There is a sample of size 𝑛=100 of counties in U.S. where for each county a total population and amount of veterans are given. The sample is drawn using simple random sampling. What would be the best estimation method to use, in case I'd like to estimate the total amount of veterans in U.S. As I understand the ratio estimation is used when there is a correlation between the estimated variable and auxiliary variable. And regression estimation is used, when there is a linear relationship between variables, but there isn't proportionality. In this case, I suppose there should be both linear correlation between total population in a county and the amount of veterans and proportionality. Ratio estimation in this instance seems like a valid method to use, but I am not sure. Because I don't have a clear idea of what proportionality means in this instance. The total amount of all population is given as well. Can someone help? I tried to estimate the total amount of veterans using formula: $\frac{y_T}{x_T} \cdot X_T$, where $y_T$ is total amount of veterans from all the sample counties and $x_T$ - the total sample population. $X_T$ is total population in U.S. at the time. I also computed the regular total population estimator: $N\bar{y}$ and it was much larger than the ratio estimator. In what instances the ratio estimator is more precise than the regular one? Also, how to get standard error for the ratio estimate?
1 Answer
I don't have a clear idea of what proportionality means in this instance.
If you are considering the Total county population to be an auxiliary variable for the county Veteran population, then you can examine proportionality as in these course notes from Penn State. Start with a plot of the Veteran numbers against the Population numbers for each County in your sample. Do a linear regression of Veteran population against Total population. If the intercept is substantially different from 0 then you do not have strict proportionality.
But also think about whether non-proportionality makes sense in your particular case. In the example in the web page linked above, the comparison was between two different types of test scores on different types of scales. A non-zero intercept is quite possible in that case. But a non-zero intercept for Veteran versus Total population would mean a non-zero number of Veterans at a Total population of 0. Would that make sense? Probably not, suggesting that a ratio estimate is more appropriate here.
The 3142 counties (or equivalents) in the US show a lot of heterogeneity both in Total and in Veteran population. The Veterans Administration has estimated Veteran population by county for 2018 (with predictions going out for a couple of decades) and the Census Bureau has estimated Total population by county for 2010-2019 from which you can extract the 2018 estimates.*
If you play with those data, you will find populations ranging from Kalawao County, Hawaii (5 Veterans among 86 Total) to Los Angeles County, California (294,456 Veterans among 10,073,906 Total). The percentage of the Total population that is Veterans ranges from 1.3% in Starr County, Texas to 35.4% in Geary County, Kansas. Now think about what you have presumably learned about sampling strategies, for example see Section 3.1 of the Penn State course:
In simple random sampling, the probability that each unit will be sampled is the same. Sometimes, estimates can be improved by varying the probabilities with which units are sampled.
For example, we want to estimate the number of job openings in a city by sampling firms in that city. Many of the firms in the city are small firms. If one uses [simple random sampling] size of a firm is not taken into consideration and a typical sample will consist of mostly small firms. However, the number of job openings is heavily influenced by large firms.
With the major differences in populations among counties, simple random sampling of them similarly is unlikely to be a good way to estimate nationwide Veteran population numbers. That's one reason why comments focused so much on the nature of the sampling of counties.
As a teaching exercise, though, these data are rich in lessons about sampling. Your observations agree with what I could glean from the data sets: pooling the Veteran numbers and Total population numbers among counties to get the Veteran/Total ratio gives a much lower ratio than you get by averaging the individual-county ratios. Given the data you have, the first method comes closest to an average among individuals in the population, which is what you want. The second (what you call the "regular" estimator) is an average among counties, not among individuals, that greatly overweights small counties. As a hint to what method is working better in this case, the whole-US 2018 values from the above data were 20,163,806 Veterans among 326,691,703 Total population.
Variances and Confidence intervals around the estimates are based on standard procedures, with potential correction for finite populations; see for example Section 2.2 of the Penn State course notes.
Added after reflection
Although a regression estimate doesn't "make sense" logically in that it predicts "a non-zero number of Veterans at a Total population of 0," it might help counteract a bias in the ratio estimate that seems to arise in this case from a tendency of smaller-population counties to have a higher percentage of Veterans. Ratio estimates from sampling by county tend to over-estimate the percentage of Veterans in the population, as the per-county samples are weighted toward smaller counties.
*These values aren't strictly enumerations. The only full population enumeration in the US is the Census every 10 years. Only a subset of all cases are asked detailed questions about things like veteran status during that Census, with other surveys done during intermediate years. So the data you are working with are probably themselves estimates based on carefully designed surveys.
self-studytag to this question. $\endgroup$self-studytag. In your formula for the "regular total population estimator" you have a mean value $\bar y$. Is that the average number of veterans per county, or an average over something else? $\endgroup$