# how to calculate the MOE of a ratio of derived proportions

Location quotients are used to quickly explain the concentration of something in a small area compared with the whole.

I'd like to use these statistics to compare the concentration of various immigrant groups in a metro with the country overall. The basic quotient is simple to calculate, but I'm not sure I'm calculating the MoE correctly.

Here's a simple example. I calculate the prop_moe following the formula described here, under "Calculating Measures of Error for User-Derived Proportions and Percentages."

 subgroup subgroup_moe pop pop_moe prop prop_moe local 3449 727 1571784 0 0.00219 0.00046 nation 155955 5145 329725481 0 0.00047 0.00002

The location quotient is the ratio of the local proportion to the national proportion, or 0.00219 / 0.00047 = 4.659574.

I calculated the MoE for this ratio using the formula under "Calculating Measures of Error for Derived Ratios" (from the same ACS handbook as linked above).

The result is 0.9898, which passes the smell test.

In the ACS handbook example of this formula, X and Y are both counts, not proportions.

Is it still correct to use this formula to calculate the MoE of a ratio of proportions, as I have done above?

• The formula is approximate and it applies to arbitrary random variables. However when Y is near zero there can be problems.  The formula is based on the delta method  https://en.wikipedia.org/wiki/Delta_method  When applied to counts in small samples there are also issues when x and or y are based on small counts.  But this issue applies generally to sampling statistics.  The formula also assumes that x and y are uncorrelated which is not a good approximation when one area is a subarea of the other.   If you work with only PUMS data for both the large and small areas then you can take care of this problem using replicate weights which are part of the PUMS datasets. There are several YouTube videos and some slide decks that explain how to use replicate weights. You will need a statistical analysis system/program to make the calculations.  I use open source R.

Dave