# Imputation for small cells?

I have a question about the data requirements for imputation. Specifically, is there a rule of thumb about what proportion of the data have to be non-missing for the imputation to be "valid?" I am working with some data where values are missing for upwards of 50% of the observations in the dataset. The problem is discussed in greater detail below.

I want to calculate the price elasticity of demand for facility/provider (henceforth, "facility") of a curative care procedure, $X$, where $X$ can take on the following values:

2. Village health post (lower-level health worker, e.g., medical officers, medical technicians)
3. District health center (nurses/GP)
4. Teaching hospital (Specialists)

In order to calculate elasticity of demand, I have linked the prices listed by each provider type for each individual decision maker, as the survey I am using has data collected at two levels:

1. the individual decision maker ($i$), collected using a household survey, and
2. the facility ($s$), collected using a health facility survey

$i$ and $s$ can be linked via a geographic identifier $g$ equivalent to a county in the U.S.

The individual-level dataset has information about the provider chosen by $i$ along with other demographic variables, such as age, education, and income. The facility-level dataset has information about the facility, including whether the procedure ($X$) is performed at the facility, the price of $X$, and staffing capacity. The problem I am facing is that for each of the four types of facilities, I have missing information on service price data. The problem is exacerbated by the fact that price of service varies by "healthcare market" (here, the market is defined as the county). The number of the different types of facilities also varies by market--in some instances, the facility in the market that is missing a value on price is the only facility in the market surveyed, so that the source of data for imputation is limited. Because I am interested in calculating demand elasticity, I need to have an actual value for price.

I will use just one type of provider to illustrate the extent of the problem for the sake of simplicity.

Of 351 facilities of type 2 (village health post), 70 (20%) are missing price information. Table 1, run on the facility dataset, shows the first 20 rows of a crosstabulation between market ($COUNTY$) and whether price was missing for village health posts in the county that were surveyed. Of 136 counties, 48 had at least one provider of type A who did not report service price.

Given the problem, what is the best way to handle missing values? Is multiple imputation suitable if for some of counties, the provider missing information is the only provider of that type in the county? That is, if the only potential source of price information in that county is missing price.

I had started reading Little and Rubin's (2002) Statistical Analysis with Missing Data, but I think I am missing an even more fundamental issue of whether my data are suited for this kind of imputation.

 Table 1. distribution of missingness of price information for Service X by Geographic Region sorted by percent missing Price is missing No Yes TOTAL COUNTY n (%) n (%) 12 9 0 0% 1 100% 1 12 14 0 0% 1 100% 1 32 11 0 0% 1 100% 1 32 12 0 0% 2 100% 2 32 75 0 0% 1 100% 1 33 16 0 0% 1 100% 1 33 76 0 0% 1 100% 1 33 8 0 0% 1 100% 1 35 17 0 0% 3 100% 3 35 19 0 0% 1 100% 1 35 6 0 0% 2 100% 2 35 72 0 0% 1 100% 1 52 2 0 0% 2 100% 2 63 1 0 0% 1 100% 1 63 4 0 0% 1 100% 1 63 71 0 0% 1 100% 1 63 72 0 0% 1 100% 1 63 9 0 0% 1 100% 1 73 2 0 0% 1 100% 1 52 5 1 20% 4 80% 5 .. .. .. .. .. Total 281 80% 70 20% 351