Provider Retention in High Need Areas. Cohort Rates with Two Locations

12/22/2014

Suppose that a cohort of individuals enters the labor market after schooling and we wish to know what fractions of this cohort will make different choices over time.  Again assume there are two locations and two time periods.  The individuals in this cohort can choose location 1 in both periods, location 2 in both periods, location 1 in the first period and location 2 in the second period, and location 2 in the first period and location 2 in the second period.  Define six expected fractions as follows:

  • 1= fraction choosing location 1 in period 1 =
  • 2 = fraction choosing location 2 in period 1 =
  • 3 = fraction choosing location 1 in period 1 and in period 2 =
  • 4 = fraction choosing location 2 in period 1 and in period 2 =
  • 5 = fraction choosing location 1 in period 1 and location 2 in period 2 =
  • 6 = fraction choosing location 2 in period 1 and location 1 in period 2 =

 Conceptually, these expected fractions are constructed as weighted averages of the individual probabilities of choosing a given location in each time period, as given by equation (4).  In the case of two locations, the weights are based on the distribution of in the cohort.  To obtain actual expected fractions, it is necessary to make an assumption about the distribution of  in the population.  In simulations presented below, we assume that location preferences are independently normally distributed with means  and , respectively, and common standard deviation .30  With these assumptions, the distribution of is normal with mean  -  and standard deviation equal to

The retention rate in a location is the fraction of individuals who chose to locate there previously who choose to remain there.  Based on the above discussion, the period 2 retention rate in location 1 is given by   and the period 2 retention rate in location 2 is given by .  It is important to recognize that retention rates are conditional on prior choices and will depend on the composition of the cohort making the prior choices.


30 With these assumptions, the distribution of is normal with mean  -  and standard deviation

 

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