Effect of PACE on Costs, Nursing Home Admissions, and Mortality: 2006-2011. APPENDIX B. TESTING THE SIGNIFICANCE OF THE EFFECT OF PACE ON EXPENDITURES

03/01/2015

We denote the mean Medicare expenditure over a six-month period for a sample of n PACE enrollees by Ē and the estimated sample variance of E over this same period by g2. We wish to compare this mean with the mean predicted value for the same PACE enrollees had they been in the matched comparison group (HCBS enrollee or NH entrant) instead. That mean predicted value is generated using coefficients from the regression of Medicare expenditure on beneficiary characteristics for a sample of beneficiaries in the matched comparison group. We denote the mean of the predicted values as y caret/overbar, which emphasizes that this mean is generated via regression coefficients. We can denote its sample variance as h2. But the formula for h2 is more complicated than that of g2 because there are two sources of error in y caret/overbar: natural sampling variation and error in estimating the regression coefficients used to compute it. Given a formula for h2, then we can form the test statistic:23

[1]   t = E overbar minus y caret/overbar divided by (n superscript -1 g superscript 2 + h superscript 2)superscript 1/2.
Skipping the derivation, we present the formula for h, which leads to the t-statistic above.
  h superscript 2 = EstVar(y caret/overbar) = s superscript 2 [x singlequote subscript 0 (X singlequote X)superscript -1 x subscript i0 = 1/n]
where the x overbar singlequote subscript 0 are the mean values of the covariates for the PACE sample.
  h superscript 2 = s superscript 2 divided by n [1 divided by n summation over i x singlequote subscript i0 (X singlequote X)superscript -1 x subscript i0 = 1]
The first term in square brackets is an addition to the usual variance estimate. It tends to inflate the t-statistic in [1].  

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