I need some statistical advice. Paging Stas Kolenikov !
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
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.
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…
Glenn Rice the average is the ratio: you have the total (across the U.S.) of cash payments in the numerator, and you have the total (# of households that are renters) in the denominator. So the calculations…
Just another note formula (6) with the - sign applies to the case where the data is counts and the numerator is a subset of the denominator. Formula (7) applies to a ratio of any 2 things. The 2 "things…
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
Responding to my own second question: JUST COMPLETELY WRONG
Just another note formula (6) with the - sign applies to the case where the data is counts and the numerator is a subset of the denominator. Formula (7) applies to a ratio of any 2 things. The 2 "things" can even come from different tables, as they do in your case.
If you use R, I have some code that computes ratios, products and linear combinations (sums of variables with a fixed coefficient for each term ) taking into account the MoEs of the terms. The ratio code uses a + sign under the square root. I use formula 7 which is conservative (larger MoE) when compared to formula 6. All these formulas are approximate and are based on the variance (or standard deviation which is the square root of the variance) and the rest comes form the "delta method." https://en.wikipedia.org/wiki/Delta_method. The delta method (multivariate version) for x/y depends on the derivative with respect to x == (1/y) and the derivative with respect to y == -x/(y * y). With these facts about the derivatives you can kind of see where the formulas in chapter 8 come from.