
Method of Calculating the Predicted Proportion of Officebased Physicians Accepting New Medicaid Patience as a Function of the MedicaidtoMedicare Fee Index

Consider that:

I0 is the observed value of the MedicaidtoMedicare fee index 2008.

I1 is the new value of the MedicaidtoMedicare fee index (e.g. I1 could be 1 percent or 10 percent or x percent higher than I0 where x is a chosen value).

P0 is the observed proportion of officebased physicians accepting Medicaid patients in 2011.

P1 is the predicted proportion of officebased physicians accepting Medicaid patients after the x percent increase in MedicaidtoMedicare fee index.

M1 is the estimated marginal effect of the MedicaidtoMedicare fee ratio (2008) = 0.4 (reported in the study).
B1 is the estimated logit coefficient of the MedicaidtoMedicare fee ratio (2008).
Given the fact that marginal is equal to the estimated logit coefficient times P times (1P) where P is the mean proportion of officebased physicians accepting Medicaid patients in the sample, equation (12) shows the estimated logit coefficient of the MedicaidtoMedicare fee ratio (2008).
(12)
Since Decker (2012) reports that P is 0.694, this implies B1 is equal to 0.4/(0.694*(10.694)) which is approximately equal to 1.88.
Let’s name the part of the estimated logit model which incorporates the effect of other independent variables (other than MedicaidtoMedicare fee ratio) and the intercept term in the model as “Other Effects” or OE. Equation (13) below shows the expression of “Other Effects” (OE) based on a given value of P0 and a known value of B1 (where “ln” is the natural log).
(13)
Then based on the estimated logit model, we can predict the logodds of the proportion of officebased physicians accepting new Medicaid patients for a new value of the MedicaidtoMedicare index (say, I1). Equation (14) shows the logodds of the proportion of officebased physicians accepting new Medicaid patients.
(14)
Therefore, equation (15) shows the implied predicted proportion of officebased physicians accepting new Medicaid patients (where “exp” is the exponential function operation).
(15)



Example:

Based on Decker (2012):

The national average of observed proportion of officebased physicians accepting Medicaid patients in 2011 (P0) is 0.694.

The Observed value of the MedicaidtoMedicare fee index (I0) 2008 is 0.742.
We have already derived that the logit coefficient of the MedicaidtoMedicare fee ratio (B1) is 1.88. This implies that OE is approximately 0.576 (based on equation (13)).
Now consider a 10 percent increase in the MedicaidtoMedicare fee index from the national average in 2008. This implies that the new value of the MedicaidtoMedicare fee index (I1) is 0.8162.
This further implies that the predicted logodds of the proportion of officebased physicians accepting new Medicaid patients is equal to about 0.958 (using equation (14)).
Thus, using the expression in equation (15) the predicted proportion of officebased physicians accepting new Medicaid patients (P1) is equal to 0.723 or 72.3 percent.
The same methodology is applied while calculating the predicted proportions for each state and the US in Exhibit 25.


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