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United States Marshals Service's Prisoner Medical Care

Report No. 04-14
February 2004
Office of the Inspector General

Appendix XI
Statistical Sampling Model

The statistical sampling universe for our model was defined as vouchers submitted during FY 2002 for United States Marshal Service (USMS) prisoner medical services from 14 of the 94 USMS districts. The 14 districts selected accounted for 91.3 percent of the total dollar value of vouchers submitted. Our sample test results were projected only to the prisoner medical service activities in the 14 districts tested.

We used a random sampling method with stratified design to provide effective coverage of the units and to obtain precise estimates of the characteristics tested. Each unit was tested for multiple characteristics as discrete variables involving nominal measures. Statistical analysis was conducted on the test results of four variables. An explanation of the audit test results and relevance of the tests to the audit's objectives is provided in the body of the audit report. We present 95 percent confidence limits on the expected value of the proportions by using the formulae given at the end of this appendix.

From the universe of 6,525 vouchers, we selected as first stage sample units a random sample of 900 vouchers (an average of 64 vouchers per district). The random sample of 900 vouchers out of 6,525 provided a sampling fraction of 13.8 percent. From each of the randomly selected vouchers a random sample of up to 10 transactions (second stage sampling units) was tested. The sample test results were projected to the universe of transactions at the 14 USMS districts.

The table below provides the test results and projections for the random variables tested. Following the table are the mathematical model notations, and formulae used to compute the estimates of expected values and variances.

Test Results Projections for the Random Variables Tested

Question
(Variable Tested)		 Answer
(Results of Test)		 Rate of
Occurrence
(%)		 Projection at
95 Percent Lower
Confidence Limit*
(%)
1. Was the prisoner in USMS custody during the treatment? Yes 94.5  
No 0.2 0.19
Unk 5.3 5.2
2. Was the transaction accurately recorded? Yes 51.9  
No 1.0 0.99
Unk 47.1 47.0
3. Was the transaction fully supported? Yes 67.8  
No 27.3 26.9
Unk 4.9 4.5
4. Were the procedures necessary? Yes 69.7  
No 3.3 3.1
Unk 27.0 26.4
*This is the most conservative projection at the 95 percent confidence level. In other words, the projected percentage is at least the percentage of occurrence present in the corresponding universe.

Mathematical Model Notations and Formulae used to compute the Estimates of Expected Values and Variances

The mathematical model notations, and formulae used to compute the estimates of expected values, and the variances are as follows.

H The number of strata


Nh The number of units in the stratum h, where N is equal to the summation of N sub h over h


nh The number of units sampled from the stratum h
 
yhijk Variable k corresponding to jth selected item within the ith sampled item from the hth stratum


yhijkl = The variable Y sub hijkl is defined as the variable that takes the value 1 if randomly selected 2nd stage unit j has value of the type l for the random variable k; other wise the variable Y sub hijkl takes the value 0.

The summation the variable Y sub hijkl with respect to the index l is equal to the number 1 for all the indices h,i,j, and k.

yhikl The variable Y sub hikl is equal to the summation of Y sub hijkl over the index j where the limits of the summation are from 1 to n sub hi.


phikl Sample proportion of hits of lth type of the kth variable in the ith sampled item in hth stratum


p hat sub hikl is equal to y bar hat sub hikl which is again equal to the ratio of the summation of the variable Y sub hijkl on the index j from 1 to n sub hi over n sub hi. , p hat sub hkl is equal to the ratio of the summation over i of p hat of hikl times n sub hi over the summation of n sub hi over i. , and p hat sub kl is equal to the summation over h of the ratio of N sub h times p hat sub hkl over N


To compute the variance of the estimate p hat sub kl the formulae used are as follows.

V of p hat sub kl is equal to one over N square times the summation over h of N sub h times open parenthesis upper case N sub h minus lower case n sub h parenthesis closed times s sub L square over lower case n sub h

Where square of s sub L is the variance of lower level terms.

The 95 percent lower confidence limits on the estimate is given by

p hat sub kl plus or minus 2 times square root of V open parentheses p hat sub kl close parentheses.