) has a  + sign.

    The MoE given by (6) will be lower than the MoE given by (7).

Reply
  • My comments refer to this document

    https://www.census.gov/content/dam/Census/library/publications/2020/acs/acs_general_handbook_2020_ch08.pdf

    The 2 formulas are (6) and (7) in this document.

    Formula (6) has a - sign under the square root.

    The caveat for that formula is that

    "Users should note that if the value under the square root is negative, then substitute a “plus” for the “minus” sign
    under the square root in formula (6). This modified formula is the same as the formula for the MOE of a ratio,
    which will be discussed in the next section."  This occurs when either the proportion/ratio  is large or the MoE or the denominator is large when compared to the numerator MoE

    Formula (7) has a  + sign.

    The MoE given by (6) will be lower than the MoE given by (7).

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